Optimal. Leaf size=40 \[ \frac{\left (a^2-b^2\right ) \log (a+b \cosh (x))}{b^3}-\frac{a \cosh (x)}{b^2}+\frac{\cosh ^2(x)}{2 b} \]
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Rubi [A] time = 0.0689581, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2668, 697} \[ \frac{\left (a^2-b^2\right ) \log (a+b \cosh (x))}{b^3}-\frac{a \cosh (x)}{b^2}+\frac{\cosh ^2(x)}{2 b} \]
Antiderivative was successfully verified.
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Rule 2668
Rule 697
Rubi steps
\begin{align*} \int \frac{\sinh ^3(x)}{a+b \cosh (x)} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{b^2-x^2}{a+x} \, dx,x,b \cosh (x)\right )}{b^3}\\ &=-\frac{\operatorname{Subst}\left (\int \left (a-x+\frac{-a^2+b^2}{a+x}\right ) \, dx,x,b \cosh (x)\right )}{b^3}\\ &=-\frac{a \cosh (x)}{b^2}+\frac{\cosh ^2(x)}{2 b}+\frac{\left (a^2-b^2\right ) \log (a+b \cosh (x))}{b^3}\\ \end{align*}
Mathematica [A] time = 0.0528166, size = 40, normalized size = 1. \[ \frac{\left (a^2-b^2\right ) \log (a+b \cosh (x))}{b^3}-\frac{a \cosh (x)}{b^2}+\frac{\cosh (2 x)}{4 b} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.02, size = 283, normalized size = 7.1 \begin{align*}{\frac{{a}^{3}}{{b}^{3} \left ( a-b \right ) }\ln \left ( a \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}- \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}b-a-b \right ) }-{\frac{{a}^{2}}{{b}^{2} \left ( a-b \right ) }\ln \left ( a \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}- \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}b-a-b \right ) }-{\frac{a}{ \left ( a-b \right ) b}\ln \left ( a \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}- \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}b-a-b \right ) }+{\frac{1}{a-b}\ln \left ( a \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}- \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}b-a-b \right ) }+{\frac{1}{2\,b} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-2}}-{\frac{a}{{b}^{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-1}}-{\frac{1}{2\,b} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-1}}-{\frac{{a}^{2}}{{b}^{3}}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) }+{\frac{1}{b}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) }+{\frac{1}{2\,b} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-2}}+{\frac{a}{{b}^{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-1}}+{\frac{1}{2\,b} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-1}}-{\frac{{a}^{2}}{{b}^{3}}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) }+{\frac{1}{b}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.02369, size = 113, normalized size = 2.82 \begin{align*} -\frac{{\left (4 \, a e^{\left (-x\right )} - b\right )} e^{\left (2 \, x\right )}}{8 \, b^{2}} - \frac{4 \, a e^{\left (-x\right )} - b e^{\left (-2 \, x\right )}}{8 \, b^{2}} + \frac{{\left (a^{2} - b^{2}\right )} x}{b^{3}} + \frac{{\left (a^{2} - b^{2}\right )} \log \left (2 \, a e^{\left (-x\right )} + b e^{\left (-2 \, x\right )} + b\right )}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.94193, size = 629, normalized size = 15.72 \begin{align*} \frac{b^{2} \cosh \left (x\right )^{4} + b^{2} \sinh \left (x\right )^{4} - 4 \, a b \cosh \left (x\right )^{3} - 8 \,{\left (a^{2} - b^{2}\right )} x \cosh \left (x\right )^{2} + 4 \,{\left (b^{2} \cosh \left (x\right ) - a b\right )} \sinh \left (x\right )^{3} - 4 \, a b \cosh \left (x\right ) + 2 \,{\left (3 \, b^{2} \cosh \left (x\right )^{2} - 6 \, a b \cosh \left (x\right ) - 4 \,{\left (a^{2} - b^{2}\right )} x\right )} \sinh \left (x\right )^{2} + b^{2} + 8 \,{\left ({\left (a^{2} - b^{2}\right )} \cosh \left (x\right )^{2} + 2 \,{\left (a^{2} - b^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right ) +{\left (a^{2} - b^{2}\right )} \sinh \left (x\right )^{2}\right )} \log \left (\frac{2 \,{\left (b \cosh \left (x\right ) + a\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + 4 \,{\left (b^{2} \cosh \left (x\right )^{3} - 3 \, a b \cosh \left (x\right )^{2} - 4 \,{\left (a^{2} - b^{2}\right )} x \cosh \left (x\right ) - a b\right )} \sinh \left (x\right )}{8 \,{\left (b^{3} \cosh \left (x\right )^{2} + 2 \, b^{3} \cosh \left (x\right ) \sinh \left (x\right ) + b^{3} \sinh \left (x\right )^{2}\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 [A] time = 1.18362, size = 76, normalized size = 1.9 \begin{align*} \frac{b{\left (e^{\left (-x\right )} + e^{x}\right )}^{2} - 4 \, a{\left (e^{\left (-x\right )} + e^{x}\right )}}{8 \, b^{2}} + \frac{{\left (a^{2} - b^{2}\right )} \log \left ({\left | b{\left (e^{\left (-x\right )} + e^{x}\right )} + 2 \, a \right |}\right )}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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