Optimal. Leaf size=21 \[ \frac{1}{6} \log \left (3-4 \cosh ^2(x)\right )-\frac{1}{3} \log (\cosh (x)) \]
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Rubi [A] time = 0.0301864, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.714, Rules used = {4357, 266, 36, 29, 31} \[ \frac{1}{6} \log \left (3-4 \cosh ^2(x)\right )-\frac{1}{3} \log (\cosh (x)) \]
Antiderivative was successfully verified.
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Rule 4357
Rule 266
Rule 36
Rule 29
Rule 31
Rubi steps
\begin{align*} \int \text{sech}(3 x) \sinh (x) \, dx &=\operatorname{Subst}\left (\int \frac{1}{x \left (-3+4 x^2\right )} \, dx,x,\cosh (x)\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x (-3+4 x)} \, dx,x,\cosh ^2(x)\right )\\ &=-\left (\frac{1}{6} \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,\cosh ^2(x)\right )\right )+\frac{2}{3} \operatorname{Subst}\left (\int \frac{1}{-3+4 x} \, dx,x,\cosh ^2(x)\right )\\ &=-\frac{1}{3} \log (\cosh (x))+\frac{1}{6} \log \left (3-4 \cosh ^2(x)\right )\\ \end{align*}
Mathematica [A] time = 0.0086202, size = 17, normalized size = 0.81 \[ -\frac{1}{3} \tanh ^{-1}\left (\frac{1}{3} \left (8 \sinh ^2(x)+5\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.034, size = 26, normalized size = 1.2 \begin{align*} -{\frac{\ln \left ({{\rm e}^{2\,x}}+1 \right ) }{3}}+{\frac{\ln \left ({{\rm e}^{4\,x}}-{{\rm e}^{2\,x}}+1 \right ) }{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.56583, size = 61, normalized size = 2.9 \begin{align*} \frac{1}{6} \, \log \left (\sqrt{3} e^{\left (-x\right )} + e^{\left (-2 \, x\right )} + 1\right ) + \frac{1}{6} \, \log \left (-\sqrt{3} e^{\left (-x\right )} + e^{\left (-2 \, x\right )} + 1\right ) - \frac{1}{3} \, \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 = 2.05395, size = 171, normalized size = 8.14 \begin{align*} \frac{1}{6} \, \log \left (\frac{2 \, \cosh \left (x\right )^{2} + 2 \, \sinh \left (x\right )^{2} - 1}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}\right ) - \frac{1}{3} \, \log \left (\frac{2 \, \cosh \left (x\right )}{\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sinh{\left (x \right )} \operatorname{sech}{\left (3 x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.15759, size = 55, normalized size = 2.62 \begin{align*} \frac{1}{6} \, \log \left (\sqrt{3} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + \frac{1}{6} \, \log \left (-\sqrt{3} e^{x} + e^{\left (2 \, x\right )} + 1\right ) - \frac{1}{3} \, \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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