Optimal. Leaf size=7 \[ -\frac {1}{2} \tanh ^{-1}(\cosh (x)) \]
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Rubi [A] time = 0.02, antiderivative size = 7, normalized size of antiderivative = 1.00, 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 = {4287, 3770} \[ -\frac {1}{2} \tanh ^{-1}(\cosh (x)) \]
Antiderivative was successfully verified.
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Rule 3770
Rule 4287
Rubi steps
\begin {align*} \int \cosh (x) \text {csch}(2 x) \, dx &=\frac {1}{2} \int \text {csch}(x) \, dx\\ &=-\frac {1}{2} \tanh ^{-1}(\cosh (x))\\ \end {align*}
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Mathematica [A] time = 0.00, size = 11, normalized size = 1.57 \[ \frac {1}{2} \log \left (\tanh \left (\frac {x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 19, normalized size = 2.71 \[ -\frac {1}{2} \, \log \left (\cosh \relax (x) + \sinh \relax (x) + 1\right ) + \frac {1}{2} \, \log \left (\cosh \relax (x) + \sinh \relax (x) - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.13, size = 16, normalized size = 2.29 \[ -\frac {1}{2} \, \log \left (e^{x} + 1\right ) + \frac {1}{2} \, \log \left ({\left | e^{x} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.15, size = 6, normalized size = 0.86 \[ -\arctanh \left ({\mathrm e}^{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.34, size = 19, normalized size = 2.71 \[ -\frac {1}{2} \, \log \left (e^{\left (-x\right )} + 1\right ) + \frac {1}{2} \, \log \left (e^{\left (-x\right )} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 19, normalized size = 2.71 \[ \frac {\ln \left (1-{\mathrm {e}}^x\right )}{2}-\frac {\ln \left (-{\mathrm {e}}^x-1\right )}{2} \]
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 \[ \int \cosh {\relax (x )} \operatorname {csch}{\left (2 x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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