Optimal. Leaf size=19 \[ -\frac {x}{2}+\frac {1}{2 (\tanh (x)+1)}+\log (\sinh (x)) \]
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Rubi [A] time = 0.04, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {3551, 3479, 8, 3475} \[ -\frac {x}{2}+\frac {1}{2 (\tanh (x)+1)}+\log (\sinh (x)) \]
Antiderivative was successfully verified.
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Rule 8
Rule 3475
Rule 3479
Rule 3551
Rubi steps
\begin {align*} \int \frac {\coth (x)}{1+\tanh (x)} \, dx &=\int \coth (x) \, dx-\int \frac {1}{1+\tanh (x)} \, dx\\ &=\log (\sinh (x))+\frac {1}{2 (1+\tanh (x))}-\frac {\int 1 \, dx}{2}\\ &=-\frac {x}{2}+\log (\sinh (x))+\frac {1}{2 (1+\tanh (x))}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 23, normalized size = 1.21 \[ \frac {1}{4} (-2 x-\sinh (2 x)+\cosh (2 x)+4 \log (\sinh (x))) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.77, size = 73, normalized size = 3.84 \[ -\frac {6 \, x \cosh \relax (x)^{2} + 12 \, x \cosh \relax (x) \sinh \relax (x) + 6 \, x \sinh \relax (x)^{2} - 4 \, {\left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2}\right )} \log \left (\frac {2 \, \sinh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}\right ) - 1}{4 \, {\left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 18, normalized size = 0.95 \[ -\frac {3}{2} \, x + \frac {1}{4} \, e^{\left (-2 \, x\right )} + \log \left ({\left | e^{\left (2 \, x\right )} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.08, size = 43, normalized size = 2.26 \[ -\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{2}-\frac {1}{\tanh \left (\frac {x}{2}\right )+1}+\frac {1}{\left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {3 \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{2}+\ln \left (\tanh \left (\frac {x}{2}\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 24, normalized size = 1.26 \[ \frac {1}{2} \, x + \frac {1}{4} \, e^{\left (-2 \, x\right )} + \log \left (e^{\left (-x\right )} + 1\right ) + \log \left (e^{\left (-x\right )} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 17, normalized size = 0.89 \[ \ln \left ({\mathrm {e}}^{2\,x}-1\right )-\frac {3\,x}{2}+\frac {{\mathrm {e}}^{-2\,x}}{4} \]
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 \[ \int \frac {\coth {\relax (x )}}{\tanh {\relax (x )} + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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