Optimal. Leaf size=18 \[ \frac{\cosh ^4(x)}{4}-\cosh ^2(x)+\log (\cosh (x)) \]
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Rubi [A] time = 0.0225872, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {2590, 266, 43} \[ \frac{\cosh ^4(x)}{4}-\cosh ^2(x)+\log (\cosh (x)) \]
Antiderivative was successfully verified.
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Rule 2590
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \sinh ^4(x) \tanh (x) \, dx &=\operatorname{Subst}\left (\int \frac{\left (1-x^2\right )^2}{x} \, dx,x,\cosh (x)\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(1-x)^2}{x} \, dx,x,\cosh ^2(x)\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-2+\frac{1}{x}+x\right ) \, dx,x,\cosh ^2(x)\right )\\ &=-\cosh ^2(x)+\frac{\cosh ^4(x)}{4}+\log (\cosh (x))\\ \end{align*}
Mathematica [A] time = 0.0060303, size = 18, normalized size = 1. \[ \frac{\cosh ^4(x)}{4}-\cosh ^2(x)+\log (\cosh (x)) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 17, normalized size = 0.9 \begin{align*}{\frac{ \left ( \sinh \left ( x \right ) \right ) ^{4}}{4}}-{\frac{ \left ( \sinh \left ( x \right ) \right ) ^{2}}{2}}+\ln \left ( \cosh \left ( x \right ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.40772, size = 47, normalized size = 2.61 \begin{align*} -\frac{1}{64} \,{\left (12 \, e^{\left (-2 \, x\right )} - 1\right )} e^{\left (4 \, x\right )} + x - \frac{3}{16} \, e^{\left (-2 \, x\right )} + \frac{1}{64} \, e^{\left (-4 \, x\right )} + \log \left (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 = 2.1511, size = 863, normalized size = 47.94 \begin{align*} \frac{\cosh \left (x\right )^{8} + 8 \, \cosh \left (x\right ) \sinh \left (x\right )^{7} + \sinh \left (x\right )^{8} + 4 \,{\left (7 \, \cosh \left (x\right )^{2} - 3\right )} \sinh \left (x\right )^{6} - 12 \, \cosh \left (x\right )^{6} + 8 \,{\left (7 \, \cosh \left (x\right )^{3} - 9 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{5} - 64 \, x \cosh \left (x\right )^{4} + 2 \,{\left (35 \, \cosh \left (x\right )^{4} - 90 \, \cosh \left (x\right )^{2} - 32 \, x\right )} \sinh \left (x\right )^{4} + 8 \,{\left (7 \, \cosh \left (x\right )^{5} - 30 \, \cosh \left (x\right )^{3} - 32 \, x \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 4 \,{\left (7 \, \cosh \left (x\right )^{6} - 45 \, \cosh \left (x\right )^{4} - 96 \, x \cosh \left (x\right )^{2} - 3\right )} \sinh \left (x\right )^{2} - 12 \, \cosh \left (x\right )^{2} + 64 \,{\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right )^{3} \sinh \left (x\right ) + 6 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4}\right )} \log \left (\frac{2 \, \cosh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + 8 \,{\left (\cosh \left (x\right )^{7} - 9 \, \cosh \left (x\right )^{5} - 32 \, x \cosh \left (x\right )^{3} - 3 \, \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1}{64 \,{\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right )^{3} \sinh \left (x\right ) + 6 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tanh ^{5}{\left (x \right )}}{\operatorname{sech}^{4}{\left (x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.1268, size = 58, normalized size = 3.22 \begin{align*} \frac{1}{64} \,{\left (48 \, e^{\left (4 \, x\right )} - 12 \, e^{\left (2 \, x\right )} + 1\right )} e^{\left (-4 \, x\right )} - x + \frac{1}{64} \, e^{\left (4 \, x\right )} - \frac{3}{16} \, e^{\left (2 \, x\right )} + \log \left (e^{\left (2 \, x\right )} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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