Optimal. Leaf size=25 \[ \frac{\log (a+b \sinh (x))}{b}-\frac{x \cosh (x)}{a+b \sinh (x)} \]
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Rubi [A] time = 0.057565, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {5636, 2668, 31} \[ \frac{\log (a+b \sinh (x))}{b}-\frac{x \cosh (x)}{a+b \sinh (x)} \]
Antiderivative was successfully verified.
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Rule 5636
Rule 2668
Rule 31
Rubi steps
\begin{align*} \int \frac{x (b-a \sinh (x))}{(a+b \sinh (x))^2} \, dx &=-\frac{x \cosh (x)}{a+b \sinh (x)}+\int \frac{\cosh (x)}{a+b \sinh (x)} \, dx\\ &=-\frac{x \cosh (x)}{a+b \sinh (x)}+\frac{\operatorname{Subst}\left (\int \frac{1}{a+x} \, dx,x,b \sinh (x)\right )}{b}\\ &=\frac{\log (a+b \sinh (x))}{b}-\frac{x \cosh (x)}{a+b \sinh (x)}\\ \end{align*}
Mathematica [A] time = 0.168827, size = 25, normalized size = 1. \[ \frac{\log (a+b \sinh (x))}{b}-\frac{x \cosh (x)}{a+b \sinh (x)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.175, size = 58, normalized size = 2.3 \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.
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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.
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Fricas [B] time = 2.4049, size = 396, normalized size = 15.84 \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 \sinh \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.
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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.
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Giac [B] time = 1.19389, size = 130, normalized size = 5.2 \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.
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