Optimal. Leaf size=13 \[ \cosh (x) \log \left (\cosh ^2(x)\right )-2 \cosh (x) \]
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Rubi [A] time = 0.0191519, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {2638, 2554, 12} \[ \cosh (x) \log \left (\cosh ^2(x)\right )-2 \cosh (x) \]
Antiderivative was successfully verified.
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Rule 2638
Rule 2554
Rule 12
Rubi steps
\begin{align*} \int \log \left (\cosh ^2(x)\right ) \sinh (x) \, dx &=\cosh (x) \log \left (\cosh ^2(x)\right )-\int 2 \sinh (x) \, dx\\ &=\cosh (x) \log \left (\cosh ^2(x)\right )-2 \int \sinh (x) \, dx\\ &=-2 \cosh (x)+\cosh (x) \log \left (\cosh ^2(x)\right )\\ \end{align*}
Mathematica [A] time = 0.0078101, size = 13, normalized size = 1. \[ \cosh (x) \log \left (\cosh ^2(x)\right )-2 \cosh (x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 14, normalized size = 1.1 \begin{align*} -2\,\cosh \left ( x \right ) +\cosh \left ( x \right ) \ln \left ( \left ( \cosh \left ( x \right ) \right ) ^{2} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02456, size = 16, normalized size = 1.23 \begin{align*} 2 \, \cosh \left (x\right ) \log \left (\cosh \left (x\right )\right ) - 2 \, \cosh \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.79697, size = 228, normalized size = 17.54 \begin{align*} -\frac{2 \, \cosh \left (x\right )^{2} -{\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 1\right )} \log \left (\frac{1}{2} \, \cosh \left (x\right )^{2} + \frac{1}{2} \, \sinh \left (x\right )^{2} + \frac{1}{2}\right ) + 4 \, \cosh \left (x\right ) \sinh \left (x\right ) + 2 \, \sinh \left (x\right )^{2} + 2}{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 [A] time = 1.00068, size = 14, normalized size = 1.08 \begin{align*} 2 \log{\left (\cosh{\left (x \right )} \right )} \cosh{\left (x \right )} - 2 \cosh{\left (x \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.25137, size = 42, normalized size = 3.23 \begin{align*}{\left (e^{\left (-x\right )} + e^{x}\right )} \log \left (\frac{1}{2} \, e^{\left (-x\right )} + \frac{1}{2} \, e^{x}\right ) - e^{\left (-x\right )} - e^{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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