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 = 0.03, antiderivative size = 26, normalized size of antiderivative = 1.00, 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 206
Rule 1093
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*}
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Mathematica [C] time = 0.31, 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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fricas [B] time = 0.50, size = 54, normalized size = 2.08 \[ \frac {1}{8} \, \sqrt {2} \log \left (\frac {\cosh \relax (x)^{2} + \sinh \relax (x)^{2} + 2 \, \sqrt {2} \cosh \relax (x) + 2}{\cosh \relax (x)^{2} + \sinh \relax (x)^{2}}\right ) - \frac {1}{4} \, \log \left (\cosh \relax (x) + \sinh \relax (x) + 1\right ) + \frac {1}{4} \, \log \left (\cosh \relax (x) + \sinh \relax (x) - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.12, size = 57, normalized size = 2.19 \[ -\frac {1}{8} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - e^{\left (-x\right )} - e^{x}}{\sqrt {2} + e^{\left (-x\right )} + e^{x}}\right ) - \frac {1}{8} \, \log \left (e^{\left (-x\right )} + e^{x} + 2\right ) + \frac {1}{8} \, \log \left (e^{\left (-x\right )} + e^{x} - 2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.22, size = 53, normalized size = 2.04 \[ -\frac {\ln \left ({\mathrm e}^{x}+1\right )}{4}+\frac {\ln \left ({\mathrm e}^{x}-1\right )}{4}+\frac {\ln \left (1+{\mathrm e}^{2 x}+{\mathrm e}^{x} \sqrt {2}\right ) \sqrt {2}}{8}-\frac {\ln \left (1+{\mathrm e}^{2 x}-{\mathrm e}^{x} \sqrt {2}\right ) \sqrt {2}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.44, size = 60, normalized size = 2.31 \[ \frac {1}{8} \, \sqrt {2} \log \left (\sqrt {2} e^{\left (-x\right )} + e^{\left (-2 \, x\right )} + 1\right ) - \frac {1}{8} \, \sqrt {2} \log \left (-\sqrt {2} e^{\left (-x\right )} + e^{\left (-2 \, x\right )} + 1\right ) - \frac {1}{4} \, \log \left (e^{\left (-x\right )} + 1\right ) + \frac {1}{4} \, \log \left (e^{\left (-x\right )} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 61, normalized size = 2.35 \[ \frac {\ln \left (\frac {1}{2}-\frac {{\mathrm {e}}^x}{2}\right )}{4}-\frac {\ln \left (-\frac {{\mathrm {e}}^x}{2}-\frac {1}{2}\right )}{4}+\frac {\sqrt {2}\,\ln \left (-\frac {{\mathrm {e}}^{2\,x}}{8}-\frac {\sqrt {2}\,{\mathrm {e}}^x}{8}-\frac {1}{8}\right )}{8}-\frac {\sqrt {2}\,\ln \left (\frac {\sqrt {2}\,{\mathrm {e}}^x}{8}-\frac {{\mathrm {e}}^{2\,x}}{8}-\frac {1}{8}\right )}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \cosh {\relax (x )} \operatorname {csch}{\left (4 x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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