Table 11.

Complementary set of Leal-Functions.

Transcendental function	Transcendental equation $g(y)=x$	y′(x)	∫y(x)dx
y(x) = Lsin(x)	$y(x)\sin (y(x))=x$	$\frac{1}{\sin (\text{Lsin}(x))+\text{Lsin}(x)\cos (\text{Lsin}(x))}$	$-\sin (\text{Lsin}(x))+\text{Lsin}(x)\cos (\text{Lsin}(x))+\text{Lsin}{(x)}^2\sin (\text{Lsin}(x))+C$
y(x) = Lcos(x)	$y(x)\cos (y(x))=x$	$\frac{1}{\cos (\text{Lcos}(x))-\text{Lcos}(x)\sin (\text{Lcos}(x))}$	$-\cos (\text{Lcos}(x))-\text{Lcos}(x)\sin (\text{Lcos}(x))+\text{Lcos}{(x)}^2\cos (\text{Lcos}(x))+C$
y(x) = Lsec(x)	$y(x)\sec (y(x))=x$	$\frac{1}{\sec (\text{Lsec}(x))(\text{Lsec}(x)\tan (\text{ Lsec}(x))+1)}$	$\text{Lsec}{(x)}^2\text{sec(Lsec(}\mathit{\text{x}}\text{)) }-\left(\frac{{\sum }_{n=0}^{\infty }{(-1)}^n{E}_{2n}{(\text{Lsec}(x))}^{2n+2}}{(2n+2)(2n)!}\right)+C$
y(x) = Lcsc(x)	$y(x)\csc (y(x))=x$	$\frac{-1}{\csc (\text{Lcsc}(x))(\text{ Lcsc}(x)\cot (\text{Lcsc}(x))-1)}$	$\text{Lcsc}{(x)}^2\text{csch}(\text{Lcsc}(x))-\left(\frac{{\sum }_{n=0}^{\infty }2{(-1)}^{n+1}(2^{2n-1}-1)B_{2n}{(\text{Lcsc}(x))}^{2n+1}}{(2n+1)!}\right)+C$
y(x) = Lcot(x)	$y(x)\cot (y(x))=x$	$\frac{-1}{\cot {(\text{Lcot}(x))}^2\text{Lcot }(x)-\cot (\text{Lcot}(x))+\text{Lcot}(x)}$	$\text{Lcot}{(x)}^2\text{coth(Lcot(}\mathit{\text{x}}\text{))}-\left(\frac{{\sum }_{n=0}^{\infty }{2}^{2n}{(-1)}^n{B}_{2n}{(\text{Lcot}(x))}^{2n+1}}{(2n+1)!}\right)+C$
y(x) = Ltanh2(x)	$y(x)+\tanh (y(x))=x$	$\frac{-1}{\tanh {({\text{Ltanh}}_2(x))}^2-2}$	$\frac{1}{2}{\text{Ltanh}}_2{(x)}^2+{\text{Ltanh}}_2(x)\tanh ({\text{Ltanh}}_2(x))-\ln (\text{cosh}({\text{Ltanh}}_2(x)))+C$
y(x) = Lcsch2(x)	y(x)+ csch(y(x))=x	$\frac{-1}{\text{ csch(Lcsch}{}_2(x)\text{)coth(Lcsch}{}_2(x)\text{)}-1}$	$\frac{1}{2}{\text{Lcsch}}_2{(x)}^2+{\text{Lcsch}}_2(x)\text{csch}({\text{Lcsch}}_2(x))-\ln \left(\frac{\text{cosh}({\text{Lcsch}}_2(x)-1)}{\text{sinh}({\text{Lcsch}}_2(x))}\right)+C$
y(x) = Lsech2(x)	y(x)+sech(y(x))=x	$\frac{1}{\text{sech(Lsech}{}_2(x))\text{tanh(Lsech}{}_2(x)\text{) }-1}$	$\frac{1}{2}{\text{Lsech}}_2{(x)}^2+{\text{Lsech}}_2(x)\text{sech}({\text{Lsech}}_2(x))-\arctan (\text{arcsinh}({\text{Lsech}}_2(x)))+C$
y(x) = Lcoth2(x)	$y(x)+\coth (y(x))=x$	$\frac{-1}{\text{ coth(Lcoth}{{}_2(x))}^2-2}$	$\frac{1}{2}{\text{Lcoth}}_2{(x)}^2+{\text{Lcoth}}_2(x)\coth ({\text{Lcoth}}_2(x))-\ln ({\text{Lcoth}}_2(x))+C$
y(x) = Ltan2(x)	$y(x)+\tan (y(x))=x$	$\frac{1}{\tan {({\text{Ltan}}_2(x))}^2+2}$	$\frac{1}{2}{\text{Ltan}}_2{(x)}^2+{\text{Ltan}}_2(x)\tan ({\text{Ltan}}_2(x))+\ln (\text{cos}({\text{Ltan}}_2(x)))+C$
y(x) = Lsin2(x)	$y(x)+\sinh (y(x))=x$	$\frac{1}{1+\cos ({\text{Lsin}}_2(x))}$	$\frac{1}{2}{\text{Lsin}}_2{(x)}^2+\cos ({\text{Lsin}}_2(x))+{\text{Lsin}}_2(x)\sin ({\text{Lsin}}_2(x))+C$
y(x) = Lcos2(x)	$y(x)+\cos (y(x))=x$	$\frac{-1}{\sin ({\text{Lcos}}_2(x)-1)}$	$\frac{1}{2}{\text{Lcos}}_2{(x)}^2-\sin ({\text{Lcos}}_2(x))+{\text{Lcos}}_2(x)\cos ({\text{Lcos}}_2(x))+C$
y(x) = Lsec2(x)	$y(x)+\sec (y(x))=x$	$\frac{1}{1+\sec ({\text{Lsec}}_2(x))\tan ({\text{Lsec}}_2(x))}$	$\frac{1}{2}{\text{Lcsc}}_2{(x)}^2+\frac{{\text{Lcsc}}_2(x)}{\sin ({\text{Lcsc}}_2(x))}-\ln (\csc ({\text{Lcsc}}_2(x))-\cot ({\text{Lcsc}}_2(x)))+C$
y(x) = Lcsc2(x)	$y(x)+\csc (y(x))=x$	$\frac{-1}{\csc ({\text{Lcsc}}_2(x))\cot ({\text{Lcsc}}_2(x))-1}$	$\frac{1}{2}{\text{Lcsc}}_2{(x)}^2+\frac{{\text{Lcsc}}_2(x)}{\sin ({\text{Lcsc}}_2(x))}-\ln (\csc ({\text{Lcsc}}_2(x))-\cot ({\text{Lcsc}}_2(x)))+C$
y(x) = Lcot2(x)	$y(x)+\cot (y(x))=x$	$\frac{-1}{\cot {({\text{Lcot}}_2(x))}^2}$	$\frac{1}{2}{\text{Lcot}}_2{(x)}^2-({\text{Lcot}}_2(x))\cot ({\text{Lcot}}_2(x))-\ln (\sin ({\text{Lcot}}_2(x)))+C$