Optimal. Leaf size=26 \[ \frac{\sinh (a-c) \log (\sinh (b x+c))}{b}+x \cosh (a-c) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0149365, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {5626, 3475, 8} \[ \frac{\sinh (a-c) \log (\sinh (b x+c))}{b}+x \cosh (a-c) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5626
Rule 3475
Rule 8
Rubi steps
\begin{align*} \int \text{csch}(c+b x) \sinh (a+b x) \, dx &=\cosh (a-c) \int 1 \, dx+\sinh (a-c) \int \coth (c+b x) \, dx\\ &=x \cosh (a-c)+\frac{\log (\sinh (c+b x)) \sinh (a-c)}{b}\\ \end{align*}
Mathematica [A] time = 0.111104, size = 26, normalized size = 1. \[ \frac{\sinh (a-c) \log (\sinh (b x+c))}{b}+x \cosh (a-c) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.036, size = 150, normalized size = 5.8 \begin{align*} x{{\rm e}^{a-c}}-{{\rm e}^{-a-c}}{{\rm e}^{2\,a}}x+{{\rm e}^{-a-c}}{{\rm e}^{2\,c}}x-{\frac{{{\rm e}^{-a-c}}{{\rm e}^{2\,a}}a}{b}}+{\frac{{{\rm e}^{-a-c}}{{\rm e}^{2\,c}}a}{b}}+{\frac{\ln \left ({{\rm e}^{2\,bx+2\,a}}-{{\rm e}^{2\,a-2\,c}} \right ){{\rm e}^{-a-c}}{{\rm e}^{2\,a}}}{2\,b}}-{\frac{\ln \left ({{\rm e}^{2\,bx+2\,a}}-{{\rm e}^{2\,a-2\,c}} \right ){{\rm e}^{-a-c}}{{\rm e}^{2\,c}}}{2\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.26277, size = 113, normalized size = 4.35 \begin{align*} \frac{{\left (e^{\left (2 \, a\right )} - e^{\left (2 \, c\right )}\right )} e^{\left (-a - c\right )} \log \left (e^{\left (-b x\right )} + e^{c}\right )}{2 \, b} + \frac{{\left (e^{\left (2 \, a\right )} - e^{\left (2 \, c\right )}\right )} e^{\left (-a - c\right )} \log \left (e^{\left (-b x\right )} - e^{c}\right )}{2 \, b} + \frac{{\left (b x + a\right )} e^{\left (a - c\right )}}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.12788, size = 230, normalized size = 8.85 \begin{align*} \frac{2 \, b x +{\left (\cosh \left (-a + c\right )^{2} - 2 \, \cosh \left (-a + c\right ) \sinh \left (-a + c\right ) + \sinh \left (-a + c\right )^{2} - 1\right )} \log \left (\frac{2 \, \sinh \left (b x + c\right )}{\cosh \left (b x + c\right ) - \sinh \left (b x + c\right )}\right )}{2 \,{\left (b \cosh \left (-a + c\right ) - b \sinh \left (-a + c\right )\right )}} \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*} \int \sinh{\left (a + b x \right )} \operatorname{csch}{\left (b x + c \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.20695, size = 69, normalized size = 2.65 \begin{align*} \frac{2 \, b x e^{\left (-a + c\right )} +{\left (e^{\left (2 \, a + c\right )} - e^{\left (3 \, c\right )}\right )} e^{\left (-a - 2 \, c\right )} \log \left ({\left | e^{\left (2 \, b x + 2 \, c\right )} - 1 \right |}\right )}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]