Optimal. Leaf size=10 \[ 2 \tanh ^{-1}(\sinh (x))-\sinh (x) \]
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Rubi [A] time = 0.0389517, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3190, 388, 206} \[ 2 \tanh ^{-1}(\sinh (x))-\sinh (x) \]
Antiderivative was successfully verified.
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Rule 3190
Rule 388
Rule 206
Rubi steps
\begin{align*} \int \frac{\cosh ^3(x)}{1-\sinh ^2(x)} \, dx &=\operatorname{Subst}\left (\int \frac{1+x^2}{1-x^2} \, dx,x,\sinh (x)\right )\\ &=-\sinh (x)+2 \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\sinh (x)\right )\\ &=2 \tanh ^{-1}(\sinh (x))-\sinh (x)\\ \end{align*}
Mathematica [A] time = 0.0095023, size = 14, normalized size = 1.4 \[ -2 \left (\frac{\sinh (x)}{2}-\tanh ^{-1}(\sinh (x))\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.028, size = 50, normalized size = 5. \begin{align*} -\ln \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}+2\,\tanh \left ( x/2 \right ) -1 \right ) + \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-1}+\ln \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}-2\,\tanh \left ( x/2 \right ) -1 \right ) + \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.10616, size = 53, normalized size = 5.3 \begin{align*} \frac{1}{2} \, e^{\left (-x\right )} - \frac{1}{2} \, e^{x} - \log \left (2 \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} - 1\right ) + \log \left (-2 \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.49507, size = 275, normalized size = 27.5 \begin{align*} -\frac{\cosh \left (x\right )^{2} - 2 \,{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} \log \left (\frac{2 \,{\left (\sinh \left (x\right ) + 1\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + 2 \,{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} \log \left (\frac{2 \,{\left (\sinh \left (x\right ) - 1\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 1}{2 \,{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 2.30906, size = 129, normalized size = 12.9 \begin{align*} \frac{\log{\left (\tanh ^{2}{\left (\frac{x}{2} \right )} - 2 \tanh{\left (\frac{x}{2} \right )} - 1 \right )} \tanh ^{2}{\left (\frac{x}{2} \right )}}{\tanh ^{2}{\left (\frac{x}{2} \right )} - 1} - \frac{\log{\left (\tanh ^{2}{\left (\frac{x}{2} \right )} - 2 \tanh{\left (\frac{x}{2} \right )} - 1 \right )}}{\tanh ^{2}{\left (\frac{x}{2} \right )} - 1} - \frac{\log{\left (\tanh ^{2}{\left (\frac{x}{2} \right )} + 2 \tanh{\left (\frac{x}{2} \right )} - 1 \right )} \tanh ^{2}{\left (\frac{x}{2} \right )}}{\tanh ^{2}{\left (\frac{x}{2} \right )} - 1} + \frac{\log{\left (\tanh ^{2}{\left (\frac{x}{2} \right )} + 2 \tanh{\left (\frac{x}{2} \right )} - 1 \right )}}{\tanh ^{2}{\left (\frac{x}{2} \right )} - 1} + \frac{2 \tanh{\left (\frac{x}{2} \right )}}{\tanh ^{2}{\left (\frac{x}{2} \right )} - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.1708, size = 50, normalized size = 5. \begin{align*} \frac{1}{2} \, e^{\left (-x\right )} - \frac{1}{2} \, e^{x} + \log \left ({\left | -e^{\left (-x\right )} + e^{x} + 2 \right |}\right ) - \log \left ({\left | -e^{\left (-x\right )} + e^{x} - 2 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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