Optimal. Leaf size=18 \[ \text {csch}(x)-\frac {1}{2} \tanh ^{-1}(\cosh (x))-\frac {1}{2} \coth (x) \text {csch}(x) \]
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Rubi [A] time = 0.16, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {3518, 3108, 3107, 2606, 8, 2611, 3770} \[ \text {csch}(x)-\frac {1}{2} \tanh ^{-1}(\cosh (x))-\frac {1}{2} \coth (x) \text {csch}(x) \]
Antiderivative was successfully verified.
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Rule 8
Rule 2606
Rule 2611
Rule 3107
Rule 3108
Rule 3518
Rule 3770
Rubi steps
\begin {align*} \int \frac {\text {csch}^3(x)}{1+\tanh (x)} \, dx &=\int \frac {\coth (x) \text {csch}^2(x)}{\cosh (x)+\sinh (x)} \, dx\\ &=i \int \coth (x) \text {csch}^2(x) (-i \cosh (x)+i \sinh (x)) \, dx\\ &=\int \left (-\coth (x) \text {csch}(x)+\coth ^2(x) \text {csch}(x)\right ) \, dx\\ &=-\int \coth (x) \text {csch}(x) \, dx+\int \coth ^2(x) \text {csch}(x) \, dx\\ &=-\frac {1}{2} \coth (x) \text {csch}(x)+i \operatorname {Subst}(\int 1 \, dx,x,-i \text {csch}(x))+\frac {1}{2} \int \text {csch}(x) \, dx\\ &=-\frac {1}{2} \tanh ^{-1}(\cosh (x))+\text {csch}(x)-\frac {1}{2} \coth (x) \text {csch}(x)\\ \end {align*}
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Mathematica [A] time = 0.08, size = 20, normalized size = 1.11 \[ \frac {1}{2} \left (\log \left (\tanh \left (\frac {x}{2}\right )\right )-(\coth (x)-2) \text {csch}(x)\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 209, normalized size = 11.61 \[ \frac {2 \, \cosh \relax (x)^{3} + 6 \, \cosh \relax (x) \sinh \relax (x)^{2} + 2 \, \sinh \relax (x)^{3} - {\left (\cosh \relax (x)^{4} + 4 \, \cosh \relax (x) \sinh \relax (x)^{3} + \sinh \relax (x)^{4} + 2 \, {\left (3 \, \cosh \relax (x)^{2} - 1\right )} \sinh \relax (x)^{2} - 2 \, \cosh \relax (x)^{2} + 4 \, {\left (\cosh \relax (x)^{3} - \cosh \relax (x)\right )} \sinh \relax (x) + 1\right )} \log \left (\cosh \relax (x) + \sinh \relax (x) + 1\right ) + {\left (\cosh \relax (x)^{4} + 4 \, \cosh \relax (x) \sinh \relax (x)^{3} + \sinh \relax (x)^{4} + 2 \, {\left (3 \, \cosh \relax (x)^{2} - 1\right )} \sinh \relax (x)^{2} - 2 \, \cosh \relax (x)^{2} + 4 \, {\left (\cosh \relax (x)^{3} - \cosh \relax (x)\right )} \sinh \relax (x) + 1\right )} \log \left (\cosh \relax (x) + \sinh \relax (x) - 1\right ) + 6 \, {\left (\cosh \relax (x)^{2} - 1\right )} \sinh \relax (x) - 6 \, \cosh \relax (x)}{2 \, {\left (\cosh \relax (x)^{4} + 4 \, \cosh \relax (x) \sinh \relax (x)^{3} + \sinh \relax (x)^{4} + 2 \, {\left (3 \, \cosh \relax (x)^{2} - 1\right )} \sinh \relax (x)^{2} - 2 \, \cosh \relax (x)^{2} + 4 \, {\left (\cosh \relax (x)^{3} - \cosh \relax (x)\right )} \sinh \relax (x) + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.11, size = 34, normalized size = 1.89 \[ \frac {e^{\left (3 \, x\right )} - 3 \, e^{x}}{{\left (e^{\left (2 \, x\right )} - 1\right )}^{2}} - \frac {1}{2} \, \log \left (e^{x} + 1\right ) + \frac {1}{2} \, \log \left ({\left | e^{x} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.10, size = 39, normalized size = 2.17 \[ \frac {\left (\tanh ^{2}\left (\frac {x}{2}\right )\right )}{8}-\frac {\tanh \left (\frac {x}{2}\right )}{2}+\frac {1}{2 \tanh \left (\frac {x}{2}\right )}-\frac {1}{8 \tanh \left (\frac {x}{2}\right )^{2}}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.31, size = 48, normalized size = 2.67 \[ -\frac {e^{\left (-x\right )} - 3 \, e^{\left (-3 \, x\right )}}{2 \, e^{\left (-2 \, x\right )} - e^{\left (-4 \, x\right )} - 1} - \frac {1}{2} \, \log \left (e^{\left (-x\right )} + 1\right ) + \frac {1}{2} \, \log \left (e^{\left (-x\right )} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.09, size = 48, normalized size = 2.67 \[ \frac {\ln \left (1-{\mathrm {e}}^x\right )}{2}-\frac {\ln \left (-{\mathrm {e}}^x-1\right )}{2}+\frac {{\mathrm {e}}^x}{{\mathrm {e}}^{2\,x}-1}-\frac {2\,{\mathrm {e}}^x}{{\mathrm {e}}^{4\,x}-2\,{\mathrm {e}}^{2\,x}+1} \]
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 \[ \int \frac {\operatorname {csch}^{3}{\relax (x )}}{\tanh {\relax (x )} + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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