Optimal. Leaf size=8 \[ \tanh ^{-1}(\cosh (x))-\text {csch}(x) \]
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Rubi [A] time = 0.04, antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3501, 3770} \[ \tanh ^{-1}(\cosh (x))-\text {csch}(x) \]
Antiderivative was successfully verified.
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Rule 3501
Rule 3770
Rubi steps
\begin {align*} \int \frac {\text {csch}^3(x)}{1+\coth (x)} \, dx &=-\text {csch}(x)-\int \text {csch}(x) \, dx\\ &=\tanh ^{-1}(\cosh (x))-\text {csch}(x)\\ \end {align*}
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Mathematica [A] time = 0.04, size = 14, normalized size = 1.75 \[ -\text {csch}(x)-\log \left (\tanh \left (\frac {x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.39, size = 77, normalized size = 9.62 \[ \frac {{\left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} - 1\right )} \log \left (\cosh \relax (x) + \sinh \relax (x) + 1\right ) - {\left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} - 1\right )} \log \left (\cosh \relax (x) + \sinh \relax (x) - 1\right ) - 2 \, \cosh \relax (x) - 2 \, \sinh \relax (x)}{\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.13, size = 26, normalized size = 3.25 \[ -\frac {2 \, e^{x}}{e^{\left (2 \, x\right )} - 1} + \log \left (e^{x} + 1\right ) - \log \left ({\left | e^{x} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.09, size = 23, normalized size = 2.88 \[ \frac {\tanh \left (\frac {x}{2}\right )}{2}-\frac {1}{2 \tanh \left (\frac {x}{2}\right )}-\ln \left (\tanh \left (\frac {x}{2}\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.30, size = 31, normalized size = 3.88 \[ \frac {2 \, e^{\left (-x\right )}}{e^{\left (-2 \, x\right )} - 1} + \log \left (e^{\left (-x\right )} + 1\right ) - \log \left (e^{\left (-x\right )} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 29, normalized size = 3.62 \[ \ln \left (2\,{\mathrm {e}}^x+2\right )-\ln \left (2\,{\mathrm {e}}^x-2\right )-\frac {2\,{\mathrm {e}}^x}{{\mathrm {e}}^{2\,x}-1} \]
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 \[ \int \frac {\operatorname {csch}^{3}{\relax (x )}}{\coth {\relax (x )} + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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