Optimal. Leaf size=12 \[ -\sinh (x)+\cosh (x)-\tanh ^{-1}(\cosh (x)) \]
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Rubi [A] time = 0.11, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.778, Rules used = {3518, 3108, 3107, 2637, 2592, 321, 206} \[ -\sinh (x)+\cosh (x)-\tanh ^{-1}(\cosh (x)) \]
Antiderivative was successfully verified.
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Rule 206
Rule 321
Rule 2592
Rule 2637
Rule 3107
Rule 3108
Rule 3518
Rubi steps
\begin {align*} \int \frac {\text {csch}(x)}{1+\tanh (x)} \, dx &=\int \frac {\coth (x)}{\cosh (x)+\sinh (x)} \, dx\\ &=i \int \coth (x) (-i \cosh (x)+i \sinh (x)) \, dx\\ &=-\int (\cosh (x)-\cosh (x) \coth (x)) \, dx\\ &=-\int \cosh (x) \, dx+\int \cosh (x) \coth (x) \, dx\\ &=-\sinh (x)-\operatorname {Subst}\left (\int \frac {x^2}{1-x^2} \, dx,x,\cosh (x)\right )\\ &=\cosh (x)-\sinh (x)-\operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\cosh (x)\right )\\ &=-\tanh ^{-1}(\cosh (x))+\cosh (x)-\sinh (x)\\ \end {align*}
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Mathematica [A] time = 0.02, size = 14, normalized size = 1.17 \[ -\sinh (x)+\cosh (x)+\log \left (\tanh \left (\frac {x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.58, size = 38, normalized size = 3.17 \[ -\frac {{\left (\cosh \relax (x) + \sinh \relax (x)\right )} \log \left (\cosh \relax (x) + \sinh \relax (x) + 1\right ) - {\left (\cosh \relax (x) + \sinh \relax (x)\right )} \log \left (\cosh \relax (x) + \sinh \relax (x) - 1\right ) - 1}{\cosh \relax (x) + \sinh \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.11, size = 18, normalized size = 1.50 \[ e^{\left (-x\right )} - \log \left (e^{x} + 1\right ) + \log \left ({\left | e^{x} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 17, normalized size = 1.42 \[ \frac {2}{\tanh \left (\frac {x}{2}\right )+1}+\ln \left (\tanh \left (\frac {x}{2}\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 21, normalized size = 1.75 \[ e^{\left (-x\right )} - \log \left (e^{\left (-x\right )} + 1\right ) + \log \left (e^{\left (-x\right )} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 21, normalized size = 1.75 \[ \ln \left (2-2\,{\mathrm {e}}^x\right )-\ln \left (-2\,{\mathrm {e}}^x-2\right )+{\mathrm {e}}^{-x} \]
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 \[ \int \frac {\operatorname {csch}{\relax (x )}}{\tanh {\relax (x )} + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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