Optimal. Leaf size=46 \[ \frac{\log \left (1-e^{2 c (a+b x)}\right ) \sinh (a c+b c x)}{b c \sqrt{\sinh ^2(a c+b c x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.129109, 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} \[ \frac{\log \left (1-e^{2 c (a+b x)}\right ) \sinh (a c+b c x)}{b c \sqrt{\sinh ^2(a c+b c x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6720
Rule 2282
Rule 12
Rule 260
Rubi steps
\begin{align*} \int \frac{e^{c (a+b x)}}{\sqrt{\sinh ^2(a c+b c x)}} \, dx &=\frac{\sinh (a c+b c x) \int e^{c (a+b x)} \text{csch}(a c+b c x) \, dx}{\sqrt{\sinh ^2(a c+b c x)}}\\ &=\frac{\sinh (a c+b c x) \operatorname{Subst}\left (\int \frac{2 x}{-1+x^2} \, dx,x,e^{c (a+b x)}\right )}{b c \sqrt{\sinh ^2(a c+b c x)}}\\ &=\frac{(2 \sinh (a c+b c x)) \operatorname{Subst}\left (\int \frac{x}{-1+x^2} \, dx,x,e^{c (a+b x)}\right )}{b c \sqrt{\sinh ^2(a c+b c x)}}\\ &=\frac{\log \left (1-e^{2 c (a+b x)}\right ) \sinh (a c+b c x)}{b c \sqrt{\sinh ^2(a c+b c x)}}\\ \end{align*}
Mathematica [A] time = 0.050542, size = 44, normalized size = 0.96 \[ \frac{\log \left (1-e^{2 c (a+b x)}\right ) \sinh (c (a+b x))}{b c \sqrt{\sinh ^2(c (a+b x))}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 180., size = 0, normalized size = 0. \begin{align*} \int{{{\rm e}^{c \left ( bx+a \right ) }}{\frac{1}{\sqrt{ \left ( \sinh \left ( bcx+ac \right ) \right ) ^{2}}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.60244, 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 = 1.82143, 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 \frac{e^{b c x}}{\sqrt{\sinh ^{2}{\left (a c + b c x \right )}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.21016, size = 115, normalized size = 2.5 \begin{align*} \frac{\log \left (e^{\left (b c x\right )} + e^{\left (-a c\right )}\right ) \mathrm{sgn}\left (e^{\left (b c x + a c\right )} - e^{\left (-b c x - a c\right )}\right ) + \log \left ({\left | e^{\left (b c x\right )} - e^{\left (-a c\right )} \right |}\right ) \mathrm{sgn}\left (e^{\left (b c x + a c\right )} - e^{\left (-b c x - a c\right )}\right )}{b c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]