Explicit Expressions for Most Common Entropies

Abstract

Entropies are useful measures of variation. However, explicit expressions for entropies available in the literature are limited. In this paper, we provide a comprehensive collection of explicit expressions for four of the most common entropies for over sixty continuous univariate distributions. Most of the derived expressions are new. The explicit expressions involve known special functions.

1. Introduction

Let X denote a continuous random variable with probability density and cumulative distribution functions specified by $f_X(·)$ and $F_X(·)$, respectively. Four of the most popular entropies are the geometric mean [1,2], Shannon entropy ([3], pp. 379–423; [3], pp. 623–656), Rényi entropy [4] and the cumulative residual entropy [5], defined by

$$\begin{array}{c}\hfill {\displaystyle GM\left(X\right)=\int logx{f}_X\left(x\right)dx,}\end{array}$$ (1)

$$\begin{array}{c}\hfill {\displaystyle S\left(X\right)=-\int log{f}_X\left(x\right)f_X\left(x\right)dx,}\end{array}$$ (2)

$$\begin{array}{c}\hfill {\displaystyle R\left(X\right)=\frac{1}{1-\gamma }log\left\{\int {\left[f_X\left(x\right)\right]}^{\gamma }dx\right\}}\end{array}$$ (3)

and

$$\begin{array}{c}\hfill {\displaystyle CE\left(X\right)=-\int \left[1-F_X\left(x\right)\right]log\left[1-F_X\left(x\right)\right]dx,}\end{array}$$ (4)

respectively, for $\gamma \ge 0$ and $\gamma \ne 1$.

There have been several papers giving explicit expressions for entropies. Ref. [6] derived expressions for $S\left(X\right)$ for twenty univariate distributions. Ref. [7] derived expressions for $S\left(X\right)$ for five multivariate distributions. Ref. [8] derived expressions for $S\left(X\right)$ and mutual information for eight multivariate distributions. Ref. [9] derived expressions for $S\left(X\right)$ and $R\left(X\right)$ for fifteen bivariate distributions. Ref. [10] derived expressions for $S\left(X\right)$ and $R\left(X\right)$ for fifteen multivariate distributions. Ref. [11] derived expressions for $S\left(X\right)$, $R\left(X\right)$ and the q-entropy for the Dagum distribution. Ref. [12] derived expressions for $S\left(X\right)$ for certain binomial type distributions. Ref. [13] derived expressions for $GM\left(X\right)$ and $CE\left(X\right)$ for three Lindley type distributions.

All of these and other papers are restrictive in terms of the entropies considered and the number of distributions considered. In this paper, we derive expressions for (1)–(4) for more than sixty continuous univariate distributions, see Section 3. Most of the derived expressions are new. Some technicalities used in the derivations are given in Section 2. The derivations themselves are not given and can be obtained from the corresponding author. Some conclusions and future work are noted in Section 4.

The calculations of this paper involve several special functions, including the the exponential integral defined by

$$\begin{array}{c}\hfill {\displaystyle \mathrm{Ei}\left(a\right)={\int }_{-\infty }^a\frac{exp\left(t\right)}{t}dt;}\end{array}$$

the gamma function defined by

$$\begin{array}{c}\hfill {\displaystyle \Gamma \left(a\right)={\int }_0^{\infty }{t}^{a-1}exp(-t)dt;}\end{array}$$

the lower incomplete gamma function defined by

$$\begin{array}{c}\hfill {\displaystyle \Gamma (a,x)={\int }_x^{\infty }{t}^{a-1}exp(-t)dt;}\end{array}$$

the upper incomplete gamma function defined by

$$\begin{array}{c}\hfill {\displaystyle \gamma (a,x)={\int }_0^x{t}^{a-1}exp(-t)dt;}\end{array}$$

the digamma function defined by

$$\begin{array}{c}\hfill {\displaystyle \psi \left(a\right)=\frac{log\Gamma \left(a\right)}{da};}\end{array}$$

the standard normal distribution function defined by

$$\begin{array}{c}\hfill {\displaystyle \Phi \left(a\right)=\frac{1}{\sqrt{2\pi }}{\int }_{-\infty }^{x}exp\left(-\frac{t^2}{2}\right)dt;}\end{array}$$

the error function defined by

$$\begin{array}{c}\hfill {\displaystyle \mathrm{erf}\left(a\right)=\frac{2}{\sqrt{\pi }}{\int }_0^{a}exp\left(-t^2\right)dt;}\end{array}$$

the complementary error function defined by

$$\begin{array}{c}\hfill {\displaystyle \mathrm{erfc}\left(a\right)=\frac{2}{\sqrt{\pi }}{\int }_a^{\infty }exp\left(-t^2\right)dt;}\end{array}$$

the beta function defined by

$$\begin{array}{c}\hfill {\displaystyle B(a,b)={\int }_0^1{t}^{a-1}{(1-t)}^{b-1}dt;}\end{array}$$

the incomplete beta function defined by

$$\begin{array}{c}\hfill {\displaystyle B_x(a,b)={\int }_0^x{t}^{a-1}{(1-t)}^{b-1}dt;}\end{array}$$

the incomplete beta function ratio defined by

$$\begin{array}{c}\hfill {\displaystyle I_x(a,b)=\frac{B_x(a,b)}{B(a,b)};}\end{array}$$

the modified Bessel function of the first kind of order $\nu$ defined by

$$\begin{array}{c}\hfill {\displaystyle I_{\nu }\left(x\right)=\sum _{k=0}^{\infty }\frac{1}{\Gamma (k+\nu +1)k!}{\left(\frac{x}{2}\right)}^{2k+\nu };}\end{array}$$

the modified Bessel function of the second kind defined by

$$\begin{array}{c}\hfill {\displaystyle K_{\nu }\left(x\right)=\left\{\begin{array}{cc}{\displaystyle \frac{\pi }{2sin\left(\pi \nu \right)}\left[I_{-\nu }\left(x\right)-I_{\nu }\left(x\right)\right],}\hfill & \mathrm{if} \nu \notin \mathbb{Z},\hfill \\ \\ {\displaystyle \underset{\mu \to \nu }{lim}{K}_{\mu }\left(x\right),}\hfill & \mathrm{if} \nu \in \mathbb{Z};\hfill \end{array}\right.}\end{array}$$

the confluent hypergeometric function defined by

$$\begin{array}{c}\hfill {\displaystyle {}_1{F}_1\left(a;b;x\right)=\sum _{k=0}^{\infty }\frac{{\left(a\right)}_k}{{\left(b\right)}_k}\frac{x^k}{k!},}\end{array}$$

where ${\left(a\right)}_k=a(a+1)\cdots (a+k-1)$ denotes the ascending factorial; the Kummer function defined by

$$\begin{array}{c}\hfill {\displaystyle \Psi \left(a;b;x\right)=\frac{\Gamma (1-b)}{\Gamma (1+a-b)}{}_1{F}_1\left(a;b;x\right)+\frac{\Gamma (b-1)}{\Gamma \left(a\right)}{x}^{1-b}{}_1{F}_1\left(1+a-b;1-b;x\right);}\end{array}$$

the Gauss hypergeometric function defined by

$$\begin{array}{c}\hfill {\displaystyle {}_2{F}_1\left(a,b;c;x\right)=\sum _{k=0}^{\infty }\frac{{\left(a\right)}_k{\left(b\right)}_k}{{\left(c\right)}_k}\frac{x^k}{k!};}\end{array}$$

the degenerate hypergeometric series of two variables defined by

$$\begin{array}{c}\hfill {\displaystyle {\Phi }_1\left(a,b,c,x,y\right)=\sum _{m=0}^{\infty }\sum _{n=0}^{\infty }\frac{{\displaystyle {\left(a\right)}_{m+n}{\left(b\right)}_n{x}^m{y}^n}}{{\displaystyle {\left(c\right)}_{m+n}m!n!}};}\end{array}$$

the degenerate hypergeometric function of two variables defined by

$$\begin{array}{c}\hfill {\displaystyle F_1\left(a,b,c;d;x,y\right)=\sum _{m=0}^{\infty }\sum _{n=0}^{\infty }\frac{{\displaystyle {\left(a\right)}_{m+n}{\left(b\right)}_m{\left(c\right)}_n{x}^m{y}^n}}{{\displaystyle {\left(d\right)}_{m+n}m!n!}}.}\end{array}$$

The properties of these special functions can be found in [14,15].

2. Technical Lemmas

The derivations in Section 3 use the following two lemmas.

Lemma 1.

The geometric mean defined by (1) can be calculated using

$$\begin{array}{c}\hfill {\displaystyle GM\left(X\right)={\left.\frac{d}{d\alpha }E\left(X^{\alpha }\right)\right|}_{\alpha =0},}\end{array}$$ (5)

where $E(·)$ denotes the expectation defined by

$$\begin{array}{c}\hfill {\displaystyle E\left(X^{\alpha }\right)=\int x^{\alpha }{f}_X\left(x\right)dx.}\end{array}$$

Proof. 

Note that

$$\begin{array}{c}\hfill {\displaystyle GM\left(X\right)=\int {\left.\frac{d}{d\alpha }{x}^{\alpha }\right|}_{\alpha =0}{f}_X\left(x\right)dx={\left.\frac{d}{d\alpha }\left[\int x^{\alpha }{f}_X\left(x\right)dx\right]\right|}_{\alpha =0}.}\end{array}$$

Hence, the result. □

Lemma 2.

The cumulative residual entropy defined by (4) can be calculated using

$$\begin{array}{c}\hfill {\displaystyle CE\left(X\right)=\sum _{k=1}^{\infty }\frac{1}{k}\int {\left[F_X\left(x\right)\right]}^{k}dx-\sum _{k=1}^{\infty }\frac{1}{k}\int {\left[F_X\left(x\right)\right]}^{k+1}dx.}\end{array}$$ (6)

Proof. 

Using the Taylor series expansion for $log(1-z)$, we can write

$$\begin{array}{c}\hfill {\displaystyle CE\left(X\right)=\int \left[1-F_X\left(x\right)\right]\sum _{k=1}^{\infty }\frac{1}{k}{\left[F_X\left(x\right)\right]}^{k}dx=\int \sum _{k=1}^{\infty }\frac{1}{k}{\left[F\left(x\right)\right]}^{k}dx-\int \sum _{k=1}^{\infty }\frac{1}{k}{\left[F\left(x\right)\right]}^{k+1}dx.}\end{array}$$

Hence, the result. □

3. The Tabulation

In this section, we give expressions for $f_X\left(x\right)$ (the probability density function), $F_X\left(x\right)$ (the cumulative distribution function), $GM\left(X\right)$ (the geometric mean), $S\left(X\right)$ (Shannon entropy), $R\left(X\right)$ (Rényi entropy), and $CE\left(X\right)$ (the cumulative residual entropy) for over sixty continuous univariate distributions.

1. Gauss hypergeometric beta distribution [16]: for this distribution,

$$\begin{array}{c}\hfill {\displaystyle f_X\left(x\right)=\frac{K{x}^{a-1}{(1-x)}^{b-1}}{{(1+dx)}^c},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle F_X\left(x\right)=\frac{K{x}^a}{a}{F}_1\left(a,c,1-b,a+1;-dx,x\right),}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle GM\left(X\right)=exp\left[\frac{{\Gamma }^{{}^{\prime }}\left(a\right)}{\Gamma \left(a\right)}+{\left.\frac{1}{{}_2{F}_1\left(c,a;a+b;-d\right)}\frac{\partial }{\partial \alpha }{}_2{F}_1\left(c,\alpha +a;\alpha +a+b;-d\right)\right|}_{\alpha =0}-\frac{{\Gamma }^{{}^{\prime }}(a+b)}{\Gamma (a+b)}\right],}\end{array}$$

$$\begin{array}{ccc}& & {\displaystyle S\left(X\right)=-\frac{{\Gamma }^{{}^{\prime }}\left(a\right)}{\Gamma \left(a\right)}-{\left.\frac{1}{{}_2{F}_1\left(c,a;a+b;-d\right)}\frac{\partial }{\partial \alpha }{}_2{F}_1\left(c,\alpha +a;\alpha +a+b;-d\right)\right|}_{\alpha =0}}\hfill \\ & & {\displaystyle +\frac{2{\Gamma }^{{}^{\prime }}(a+b)}{\Gamma (a+b)}-\frac{{\Gamma }^{{}^{\prime }}\left(b\right)}{\Gamma \left(b\right)}}\hfill \\ & & {\displaystyle -{\left.\frac{1}{{}_2{F}_1\left(c,a;a+b;-d\right)}\frac{\partial }{\partial \alpha }{}_2{F}_1\left(c,a;\alpha +a+b;-d\right)\right|}_{\alpha =0}}\hfill \\ & & {\displaystyle -\frac{1}{{}_2{F}_1\left(c,a;a+b;-d\right)}\frac{d}{d\alpha }{}_2{F}_1\left(c-\alpha ,a;a+b;-d\right)-logB(a,b)}\hfill \\ & & {\displaystyle -log{}_2{F}_1\left(c-\alpha ,a;a+b;-d\right),}\hfill \end{array}$$

$$\begin{array}{ccc}& & {\displaystyle R\left(X\right)=\frac{1}{1-\gamma }logB\left(a\gamma -\gamma +1,b\gamma -\gamma +1\right)}\hfill \\ & & {\displaystyle +\frac{1}{1-\gamma }log{}_2{F}_1\left(c\gamma ,a\gamma -\gamma +1;a\gamma +b\gamma -2\gamma +2;-d\right)}\hfill \\ & & {\displaystyle -\frac{\gamma }{1-\gamma }logB(a,b)-\frac{\gamma }{1-\gamma }log{}_2{F}_1\left(c,a;a+b;-d\right)}\hfill \end{array}$$

and

$$\begin{array}{c}\hfill {\displaystyle CE\left(X\right)=-{\int }_0^1\left[1-\frac{K{x}^a}{a}{F}_1\left(a,c,1-b,a+1;-dx,x\right)\right]log\left[1-\frac{K{x}^a}{a}{F}_1\left(a,c,1-b,a+1;-dx,x\right)\right]dx}\end{array}$$

for $0<x<1$, $a>0$, $b>0$, $-\infty <c<\infty$ and $d>-1$, where ${\Gamma }^{{}^{\prime }}\left(x\right)=\frac{d\Gamma \left(x\right)}{dx}$ and $\frac{1}{K}=B(a,b){}_2{F}_1\left(c,a;a+b;-d\right)$.

2. q Weibull distribution [17]: for this distribution,

$$\begin{array}{c}\hfill {\displaystyle f_X\left(x\right)=(2-q)ab{x}^{a-1}{\left[1-(1-q)b{x}^a\right]}^{\frac{1}{1-q}},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle F_X\left(x\right)=1-{\left[1-(1-q)b{x}^a\right]}^{\frac{2-q}{1-q}},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle GM\left(X\right)=\left\{\begin{array}{c}{\displaystyle -\frac{2-q}{a(q-1)}log\left[(q-1)b\right]\frac{\Gamma \left(\frac{2-q}{q-1}\right)}{\Gamma \left(\frac{1}{q-1}\right)}-\frac{2-q}{a(q-1)}\frac{{\Gamma }^{{}^{\prime }}\left(\frac{2-q}{q-1}\right)}{\Gamma \left(\frac{1}{q-1}\right)}}\hfill \\ {\displaystyle +\frac{2-q}{a(q-1)}\frac{{\Gamma }^{{}^{\prime }}\left(1\right)\Gamma \left(\frac{2-q}{q-1}\right)}{\Gamma \left(\frac{1}{q-1}\right)},}\hfill & \mathrm{if} 1<q<2,\hfill \\ \\ {\displaystyle \frac{2-q}{a(q-1)}log\left[(1-q)b\right]\frac{\Gamma \left(\frac{2-q}{1-q}\right)}{\Gamma \left(\frac{3-2q}{1-q}\right)}+\frac{2-q}{a(q-1)}\frac{{\Gamma }^{{}^{\prime }}\left(\frac{3-2q}{1-q}\right)}{{\left[\Gamma \left(\frac{3-2q}{1-q}\right)\right]}^2}}\hfill \\ {\displaystyle -\frac{2-q}{a(q-1)}\frac{{\Gamma }^{{}^{\prime }}\left(1\right)\Gamma \left(\frac{2-q}{1-q}\right)}{\Gamma \left(\frac{3-2q}{1-q}\right)},}\hfill & \mathrm{if} q<1,\hfill \end{array}\right.}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle S\left(X\right)=\left\{\begin{array}{c}{\displaystyle -log\left[(2-q)ab\right]+\frac{1}{2-q}-\frac{(1-a)(2-q)}{a(q-1)}log\left[(q-1)b\right]\frac{\Gamma \left(\frac{2-q}{q-1}\right)}{\Gamma \left(\frac{1}{q-1}\right)}}\hfill \\ {\displaystyle -\frac{(1-a)2-q}{a(q-1)}\frac{{\Gamma }^{{}^{\prime }}\left(\frac{2-q}{q-1}\right)}{\Gamma \left(\frac{1}{q-1}\right)}}\hfill \\ {\displaystyle +\frac{(1-a)(2-q)}{a(q-1)}\frac{{\Gamma }^{{}^{\prime }}\left(1\right)\Gamma \left(\frac{2-q}{q-1}\right)}{\Gamma \left(\frac{1}{q-1}\right)},}\hfill & \mathrm{if} 1<q<2,\hfill \\ \\ {\displaystyle -log\left[(2-q)ab\right]+\frac{1}{2-q}+\frac{(1-a)(2-q)}{a(q-1)}log\left[(1-q)b\right]\frac{\Gamma \left(\frac{2-q}{1-q}\right)}{\Gamma \left(\frac{3-2q}{1-q}\right)}}\hfill \\ {\displaystyle +\frac{(1-a)(2-q)}{a(q-1)}\frac{{\Gamma }^{{}^{\prime }}\left(\frac{3-2q}{1-q}\right)}{{\left[\Gamma \left(\frac{3-2q}{1-q}\right)\right]}^2}}\hfill \\ {\displaystyle -\frac{(1-a)(2-q)}{a(q-1)}\frac{{\Gamma }^{{}^{\prime }}\left(1\right)\Gamma \left(\frac{2-q}{1-q}\right)}{\Gamma \left(\frac{3-2q}{1-q}\right)},}\hfill & \mathrm{if} q<1,\hfill \end{array}\right.}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle R\left(X\right)=\left\{\begin{array}{cc}{\displaystyle \frac{1}{1-\gamma }log\left[\frac{{(2-q)}^{\gamma }{a}^{\gamma -1}{b}^{\frac{\gamma -1}{a}}}{{(q-1)}^{\gamma +\frac{1-\gamma }{a}}}B\left(\gamma +\frac{1-\gamma }{a},-\frac{\gamma }{1-q}-\gamma -\frac{1-\gamma }{a}\right)\right],}\hfill & \mathrm{if} 1<q<2,\hfill \\ \\ {\displaystyle log\left[\frac{{(2-q)}^{\gamma }{a}^{\gamma -1}{b}^{\frac{\gamma -1}{a}}}{{(q-1)}^{\gamma +\frac{1-\gamma }{a}}}B\left(\gamma +\frac{1-\gamma }{a},1+\frac{\gamma }{1-q}\right)\right],}\hfill & \mathrm{if} q<1\hfill \end{array}\right.}\end{array}$$

and

$$\begin{array}{c}\hfill {\displaystyle CE\left(X\right)=\left\{\begin{array}{c}{\displaystyle -\frac{2-q}{a(q-1)}\frac{\Gamma \left(\frac{1}{a}\right)}{{\left[(q-1)b\right]}^{\frac{1}{a}}}\frac{{\Gamma }^{{}^{\prime }}\left(\frac{2-q}{q-1}-\frac{1}{a}\right)}{\Gamma \left(\frac{2-q}{q-1}\right)}}\hfill \\ {\displaystyle +\frac{2-q}{a(q-1)}\frac{\Gamma \left(\frac{1}{a}\right)}{{\left[(q-1)b\right]}^{\frac{1}{a}}}\frac{\Gamma \left(\frac{2-q}{q-1}-\frac{1}{a}\right){\Gamma }^{{}^{\prime }}\left(\frac{2-q}{q-1}\right)}{{\left[\Gamma \left(\frac{2-q}{q-1}\right)\right]}^2},}\hfill & \mathrm{if} 1<q<2,\hfill \\ \\ {\displaystyle \frac{2-q}{a(q-1)}\frac{\Gamma \left(\frac{1}{a}\right)}{{\left[(1-q)b\right]}^{\frac{1}{a}}}\frac{{\Gamma }^{{}^{\prime }}\left(\frac{2-q}{1-q}+1\right)}{\Gamma \left(\frac{2-q}{1-q}+1+\frac{1}{a}\right)}}\hfill \\ {\displaystyle -\frac{2-q}{a(q-1)}\frac{\Gamma \left(\frac{1}{a}\right)}{{\left[(1-q)b\right]}^{\frac{1}{a}}}\frac{\Gamma \left(\frac{2-q}{1-q}+1\right){\Gamma }^{{}^{\prime }}\left(\frac{2-q}{1-q}+1+\frac{1}{a}\right)}{{\left[\Gamma \left(\frac{2-q}{1-q}+1+\frac{1}{a}\right)\right]}^2},}\hfill & \mathrm{if} q<1\hfill \end{array}\right.}\end{array}$$

for $a>0$, $b>0$, $0<x<\infty$ if $1<q<2$ and $0<x<{\left[(1-q)b\right]}^{\frac{1}{a}}$ if $q<1$.

3. q exponential distribution [17]: for this distribution,

$$\begin{array}{c}\hfill {\displaystyle f_X\left(x\right)=(2-q)b{\left[1-(1-q)bx\right]}^{\frac{1}{1-q}},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle F_X\left(x\right)=1-{\left[1-(1-q)bx\right]}^{\frac{2-q}{1-q}},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle GM\left(X\right)=\left\{\begin{array}{cc}{\displaystyle -\frac{2-q}{q-1}\frac{{\Gamma }^{{}^{\prime }}\left(\frac{q}{q-1}\right)}{\Gamma \left(\frac{2q-1}{q-1}\right)}+\frac{2-q}{q}{\Gamma }^{{}^{\prime }}\left(1\right)-\frac{2-q}{q}log\left[(q-1)b\right],}\hfill & \mathrm{if} 1<q<2,\hfill \\ \\ {\displaystyle -\frac{{\Gamma }^{{}^{\prime }}\left(\frac{3-2q}{1-q}\right)}{\Gamma \left(\frac{3-2q}{1-q}\right)}+{\Gamma }^{{}^{\prime }}\left(1\right)-log\left[(1-q)b\right],}\hfill & \mathrm{if} q<1,\hfill \end{array}\right.}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle S\left(X\right)=-log\left[(2-q)b\right]-\frac{2-q}{1-q}\frac{1}{1+(\alpha +1)(1-q)},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle R\left(X\right)=-logb+\frac{1}{1-\gamma }log\frac{{(2-q)}^{\gamma }}{\gamma +1-q}}\end{array}$$

and

$$\begin{array}{c}\hfill {\displaystyle CE\left(X\right)=\frac{2-q}{b{\left(3-2q\right)}^2}}\end{array}$$

for $b>0$, $0<x<\infty$ if $1<q<2$ and $0<x<(1-q)b$ if $q<1$.

4. Weighted exponential distribution: for this distribution,

$$\begin{array}{c}\hfill {\displaystyle f_X\left(x\right)=\frac{a+1}{a}bexp(-bx)\left[1-exp(-abx)\right],}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle F_X\left(x\right)=\frac{1}{a}exp(-bx)\left[exp(-abx)-a-1\right],}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle GM\left(X\right)=a{\Gamma }^{{}^{\prime }}\left(1\right)-alogb+(a+1)log(a+1),}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle S\left(X\right)=-log\frac{(a+1)b}{a}+a+1-\frac{1}{a+1}-{\Gamma }^{{}^{\prime }}\left(2\right)+\frac{{\Gamma }^{{}^{\prime }}\left(\frac{1}{a}+2\right)}{\Gamma \left(\frac{1}{a}+2\right)},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle R\left(X\right)=-logb-\frac{1+\gamma }{1-\gamma }loga+\frac{\gamma }{1-\gamma }log(a+1)+\frac{1}{1-\gamma }logB\left(\frac{\gamma }{a},\gamma +1\right)}\end{array}$$

and

$$\begin{array}{c}\hfill {\displaystyle CE\left(X\right)=\frac{a+1}{ab}\left[1-\frac{1}{{(1+a)}^3}\right]+\frac{(a+1)loga}{ab}\left[1-\frac{1}{{(1+a)}^2}\right]}\end{array}$$

for $x>0$, $a>0$ and $b>0$.

5. Teissier distribution [18]: for this distribution,

$$\begin{array}{c}\hfill {\displaystyle f_X\left(x\right)=\left[exp\left(ax\right)-1\right]exp\left[ax-exp\left(ax\right)+1\right],}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle F_X\left(x\right)=1-exp\left[ax-exp\left(ax\right)+1\right],}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle GM\left(X\right)=e{\left.\frac{\partial }{\partial \alpha }\left[a^{-\alpha }{\int }_1^{\infty }{(logy)}^{\alpha }(y-1)exp(-y)dy\right]\right|}_{\alpha =0},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle S\left(X\right)=-loga-ae{\left.\frac{\partial }{\partial \alpha }{\int }_1^{\infty }y{(y-1)}^{\alpha +1}exp(-y)dy\right|}_{\alpha =0}-e{\left.\frac{\partial }{\partial \alpha }\left[\Gamma (\alpha +2,1)-\Gamma (\alpha +1,1)\right]\right|}_{\alpha =0}+2,}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle R\left(X\right)=a^{\gamma }\Gamma (\gamma +1)\Psi \left(\gamma +2,2\gamma +1;\gamma \right)}\end{array}$$

and

$$\begin{array}{c}\hfill {\displaystyle CE\left(X\right)=-\frac{e}{a}\left[\mathrm{Ei}\left(1\right)-exp(-1)\right]}\end{array}$$

for $x>0$ and $a>0$.

6. Maxwell distribution [19,20]: for this distribution,

$$\begin{array}{c}\hfill {\displaystyle f_X\left(x\right)=\frac{4a^{\frac{3}{2}}}{\sqrt{\pi }}{x}^{2}exp\left(-a{x}^2\right),}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle F_X\left(x\right)=\frac{2}{\sqrt{\pi }}\gamma \left(\frac{3}{2},a{x}^2\right),}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle GM\left(X\right)=\frac{1-loga}{2},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle S\left(X\right)=\frac{-\Gamma \left(\frac{3}{2}\right)loga+{\Gamma }^{{}^{\prime }}\left(\frac{3}{2}\right)}{\sqrt{\pi }},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle R\left(X\right)=-log\left(2\sqrt{a}\right)+\frac{log2}{1-\gamma }-\frac{\gamma log\pi }{2(1-\gamma )}-\left(\gamma +\frac{1}{2}\right)\frac{log\gamma }{1-\gamma }+\frac{1}{1-\gamma }log\Gamma \left(\gamma +\frac{1}{2}\right)}\end{array}$$

and

$$\begin{array}{c}\hfill {\displaystyle CE\left(X\right)=-\frac{2}{\sqrt{\pi }}{\int }_0^{\infty }\left[log2-log\sqrt{\pi }+log\Gamma \left(\frac{3}{2},a{x}^2\right)\right]\Gamma \left(\frac{3}{2},a{x}^2\right)dx}\end{array}$$

for $x>0$ and $a>0$.

7. Inverse Maxwell distribution: for this distribution,

$$\begin{array}{c}\hfill {\displaystyle f_X\left(x\right)=\frac{4a^{\frac{3}{2}}}{\sqrt{\pi }}{x}^{-4}exp\left(-a{x}^{-2}\right),}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle F_X\left(x\right)=\frac{2}{\sqrt{\pi }}\Gamma \left(\frac{3}{2},a{x}^{-2}\right),}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle GM\left(X\right)=\frac{\Gamma \left(\frac{3}{2}\right)loga-{\Gamma }^{{}^{\prime }}\left(\frac{3}{2}\right)}{\sqrt{\pi }},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle S\left(X\right)=-log\left(\frac{4}{\sqrt{\pi }}\right)+\frac{1}{2}loga-\frac{4}{\sqrt{\pi }}{\Gamma }^{{}^{\prime }}\left(\frac{3}{2}\right),}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle R\left(X\right)=\frac{1}{1-\gamma }log\frac{2a^{\frac{5}{2}}}{\sqrt{\pi }}+\frac{\gamma }{1-\gamma }log\frac{4}{a\sqrt{\pi }}+\frac{1}{1-\gamma }log\Gamma \left(2\gamma -\frac{1}{2}\right)}\end{array}$$

and

$$\begin{array}{c}\hfill {\displaystyle CE\left(X\right)=-\frac{2\sqrt{a}}{\sqrt{\pi }}log\frac{2}{\sqrt{\pi }}-\frac{2}{\sqrt{\pi }}{\int }_0^{\infty }\gamma \left(\frac{3}{2},a{x}^{-2}\right)log\gamma \left(\frac{3}{2},a{x}^{-2}\right)dx}\end{array}$$

for $x>0$ and $a>0$.

8. Power Maxwell distribution [21]: for this distribution,

$$\begin{array}{c}\hfill {\displaystyle f_X\left(x\right)=\frac{4a{b}^{\frac{3}{2}}}{\sqrt{\pi }}{x}^{3a-1}exp\left(-b{x}^{2a}\right),}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle F_X\left(x\right)=\frac{2}{\sqrt{\pi }}\gamma \left(\frac{3}{2},b{x}^{2a}\right),}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle GM\left(X\right)=\frac{-\Gamma \left(\frac{3}{2}\right)logb+{\Gamma }^{{}^{\prime }}\left(\frac{3}{2}\right)}{a\sqrt{\pi }},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle S\left(X\right)=\frac{3}{2}-\frac{logb}{2a}+\frac{1-3a}{a\sqrt{\pi }}{\Gamma }^{{}^{\prime }}\left(\frac{3}{2}\right)-log\frac{4a}{\sqrt{\pi }},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle R\left(X\right)=-log\left[2a{b}^{\frac{1}{2a}}\right]+\frac{log2}{1-\gamma }-\frac{\gamma log\pi }{2(1-\gamma )}-\left(\frac{3\gamma }{2}+\frac{1-\gamma }{2a}\right)\frac{log\gamma }{1-\gamma }+\frac{1}{1-\gamma }log\Gamma \left(\frac{3\gamma }{2}+\frac{1-\gamma }{2a}\right)}\end{array}$$

and

$$\begin{array}{c}\hfill {\displaystyle CE\left(X\right)=-\frac{2}{\sqrt{\pi }{b}^{\frac{1}{2a}}}log\left(\frac{2}{\sqrt{\pi }}\right)\Gamma \left(\frac{1}{2a}+\frac{3}{2}\right)-\frac{2}{\sqrt{\pi }}{\int }_0^{\infty }\gamma \left(\frac{3}{2},b{x}^{2a}\right)log\Gamma \left(\frac{3}{2},b{x}^{2a}\right)dx}\end{array}$$

for $x>0$, $a>0$ and $b>0$.

9. Inverse power Maxwell distribution [22]: for this distribution,

$$\begin{array}{c}\hfill {\displaystyle f_X\left(x\right)=\frac{4a{b}^{\frac{3}{2}}}{\sqrt{\pi }}{x}^{-3a-1}exp\left(-b{x}^{-2a}\right),}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle F_X\left(x\right)=\frac{2}{\sqrt{\pi }}\Gamma \left(\frac{3}{2},b{x}^{-2a}\right),}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle GM\left(X\right)=\frac{\Gamma \left(\frac{3}{2}\right)logb-{\Gamma }^{{}^{\prime }}\left(\frac{3}{2}\right)}{a\sqrt{\pi }},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle S\left(X\right)=-log\left(\frac{4a}{\sqrt{\pi }}\right)+\frac{1}{2a}logb-\frac{3a+1}{a\sqrt{\pi }}{\Gamma }^{{}^{\prime }}\left(\frac{3}{2}\right),}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle R\left(X\right)=\frac{1}{1-\gamma }log\frac{2b^{\frac{5}{2}}}{\sqrt{\pi }}+\frac{\gamma }{1-\gamma }log\frac{4a}{b\sqrt{\pi }}+\frac{1}{1-\gamma }log\Gamma \left(\frac{3\gamma }{2}+\frac{\gamma -1}{2a}\right)}\end{array}$$

and

$$\begin{array}{c}\hfill {\displaystyle CE\left(X\right)=-\frac{2}{\sqrt{\pi }}{b}^{\frac{1}{2a}}log\left(\frac{2}{\sqrt{\pi }}\right)\Gamma \left(\frac{3}{2}-\frac{1}{2a}\right)-\frac{2}{\sqrt{\pi }}{\int }_0^{\infty }\gamma \left(\frac{3}{2},b{x}^{-2a}\right)log\gamma \left(\frac{3}{2},b{x}^{-2a}\right)dx}\end{array}$$

for $x>0$, $a>0$ and $b>0$.

10. Omega distribution [23]: for this distribution,

$$\begin{array}{c}\hfill {\displaystyle f_X\left(x\right)=\frac{ab{x}^{b-1}}{1-x^{2b}}{\left(\frac{1+x^b}{1-x^b}\right)}^{-\frac{a}{2}},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle F_X\left(x\right)=1-{\left(\frac{1+x^b}{1-x^b}\right)}^{-\frac{a}{2}},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle GM\left(X\right)=a{\left.\frac{\partial }{\partial \alpha }\left[B\left(\frac{\alpha }{b}+1,\frac{a}{2}\right){}_2{F}_1\left(\frac{\alpha }{b}+1,\frac{a}{2}+1;\frac{\alpha }{b}+1+\frac{a}{2};-1\right)\right]\right|}_{\alpha =0},}\end{array}$$

$$\begin{array}{ccc}& & {\displaystyle S\left(X\right)=-log\left(ab\right)+a(1-b){\left.\frac{\partial }{\partial \alpha }\left[B\left(\frac{\alpha }{b}+1,\frac{a}{2}\right){}_2{F}_1\left(\frac{\alpha }{b}+1,\frac{a}{2}+1;\frac{\alpha }{b}+1+\frac{a}{2};-1\right)\right]\right|}_{\alpha =0}}\hfill \\ & & {\displaystyle +(a+2){\left.\frac{\partial }{\partial \alpha }{}_2{F}_1\left(1,\frac{a}{2}+1-\alpha ;\frac{\alpha }{2}+1;-1\right)\right|}_{\alpha =0}}\hfill \\ & & {\displaystyle -a(a-2){\left.\frac{\partial }{\partial \alpha }\left[\frac{1}{a+2\alpha }{}_2{F}_1\left(1,\frac{a}{2}+1;\alpha +1+\frac{a}{2};-1\right)\right]\right|}_{\alpha =0},}\hfill \end{array}$$

$$\begin{array}{c}\hfill {\displaystyle R\left(X\right)=\frac{1}{1-\gamma }log\left[a^{\gamma -1}{b}^{\gamma }B\left(\gamma +\frac{1-\gamma }{b},1-\gamma +\frac{a\gamma }{2}\right){}_2{F}_1\left(\gamma +\frac{1-\gamma }{b},\frac{\gamma a}{2}+\gamma ;\frac{a\gamma }{2}+\frac{1-\gamma }{b}+1;-1\right)\right]}\end{array}$$

and

$$\begin{array}{ccc}& & {\displaystyle CE\left(X\right)=\frac{a}{2b}{\left.\frac{\partial }{\partial \alpha }\left[B\left(b,\frac{a}{2}+1\right){}_2{F}_1\left(b,\frac{a}{2}-\alpha ;b+\frac{a}{2}+1;-1\right)\right]\right|}_{\alpha =0}}\hfill \\ & & {\displaystyle -\frac{a}{2b}{\left.\frac{\partial }{\partial \alpha }\left[B\left(b,\alpha +\frac{a}{2}+1\right){}_2{F}_1\left(b,\frac{a}{2};b+\alpha +\frac{a}{2}+1;-1\right)\right]\right|}_{\alpha =0}}\hfill \end{array}$$

for $x>0$, $a>0$ and $b>0$.

11. Colak et al.’s distribution [24]: for this distribution,

$$\begin{array}{c}\hfill {\displaystyle f_X\left(x\right)=\frac{a(b+1){(1-x)}^{a-1}}{{(1+bx)}^{a+1}},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle F_X\left(x\right)={\left(\frac{1-x}{1+bx}\right)}^a,}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle GM\left(X\right)=a(b+1){\left.\frac{\partial }{\partial \alpha }\left[B(\alpha +1,a){}_2{F}_1\left(\alpha +1,a+1;\alpha +1+a;-b\right)\right]\right|}_{\alpha =0},}\end{array}$$

$$\begin{array}{ccc}& & {\displaystyle S\left(X\right)=-log\left[a(b+1)\right]+a(1-a)(b+1){\left.\frac{\partial }{\partial \alpha }\left[\frac{1}{\alpha +a}{}_2{F}_1\left(1,a+1;\alpha +a+1;-b\right)\right]\right|}_{\alpha =0}}\hfill \\ & & {\displaystyle +(a+1)(b+1){\left.\frac{\partial }{\partial \alpha }{}_2{F}_1\left(a,a+1-\alpha ;a+1;-b\right)\right|}_{\alpha =0},}\hfill \end{array}$$

$$\begin{array}{c}\hfill {\displaystyle R\left(X\right)=\frac{1}{1-\gamma }log\left\{\frac{{\left[a(b+1)\right]}^{\gamma }}{\gamma a-\gamma +1}{}_2{F}_1\left(1,\gamma a+\gamma ;\gamma a-\gamma +2;-b\right)\right\}}\end{array}$$

and

$$\begin{array}{c}\hfill {\displaystyle CE\left(X\right)=\frac{a}{a+1}{\left.\frac{\partial }{\partial \alpha }{}_2{F}_1\left(1,a-\alpha ;a+2;-b\right)\right|}_{\alpha =0}-{\left.\frac{\partial }{\partial \alpha }\left[\frac{a}{a+\alpha +1}{}_2{F}_1\left(1,a;a+\alpha +2;-b\right)\right]\right|}_{\alpha =0}}\end{array}$$

for $x>0$, $a>0$ and $b>0$.

12. Bimodal beta distribution [25]: for this distribution,

$$\begin{array}{c}\hfill {\displaystyle f_X\left(x\right)=\frac{\rho +{(1-\delta x)}^2}{CB(\alpha ,\beta )}{x}^{\alpha -1}{(1-x)}^{\beta -1},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle F_X\left(x\right)=\frac{1}{C}\left[(1+\rho )I_x(\alpha ,\beta )-2\delta \frac{B_x(\alpha +1,\beta )}{B_x(\alpha ,\beta )}+{\delta }^2\frac{B_x(\alpha +2,\beta )}{B_x(\alpha ,\beta )}\right],}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle GM\left(X\right)=\frac{\Gamma (\alpha +\beta )}{C\Gamma \left(\alpha \right)}\sum _{i=0}^2{c}_i\left\{\frac{{\Gamma }^{{}^{\prime }}(\alpha +i)}{\Gamma (\alpha +\beta +i)}-\frac{\Gamma (\alpha +i){\Gamma }^{{}^{\prime }}(\alpha +\beta +i)}{{\left[\Gamma (\alpha +\beta +i)\right]}^2}\right\},}\end{array}$$

$$\begin{array}{ccc}& & {\displaystyle S\left(X\right)=log\left[CB(\alpha ,\beta )\right]+\frac{(1-\alpha )\Gamma (\alpha +\beta )}{C\Gamma \left(\alpha \right)}\sum _{i=0}^2{c}_i\left\{\frac{{\Gamma }^{{}^{\prime }}(\alpha +i)}{\Gamma (\alpha +\beta +i)}-\frac{\Gamma (\alpha +i){\Gamma }^{{}^{\prime }}(\alpha +\beta +i)}{{\left[\Gamma (\alpha +\beta +i)\right]}^2}\right\}}\hfill \\ & & {\displaystyle +\frac{(1-\beta )\Gamma (\alpha +\beta )}{C\Gamma \left(\alpha \right)}\sum _{i=0}^2{c}_i\left\{\frac{\Gamma (\alpha +i){\Gamma }^{{}^{\prime }}\left(\beta \right)}{\Gamma (\alpha +\beta +i)}-\frac{\Gamma (\alpha +i)\Gamma \left(\beta \right){\Gamma }^{{}^{\prime }}(\alpha +\beta +i)}{{\left[\Gamma (\alpha +\beta +i)\right]}^2}\right\}}\hfill \\ & & {\displaystyle -{\left.\frac{\partial }{\partial a}\left[\frac{{\left(1+\rho \right)}^{a+1}}{C}{F}_1\left(\alpha ,-a-1,-a-1;\alpha +\beta ;\frac{\delta }{1+\mathrm{i}\sqrt{\rho }},\frac{\delta }{1-\mathrm{i}\sqrt{\rho }}\right)\right]\right|}_{a=0},}\hfill \end{array}$$

$$\begin{array}{c}\hfill {\displaystyle R\left(X\right)=\frac{B\left(\alpha \gamma -\gamma +1,\beta \gamma -\gamma +1\right){(1+\rho )}^{\gamma }}{C^{\gamma }{\left[B(\alpha ,\beta )\right]}^{\gamma }}{F}_1\left(\alpha \gamma -\gamma +1,-\gamma ,-\gamma ,\alpha \gamma +\beta \gamma -2\gamma +2;\frac{\delta }{1+\mathrm{i}\sqrt{\rho }},\frac{\delta }{1-\mathrm{i}\sqrt{\rho }}\right)}\end{array}$$

and

$$\begin{array}{ccc}& & {\displaystyle CE\left(X\right)=-{\int }_0^1\left\{1-\frac{1}{C}\left[(1+\rho )I_x(\alpha ,\beta )-2\delta \frac{B_x(\alpha +1,\beta )}{B_x(\alpha ,\beta )}+{\delta }^2\frac{B_x(\alpha +2,\beta )}{B_x(\alpha ,\beta )}\right]\right\}}\hfill \\ & & {\displaystyle ·log\left\{1-\frac{1}{C}\left[(1+\rho )I_x(\alpha ,\beta )-2\delta \frac{B_x(\alpha +1,\beta )}{B_x(\alpha ,\beta )}+{\delta }^2\frac{B_x(\alpha +2,\beta )}{B_x(\alpha ,\beta )}\right]\right\}dx}\hfill \end{array}$$

for $0<x<1$, $\alpha >0$, $\beta >0$, $\rho \ge 0$ and $-\infty <\delta <\infty$, where $\mathrm{i}=\sqrt{-1}$, $c_0=1+\rho$, $c_1=-2\delta$, $c_2={\delta }^2$ and $C=1+\rho -2\delta \frac{\alpha }{\alpha +\beta }+{\delta }^2\frac{\alpha (\alpha +1)}{(\alpha +\beta )(\alpha +\beta +1)}$.

13. Confluent hypergeometric beta distribution [26]: for this distribution,

$$\begin{array}{c}\hfill {\displaystyle f_X\left(x\right)=\frac{x^{a-1}{(1-x)}^{b-1}exp(-cx)}{B(a,b){}_1{F}_1\left(a;a+b;-c\right)},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle F_X\left(x\right)=\frac{x^a{\Phi }_1\left(a,1-b,a+1;x,cx\right)}{aB(a,b){}_1{F}_1(a;a+b;-c)},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle GM\left(X\right)=\frac{\Gamma (a+b)}{\Gamma \left(a\right){}_1{F}_1\left(a;a+b;-c\right)}{\left.\frac{\partial }{\partial \alpha }\left[\frac{\Gamma (a+\alpha )}{\Gamma (a+b+\alpha )}{}_1{F}_1\left(a+\alpha ;a+b+\alpha ;-c\right)\right]\right|}_{\alpha =0},}\end{array}$$

$$\begin{array}{ccc}& & {\displaystyle S\left(X\right)=\frac{(1-a)\Gamma (a+b)}{\Gamma \left(a\right){}_1{F}_1\left(a;a+b;-c\right)}{\left.\frac{\partial }{\partial \alpha }\left[\frac{\Gamma (a+\alpha )}{\Gamma (a+b+\alpha )}{}_1{F}_1\left(a+\alpha ;a+b+\alpha ;-c\right)\right]\right|}_{\alpha =0}}\hfill \\ & & {\displaystyle +\frac{(1-b)\Gamma (a+b)}{\Gamma \left(b\right){}_1{F}_1\left(a;a+b;-c\right)}{\left.\frac{\partial }{\partial \alpha }\left[\frac{\Gamma (b+\alpha )}{\Gamma (a+b+\alpha )}{}_1{F}_1\left(a;a+b+\alpha ;-c\right)\right]\right|}_{\alpha =0}}\hfill \\ & & {\displaystyle +\frac{ca}{a+b}\frac{{}_1{F}_1\left(a+1;a+b+1;-c\right)}{{}_1{F}_1\left(a;a+b;-c\right)}+logB(a,b)+log{}_1{F}_1\left(a;a+b;-c\right),}\hfill \end{array}$$

$$\begin{array}{c}\hfill {\displaystyle R\left(X\right)=\frac{1}{1-\gamma }log\left\{\frac{B\left(a\gamma -\gamma +1,b\gamma -\gamma +1\right){}_1{F}_1\left(a\gamma -\gamma +1;a\gamma +b\gamma -2\gamma +2;-c\gamma \right)}{{\left[B\left(a,b\right)\right]}^{\gamma }{\left[{}_1{F}_1\left(a;a+b;-c\right)\right]}^{\gamma }}\right\}}\end{array}$$

and

$$\begin{array}{c}\hfill {\displaystyle CE\left(X\right)=-{\int }_0^1\left[1-\frac{x^a{\Phi }_1\left(a,1-b,a+1;x,cx\right)}{aB(a,b){}_1{F}_1(a;a+b;-c)}\right]log\left[1-\frac{x^a{\Phi }_1\left(a,1-b,a+1;x,cx\right)}{aB(a,b){}_1{F}_1(a;a+b;-c)}\right]dx}\end{array}$$

for $0<x<1$, $a>0$, $b>0$ and $c>0$.

14. Libby and Novick’s beta distribution [27]: for this distribution,

$$\begin{array}{c}\hfill {\displaystyle f_X\left(x\right)=\frac{c^a{x}^{a-1}{(1-x)}^{b-1}}{B(a,b){\left[1-(1-c)x\right]}^{a+b}},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle F_X\left(x\right)=I_{\frac{1-x}{1+cx-x}}(b,a),}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle GM\left(X\right)=\frac{c^a}{B(a,b)}{\left.\frac{\partial }{\partial \alpha }\left[B(\alpha +a,b){}_2{F}_1\left(\alpha +a,a+b;\alpha +a+b;1-c\right)\right]\right|}_{\alpha =0},}\end{array}$$

$$\begin{array}{ccc}& & {\displaystyle S\left(X\right)=-alogc+\frac{(1-a)c^a}{B(a,b)}{\left.\frac{\partial }{\partial \alpha }\left[B(\alpha +a,b){}_2{F}_1\left(\alpha +a,a+b;\alpha +a+b;1-c\right)\right]\right|}_{\alpha =0}}\hfill \\ & & {\displaystyle +\frac{(1-b)c^a}{B(a,b)}{\left.\frac{\partial }{\partial \alpha }\left[B(a,b+\alpha ){}_2{F}_1\left(a,a+b;\alpha +a+b;1-c\right)\right]\right|}_{\alpha =0}}\hfill \\ & & {\displaystyle +c^a(a+b){\left.\frac{\partial }{\partial \alpha }{}_2{F}_1\left(a,a+b-\alpha ;a+b;1-c\right)\right|}_{\alpha =0},}\hfill \end{array}$$

$$\begin{array}{c}\hfill {\displaystyle R\left(X\right)=\frac{1}{1-\gamma }log\left\{\frac{c^{a\gamma }B\left(a\gamma -\gamma +1,b\gamma -\gamma +1\right)}{B{\left[(a,b)\right]}^{\gamma }}{}_2{F}_1\left(a\gamma -\gamma +1,a+b;a\gamma +b\gamma -2\gamma +2;1-c\right)\right\}}\end{array}$$

and

$$\begin{array}{c}\hfill {\displaystyle CE\left(X\right)=-{\int }_0^1{I}_{\frac{cx}{1+cx-x}}(a,b)log{I}_{\frac{cx}{1+cx-x}}(a,b)dx}\end{array}$$

for $0<x<1$, $a>0$, $b>0$, and $c>0$.

15. Generalized beta distribution [28]: for this distribution,

$$\begin{array}{c}\hfill {\displaystyle f_X\left(x\right)=\frac{\mid a\mid x^{ap-1}{\left[1-(1-c){\left(\frac{x}{b}\right)}^a\right]}^{q-1}}{b^{ap}B(p,q){\left[1+c{\left(\frac{x}{b}\right)}^a\right]}^{p+q}},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle F_X\left(x\right)=\frac{x^{ap}}{pB(p,q)b^{ap}}{F}_1\left(p,1-q,p+q,p+1;(1-c){\left(\frac{x}{b}\right)}^a,-c{\left(\frac{x}{b}\right)}^a\right),}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle GM\left(X\right)={\left.\frac{\partial }{\partial \alpha }\left[\frac{b^{\alpha }B\left(p+\frac{\alpha }{a},q\right)}{B(p,q)}{}_2{F}_1\left(p+\frac{\alpha }{a},\frac{\alpha }{a};p+q+\frac{\alpha }{a};c\right)\right]\right|}_{\alpha =0},}\end{array}$$

$$\begin{array}{ccc}& & {\displaystyle S\left(X\right)=-log\mid a\mid +aplogb+logB(p,q)}\hfill \\ & & {\displaystyle +(1-ap){\left.\frac{\partial }{\partial \alpha }\left[\frac{b^{\alpha }B\left(p+\frac{\alpha }{a},q\right)}{B(p,q)}{}_2{F}_1\left(p+\frac{\alpha }{a},\frac{\alpha }{a};p+q+\frac{\alpha }{a};c\right)\right]\right|}_{\alpha =0}}\hfill \\ & & {\displaystyle -(1-q){\left.\frac{\partial }{\partial \alpha }\left[\frac{B\left(p,q+\alpha \right)}{B(p,q)}{}_2{F}_1\left(p,\alpha ;p+q+\alpha ;c\right)\right]\right|}_{\alpha =0}}\hfill \\ & & {\displaystyle +(p+q){\left.\frac{\partial }{\partial \alpha }{}_2{F}_1\left(p,\alpha ;p+q;c\right)\right|}_{\alpha =0},}\hfill \end{array}$$

$$\begin{array}{ccc}& & {\displaystyle R\left(X\right)=\frac{1}{1-\gamma }log\{\frac{b^{1-\gamma }B\left(p\gamma +\frac{1-\gamma }{a},q\gamma -\gamma +1\right)}{{\left[B(p,q)\right]}^{\gamma }}}\hfill \\ & & {\displaystyle ·{}_2{F}_1\left(p\gamma +\frac{1-\gamma }{a},\frac{(a+1)(1-\gamma )}{a};p\gamma +q\gamma +(1-\gamma )\left(\frac{1}{a}+1\right);c\right)\}}\hfill \end{array}$$

and

$$\begin{array}{ccc}& & {\displaystyle CE\left(X\right)=-{\int }_0^1\left[1-\frac{x^{ap}}{pB(p,q)b^{ap}}{F}_1\left(p,1-q,p+q,p+1;(1-c){\left(\frac{x}{b}\right)}^a,-c{\left(\frac{x}{b}\right)}^a\right)\right]}\hfill \\ & & {\displaystyle ·log\left[1-\frac{x^{ap}}{pB(p,q)b^{ap}}{F}_1\left(p,1-q,p+q,p+1;(1-c){\left(\frac{x}{b}\right)}^a,-c{\left(\frac{x}{b}\right)}^a\right)\right]dx}\hfill \end{array}$$

for $0<x^a<\frac{b^a}{1-c}$, $b>0$, $0<c<1$, $p>0$ and $q>0$.

16. Log-logistic distribution: for this distribution,

$$\begin{array}{c}\hfill {\displaystyle f_X\left(x\right)=\frac{b{a}^b{x}^{b-1}}{{\left(a^b+x^b\right)}^2},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle F_X\left(x\right)=\frac{x^b}{a^b+x^b},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle GM\left(X\right)=loga,}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle S\left(X\right)=loga-logb+2,}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle R\left(X\right)=loga-logb+\frac{2b\gamma loga}{1-\gamma }+\frac{1}{1-\gamma }logB\left(\gamma +\frac{\gamma -1}{b},\gamma +\frac{1-\gamma }{b}\right)}\end{array}$$

and

$$\begin{array}{c}\hfill {\displaystyle CE\left(X\right)=-\frac{a}{b}\Gamma \left(\frac{1}{b}\right)\left[{\Gamma }^{{}^{\prime }}\left(1-\frac{1}{b}\right)-{\Gamma }^{{}^{\prime }}\left(1\right)\Gamma \left(1-\frac{1}{b}\right)\right]}\end{array}$$

for $x>0$, $a>0$ and $b>0$.

17. Inverse Gaussian distribution [29]: for this distribution,

$$\begin{array}{c}\hfill {\displaystyle f_X\left(x\right)=\sqrt{\frac{a}{2\pi x^3}}exp\left[-\frac{a{(x-b)}^2}{2b^{2}x}\right],}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle F_X\left(x\right)=\Phi \left(\sqrt{\frac{a}{x}}\left(\frac{x}{b}-1\right)\right)-\Phi \left(-\sqrt{\frac{a}{x}}\left(\frac{x}{b}+1\right)\right)exp\left(\frac{2a}{b}\right),}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle GM\left(X\right)=\sqrt{\frac{2alogb}{\pi b}}exp\left(\frac{a}{b}\right)K_{-\frac{1}{2}}exp\left(\frac{a}{b}\right)+\sqrt{\frac{2a}{\pi b}}exp\left(\frac{a}{b}\right){\left.\frac{\partial }{\partial \alpha }{K}_{\alpha -\frac{1}{2}}exp\left(\frac{a}{b}\right)\right|}_{\alpha =0},}\end{array}$$

$$\begin{array}{ccc}& & {\displaystyle S\left(X\right)=\frac{1}{2}-\frac{1}{2}log\left(\frac{a}{2\pi }\right)+3\sqrt{\frac{alogb}{2\pi b}}exp\left(\frac{a}{b}\right)K_{-\frac{1}{2}}exp\left(\frac{a}{b}\right)}\hfill \\ & & {\displaystyle +3\sqrt{\frac{a}{2\pi b}}exp\left(\frac{a}{b}\right){\left.\frac{\partial }{\partial \alpha }{K}_{\alpha -\frac{1}{2}}exp\left(\frac{a}{b}\right)\right|}_{\alpha =0},}\hfill \end{array}$$

$$\begin{array}{c}\hfill {\displaystyle R\left(X\right)=\frac{\gamma }{2(1-\gamma )}log\frac{a}{2\pi b^3}+\frac{1}{1-\gamma }log\frac{2b}{\sqrt{\gamma }}+\frac{a}{(1-\gamma )b}+\frac{1}{1-\gamma }{K}_{1-\frac{3\gamma }{2}}\left(\frac{a}{b}\right)}\end{array}$$

and

$$\begin{array}{ccc}& & {\displaystyle CE\left(X\right)=-{\int }_0^{\infty }\left[\Phi \left(\sqrt{\frac{a}{x}}\left(1-\frac{x}{b}\right)\right)-\Phi \left(-\sqrt{\frac{a}{x}}\left(\frac{x}{b}+1\right)\right)exp\left(\frac{2a}{b}\right)\right]}\hfill \\ & & {\displaystyle ·log\left[\Phi \left(\sqrt{\frac{a}{x}}\left(1-\frac{x}{b}\right)\right)-\Phi \left(-\sqrt{\frac{a}{x}}\left(\frac{x}{b}+1\right)\right)exp\left(\frac{2a}{b}\right)\right]dx}\hfill \end{array}$$

for $x>0$, $a>0$ and $b>0$.

18. Gompertz distribution [30]: for this distribution,

$$\begin{array}{c}\hfill {\displaystyle f_X\left(x\right)=abexp\left[a+bx-aexp\left(bx\right)\right],}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle F_X\left(x\right)=1-exp\left[a-aexp\left(bx\right)\right],}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle GM\left(X\right)=-logb+aexp\left(a\right){\int }_1^{\infty }loglogyexp(-ay)dy,}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle S\left(X\right)=-a-log\left(ab\right)-\mathrm{Ei}(-a)exp\left(a\right)+a^{2}exp\left(a\right){\int }_1^{\infty }texp\left(at\right)dt,}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle R\left(X\right)=-logb-\frac{\gamma log\gamma }{1-\gamma }+\frac{a\gamma }{1-\gamma }+\frac{log\Gamma (\gamma ,a\gamma )}{1-\gamma }}\end{array}$$

and

$$\begin{array}{c}\hfill {\displaystyle CE\left(X\right)=\frac{1-aexp\left(a\right)\mathrm{Ei}(-a)}{b}}\end{array}$$

for $x>0$, $a>0$ and $b>0$.

19. Exponential distribution: for this distribution,

$$\begin{array}{c}\hfill {\displaystyle f_X\left(x\right)=aexp(-ax),}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle F_X\left(x\right)=1-exp(-ax),}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle GM\left(X\right)={\Gamma }^{{}^{\prime }}\left(1\right)-loga,}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle S\left(X\right)=1-loga,}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle R\left(X\right)=-loga-\frac{log\gamma }{1-\gamma }}\end{array}$$

and

$$\begin{array}{c}\hfill {\displaystyle CE\left(X\right)=\frac{1}{a}}\end{array}$$

for $x>0$ and $a>0$.

20. Inverse exponential distribution: for this distribution,

$$\begin{array}{c}\hfill {\displaystyle f_X\left(x\right)=b{x}^{-2}exp\left(-\frac{b}{x}\right),}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle F_X\left(x\right)=exp\left(-\frac{b}{x}\right),}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle GM\left(X\right)=logb-{\Gamma }^{{}^{\prime }}\left(1\right),}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle S\left(X\right)=logb-2{\Gamma }^{{}^{\prime }}\left(1\right)+1,}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle R\left(X\right)=logb+\frac{1-2\gamma }{1-\gamma }log\gamma +\frac{log\Gamma (2\gamma -1)}{1-\gamma }}\end{array}$$

and

$$\begin{array}{c}\hfill {\displaystyle CE\left(X\right)=-{\int }_0^{\infty }\left[1-exp\left(-\frac{b}{x}\right)\right]log\left[1-exp\left(-\frac{b}{x}\right)\right]dx}\end{array}$$

for $x>0$ and $b>0$.

21. Exponentiated exponential distribution [31]: for this distribution,

$$\begin{array}{c}\hfill {\displaystyle f_X\left(x\right)=abexp(-bx){\left[1-exp(-bx)\right]}^{a-1},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle F_X\left(x\right)={\left[1-exp(-bx)\right]}^a,}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle GM\left(X\right)=a{\int }_0^1{(1-y)}^{a-1}log(-logy)dy-logb,}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle S\left(X\right)=-log\left(ab\right)-{\Gamma }^{{}^{\prime }}\left(1\right)+\frac{{\Gamma }^{{}^{\prime }}(a+1)}{\Gamma (a+1)}+\frac{a-1}{a},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle R\left(X\right)=logb+\frac{\gamma loga}{1-\gamma }+\frac{1}{1-\gamma }logB\left(\gamma ,a\gamma -\gamma +1\right)}\end{array}$$

and

$$\begin{array}{c}\hfill {\displaystyle CE\left(X\right)=\frac{1}{b}\sum _{k=1}^{\infty }\frac{1}{k}B(0,ak+1)-\frac{1}{b}\sum _{k=1}^{\infty }\frac{1}{k}B(0,ak+a+1)}\end{array}$$

for $x>0$ and $a>0$.

22. Gamma distribution: for this distribution,

$$\begin{array}{c}\hfill {\displaystyle f_X\left(x\right)=\frac{b^a{x}^{a-1}exp\left(-bx\right)}{\Gamma \left(a\right)},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle F_X\left(x\right)=\frac{\gamma \left(a,bx\right)}{\Gamma \left(a\right)},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle GM\left(X\right)=\frac{{\Gamma }^{{}^{\prime }}\left(a\right)-\Gamma \left(a\right)logb}{{\left[\Gamma \left(a\right)\right]}^2},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle S\left(X\right)=a-alogb+log\Gamma \left(a\right)+(1-a)\left\{\frac{{\Gamma }^{{}^{\prime }}\left(a\right)}{{\left[\Gamma \left(a\right)\right]}^2}-\frac{logb}{\Gamma \left(a\right)}\right\},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle R\left(X\right)=logb+\frac{\gamma -a\gamma -1}{1-\gamma }log\gamma +\frac{1}{1-\gamma }log\Gamma (a\gamma -\gamma +1)-\frac{\gamma }{1-\gamma }log\Gamma \left(a\right)}\end{array}$$

and

$$\begin{array}{c}\hfill {\displaystyle CE\left(X\right)=-{\int }_0^{\infty }\frac{\Gamma \left(a,bx\right)}{\Gamma \left(a\right)}log\frac{\Gamma \left(a,bx\right)}{\Gamma \left(a\right)}dx}\end{array}$$

for $x>0$, $a>0$ and $b>0$.

23. Chisquare distribution: for this distribution,

$$\begin{array}{c}\hfill {\displaystyle f_X\left(x\right)=\frac{x^{\frac{k}{2}-1}exp\left(-\frac{x}{2}\right)}{2^{\frac{k}{2}}\Gamma \left(\frac{k}{2}\right)},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle F_X\left(x\right)=\frac{\gamma \left(\frac{k}{2},\frac{x}{2}\right)}{\Gamma \left(\frac{k}{2}\right)},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle GM\left(X\right)=\frac{{\Gamma }^{{}^{\prime }}\left(\frac{k}{2}\right)+\Gamma \left(\frac{k}{2}\right)log2}{{\left[\Gamma \left(\frac{k}{2}\right)\right]}^2},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle S\left(X\right)=\frac{k}{2}+\frac{k}{2}log2+log\Gamma \left(\frac{k}{2}\right)+\frac{k}{2}\left\{\frac{{\Gamma }^{{}^{\prime }}\left(\frac{k}{2}\right)}{{\left[\Gamma \left(\frac{k}{2}\right)\right]}^2}+\frac{log2}{\Gamma \left(\frac{k}{2}\right)}\right\},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle R\left(X\right)=-log2+\frac{\gamma -\frac{k}{2}\gamma -1}{1-\gamma }log\gamma +\frac{1}{1-\gamma }log\Gamma \left(\frac{k}{2}\gamma -\gamma +1\right)-\frac{\gamma }{1-\gamma }log\Gamma \left(\frac{k}{2}\right)}\end{array}$$

and

$$\begin{array}{c}\hfill {\displaystyle CE\left(X\right)=-{\int }_0^{\infty }\frac{\Gamma \left(\frac{k}{2},\frac{x}{2}\right)}{\Gamma \left(\frac{k}{2}\right)}log\frac{\Gamma \left(\frac{k}{2},\frac{x}{2}\right)}{\Gamma \left(\frac{k}{2}\right)}dx}\end{array}$$

for $x>0$ and $k>0$.

24. Chi distribution: for this distribution,

$$\begin{array}{c}\hfill {\displaystyle f_X\left(x\right)=\frac{x^{k-1}exp\left(-\frac{x^2}{2}\right)}{2^{\frac{k}{2}-1}\Gamma \left(\frac{k}{2}\right)},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle F_X\left(x\right)=\frac{\gamma \left(\frac{k}{2},\frac{x^2}{2}\right)}{\Gamma \left(\frac{k}{2}\right)},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle GM\left(X\right)=\frac{\Gamma \left(\frac{k}{2}\right)log2+{\Gamma }^{{}^{\prime }}\left(\frac{k}{2}\right)}{2\Gamma \left(\frac{k}{2}\right)},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle S\left(X\right)=-\frac{1}{2}log2+\frac{k}{2}-\frac{1-k}{2}\frac{{\Gamma }^{{}^{\prime }}\left(\frac{k}{2}\right)}{\Gamma \left(\frac{k}{2}\right)}+log\Gamma \left(\frac{k}{2}\right),}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle R\left(X\right)=\frac{1}{1-\gamma }log\left\{\frac{2^{\frac{\gamma -1}{2}}{\gamma }^{\frac{\gamma -\gamma k-1}{2}}}{{\left[\Gamma \left(\frac{k}{2}\right)\right]}^{\gamma }}\Gamma \left(\frac{\gamma k-\gamma +1}{2}\right)\right\}}\end{array}$$

and

$$\begin{array}{c}\hfill {\displaystyle CE\left(X\right)=-{\int }_0^{\infty }\frac{\Gamma \left(\frac{k}{2},\frac{x^2}{2}\right)}{\Gamma \left(\frac{k}{2}\right)}log\frac{\Gamma \left(\frac{k}{2},\frac{x^2}{2}\right)}{\Gamma \left(\frac{k}{2}\right)}dx}\end{array}$$

for $x>0$ and $k>0$.

25. Inverse gamma distribution: for this distribution,

$$\begin{array}{c}\hfill {\displaystyle f_X\left(x\right)=\frac{b^a{x}^{-a-1}exp\left(-\frac{b}{x}\right)}{\Gamma \left(a\right)},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle F_X\left(x\right)=\frac{\Gamma \left(a,\frac{b}{x}\right)}{\Gamma \left(a\right)},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle GM\left(X\right)=\frac{\Gamma \left(a\right)logb-{\Gamma }^{{}^{\prime }}\left(a\right)}{\Gamma \left(a\right)},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle S\left(X\right)=-alogb+(a+1)\frac{logb-{\Gamma }^{{}^{\prime }}\left(1\right)}{\Gamma \left(a\right)}+a+log\Gamma \left(a\right),}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle R\left(X\right)=logb+\frac{1-\gamma -a\gamma }{1-\gamma }log\gamma -\frac{\gamma }{1-\gamma }log\Gamma \left(a\right)+\frac{log\Gamma (a\gamma +\gamma -1)}{1-\gamma }}\end{array}$$

and

$$\begin{array}{c}\hfill {\displaystyle CE\left(X\right)=-{\int }_0^{\infty }\frac{\gamma \left(a,\frac{b}{x}\right)}{\Gamma \left(a\right)}log\frac{\gamma \left(a,\frac{b}{x}\right)}{\Gamma \left(a\right)}dx}\end{array}$$

for $x>0$, $a>0$ and $b>0$.

26. Inverse chisquare distribution: for this distribution,

$$\begin{array}{c}\hfill {\displaystyle f_X\left(x\right)=\frac{x^{-\frac{k}{2}-1}exp\left(-\frac{1}{2x}\right)}{2^{\frac{k}{2}}\Gamma \left(\frac{k}{2}\right)},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle F_X\left(x\right)=\frac{\Gamma \left(\frac{k}{2},\frac{1}{2x}\right)}{\Gamma \left(\frac{k}{2}\right)},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle GM\left(X\right)=\frac{-\Gamma \left(\frac{k}{2}\right)log2-{\Gamma }^{{}^{\prime }}\left(\frac{k}{2}\right)}{\Gamma \left(\frac{k}{2}\right)},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle S\left(X\right)=\frac{k}{2}log2-\left(\frac{k}{2}+1\right)\frac{log2+{\Gamma }^{{}^{\prime }}\left(1\right)}{\Gamma \left(\frac{k}{2}\right)}+\frac{k}{2}+log\Gamma \left(\frac{k}{2}\right),}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle R\left(X\right)=-log2+\frac{1-\gamma -\frac{k\gamma }{2}}{1-\gamma }log\gamma -\frac{\gamma }{1-\gamma }log\Gamma \left(\frac{k}{2}\right)+\frac{log\Gamma \left(\frac{k}{2}\gamma +\gamma -1\right)}{1-\gamma }}\end{array}$$

and

$$\begin{array}{c}\hfill {\displaystyle CE\left(X\right)=-{\int }_0^{\infty }\frac{\gamma \left(\frac{k}{2},\frac{1}{2x}\right)}{\Gamma \left(\frac{k}{2}\right)}log\frac{\gamma \left(\frac{k}{2},\frac{1}{2x}\right)}{\Gamma \left(\frac{k}{2}\right)}dx}\end{array}$$

for $x>0$ and $k>0$.

27. Inverse chi distribution: for this distribution,

$$\begin{array}{c}\hfill {\displaystyle f_X\left(x\right)=\frac{x^{-k-1}exp\left(-\frac{1}{2x^2}\right)}{2^{\frac{k}{2}-1}\Gamma \left(\frac{k}{2}\right)},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle F_X\left(x\right)=\frac{\Gamma \left(\frac{k}{2},\frac{1}{2x^2}\right)}{\Gamma \left(\frac{k}{2}\right)},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle GM\left(X\right)=-\frac{\Gamma \left(\frac{k}{2}\right)log2+{\Gamma }^{{}^{\prime }}\left(\frac{k}{2}\right)}{2\Gamma \left(\frac{k}{2}\right)},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle S\left(X\right)=-\frac{3}{2}log2-\frac{k}{2}-\frac{1+k}{2}\frac{{\Gamma }^{{}^{\prime }}\left(\frac{k}{2}\right)}{\Gamma \left(\frac{k}{2}\right)}+log\Gamma \left(\frac{k}{2}\right),}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle R\left(X\right)=\frac{1}{1-\gamma }log\left\{\frac{2^{\frac{3(\gamma -1)}{2}}{\gamma }^{\frac{1-\gamma -\gamma k}{2}}}{{\left[\Gamma \left(\frac{k}{2}\right)\right]}^{\gamma }}\Gamma \left(\frac{\gamma +\gamma k-1}{2}\right)\right\}}\end{array}$$

and

$$\begin{array}{c}\hfill {\displaystyle CE\left(X\right)=-{\int }_0^{\infty }\frac{\gamma \left(\frac{k}{2},\frac{1}{2x^2}\right)}{\Gamma \left(\frac{k}{2}\right)}log\frac{\gamma \left(\frac{k}{2},\frac{1}{2x^2}\right)}{\Gamma \left(\frac{k}{2}\right)}dx}\end{array}$$

for $x>0$ and $k>0$.

28. Rayleigh distribution: for this distribution,

$$\begin{array}{c}\hfill {\displaystyle f_X\left(x\right)=2b^{2}xexp\left[-{\left(bx\right)}^2\right],}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle F_X\left(x\right)=1-exp\left[-{\left(bx\right)}^2\right],}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle GM\left(X\right)=\frac{1}{2}{\Gamma }^{{}^{\prime }}\left(1\right)-logb,}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle S\left(X\right)=1-log\left(2b\right)-\frac{1}{2}{\Gamma }^{{}^{\prime }}\left(1\right),}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle R\left(X\right)=-log\left(2b\right)-\frac{log\gamma }{2}-\frac{\gamma log\gamma }{1-\gamma }+\frac{1}{1-\gamma }log\left(\frac{1+\gamma }{2}\right)}\end{array}$$

and

$$\begin{array}{c}\hfill {\displaystyle CE\left(X\right)=\frac{\sqrt{\pi }}{4b}}\end{array}$$

for $x>0$ and $b>0$.

29. Weibull distribution [32]: for this distribution,

$$\begin{array}{c}\hfill {\displaystyle f_X\left(x\right)=a{b}^a{x}^{a-1}exp\left[-{\left(bx\right)}^a\right],}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle F_X\left(x\right)=1-exp\left[-{\left(bx\right)}^a\right],}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle GM\left(X\right)=\frac{1}{a}{\Gamma }^{{}^{\prime }}\left(1\right)-logb,}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle S\left(X\right)=1-log\left(ab\right)+\frac{1-a}{a}{\Gamma }^{{}^{\prime }}\left(1\right),}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle R\left(X\right)=-log\left(ab\right)-\frac{log\gamma }{a}-\frac{\gamma log\gamma }{1-\gamma }+\frac{1}{1-\gamma }log\left(\frac{1-\gamma }{a}+\gamma \right)}\end{array}$$

and

$$\begin{array}{c}\hfill {\displaystyle CE\left(X\right)=\frac{1}{ab}\Gamma \left(1+\frac{1}{a}\right)}\end{array}$$

for $x>0$, $a>0$ and $b>0$.

30. Inverse Rayleigh distribution: for this distribution,

$$\begin{array}{c}\hfill {\displaystyle f_X\left(x\right)=2a{x}^{-3}exp\left(-a{x}^{-2}\right),}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle F_X\left(x\right)=exp\left(-a{x}^{-2}\right),}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle GM\left(X\right)=\frac{loga-{\Gamma }^{{}^{\prime }}\left(1\right)}{2},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle S\left(X\right)=1+\frac{loga}{2}-log2-\frac{{\Gamma }^{{}^{\prime }}\left(1\right)}{b},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle R\left(X\right)=\frac{1}{2}loga-log2+\frac{3\gamma -1}{2}\frac{log\gamma }{1-\gamma }+\frac{1}{1-\gamma }log\Gamma \left(\frac{3\gamma -1}{2}\right)}\end{array}$$

and

$$\begin{array}{c}\hfill {\displaystyle CE\left(X\right)=\frac{\sqrt{a}}{2}\Gamma \left(-\frac{1}{2}\right)\sum _{k=1}^{\infty }{k}^{-\frac{1}{2}}-\frac{\sqrt{a}}{2}\Gamma \left(-\frac{1}{2}\right)\sum _{k=1}^{\infty }\frac{\sqrt{k+1}}{k}}\end{array}$$

for $x>0$ and $a>0$.

31. Inverse Weibull distribution: for this distribution,

$$\begin{array}{c}\hfill {\displaystyle f_X\left(x\right)=ab{x}^{-b-1}exp\left(-a{x}^{-b}\right),}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle F_X\left(x\right)=exp\left(-a{x}^{-b}\right),}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle GM\left(X\right)=\frac{loga-{\Gamma }^{{}^{\prime }}\left(1\right)}{b},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle S\left(X\right)=1+\frac{loga}{b}-logb-\frac{{\Gamma }^{{}^{\prime }}\left(1\right)}{b},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle R\left(X\right)=\frac{1}{b}loga-logb+\left(\gamma +\frac{\gamma -1}{b}\right)\frac{log\gamma }{1-\gamma }+\frac{1}{1-\gamma }log\Gamma \left(\gamma +\frac{\gamma -1}{b}\right)}\end{array}$$

and

$$\begin{array}{c}\hfill {\displaystyle CE\left(X\right)=\frac{a^{\frac{1}{b}}}{b}\Gamma \left(-\frac{1}{b}\right)\sum _{k=1}^{\infty }{k}^{\frac{1}{b}-1}-\frac{a^{\frac{1}{b}}}{b}\Gamma \left(-\frac{1}{b}\right)\sum _{k=1}^{\infty }\frac{{(k+1)}^{\frac{1}{b}}}{k}}\end{array}$$

for $x>0$, $a>0$ and $b>0$.

32. Gumbel distribution [33]: for this distribution,

$$\begin{array}{c}\hfill {\displaystyle f_X\left(x\right)=\frac{1}{a}exp\left(-\frac{x-b}{a}\right)exp\left[-exp\left(-\frac{x-b}{a}\right)\right],}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle F_X\left(x\right)=exp\left[-exp\left(-\frac{x-b}{a}\right)\right],}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle GM\left(X\right)=\frac{1}{a}{\int }_{-\infty }^{\infty }logxexp\left(-\frac{x-b}{a}\right)exp\left[-exp\left(-\frac{x-b}{a}\right)\right]dx,}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle S\left(X\right)=1+loga-{\Gamma }^{{}^{\prime }}\left(1\right),}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle R\left(X\right)=\frac{1}{1-\gamma }log\left[\frac{\Gamma \left(\gamma \right)}{a^{\gamma -1}{\gamma }^{\gamma }}\right]}\end{array}$$

and

$$\begin{array}{c}\hfill {\displaystyle CE\left(X\right)={\int }_{-\infty }^{\infty }\left\{1-exp\left[-exp\left(-\frac{x-b}{a}\right)\right]\right\}log\left\{1-exp\left[-exp\left(-\frac{x-b}{a}\right)\right]\right\}dx}\end{array}$$

for $-\infty <x<\infty$, $a>0$ and $-\infty <b<\infty$.

33. Generalized extreme value distribution [34]: for this distribution,

$$\begin{array}{c}\hfill {\displaystyle f_X\left(x\right)=\frac{1}{a}{\left(1+\xi \frac{x-b}{a}\right)}^{-\frac{\xi +1}{\xi }}exp\left[-{\left(1+\xi \frac{x-b}{a}\right)}^{-\frac{1}{\xi }}\right],}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle F_X\left(x\right)=exp\left[-{\left(1+\xi \frac{x-b}{a}\right)}^{-\frac{1}{\xi }}\right],}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle GM\left(X\right)=\frac{1}{a}{\int }_{-\infty }^{\infty }logx{\left(1+\xi \frac{x-b}{a}\right)}^{-\frac{\xi +1}{\xi }}exp\left[-{\left(1+\xi \frac{x-b}{a}\right)}^{-\frac{1}{\xi }}\right]dx,}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle S\left(X\right)=1+loga-(\xi +1){\Gamma }^{{}^{\prime }}\left(1\right),}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle R\left(X\right)=\frac{1}{1-\gamma }log\left[\frac{\Gamma \left(\gamma \xi -\xi +\gamma \right)}{a^{\gamma -1}{\gamma }^{\gamma \xi -\xi +\gamma }}\right]}\end{array}$$

and

$$\begin{array}{c}\hfill {\displaystyle CE\left(X\right)=a\Gamma \left(-\xi \right)\sum _{k=1}^{\infty }{k}^{\xi }-a\Gamma \left(-\xi \right)\sum _{k=1}^{\infty }\frac{{(k+1)}^{\xi +1}}{k}}\end{array}$$

for $b-\frac{a}{\xi }<x<\infty$ if $\xi >0$, $-\infty <x<b-\frac{a}{\xi }$ if $\xi <0$, $-\infty <b<\infty$ and $a>0$.

34. Generalized gamma distribution [35]: for this distribution,

$$\begin{array}{c}\hfill {\displaystyle f_X\left(x\right)=\frac{p{a}^d{x}^{d-1}exp\left[-{\left(ax\right)}^p\right]}{\Gamma \left(\frac{d}{p}\right)},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle F_X\left(x\right)=\frac{\gamma \left(\frac{d}{p},{\left(ax\right)}^p\right)}{\Gamma \left(\frac{d}{p}\right)},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle GM\left(X\right)=\frac{\frac{1}{p}{\Gamma }^{{}^{\prime }}\left(\frac{d}{p}\right)-\Gamma \left(\frac{d}{p}\right)loga}{{\left[\Gamma \left(\frac{d}{p}\right)\right]}^3},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle S\left(X\right)=-logp-dloga+\Gamma \left(1+\frac{d}{p}\right)+log\Gamma \left(\frac{d}{p}\right)+\frac{1-d}{p}\frac{{\Gamma }^{{}^{\prime }}\left(\frac{d}{p}\right)}{{\left[\Gamma \left(\frac{d}{p}\right)\right]}^3}-\frac{(1-d)loga}{{\left[\Gamma \left(\frac{d}{p}\right)\right]}^2},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle R\left(X\right)=-\frac{logp}{1-\gamma }-loga-\frac{d\gamma -\gamma +1}{p}\frac{log\gamma }{1-\gamma }-\frac{\gamma }{1-\gamma }log\Gamma \left(\frac{d}{p}\right)+\frac{1}{1-\gamma }log\Gamma \left(\frac{d\gamma -\gamma +1}{p}\right)}\end{array}$$

and

$$\begin{array}{c}\hfill {\displaystyle CE\left(X\right)=-{\int }_0^{\infty }\frac{\Gamma \left(\frac{d}{p},{\left(ax\right)}^p\right)}{\Gamma \left(\frac{d}{p}\right)}log\frac{\Gamma \left(\frac{d}{p},{\left(ax\right)}^p\right)}{\Gamma \left(\frac{d}{p}\right)}dx}\end{array}$$

for $x>0$, $a>0$, $d>0$ and $p>0$.

35. Pareto distribution of type I [36]: for this distribution,

$$\begin{array}{c}\hfill {\displaystyle f_X\left(x\right)=\frac{a{K}^a}{x^{a+1}},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle F_X\left(x\right)=1-{\left(\frac{K}{x}\right)}^a,}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle GM\left(X\right)=logK+\frac{1}{a},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle S\left(X\right)=1-a+\frac{1}{a}+logK,}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle R\left(X\right)=logK+\frac{\gamma loga}{1-\gamma }-\frac{log\left(a\gamma +\gamma -1\right)}{1-\gamma }}\end{array}$$

and

$$\begin{array}{c}\hfill {\displaystyle CE\left(X\right)=-\frac{Ka}{{\left(a-1\right)}^2}}\end{array}$$

for $x\ge K$, $K>0$ and $a>0$.

36. Pareto distribution of type II [37]: for this distribution,

$$\begin{array}{c}\hfill {\displaystyle f_X\left(x\right)=\frac{a{b}^a}{{(x+b)}^{a+1}},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle F_X\left(x\right)=1-\frac{b^a}{{(x+b)}^a},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle GM\left(X\right)=logb+{\Gamma }^{{}^{\prime }}\left(1\right)-\frac{{\Gamma }^{{}^{\prime }}\left(a\right)}{\Gamma \left(a\right)},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle S\left(X\right)=-loga-alogb+(a+1)(alogb+1),}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle R\left(X\right)=logb+\frac{\gamma loga}{1-\gamma }-\frac{log(a\gamma +\gamma -1)}{1-\gamma }}\end{array}$$

and

$$\begin{array}{c}\hfill {\displaystyle CE\left(X\right)=\frac{ab}{{\left(a-1\right)}^2}}\end{array}$$

for $x>0$, $a>0$ and $b>0$.

37. Generalized Pareto distribution [38]: for this distribution,

$$\begin{array}{c}\hfill {\displaystyle f_X\left(x\right)={\left(1+\xi x\right)}^{-\frac{\xi +1}{\xi }},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle F_X\left(x\right)=1-{\left(1+\xi x\right)}^{-\frac{1}{\xi }},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle GM\left(X\right)=\left\{\begin{array}{cc}{\displaystyle -log\xi +{\Gamma }^{{}^{\prime }}\left(1\right)-\frac{{\Gamma }^{{}^{\prime }}\left(\frac{1}{\xi }\right)}{\xi \Gamma \left(1+\frac{1}{\xi }\right)},}\hfill & \mathrm{if} \xi >0,\hfill \\ \\ {\displaystyle -log(-\xi )+{\Gamma }^{{}^{\prime }}\left(1\right)-\frac{{\Gamma }^{{}^{\prime }}\left(1-\frac{1}{\xi }\right)}{\xi \Gamma \left(1-\frac{1}{\xi }\right)},}\hfill & \mathrm{if} \xi <0,\hfill \end{array}\right.}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle S\left(X\right)=\xi +1,}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle R\left(X\right)=\frac{1}{\gamma (\xi +1)-\xi }}\end{array}$$

and

$$\begin{array}{c}\hfill {\displaystyle CE\left(X\right)=\frac{1}{{\left(\xi -1\right)}^2}}\end{array}$$

for $0<x<\infty$ if $\xi >0$ and $0<x<-\frac{1}{\xi }$ if $\xi <0$.

38. Uniform distribution: for this distribution,

$$\begin{array}{c}\hfill {\displaystyle f_X\left(x\right)=\frac{1}{b-a},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle F_X\left(x\right)=\frac{x-a}{b-a},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle GM\left(X\right)=\frac{blogb-aloga-b+a}{b-a},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle S\left(X\right)=log(b-a),}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle R\left(X\right)=log(b-a)}\end{array}$$

and

$$\begin{array}{c}\hfill {\displaystyle CE\left(X\right)=-\frac{b-a}{4}}\end{array}$$

for $a<x<b$ and $\infty >b>a>-\infty$.

39. Power function distribution of type I: for this distribution,

$$\begin{array}{c}\hfill {\displaystyle f_X\left(x\right)=a{x}^{a-1},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle F_X\left(x\right)=x^a,}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle GM\left(X\right)=-\frac{1}{a},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle S\left(X\right)=1-\frac{1}{a}-loga,}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle R\left(X\right)=\frac{\gamma loga}{1-\gamma }-\frac{log\left(a\gamma -\gamma +1\right)}{1-\gamma }}\end{array}$$

and

$$\begin{array}{c}\hfill {\displaystyle CE\left(X\right)=\psi \left(\frac{1}{a}+2\right)-\psi \left(2\right)}\end{array}$$

for $0<x<1$ and $a>0$.

40. Power function distribution of type II: for this distribution,

$$\begin{array}{c}\hfill {\displaystyle f_X\left(x\right)=a{(1-x)}^{a-1},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle F_X\left(x\right)=1-{(1-x)}^a,}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle GM\left(X\right)=-\frac{1}{a},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle S\left(X\right)=1-\frac{1}{a}-loga,}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle R\left(X\right)=\frac{\gamma loga}{1-\gamma }-\frac{log\left(a\gamma -\gamma +1\right)}{1-\gamma }}\end{array}$$

and

$$\begin{array}{c}\hfill {\displaystyle CE\left(X\right)=\frac{a}{{(a+1)}^2}}\end{array}$$

for $0<x<1$ and $a>0$.

41. Arcsine distribution: for this distribution,

$$\begin{array}{c}\hfill {\displaystyle f_X\left(x\right)=\frac{1}{\pi \sqrt{x(1-x)}},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle F_X\left(x\right)=\frac{2}{\pi }\arcsin \left(\sqrt{x}\right),}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle GM\left(X\right)=\frac{{\Gamma }^{{}^{\prime }}\left(\frac{1}{2}\right)}{\sqrt{\pi }}-{\Gamma }^{{}^{\prime }}\left(1\right),}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle S\left(X\right)=log\pi +\frac{{\Gamma }^{{}^{\prime }}\left(\frac{1}{2}\right)}{\sqrt{\pi }}-{\Gamma }^{{}^{\prime }}\left(1\right),}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle R\left(X\right)=\frac{1}{1-\gamma }log\frac{B\left(\frac{\gamma }{2}-\gamma +1,\frac{\gamma }{2}-\gamma +1\right)}{{\pi }^{\gamma }}}\end{array}$$

and

$$\begin{array}{c}\hfill {\displaystyle CE\left(X\right)=-{\int }_0^{\infty }\left[1-\frac{2}{\pi }\arcsin \left(\sqrt{x}\right)\right]log\left[1-\frac{2}{\pi }\arcsin \left(\sqrt{x}\right)\right]dx}\end{array}$$

for $0<x<1$.

42. Beta distribution: for this distribution,

$$\begin{array}{c}\hfill {\displaystyle f_X\left(x\right)=\frac{x^{a-1}{(1-x)}^{b-1}}{B(a,b)},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle F_X\left(x\right)=I_x(a,b),}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle GM\left(X\right)=\frac{{\Gamma }^{{}^{\prime }}\left(a\right)}{\Gamma \left(a\right)}-\frac{{\Gamma }^{{}^{\prime }}\left(a+b\right)}{\Gamma \left(a+b\right)},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle S\left(X\right)=logB(a,b)+(1-a)\frac{{\Gamma }^{{}^{\prime }}\left(a\right)}{\Gamma \left(a\right)}+(1-b)\frac{{\Gamma }^{{}^{\prime }}\left(b\right)}{\Gamma \left(b\right)}-(2-a-b)\frac{{\Gamma }^{{}^{\prime }}(a+b)}{\Gamma (a+b)},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle R\left(X\right)=\frac{1}{1-\gamma }log\frac{B\left(a\gamma -\gamma +1,b\gamma -\gamma +1\right)}{{\left[B(a,b)\right]}^{\gamma }}}\end{array}$$

and

$$\begin{array}{c}\hfill {\displaystyle CE\left(X\right)=-{\int }_0^{\infty }{I}_{1-x}(b,a)log{I}_{1-x}(b,a)dx}\end{array}$$

for $0<x<1$, $a>0$ and $b>0$.

43. Inverted beta distribution: for this distribution,

$$\begin{array}{c}\hfill {\displaystyle f_X\left(x\right)=\frac{x^{a-1}{(1+x)}^{-a-b}}{B(a,b)},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle F_X\left(x\right)=I_{\frac{x}{1+x}}(a,b),}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle GM\left(X\right)=\frac{{\Gamma }^{{}^{\prime }}\left(a\right)}{\Gamma \left(a\right)}-\frac{{\Gamma }^{{}^{\prime }}\left(a+b\right)}{\Gamma \left(a+b\right)},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle S\left(X\right)=logB(a,b)+(1-a)\frac{{\Gamma }^{{}^{\prime }}\left(a\right)}{\Gamma \left(a\right)}+(a+b)\frac{{\Gamma }^{{}^{\prime }}(a+b)}{\Gamma (a+b)}-(1+b)\frac{{\Gamma }^{{}^{\prime }}\left(b\right)}{\Gamma \left(b\right)},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle R\left(X\right)=\frac{1}{1-\gamma }log\frac{B\left(a\gamma -\gamma +1,b\gamma +\gamma -1\right)}{{\left[B(a,b)\right]}^{\gamma }}}\end{array}$$

and

$$\begin{array}{c}\hfill {\displaystyle CE\left(X\right)=-{\int }_0^{\infty }{I}_{\frac{1}{1+x}}(b,a)log{I}_{\frac{1}{1+x}}(b,a)dx}\end{array}$$

for $0<x<1$, $a>0$ and $b>0$.

44. Kumaraswamy distribution [39]: for this distribution,

$$\begin{array}{c}\hfill {\displaystyle f_X\left(x\right)=ab{x}^{a-1}{\left(1-x^a\right)}^{b-1},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle F_X\left(x\right)=1-{\left(1-x^a\right)}^b,}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle GM\left(X\right)=b{\Gamma }^{{}^{\prime }}\left(1\right)-b\frac{{\Gamma }^{{}^{\prime }}\left(1+\frac{1}{a}\right)}{\Gamma \left(1+\frac{1}{a}\right)},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle S\left(X\right)=1-\frac{1}{b}-log\left(ab\right)+(1-a)b{\Gamma }^{{}^{\prime }}\left(1\right)-(1-a)b\frac{{\Gamma }^{{}^{\prime }}\left(\frac{1}{a}+1\right)}{\Gamma \left(\frac{1}{a}+1\right)},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle R\left(X\right)=-log\left(ab\right)+\frac{\gamma logb}{1-\gamma }+\frac{1}{1-\gamma }logB\left(\gamma +\frac{1-\gamma }{a},b\gamma +1-\gamma \right)}\end{array}$$

and

$$\begin{array}{c}\hfill {\displaystyle CE\left(X\right)=\frac{b}{a}B\left(\frac{1}{a},b+1\right)\left[\psi (b+1)-\psi \left(\frac{1}{a}+b+1\right)\right]}\end{array}$$

for $0<x<1$, $a>0$ and $b>0$.

45. Inverted Kumaraswamy distribution [40]: for this distribution,

$$\begin{array}{c}\hfill {\displaystyle f_X\left(x\right)=ab{(1+x)}^{-a-1}{\left[1-{(1+x)}^{-a}\right]}^{b-1},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle F_X\left(x\right)={\left[1-{(1+x)}^{-a}\right]}^b,}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle GM\left(X\right)=b{\int }_0^{1}log\left(y^{-\frac{1}{a}}-1\right){(1+y)}^{b-1}dy,}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle S\left(X\right)=-log\left(ab\right)+\left(1+\frac{1}{a}\right)\left[\frac{{\Gamma }^{{}^{\prime }}(b+1)}{\Gamma (b+1)}-{\Gamma }^{{}^{\prime }}\left(1\right)\right]+1-\frac{1}{b},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle R\left(X\right)=-loga+\frac{\gamma logb}{1-\gamma }+\frac{1}{1-\gamma }logB\left(\gamma +\frac{\gamma -1}{a},b\gamma -\gamma +1\right)}\end{array}$$

and

$$\begin{array}{c}\hfill {\displaystyle CE\left(X\right)=\frac{1}{a}\sum _{k=1}^{\infty }\frac{1}{k}B\left(-\frac{1}{a},kb+1\right)-\frac{1}{a}\sum _{k=1}^{\infty }\frac{1}{k}B\left(-\frac{1}{a},kb+b+1\right)}\end{array}$$

for $x>0$, $a>0$ and $b>0$.

46. Normal distribution: for this distribution,

$$\begin{array}{c}\hfill {\displaystyle f_X\left(x\right)=\frac{1}{\sqrt{2\pi }a}exp\left[-\frac{{(x-b)}^2}{2a^2}\right],}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle F_X\left(x\right)=\Phi \left(\frac{x-b}{a}\right),}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle GM\left(X\right)=\frac{1}{\sqrt{2\pi }a}{\int }_{-\infty }^{\infty }logxexp\left[-\frac{{(x-b)}^2}{2a^2}\right]dx,}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle S\left(X\right)=log\left(\sqrt{2\pi }a\right)+\frac{1}{2},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle R\left(X\right)=log\left(\sqrt{2\pi }a\right)-\frac{log\gamma }{2(1-\gamma )}}\end{array}$$

and

$$\begin{array}{c}\hfill {\displaystyle CE\left(X\right)=-\frac{\sqrt{2}a}{4}{\int }_{-\infty }^{\infty }\mathrm{erfc}\left(x\right)log\left[\mathrm{erfc}\left(x\right)\right]dx}\end{array}$$

for $-\infty <x<\infty$, $a>0$ and $-\infty <b<\infty$.

47. Lognormal distribution: for this distribution,

$$\begin{array}{c}\hfill {\displaystyle f_X\left(x\right)=\frac{1}{\sqrt{2\pi }ax}exp\left[-\frac{{(logx-b)}^2}{2a^2}\right],}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle F_X\left(x\right)=\Phi \left(\frac{logx-b}{a}\right),}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle GM\left(X\right)=b,}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle S\left(X\right)=log\left(\sqrt{2\pi }a\right)+b+\frac{1}{2},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle R\left(X\right)=\sqrt{2\pi }aexp\left(b\right)+\frac{(1-\gamma )a^2}{2\gamma }-\frac{log\gamma }{2(1-\gamma )}}\end{array}$$

and

$$\begin{array}{c}\hfill {\displaystyle CE\left(X\right)=-{\int }_0^{\infty }\Phi \left(\frac{b-logx}{a}\right)log\left[\Phi \left(\frac{b-logx}{a}\right)\right]dx}\end{array}$$

for $x>0$, $a>0$ and $-\infty <b<\infty$.

48. Half normal distribution: for this distribution,

$$\begin{array}{c}\hfill {\displaystyle f_X\left(x\right)=\frac{\sqrt{2}}{\sqrt{\pi }a}exp\left(-\frac{x^2}{2a^2}\right),}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle F_X\left(x\right)=\mathrm{erf}\left(\frac{x}{\sqrt{2}a}\right),}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle GM\left(X\right)=\frac{log2}{2}+loga+\frac{1}{\sqrt{\pi }}{\Gamma }^{{}^{\prime }}\left(\frac{1}{2}\right),}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle S\left(X\right)=\frac{1}{2}-\frac{1}{2}log\frac{2}{\pi }+loga,}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle R\left(X\right)=a\sqrt{\frac{2}{\pi }}-\frac{1}{2}\frac{log\gamma }{1-\gamma }}\end{array}$$

and

$$\begin{array}{c}\hfill {\displaystyle CE\left(X\right)=-\sqrt{2}a{\int }_{-\infty }^{\infty }\mathrm{erf}\left(x\right)log\left[\mathrm{erf}\left(x\right)\right]dx}\end{array}$$

for $x>0$ and $a>0$.

49. Student’s t distribution [41]: for this distribution,

$$\begin{array}{c}\hfill {\displaystyle f_X\left(x\right)=\frac{\Gamma \left(\frac{a+1}{2}\right)}{\sqrt{a\pi }\Gamma \left(\frac{a}{2}\right)}{\left(1+\frac{x^2}{a}\right)}^{-\frac{a+1}{2}},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle F_X\left(x\right)=\frac{1}{2}-\frac{x\Gamma \left(\frac{a+1}{2}\right)}{\sqrt{a\pi }\Gamma \left(\frac{a}{2}\right)}{}_2{F}_1\left(\frac{1}{2},\frac{a+1}{2};\frac{3}{2};-\frac{x^2}{a}\right),}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle GM\left(X\right)=\frac{\Gamma \left(\frac{a+1}{2}\right)}{\sqrt{a\pi }\Gamma \left(\frac{a}{2}\right)}{\int }_{-\infty }^{\infty }logx{\left(1+\frac{x^2}{a}\right)}^{-\frac{a+1}{2}}dx,}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle S\left(X\right)=log\frac{\sqrt{a\pi }\Gamma \left(\frac{a}{2}\right)}{\Gamma \left(\frac{a+1}{2}\right)}+\frac{\sqrt{a}(a+1)}{2}\left[\frac{{\Gamma }^{{}^{\prime }}\left(\frac{a+1}{2}\right)}{\Gamma \left(\frac{a+1}{2}\right)}-\frac{{\Gamma }^{{}^{\prime }}\left(\frac{a}{2}\right)}{\Gamma \left(\frac{a}{2}\right)}\right],}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle R\left(X\right)=\frac{1}{1-\gamma }log\left\{2{\left[\frac{\Gamma \left(\frac{a+1}{2}\right)}{\sqrt{a\pi }\Gamma \left(\frac{a}{2}\right)}\right]}^{\gamma }B\left(\frac{a\gamma +\gamma -1}{2},\frac{1}{2}\right)\right\}}\end{array}$$

and

$$\begin{array}{c}\hfill {\displaystyle CE\left(X\right)=-{\int }_{-\infty }^{\infty }\left[\frac{1}{2}+\frac{x\Gamma \left(\frac{a+1}{2}\right)}{\sqrt{a\pi }\Gamma \left(\frac{a}{2}\right)}{}_2{F}_1\left(\frac{1}{2},\frac{a+1}{2};\frac{3}{2};-\frac{x^2}{a}\right)\right]log\left[\frac{1}{2}+\frac{x\Gamma \left(\frac{a+1}{2}\right)}{\sqrt{a\pi }\Gamma \left(\frac{a}{2}\right)}{}_2{F}_1\left(\frac{1}{2},\frac{a+1}{2};\frac{3}{2};-\frac{x^2}{a}\right)\right]dx}\end{array}$$

for $-\infty <x<\infty$ and $a>0$.

50. Cauchy distribution: for this distribution,

$$\begin{array}{c}\hfill {\displaystyle f_X\left(x\right)=\frac{1}{\pi \left(1+x^2\right)},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle F_X\left(x\right)=\frac{1}{2}+\frac{1}{\pi }\arctan \left(x\right),}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle GM\left(X\right)=\frac{1}{\pi }{\int }_{-\infty }^{\infty }\frac{logx}{1+x^2}dx,}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle S\left(X\right)=log\pi +{\Gamma }^{{}^{\prime }}\left(1\right)-\frac{{\Gamma }^{{}^{\prime }}\left(\frac{1}{2}\right)}{\sqrt{\pi }},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle R\left(X\right)=\frac{1}{1-\gamma }log\left[\frac{2}{{\pi }^{\gamma }}B\left(\gamma -\frac{1}{2},\frac{1}{2}\right)\right]}\end{array}$$

and

$$\begin{array}{c}\hfill {\displaystyle CE\left(X\right)=-{\int }_{-\infty }^{\infty }\left(\frac{1}{2}-\frac{1}{\pi }\arctan \left(x\right)\right)log\left[\frac{1}{2}-\frac{1}{\pi }\arctan \left(x\right)\right]dx}\end{array}$$

for $-\infty <x<\infty$.

51. Laplace distribution [42]: for this distribution,

$$\begin{array}{c}\hfill {\displaystyle f_X\left(x\right)=\frac{1}{2a}exp\left(-\frac{\mid x-b\mid }{a}\right),}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle F_X\left(x\right)=\left\{\begin{array}{cc}{\displaystyle \frac{1}{2}exp\left(\frac{x-b}{a}\right),}\hfill & \mathrm{if} x\le b,\hfill \\ \\ {\displaystyle 1-\frac{1}{2}exp\left(-\frac{x-b}{a}\right),}\hfill & \mathrm{if} x\ge b,\hfill \end{array}\right.}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle GM\left(X\right)=\frac{1}{2a}{\int }_{-\infty }^{\infty }logxexp\left(-\frac{\mid x-b\mid }{a}\right)dx,}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle S\left(X\right)=1+log\left(2a\right),}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle R\left(X\right)=log\left(2a\right)-\frac{log\gamma }{1-\gamma }}\end{array}$$

and

$$\begin{array}{c}\hfill {\displaystyle CE\left(X\right)=\frac{a}{2}+\frac{a}{2}log2-a{\left.\frac{\partial }{\partial \alpha }{B}_{\frac{1}{2}}\left(0,2+\alpha \right)\right|}_{\alpha =0}}\end{array}$$

for $-\infty <x<\infty$, $a>0$ and $-\infty <b<\infty$.

52. Logistic distribution of type I: for this distribution,

$$\begin{array}{c}\hfill {\displaystyle f_X\left(x\right)=\frac{acexp(-cx)}{{\left[1+exp(-cx)\right]}^{a+1}},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle F_X\left(x\right)=\frac{1}{{\left[1+exp(-cx)\right]}^a},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle GM\left(X\right)=ac{\int }_{-\infty }^{\infty }logx\frac{exp(-cx)}{{\left[1+exp(-cx)\right]}^{a+1}}dx,}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle S\left(X\right)=-loga-logc+(a+1)\psi (a+1)-a\psi \left(a\right)-{\Gamma }^{{}^{\prime }}\left(1\right),}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle R\left(X\right)=-logc+\frac{1}{1-\gamma }log\left[a^{\gamma }B\left(a\gamma ,\gamma \right)\right]}\end{array}$$

and

$$\begin{array}{c}\hfill {\displaystyle CE\left(X\right)=-{\int }_{-\infty }^{\infty }\left\{1-\frac{1}{{\left[1+exp(-cx)\right]}^a}\right\}log\left\{1-\frac{1}{{\left[1+exp(-cx)\right]}^a}\right\}dx}\end{array}$$

for $-\infty <x<\infty$, $a>0$ and $c>0$.

53. Logistic distribution of type II: for this distribution,

$$\begin{array}{c}\hfill {\displaystyle f_X\left(x\right)=\frac{acexp(-acx)}{{\left[1+exp(-cx)\right]}^{a+1}},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle F_X\left(x\right)=1-\frac{1}{{\left[1+exp\left(cx\right)\right]}^a},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle GM\left(X\right)=ac{\int }_{-\infty }^{\infty }logx\frac{exp(-acx)}{{\left[1+exp(-cx)\right]}^{a+1}}dx,}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle S\left(X\right)=-loga-logc+(a+1)\psi (a+1)-{\Gamma }^{{}^{\prime }}\left(1\right)-a\psi \left(a\right),}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle R\left(X\right)=-logc+\frac{1}{1-\gamma }log\left[a^{\gamma }B\left(\gamma ,a\gamma \right)\right]}\end{array}$$

and

$$\begin{array}{c}\hfill {\displaystyle CE\left(X\right)=a{\int }_{-\infty }^{\infty }{\left[1+exp(-cx)\right]}^{-a}log\left[1+exp(-cx)\right]dx}\end{array}$$

for $-\infty <x<\infty$, $a>0$ and $c>0$.

54. Logistic distribution of type III: for this distribution,

$$\begin{array}{c}\hfill {\displaystyle f_X\left(x\right)=\frac{cexp(-acx)}{B(a,a){\left[1+exp(-cx)\right]}^{2a}},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle F_X\left(x\right)=I_{\frac{1}{1+exp(-cx)}}(a,a),}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle GM\left(X\right)=\frac{c}{B(a,a)}{\int }_{-\infty }^{\infty }logx\frac{exp(-acx)}{{\left[1+exp(-cx)\right]}^{2a}}dx,}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle S\left(X\right)=logB(a,a)-logc+2a\psi \left(2a\right)-2a\psi \left(a\right),}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle R\left(X\right)=-logc+\frac{1}{1-\gamma }log\frac{B\left(a\gamma ,a\gamma \right)}{{\left[B(a,a)\right]}^{\gamma }}}\end{array}$$

and

$$\begin{array}{c}\hfill {\displaystyle CE\left(X\right)=-{\int }_{-\infty }^{\infty }{I}_{\frac{exp(-cx)}{1+exp(-cx)}}(a,a)log{I}_{\frac{exp(-cx)}{1+exp(-cx)}}(a,a)dx}\end{array}$$

for $-\infty <x<\infty$, $a>0$ and $c>0$.

55. Logistic distribution of type IV [43]: for this distribution,

$$\begin{array}{c}\hfill {\displaystyle f_X\left(x\right)=\frac{cexp(-bcx)}{B(a,b){\left[1+exp(-cx)\right]}^{a+b}},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle F_X\left(x\right)=I_{\frac{1}{1+exp(-cx)}}(a,b),}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle GM\left(X\right)=\frac{c}{B(a,b)}{\int }_{-\infty }^{\infty }logx\frac{exp(-bcx)}{{\left[1+exp(-cx)\right]}^{a+b}}dx,}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle S\left(X\right)=logB(a,b)-logc+(a+b)\psi (a+b)-a\psi \left(a\right)-b\psi \left(b\right),}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle R\left(X\right)=-logc+\frac{1}{1-\gamma }log\frac{B\left(a\gamma ,b\gamma \right)}{{\left[B(a,b)\right]}^{\gamma }}}\end{array}$$

and

$$\begin{array}{c}\hfill {\displaystyle CE\left(X\right)=-{\int }_{-\infty }^{\infty }{I}_{\frac{exp(-cx)}{1+exp(-cx)}}(b,a)log{I}_{\frac{exp(-cx)}{1+exp(-cx)}}(b,a)dx}\end{array}$$

for $-\infty <x<\infty$, $a>0$, $b>0$ and $c>0$.

56. Burr distribution [44]: for this distribution,

$$\begin{array}{c}\hfill {\displaystyle f_X\left(x\right)=ck{x}^{c-1}{\left(1+x^c\right)}^{-k-1},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle F_X\left(x\right)=1-{\left(1+x^c\right)}^{-k},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle GM\left(X\right)=\frac{{\Gamma }^{{}^{\prime }}\left(1\right)\Gamma \left(k\right)-{\Gamma }^{{}^{\prime }}\left(k\right)}{c\Gamma \left(k\right)},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle S\left(X\right)=1+\frac{1}{k}-log\left(ck\right)+\frac{1-c}{c\Gamma \left(k\right)}\left[{\Gamma }^{{}^{\prime }}\left(1\right)\Gamma \left(k\right)-{\Gamma }^{{}^{\prime }}\left(k\right)\right],}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle R\left(X\right)=\frac{\gamma log\left(ck\right)}{1-\gamma }+\frac{1}{1-\gamma }logB\left(k\gamma +\frac{\gamma -1}{c},\gamma +\frac{1-\gamma }{c}\right)}\end{array}$$

and

$$\begin{array}{c}\hfill {\displaystyle CE\left(X\right)=-\frac{k}{c}B\left(\frac{1}{c},k-\frac{1}{c}\right)\left[\psi \left(\frac{1}{c}\right)-\psi \left(k\right)\right]}\end{array}$$

for $x>0$, $k>0$ and $c>0$.

57. Dagum distribution [45]: for this distribution,

$$\begin{array}{c}\hfill {\displaystyle f_X\left(x\right)=ap{x}^{ap-1}{\left(1+x^a\right)}^{-p-1},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle F_X\left(x\right)={\left(1+x^a\right)}^{-p},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle GM\left(X\right)=-\frac{{\Gamma }^{{}^{\prime }}\left(1\right)}{a}+\frac{{\Gamma }^{{}^{\prime }}\left(p\right)}{a\Gamma \left(p\right)},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle S\left(X\right)=-log\left(ap\right)+(p+1)\left[{\Gamma }^{{}^{\prime }}\left(1\right)-\frac{{\Gamma }^{{}^{\prime }}(p+1)}{\Gamma (p+1)}\right]+\left(\frac{1}{a}-p\right)\left[-{\Gamma }^{{}^{\prime }}\left(1\right)+\frac{{\Gamma }^{{}^{\prime }}\left(p\right)}{\Gamma \left(p\right)}\right],}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle R\left(X\right)=\frac{1}{1-\gamma }log\left[a^{\gamma -1}{p}^{\gamma }B\left(\frac{\gamma -1}{a},p\gamma +\frac{1-\gamma }{a}\right)\right]}\end{array}$$

and

$$\begin{array}{c}\hfill {\displaystyle CE\left(X\right)=\sum _{k=1}^{\infty }\frac{1}{k}B\left(pk-\frac{1}{a},\frac{1}{a}\right)-\sum _{k=1}^{\infty }\frac{1}{k}B\left(pk+p-\frac{1}{a},\frac{1}{a}\right)}\end{array}$$

for $x>0$, $a>0$ and $p>0$.

58. J shaped distribution [46]: for this distribution,

$$\begin{array}{c}\hfill {\displaystyle f_X\left(x\right)=2a(1-x){\left[x(2-x)\right]}^{a-1},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle F_X\left(x\right)={\left[x(2-x)\right]}^a,}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle GM\left(X\right)=a{2}^a{\left.\frac{\partial }{\partial \alpha }\left[\frac{1}{a+\alpha }{}_2{F}_1\left(a+\alpha ,1-a;a+\alpha +1;\frac{1}{2}\right)\right]\right|}_{\alpha =0},}\end{array}$$

$$\begin{array}{ccc}& & {\displaystyle S\left(X\right)=-log\left(2a\right)+a(1-a)2^a{\left.\frac{\partial }{\partial \alpha }\left[\frac{1}{a+\alpha }{}_2{F}_1\left(a+\alpha ,1-a;a+\alpha +1;\frac{1}{2}\right)\right]\right|}_{\alpha =0}}\hfill \\ & & {\displaystyle -a{2}^a{\left.\frac{\partial }{\partial \alpha }\left[B(a,\alpha +2){}_2{F}_1\left(a,1-a;a+\alpha +2;\frac{1}{2}\right)\right]\right|}_{\alpha =0}}\hfill \\ & & {\displaystyle +\frac{2^a(1-a)}{1+a}{\left.\frac{\partial }{\partial \alpha }\left[2^{\alpha }{}_2{F}_1\left(a,1-\alpha -a;a+1;\frac{1}{2}\right)\right]\right|}_{\alpha =0},}\hfill \end{array}$$

$$\begin{array}{c}\hfill {\displaystyle R\left(X\right)=\frac{1}{1-\gamma }log\left[a^{\gamma }{2}^{a\gamma }B\left(a\gamma -\gamma +1,\gamma +1\right){}_2{F}_1\left(a\gamma -\gamma +1,\gamma -a\gamma ;a\gamma +2;\frac{1}{2}\right)\right]}\end{array}$$

and

$$\begin{array}{ccc}& & {\displaystyle CE\left(X\right)=\sum _{k=1}^{\infty }\frac{2^{ak}}{k(ak+1)}{}_2{F}_1\left(ak+1,-ak;ak+2;\frac{1}{2}\right)}\hfill \\ & & {\displaystyle -\sum _{k=1}^{\infty }\frac{2^{a(k+1)}}{k(ak+a+1)}{}_2{F}_1\left(ak+a+1,-ak-a;ak+a+2;\frac{1}{2}\right)}\hfill \end{array}$$

for $0<x<1$ and $a>0$.

59. Nadarajah–Haghighi distribution [47]: for this distribution,

$$\begin{array}{c}\hfill {\displaystyle f_X\left(x\right)=ab{(1+bx)}^{a-1}exp\left[1-{(1+bx)}^a\right],}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle F_X\left(x\right)=1-exp\left[1-{(1+bx)}^a\right],}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle GM\left(X\right)=-elogb+e{\int }_0^{\infty }log\left(y^{\frac{1}{a}}-1\right)exp(-y)dy,}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle S\left(X\right)=-log\left(ab\right)+(1-a)e{\left.\frac{\partial }{\partial \alpha }\Gamma \left(\frac{\alpha }{a}+1,1\right)\right|}_{\alpha =0}+1,}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle R\left(X\right)=\frac{\gamma }{1-\gamma }-\frac{log\left(ab\right)}{1-\gamma }-\left(\gamma +\frac{1-\gamma }{a}\right)\frac{log\gamma }{1-\gamma }+\frac{1}{1-\gamma }log\Gamma \left(\gamma +\frac{1-\gamma }{a},\gamma \right)}\end{array}$$

and

$$\begin{array}{c}\hfill {\displaystyle CE\left(X\right)=\frac{1}{b}+\frac{e}{b}\left(\frac{1}{a}-1\right)\Gamma \left(\frac{1}{a}+1,1\right)}\end{array}$$

for $x>0$, $a>0$ and $b>0$.

60. Two-sided power distribution [48]: for this distribution,

$$\begin{array}{c}\hfill {\displaystyle f_X\left(x\right)=\left\{\begin{array}{cc}{\displaystyle a{\left(\frac{x}{\theta }\right)}^{a-1},}\hfill & \mathrm{if} 0<x\le \theta ,\hfill \\ \\ {\displaystyle a{\left(\frac{1-x}{1-\theta }\right)}^{a-1},}\hfill & \mathrm{if} \theta \le x<1,\hfill \end{array}\right.}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle F_X\left(x\right)=\left\{\begin{array}{cc}{\displaystyle \theta {\left(\frac{x}{\theta }\right)}^a,}\hfill & \mathrm{if} 0<x\le \theta ,\hfill \\ \\ {\displaystyle 1-(1-\theta ){\left(\frac{1-x}{1-\theta }\right)}^a,}\hfill & \mathrm{if} \theta \le x<1,\hfill \end{array}\right.}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle GM\left(X\right)=\theta log\theta -\frac{\theta }{a}+\frac{a}{{(1-\theta )}^{a-1}}{\left.\frac{\partial }{\partial \alpha }{B}_{1-\theta }(a,\alpha +1)\right|}_{\alpha =0},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle S\left(X\right)=-loga-\frac{1-a}{a},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle R\left(X\right)=\frac{1}{1-\gamma }log\frac{a^{\gamma }}{a\gamma -\gamma +1}}\end{array}$$

and

$$\begin{array}{ccc}& & {\displaystyle CE\left(X\right)=\sum _{k=1}^{\infty }\frac{{\theta }^{k+1}}{k(k\theta +1)}-\sum _{k=1}^{\infty }\frac{1}{k}\sum _{m=0}^k\left(\genfrac{}{}{0pt}{}{k}{m}\right){(-1)}^m\frac{{(1-\theta )}^{m+1}}{am+1}-\sum _{k=1}^{\infty }\frac{{\theta }^{k+2}}{k(k\theta +k+1)}}\hfill \\ & & {\displaystyle +\sum _{k=1}^{\infty }\frac{1}{k}\sum _{m=0}^{k+1}\left(\genfrac{}{}{0pt}{}{k+1}{m}\right){(-1)}^m\frac{{(1-\theta )}^{m+1}}{am+1}}\hfill \end{array}$$

for $a>0$.

61. Power Lindley distribution [49]: for this distribution,

$$\begin{array}{c}\hfill {\displaystyle f_X\left(x\right)=\frac{a{b}^2}{b+1}\left(1+x^a\right)x^{a-1}exp\left(-b{x}^a\right),}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle F_X\left(x\right)=1-\left(1+\frac{b{x}^a}{b+1}\right)exp\left(-b{x}^a\right),}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle GM\left(X\right)=\frac{b{\Gamma }^{{}^{\prime }}\left(1\right)+{\Gamma }^{{}^{\prime }}\left(2\right)}{a(b+1)}-\frac{logb}{a},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle S\left(X\right)=-log\frac{ab}{b+1}-\frac{logb}{a}+\frac{b(1-a)}{a(b+1)}\left[{\Gamma }^{{}^{\prime }}\left(1\right)+\frac{1}{b}{\Gamma }^{{}^{\prime }}\left(2\right)\right]+\frac{b+2}{b+1}-\frac{exp\left(b\right)}{b+1}{\left.\frac{\partial }{\partial \alpha }\Gamma (\alpha +2,b)\right|}_{\alpha =0},}\end{array}$$

$$\begin{array}{ccc}& & {\displaystyle R\left(X\right)=-loga-\frac{\gamma }{1-\gamma }log(b+1)}\hfill \\ & & {\displaystyle +\frac{1}{1-\gamma }log\left[\Gamma \left(\gamma +\frac{1-\gamma }{a}\right)\Psi \left(\gamma +\frac{1-\gamma }{a},\frac{1-\gamma }{a}+1;b\gamma \right)\right]}\hfill \end{array}$$

and

$$\begin{array}{ccc}& & {\displaystyle CE\left(X\right)=\frac{1}{a{b}^{\frac{1}{a}}}\left[\Gamma \left(\frac{1}{a}+1\right)+\frac{1}{b+1}\Gamma \left(\frac{1}{a}+2\right)\right]}\hfill \\ & & {\displaystyle -\frac{\Gamma \left(\frac{1}{a}\right){(1+b)}^{\frac{1}{a}}}{a{b}^{\frac{1}{a}}}{\left.\frac{\partial }{\partial \alpha }\Psi \left(\frac{1}{a},\frac{1}{a}+\alpha +2;b+1\right)\right|}_{\alpha =0}}\hfill \end{array}$$

for $x>0$, $a>0$ and $b>0$.

62. Modified slash Lindley–Weibull distribution [50]: for this distribution,

$$\begin{array}{c}\hfill {\displaystyle f_X\left(x\right)=\frac{2a^3{b}^2{x}^{a-1}}{a+1}\frac{(a+2)x^a+2b^a}{{\left(a{x}^a+2b^a\right)}^3},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle F_X\left(x\right)=\frac{a^2{x}^a}{a+1}\frac{(a+1)x^a+2b^a}{{\left(a{x}^a+2b^a\right)}^2},}\end{array}$$

$$\begin{array}{ccc}& & {\displaystyle GM\left(X\right)=\frac{(a+2)log2}{4(a+1)}-\frac{(a+2)loga}{4(a+1)}+\frac{(a+2)logb}{2(a+1)}{\Gamma }^{{}^{\prime }}\left(2\right)-\frac{a+2}{a(a+1)}{\Gamma }^{{}^{\prime }}\left(1\right)}\hfill \\ & & {\displaystyle +\frac{b^{1-a}{a}^{2-\frac{1}{a}}log2}{2^{\frac{5}{2}}(a+1)}\Gamma \left(\frac{1}{a}\right)\Gamma \left(3-\frac{1}{a}\right)+\frac{b^{1-a}{a}^{2-\frac{1}{a}}logb}{2^{a-\frac{1}{2}}(a+1)}\Gamma \left(\frac{1}{a}\right)\Gamma \left(3-\frac{1}{a}\right)}\hfill \\ & & {\displaystyle -\frac{b^{1-a}\sqrt{a}log2}{2^{\frac{3}{2}}(a+1)}\Gamma \left(\frac{1}{a}\right)\Gamma \left(3-\frac{1}{a}\right)+\frac{b^{1-a}{a}^{1-\frac{1}{a}}}{2^{\frac{3}{2}}(a+1)}{\Gamma }^{{}^{\prime }}\left(\frac{1}{a}\right)\Gamma \left(3-\frac{1}{a}\right)}\hfill \\ & & {\displaystyle -\frac{b^{1-a}{a}^{1-\frac{1}{a}}}{2^{\frac{3}{2}}(a+1)}\Gamma \left(\frac{1}{a}\right){\Gamma }^{{}^{\prime }}\left(3-\frac{1}{a}\right),}\hfill \end{array}$$

$$\begin{array}{ccc}& & {\displaystyle S\left(X\right)=-log\frac{2a^3{b}^a}{a+1}+\frac{(1-a)(a+2)log2}{4(a+1)}-\frac{(1-a)(a+2)loga}{4(a+1)}}\hfill \\ & & {\displaystyle +\frac{(1-a)(a+2)logb}{2(a+1)}{\Gamma }^{{}^{\prime }}\left(2\right)-\frac{(1-a)(a+2)}{a(a+1)}{\Gamma }^{{}^{\prime }}\left(1\right)}\hfill \\ & & {\displaystyle +\frac{(1-a)b^{1-a}{a}^{2-\frac{1}{a}}log2}{2^{\frac{5}{2}}(a+1)}\Gamma \left(\frac{1}{a}\right)\Gamma \left(3-\frac{1}{a}\right)+\frac{(1-a)b^{1-a}{a}^{2-\frac{1}{a}}logb}{2^{a-\frac{1}{2}}(a+1)}\Gamma \left(\frac{1}{a}\right)\Gamma \left(3-\frac{1}{a}\right)}\hfill \\ & & {\displaystyle -\frac{(1-a)b^{1-a}\sqrt{a}log2}{2^{\frac{3}{2}}(a+1)}\Gamma \left(\frac{1}{a}\right)\Gamma \left(3-\frac{1}{a}\right)+\frac{(1-a)b^{1-a}{a}^{1-\frac{1}{a}}}{2^{\frac{3}{2}}(a+1)}{\Gamma }^{{}^{\prime }}\left(\frac{1}{a}\right)\Gamma \left(3-\frac{1}{a}\right)}\hfill \\ & & {\displaystyle -\frac{(1-a)b^{1-a}{a}^{1-\frac{1}{a}}}{2^{\frac{3}{2}}(a+1)}\Gamma \left(\frac{1}{a}\right){\Gamma }^{{}^{\prime }}\left(3-\frac{1}{a}\right)}\hfill \\ & & {\displaystyle -\frac{2a{b}^a}{a+1}{\left.\frac{\partial }{\partial \alpha }\left[{\left(\frac{a+2}{a}\right)}^{\alpha +1}{\int }_0^{\infty }{\left(y+\frac{2a{b}^a}{a+2}\right)}^{\alpha +1}{\left(y+2b^a\right)}^{-3}dy\right]\right|}_{\alpha =0}}\hfill \\ & & {\displaystyle +\frac{3a}{a+1}\left\{\frac{a+2}{a}\left[1+log\left(2b^2\right)\right]-\frac{a-2}{4}\left[1+2log\left(2b^2\right)\right]\right\},}\hfill \end{array}$$

$$\begin{array}{c}\hfill {\displaystyle R\left(X\right)=\frac{1}{1-\gamma }log\left[{\left(\frac{2a^2{b}^a}{a+1}\right)}^{\gamma }{a}^{\frac{\gamma -1}{a}-1}{\int }_0^{\infty }{y}^{\gamma +\frac{1-\gamma }{a}}{\left(\frac{a+2}{a}y+2b^a\right)}^{\gamma }{\left(y+2b^a\right)}^{-3\gamma }dy\right]}\end{array}$$

and

$$\begin{array}{ccc}& & {\displaystyle CE\left(X\right)=-log\left(\frac{2b^a}{a+1}\right)\frac{2^{\frac{1}{a}}b}{a^{\frac{1}{a}+2}}\left(a+\frac{1}{a}\right)\Gamma \left(\frac{1}{a}\right)\Gamma \left(1-\frac{1}{a}\right)}\hfill \\ & & {\displaystyle -\frac{{(a+1)}^{\frac{1}{a}}{\left(2b^a\right)}^{\frac{1}{a}}}{a^{\frac{1}{a}+1}{(a+2)}^{\frac{1}{a}}}{\left.\frac{\partial }{\partial \alpha }\left\{{\left[2(a+1)b^a\right]}^{\alpha }B\left(\frac{1}{a},1-\alpha -\frac{1}{a}\right){}_2{F}_1\left(\frac{1}{a},2;1-\alpha ;\frac{1}{a+2}\right)\right\}\right|}_{\alpha =0}}\hfill \\ & & {\displaystyle -\frac{2^{\frac{1}{a}+1}blog2}{a^{\frac{1}{a}+1}}\Gamma \left(\frac{1}{a}\right)\Gamma \left(2-\frac{1}{a}\right)-\frac{2^{\frac{1}{a}+1}blogb}{a^{\frac{1}{a}}}\Gamma \left(\frac{1}{a}\right)\Gamma \left(2-\frac{1}{a}\right)}\hfill \\ & & {\displaystyle +\frac{2^{\frac{1}{a}+1}b}{a^{\frac{1}{a}}}\Gamma \left(\frac{1}{a}\right){\Gamma }^{{}^{\prime }}\left(2-\frac{1}{a}\right)-\frac{2^{\frac{1}{a}+1}b}{a^{\frac{1}{a}+1}}\Gamma \left(\frac{1}{a}\right)\Gamma \left(2-\frac{1}{a}\right){\Gamma }^{{}^{\prime }}\left(2\right)}\hfill \\ & & {\displaystyle -\frac{2^{\frac{1}{a}+1}b(a+2)log2}{a^{\frac{1}{a}+1}(a+1)}\Gamma \left(1+\frac{1}{a}\right)\Gamma \left(1-\frac{1}{a}\right)}\hfill \\ & & {\displaystyle -\frac{2^{\frac{1}{a}+1}b(a+2)logb}{a^{\frac{1}{a}}(a+1)}\Gamma \left(1+\frac{1}{a}\right)\Gamma \left(1-\frac{1}{a}\right)}\hfill \\ & & {\displaystyle +\frac{2^{\frac{1}{a}+1}b(a+2)}{a^{\frac{1}{a}+1}(a+1)}\Gamma \left(1+\frac{1}{a}\right){\Gamma }^{{}^{\prime }}\left(1-\frac{1}{a}\right)}\hfill \\ & & {\displaystyle -\frac{2^{\frac{1}{a}+1}b(a+2)log2}{a^{\frac{1}{a}+1}(a+1)}\Gamma \left(\frac{1}{a}\right)\Gamma \left(2-\frac{1}{a}\right){\Gamma }^{{}^{\prime }}\left(2\right)}\hfill \end{array}$$

for $x>0$, $a>0$ and $b>0$.

63. Reciprocal distribution: for this distribution,

$$\begin{array}{c}\hfill {\displaystyle f_X\left(x\right)=\frac{1}{x\left(logb-loga\right)},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle F_X\left(x\right)=\frac{logx-loga}{logb-loga},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle GM\left(X\right)=\frac{logb+loga}{2},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle S\left(X\right)=-log\left[logb-loga\right]+\frac{logb+loga}{2},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle R\left(X\right)=\frac{1}{1-\gamma }log\left\{\frac{b^{1-\gamma }-a^{1-\gamma }}{1-\gamma }{\left[logb-loga\right]}^{\gamma }\right\}}\end{array}$$

and

$$\begin{array}{c}\hfill {\displaystyle CE\left(X\right)=\frac{(b-2a)log\left[logb-loga\right]}{logb-loga}-\frac{b}{logb-loga}{\left.\frac{\partial }{\partial \alpha }\gamma \left(\alpha +2,logb-loga\right)\right|}_{\alpha =0}}\end{array}$$

for $0<a\le x\le b<\infty$.

4. Conclusions

We have derived the most comprehensive collection of explicit expressions for the geometric mean, Shannon entropy, Rényi entropy and the cumulative residual entropy for the following continuous univariate distributions: 1. Gauss hypergeometric beta distribution, 2. q Weibull distribution, 3. q exponential distribution, 4. Weighted exponential distribution, 5. Teissier distribution, 6. Maxwell distribution, 7. Inverse Maxwell distribution, 8. Power Maxwell distribution, 9. Inverse power Maxwell distribution, 10. Omega distribution, 11. Colak et al.’s distribution, 12. Bimodal beta distribution, 13. Confluent hypergeometric beta distribution, 14. Libby and Novick’s beta distribution, 15. Generalized beta distribution, 16. Log-logistic distribution, 17. Inverse Gaussian distribution, 18. Gompertz distribution, 19. Exponential distribution, 20. Inverse exponential distribution, 21. Exponentiated exponential distribution, 22. Gamma distribution, 23. Chisquare distribution, 24. Chi distribution, 25. Inverse gamma distribution, 26. Inverse chisquare distribution, 27. Inverse chi distribution, 28. Rayleigh distribution, 29. Weibull distribution, 30. Inverse Rayleigh distribution, 31. Inverse Weibull distribution, 32. Gumbel distribution, 33. Generalized extreme value distribution, 34. Generalized gamma distribution, 35. Pareto distribution of type I, 36. Pareto distribution of type II, 37. Generalized Pareto distribution, 38. Uniform distribution, 39. Power function distribution of type I, 40. Power function distribution of type II, 41. Arcsine distribution, 42. Beta distribution, 43. Inverted beta distribution, 44. Kumaraswamy distribution, 45. Inverted Kumaraswamy distribution, 46. Normal distribution, 47. Lognormal distribution, 48. Half normal distribution, 49. Student’s t distribution, 50. Cauchy distribution, 51. Laplace distribution, 52. Logistic distribution of type I, 53. Logistic distribution of type II, 54. Logistic distribution of type III, 55. Logistic distribution of type IV, 56. Burr distribution, 57. Dagum distribution, 58. J shaped distribution, 59. Nadarajah–Haghighi distribution, 60. Two-sided power distribution, 61. Power Lindley distribution, 62. Modified slash Lindley–Weibull distribution, 63. Reciprocal distribution. This collection could be a useful reference for both theoreticians and practitioners of entropies. Future work will be to derive similar collections of explicit expressions for entropies of discrete univariate distributions, continuous bivariate distributions, discrete bivariate distributions, continuous multivariate distributions, discrete multivariate distributions, continuous matrix variate distributions, discrete matrix variate distributions, continuous complex variate distributions, and discrete complex variate distributions.

Author Contributions

Conceptualization, S.N. and M.K.; methodology, S.N. and M.K.; writing—original draft preparation, S.N.; writing—review and editing, S.N. All authors have read and agreed to the published version of the manuscript.

Institutional Review Board Statement

Not applicable.

Data Availability Statement

Not applicable.

Conflicts of Interest

Authors declare no conflicts of interest.

Funding Statement

This research received no external funding.