Optimal. Leaf size=26 \[ \frac{\tanh ^{-1}\left (\sqrt{2} \cosh (x)\right )}{2 \sqrt{2}}-\frac{1}{4} \tanh ^{-1}(\cosh (x)) \]
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Rubi [A] time = 0.03261, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {1093, 206} \[ \frac{\tanh ^{-1}\left (\sqrt{2} \cosh (x)\right )}{2 \sqrt{2}}-\frac{1}{4} \tanh ^{-1}(\cosh (x)) \]
Antiderivative was successfully verified.
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Rule 1093
Rule 206
Rubi steps
\begin{align*} \int \cosh (x) \text{csch}(4 x) \, dx &=-\operatorname{Subst}\left (\int \frac{1}{-4+12 x^2-8 x^4} \, dx,x,\cosh (x)\right )\\ &=2 \operatorname{Subst}\left (\int \frac{1}{4-8 x^2} \, dx,x,\cosh (x)\right )-2 \operatorname{Subst}\left (\int \frac{1}{8-8 x^2} \, dx,x,\cosh (x)\right )\\ &=-\frac{1}{4} \tanh ^{-1}(\cosh (x))+\frac{\tanh ^{-1}\left (\sqrt{2} \cosh (x)\right )}{2 \sqrt{2}}\\ \end{align*}
Mathematica [C] time = 0.294351, size = 183, normalized size = 7.04 \[ \frac{4 \tanh ^{-1}\left (\sqrt{2}-i \tanh \left (\frac{x}{2}\right )\right )+2 \sqrt{2} \log \left (\sinh \left (\frac{x}{2}\right )\right )-2 \sqrt{2} \log \left (\cosh \left (\frac{x}{2}\right )\right )-\log \left (\sqrt{2}-2 \cosh (x)\right )+\log \left (2 \cosh (x)+\sqrt{2}\right )+2 i \tan ^{-1}\left (\frac{\sinh \left (\frac{x}{2}\right )+\cosh \left (\frac{x}{2}\right )}{\left (1+\sqrt{2}\right ) \cosh \left (\frac{x}{2}\right )-\left (\sqrt{2}-1\right ) \sinh \left (\frac{x}{2}\right )}\right )-2 i \tan ^{-1}\left (\frac{\sinh \left (\frac{x}{2}\right )+\cosh \left (\frac{x}{2}\right )}{\left (\sqrt{2}-1\right ) \cosh \left (\frac{x}{2}\right )-\left (1+\sqrt{2}\right ) \sinh \left (\frac{x}{2}\right )}\right )}{8 \sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.039, size = 53, normalized size = 2. \begin{align*} -{\frac{\ln \left ({{\rm e}^{x}}+1 \right ) }{4}}+{\frac{\ln \left ({{\rm e}^{x}}-1 \right ) }{4}}+{\frac{\ln \left ( 1+{{\rm e}^{2\,x}}+{{\rm e}^{x}}\sqrt{2} \right ) \sqrt{2}}{8}}-{\frac{\ln \left ( 1+{{\rm e}^{2\,x}}-{{\rm e}^{x}}\sqrt{2} \right ) \sqrt{2}}{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.68406, size = 81, normalized size = 3.12 \begin{align*} \frac{1}{8} \, \sqrt{2} \log \left (\sqrt{2} e^{\left (-x\right )} + e^{\left (-2 \, x\right )} + 1\right ) - \frac{1}{8} \, \sqrt{2} \log \left (-\sqrt{2} e^{\left (-x\right )} + e^{\left (-2 \, x\right )} + 1\right ) - \frac{1}{4} \, \log \left (e^{\left (-x\right )} + 1\right ) + \frac{1}{4} \, \log \left (e^{\left (-x\right )} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.10151, size = 211, normalized size = 8.12 \begin{align*} \frac{1}{8} \, \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 ) - \frac{1}{4} \, \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) + \frac{1}{4} \, \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\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 \cosh{\left (x \right )} \operatorname{csch}{\left (4 x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.14539, size = 77, normalized size = 2.96 \begin{align*} -\frac{1}{8} \, \sqrt{2} \log \left (-\frac{\sqrt{2} - e^{\left (-x\right )} - e^{x}}{\sqrt{2} + e^{\left (-x\right )} + e^{x}}\right ) - \frac{1}{8} \, \log \left (e^{\left (-x\right )} + e^{x} + 2\right ) + \frac{1}{8} \, \log \left (e^{\left (-x\right )} + e^{x} - 2\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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