Optimal. Leaf size=19 \[ -\frac {x}{2}-\frac {1}{2 (\coth (x)+1)}+\log (\sinh (x)) \]
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Rubi [A] time = 0.04, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3540, 3475} \[ -\frac {x}{2}-\frac {1}{2 (\coth (x)+1)}+\log (\sinh (x)) \]
Antiderivative was successfully verified.
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Rule 3475
Rule 3540
Rubi steps
\begin {align*} \int \frac {\coth ^2(x)}{1+\coth (x)} \, dx &=-\frac {1}{2 (1+\coth (x))}-\frac {1}{2} \int (1-2 \coth (x)) \, dx\\ &=-\frac {x}{2}-\frac {1}{2 (1+\coth (x))}+\int \coth (x) \, dx\\ &=-\frac {x}{2}-\frac {1}{2 (1+\coth (x))}+\log (\sinh (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 = 0.42, 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 [A] time = 0.05, size = 24, normalized size = 1.26 \[ -\frac {\ln \left (\coth \relax (x )-1\right )}{4}-\frac {1}{2 \left (1+\coth \relax (x )\right )}-\frac {3 \ln \left (1+\coth \relax (x )\right )}{4} \]
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 = 21, normalized size = 1.11 \[ \frac {x}{2}-\ln \left (\mathrm {coth}\relax (x)+1\right )-\frac {1}{2\,\left (\mathrm {coth}\relax (x)+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.62, size = 92, normalized size = 4.84 \[ \frac {x \tanh {\relax (x )}}{2 \tanh {\relax (x )} + 2} + \frac {x}{2 \tanh {\relax (x )} + 2} - \frac {2 \log {\left (\tanh {\relax (x )} + 1 \right )} \tanh {\relax (x )}}{2 \tanh {\relax (x )} + 2} - \frac {2 \log {\left (\tanh {\relax (x )} + 1 \right )}}{2 \tanh {\relax (x )} + 2} + \frac {2 \log {\left (\tanh {\relax (x )} \right )} \tanh {\relax (x )}}{2 \tanh {\relax (x )} + 2} + \frac {2 \log {\left (\tanh {\relax (x )} \right )}}{2 \tanh {\relax (x )} + 2} + \frac {1}{2 \tanh {\relax (x )} + 2} \]
Verification of antiderivative is not currently implemented for this CAS.
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