Optimal. Leaf size=25 \[ \frac{x \sinh (x)}{a+b \cosh (x)}-\frac{\log (a+b \cosh (x))}{b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0579455, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {5637, 2668, 31} \[ \frac{x \sinh (x)}{a+b \cosh (x)}-\frac{\log (a+b \cosh (x))}{b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5637
Rule 2668
Rule 31
Rubi steps
\begin{align*} \int \frac{x (b+a \cosh (x))}{(a+b \cosh (x))^2} \, dx &=\frac{x \sinh (x)}{a+b \cosh (x)}-\int \frac{\sinh (x)}{a+b \cosh (x)} \, dx\\ &=\frac{x \sinh (x)}{a+b \cosh (x)}-\frac{\operatorname{Subst}\left (\int \frac{1}{a+x} \, dx,x,b \cosh (x)\right )}{b}\\ &=-\frac{\log (a+b \cosh (x))}{b}+\frac{x \sinh (x)}{a+b \cosh (x)}\\ \end{align*}
Mathematica [A] time = 0.122942, size = 25, normalized size = 1. \[ \frac{x \sinh (x)}{a+b \cosh (x)}-\frac{\log (a+b \cosh (x))}{b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.062, size = 55, normalized size = 2.2 \begin{align*} 2\,{\frac{x}{b}}-2\,{\frac{x \left ( a{{\rm e}^{x}}+b \right ) }{b \left ( b{{\rm e}^{2\,x}}+2\,a{{\rm e}^{x}}+b \right ) }}-{\frac{1}{b}\ln \left ({{\rm e}^{2\,x}}+2\,{\frac{a{{\rm e}^{x}}}{b}}+1 \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.43145, size = 394, normalized size = 15.76 \begin{align*} \frac{2 \, b x \cosh \left (x\right )^{2} + 2 \, b x \sinh \left (x\right )^{2} + 2 \, a x \cosh \left (x\right ) -{\left (b \cosh \left (x\right )^{2} + b \sinh \left (x\right )^{2} + 2 \, a \cosh \left (x\right ) + 2 \,{\left (b \cosh \left (x\right ) + a\right )} \sinh \left (x\right ) + b\right )} \log \left (\frac{2 \,{\left (b \cosh \left (x\right ) + a\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + 2 \,{\left (2 \, b x \cosh \left (x\right ) + a x\right )} \sinh \left (x\right )}{b^{2} \cosh \left (x\right )^{2} + b^{2} \sinh \left (x\right )^{2} + 2 \, a b \cosh \left (x\right ) + b^{2} + 2 \,{\left (b^{2} \cosh \left (x\right ) + a b\right )} \sinh \left (x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.21412, size = 135, normalized size = 5.4 \begin{align*} \frac{2 \, b x e^{\left (2 \, x\right )} - b e^{\left (2 \, x\right )} \log \left (-b e^{\left (2 \, x\right )} - 2 \, a e^{x} - b\right ) - 2 \, a e^{x} \log \left (-b e^{\left (2 \, x\right )} - 2 \, a e^{x} - b\right ) - 2 \, b x - b \log \left (-b e^{\left (2 \, x\right )} - 2 \, a e^{x} - b\right )}{b^{2} e^{\left (2 \, x\right )} + 2 \, a b e^{x} + b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]