Optimal. Leaf size=46 \[ \frac{\log \left (1-e^{2 c (a+b x)}\right ) \sinh (a c+b c x) \sqrt{\text{csch}^2(a c+b c x)}}{b c} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.090595, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {6720, 2282, 12, 260} \[ \frac{\log \left (1-e^{2 c (a+b x)}\right ) \sinh (a c+b c x) \sqrt{\text{csch}^2(a c+b c x)}}{b c} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6720
Rule 2282
Rule 12
Rule 260
Rubi steps
\begin{align*} \int e^{c (a+b x)} \sqrt{\text{csch}^2(a c+b c x)} \, dx &=\left (\sqrt{\text{csch}^2(a c+b c x)} \sinh (a c+b c x)\right ) \int e^{c (a+b x)} \text{csch}(a c+b c x) \, dx\\ &=\frac{\left (\sqrt{\text{csch}^2(a c+b c x)} \sinh (a c+b c x)\right ) \operatorname{Subst}\left (\int \frac{2 x}{-1+x^2} \, dx,x,e^{c (a+b x)}\right )}{b c}\\ &=\frac{\left (2 \sqrt{\text{csch}^2(a c+b c x)} \sinh (a c+b c x)\right ) \operatorname{Subst}\left (\int \frac{x}{-1+x^2} \, dx,x,e^{c (a+b x)}\right )}{b c}\\ &=\frac{\sqrt{\text{csch}^2(a c+b c x)} \log \left (1-e^{2 c (a+b x)}\right ) \sinh (a c+b c x)}{b c}\\ \end{align*}
Mathematica [A] time = 0.0358906, size = 44, normalized size = 0.96 \[ \frac{\log \left (1-e^{2 c (a+b x)}\right ) \sinh (c (a+b x)) \sqrt{\text{csch}^2(c (a+b x))}}{b c} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.183, size = 68, normalized size = 1.5 \begin{align*}{\frac{ \left ({{\rm e}^{2\,c \left ( bx+a \right ) }}-1 \right ) \ln \left ({{\rm e}^{2\,bcx}}-{{\rm e}^{-2\,ac}} \right ){{\rm e}^{-c \left ( bx+a \right ) }}}{cb}\sqrt{{\frac{{{\rm e}^{2\,c \left ( bx+a \right ) }}}{ \left ({{\rm e}^{2\,c \left ( bx+a \right ) }}-1 \right ) ^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.54704, size = 53, normalized size = 1.15 \begin{align*} \frac{\log \left (e^{\left (b c x + a c\right )} + 1\right )}{b c} + \frac{\log \left (e^{\left (b c x + a c\right )} - 1\right )}{b c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.59559, size = 97, normalized size = 2.11 \begin{align*} \frac{\log \left (\frac{2 \, \sinh \left (b c x + a c\right )}{\cosh \left (b c x + a c\right ) - \sinh \left (b c x + a c\right )}\right )}{b c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} e^{a c} \int \sqrt{\operatorname{csch}^{2}{\left (a c + b c x \right )}} e^{b c x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.15407, size = 65, normalized size = 1.41 \begin{align*} \frac{\log \left ({\left | e^{\left (2 \, b c x + 2 \, a c\right )} - 1 \right |}\right )}{b c \mathrm{sgn}\left (e^{\left (b c x + a c\right )} - e^{\left (-b c x - a c\right )}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]