Optimal. Leaf size=16 \[ -\frac{\tanh ^{-1}\left (\sqrt{2} \cosh (x)\right )}{\sqrt{2}} \]
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Rubi [A] time = 0.0194854, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {4357, 207} \[ -\frac{\tanh ^{-1}\left (\sqrt{2} \cosh (x)\right )}{\sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 4357
Rule 207
Rubi steps
\begin{align*} \int \text{sech}(2 x) \sinh (x) \, dx &=\operatorname{Subst}\left (\int \frac{1}{-1+2 x^2} \, dx,x,\cosh (x)\right )\\ &=-\frac{\tanh ^{-1}\left (\sqrt{2} \cosh (x)\right )}{\sqrt{2}}\\ \end{align*}
Mathematica [C] time = 0.28133, size = 155, normalized size = 9.69 \[ \frac{-4 \tanh ^{-1}\left (\sqrt{2}-i \tanh \left (\frac{x}{2}\right )\right )+\log \left (\sqrt{2}-2 \cosh (x)\right )-\log \left (2 \cosh (x)+\sqrt{2}\right )-2 i \tan ^{-1}\left (\frac{\sinh \left (\frac{x}{2}\right )+\cosh \left (\frac{x}{2}\right )}{\left (1+\sqrt{2}\right ) \cosh \left (\frac{x}{2}\right )-\left (\sqrt{2}-1\right ) \sinh \left (\frac{x}{2}\right )}\right )+2 i \tan ^{-1}\left (\frac{\sinh \left (\frac{x}{2}\right )+\cosh \left (\frac{x}{2}\right )}{\left (\sqrt{2}-1\right ) \cosh \left (\frac{x}{2}\right )-\left (1+\sqrt{2}\right ) \sinh \left (\frac{x}{2}\right )}\right )}{4 \sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.033, size = 39, normalized size = 2.4 \begin{align*}{\frac{\ln \left ( 1+{{\rm e}^{2\,x}}-{{\rm e}^{x}}\sqrt{2} \right ) \sqrt{2}}{4}}-{\frac{\ln \left ( 1+{{\rm e}^{2\,x}}+{{\rm e}^{x}}\sqrt{2} \right ) \sqrt{2}}{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.52406, size = 57, normalized size = 3.56 \begin{align*} -\frac{1}{4} \, \sqrt{2} \log \left (\sqrt{2} e^{\left (-x\right )} + e^{\left (-2 \, x\right )} + 1\right ) + \frac{1}{4} \, \sqrt{2} \log \left (-\sqrt{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 = 1.98553, size = 122, normalized size = 7.62 \begin{align*} \frac{1}{4} \, \sqrt{2} \log \left (\frac{\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2} - 2 \, \sqrt{2} \cosh \left (x\right ) + 2}{\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2}}\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 (2 x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.19838, size = 51, normalized size = 3.19 \begin{align*} -\frac{1}{4} \, \sqrt{2} \log \left (\sqrt{2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + \frac{1}{4} \, \sqrt{2} \log \left (-\sqrt{2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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