Optimal. Leaf size=31 \[ \frac {3 x}{2}+\frac {\coth ^2(x)}{2 (\coth (x)+1)}-\frac {3 \coth (x)}{2}-\log (\sinh (x)) \]
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Rubi [A] time = 0.06, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {3550, 3525, 3475} \[ \frac {3 x}{2}+\frac {\coth ^2(x)}{2 (\coth (x)+1)}-\frac {3 \coth (x)}{2}-\log (\sinh (x)) \]
Antiderivative was successfully verified.
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Rule 3475
Rule 3525
Rule 3550
Rubi steps
\begin {align*} \int \frac {\coth ^3(x)}{1+\coth (x)} \, dx &=\frac {\coth ^2(x)}{2 (1+\coth (x))}-\frac {1}{2} \int (2-3 \coth (x)) \coth (x) \, dx\\ &=\frac {3 x}{2}-\frac {3 \coth (x)}{2}+\frac {\coth ^2(x)}{2 (1+\coth (x))}-\int \coth (x) \, dx\\ &=\frac {3 x}{2}-\frac {3 \coth (x)}{2}+\frac {\coth ^2(x)}{2 (1+\coth (x))}-\log (\sinh (x))\\ \end {align*}
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Mathematica [A] time = 0.04, size = 27, normalized size = 0.87 \[ \frac {1}{4} (6 x+\sinh (2 x)-\cosh (2 x)-4 \coth (x)-4 \log (\sinh (x))) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.42, size = 196, normalized size = 6.32 \[ \frac {10 \, x \cosh \relax (x)^{4} + 40 \, x \cosh \relax (x) \sinh \relax (x)^{3} + 10 \, x \sinh \relax (x)^{4} - {\left (10 \, x + 9\right )} \cosh \relax (x)^{2} + {\left (60 \, x \cosh \relax (x)^{2} - 10 \, x - 9\right )} \sinh \relax (x)^{2} - 4 \, {\left (\cosh \relax (x)^{4} + 4 \, \cosh \relax (x) \sinh \relax (x)^{3} + \sinh \relax (x)^{4} + {\left (6 \, \cosh \relax (x)^{2} - 1\right )} \sinh \relax (x)^{2} - \cosh \relax (x)^{2} + 2 \, {\left (2 \, \cosh \relax (x)^{3} - \cosh \relax (x)\right )} \sinh \relax (x)\right )} \log \left (\frac {2 \, \sinh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}\right ) + 2 \, {\left (20 \, x \cosh \relax (x)^{3} - {\left (10 \, x + 9\right )} \cosh \relax (x)\right )} \sinh \relax (x) + 1}{4 \, {\left (\cosh \relax (x)^{4} + 4 \, \cosh \relax (x) \sinh \relax (x)^{3} + \sinh \relax (x)^{4} + {\left (6 \, \cosh \relax (x)^{2} - 1\right )} \sinh \relax (x)^{2} - \cosh \relax (x)^{2} + 2 \, {\left (2 \, \cosh \relax (x)^{3} - \cosh \relax (x)\right )} \sinh \relax (x)\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 36, normalized size = 1.16 \[ \frac {5}{2} \, x - \frac {{\left (9 \, e^{\left (2 \, x\right )} - 1\right )} e^{\left (-2 \, x\right )}}{4 \, {\left (e^{\left (2 \, x\right )} - 1\right )}} - \log \left ({\left | e^{\left (2 \, x\right )} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 28, normalized size = 0.90 \[ -\coth \relax (x )-\frac {\ln \left (\coth \relax (x )-1\right )}{4}+\frac {1}{2+2 \coth \relax (x )}+\frac {5 \ln \left (1+\coth \relax (x )\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 38, normalized size = 1.23 \[ \frac {1}{2} \, x + \frac {2}{e^{\left (-2 \, x\right )} - 1} - \frac {1}{4} \, e^{\left (-2 \, x\right )} - \log \left (e^{\left (-x\right )} + 1\right ) - \log \left (e^{\left (-x\right )} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.16, size = 21, normalized size = 0.68 \[ \frac {x}{2}+\ln \left (\mathrm {coth}\relax (x)+1\right )-\mathrm {coth}\relax (x)+\frac {1}{2\,\left (\mathrm {coth}\relax (x)+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.93, size = 160, normalized size = 5.16 \[ \frac {x \tanh ^{2}{\relax (x )}}{2 \tanh ^{2}{\relax (x )} + 2 \tanh {\relax (x )}} + \frac {x \tanh {\relax (x )}}{2 \tanh ^{2}{\relax (x )} + 2 \tanh {\relax (x )}} + \frac {2 \log {\left (\tanh {\relax (x )} + 1 \right )} \tanh ^{2}{\relax (x )}}{2 \tanh ^{2}{\relax (x )} + 2 \tanh {\relax (x )}} + \frac {2 \log {\left (\tanh {\relax (x )} + 1 \right )} \tanh {\relax (x )}}{2 \tanh ^{2}{\relax (x )} + 2 \tanh {\relax (x )}} - \frac {2 \log {\left (\tanh {\relax (x )} \right )} \tanh ^{2}{\relax (x )}}{2 \tanh ^{2}{\relax (x )} + 2 \tanh {\relax (x )}} - \frac {2 \log {\left (\tanh {\relax (x )} \right )} \tanh {\relax (x )}}{2 \tanh ^{2}{\relax (x )} + 2 \tanh {\relax (x )}} - \frac {3 \tanh {\relax (x )}}{2 \tanh ^{2}{\relax (x )} + 2 \tanh {\relax (x )}} - \frac {2}{2 \tanh ^{2}{\relax (x )} + 2 \tanh {\relax (x )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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