Optimal. Leaf size=10 \[ 2 \tanh ^{-1}(\sinh (x))-\sinh (x) \]
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Rubi [A] time = 0.04, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3190, 388, 206} \[ 2 \tanh ^{-1}(\sinh (x))-\sinh (x) \]
Antiderivative was successfully verified.
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Rule 206
Rule 388
Rule 3190
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*}
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Mathematica [A] time = 0.01, size = 14, normalized size = 1.40 \[ -2 \left (\frac {\sinh (x)}{2}-\tanh ^{-1}(\sinh (x))\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 1.01, size = 71, normalized size = 7.10 \[ -\frac {\cosh \relax (x)^{2} - 2 \, {\left (\cosh \relax (x) + \sinh \relax (x)\right )} \log \left (\frac {2 \, {\left (\sinh \relax (x) + 1\right )}}{\cosh \relax (x) - \sinh \relax (x)}\right ) + 2 \, {\left (\cosh \relax (x) + \sinh \relax (x)\right )} \log \left (\frac {2 \, {\left (\sinh \relax (x) - 1\right )}}{\cosh \relax (x) - \sinh \relax (x)}\right ) + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} - 1}{2 \, {\left (\cosh \relax (x) + \sinh \relax (x)\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.14, size = 37, normalized size = 3.70 \[ \frac {1}{2} \, e^{\left (-x\right )} - \frac {1}{2} \, e^{x} + \log \left ({\left | -e^{\left (-x\right )} + e^{x} + 2 \right |}\right ) - \log \left ({\left | -e^{\left (-x\right )} + e^{x} - 2 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 50, normalized size = 5.00 \[ \ln \left (\tanh ^{2}\left (\frac {x}{2}\right )-2 \tanh \left (\frac {x}{2}\right )-1\right )+\frac {1}{\tanh \left (\frac {x}{2}\right )+1}-\ln \left (\tanh ^{2}\left (\frac {x}{2}\right )+2 \tanh \left (\frac {x}{2}\right )-1\right )+\frac {1}{\tanh \left (\frac {x}{2}\right )-1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.46, size = 39, normalized size = 3.90 \[ \frac {1}{2} \, e^{\left (-x\right )} - \frac {1}{2} \, e^{x} - \log \left (2 \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} - 1\right ) + \log \left (-2 \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 39, normalized size = 3.90 \[ \frac {{\mathrm {e}}^{-x}}{2}-\ln \left (32\,{\mathrm {e}}^{2\,x}-64\,{\mathrm {e}}^x-32\right )+\ln \left (32\,{\mathrm {e}}^{2\,x}+64\,{\mathrm {e}}^x-32\right )-\frac {{\mathrm {e}}^x}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.61, size = 129, normalized size = 12.90 \[ \frac {\log {\left (\tanh ^{2}{\left (\frac {x}{2} \right )} - 2 \tanh {\left (\frac {x}{2} \right )} - 1 \right )} \tanh ^{2}{\left (\frac {x}{2} \right )}}{\tanh ^{2}{\left (\frac {x}{2} \right )} - 1} - \frac {\log {\left (\tanh ^{2}{\left (\frac {x}{2} \right )} - 2 \tanh {\left (\frac {x}{2} \right )} - 1 \right )}}{\tanh ^{2}{\left (\frac {x}{2} \right )} - 1} - \frac {\log {\left (\tanh ^{2}{\left (\frac {x}{2} \right )} + 2 \tanh {\left (\frac {x}{2} \right )} - 1 \right )} \tanh ^{2}{\left (\frac {x}{2} \right )}}{\tanh ^{2}{\left (\frac {x}{2} \right )} - 1} + \frac {\log {\left (\tanh ^{2}{\left (\frac {x}{2} \right )} + 2 \tanh {\left (\frac {x}{2} \right )} - 1 \right )}}{\tanh ^{2}{\left (\frac {x}{2} \right )} - 1} + \frac {2 \tanh {\left (\frac {x}{2} \right )}}{\tanh ^{2}{\left (\frac {x}{2} \right )} - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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