Optimal. Leaf size=37 \[ -\frac {3 x}{2}+\frac {\coth ^3(x)}{2 (\coth (x)+1)}-\coth ^2(x)+\frac {3 \coth (x)}{2}+2 \log (\sinh (x)) \]
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Rubi [A] time = 0.07, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {3550, 3528, 3525, 3475} \[ -\frac {3 x}{2}+\frac {\coth ^3(x)}{2 (\coth (x)+1)}-\coth ^2(x)+\frac {3 \coth (x)}{2}+2 \log (\sinh (x)) \]
Antiderivative was successfully verified.
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Rule 3475
Rule 3525
Rule 3528
Rule 3550
Rubi steps
\begin {align*} \int \frac {\coth ^4(x)}{1+\coth (x)} \, dx &=\frac {\coth ^3(x)}{2 (1+\coth (x))}-\frac {1}{2} \int (3-4 \coth (x)) \coth ^2(x) \, dx\\ &=-\coth ^2(x)+\frac {\coth ^3(x)}{2 (1+\coth (x))}+\frac {1}{2} i \int (-4 i+3 i \coth (x)) \coth (x) \, dx\\ &=-\frac {3 x}{2}+\frac {3 \coth (x)}{2}-\coth ^2(x)+\frac {\coth ^3(x)}{2 (1+\coth (x))}+2 \int \coth (x) \, dx\\ &=-\frac {3 x}{2}+\frac {3 \coth (x)}{2}-\coth ^2(x)+\frac {\coth ^3(x)}{2 (1+\coth (x))}+2 \log (\sinh (x))\\ \end {align*}
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Mathematica [A] time = 0.06, size = 33, normalized size = 0.89 \[ \frac {1}{4} \left (-6 x-\sinh (2 x)+\cosh (2 x)+4 \coth (x)-2 \text {csch}^2(x)+8 \log (\sinh (x))\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.41, size = 357, normalized size = 9.65 \[ -\frac {14 \, x \cosh \relax (x)^{6} + 84 \, x \cosh \relax (x) \sinh \relax (x)^{5} + 14 \, x \sinh \relax (x)^{6} - {\left (28 \, x + 1\right )} \cosh \relax (x)^{4} + {\left (210 \, x \cosh \relax (x)^{2} - 28 \, x - 1\right )} \sinh \relax (x)^{4} + 4 \, {\left (70 \, x \cosh \relax (x)^{3} - {\left (28 \, x + 1\right )} \cosh \relax (x)\right )} \sinh \relax (x)^{3} + 2 \, {\left (7 \, x + 5\right )} \cosh \relax (x)^{2} + 2 \, {\left (105 \, x \cosh \relax (x)^{4} - 3 \, {\left (28 \, x + 1\right )} \cosh \relax (x)^{2} + 7 \, x + 5\right )} \sinh \relax (x)^{2} - 8 \, {\left (\cosh \relax (x)^{6} + 6 \, \cosh \relax (x) \sinh \relax (x)^{5} + \sinh \relax (x)^{6} + {\left (15 \, \cosh \relax (x)^{2} - 2\right )} \sinh \relax (x)^{4} - 2 \, \cosh \relax (x)^{4} + 4 \, {\left (5 \, \cosh \relax (x)^{3} - 2 \, \cosh \relax (x)\right )} \sinh \relax (x)^{3} + {\left (15 \, \cosh \relax (x)^{4} - 12 \, \cosh \relax (x)^{2} + 1\right )} \sinh \relax (x)^{2} + \cosh \relax (x)^{2} + 2 \, {\left (3 \, \cosh \relax (x)^{5} - 4 \, \cosh \relax (x)^{3} + \cosh \relax (x)\right )} \sinh \relax (x)\right )} \log \left (\frac {2 \, \sinh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}\right ) + 4 \, {\left (21 \, x \cosh \relax (x)^{5} - {\left (28 \, x + 1\right )} \cosh \relax (x)^{3} + {\left (7 \, x + 5\right )} \cosh \relax (x)\right )} \sinh \relax (x) - 1}{4 \, {\left (\cosh \relax (x)^{6} + 6 \, \cosh \relax (x) \sinh \relax (x)^{5} + \sinh \relax (x)^{6} + {\left (15 \, \cosh \relax (x)^{2} - 2\right )} \sinh \relax (x)^{4} - 2 \, \cosh \relax (x)^{4} + 4 \, {\left (5 \, \cosh \relax (x)^{3} - 2 \, \cosh \relax (x)\right )} \sinh \relax (x)^{3} + {\left (15 \, \cosh \relax (x)^{4} - 12 \, \cosh \relax (x)^{2} + 1\right )} \sinh \relax (x)^{2} + \cosh \relax (x)^{2} + 2 \, {\left (3 \, \cosh \relax (x)^{5} - 4 \, \cosh \relax (x)^{3} + \cosh \relax (x)\right )} \sinh \relax (x)\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 40, normalized size = 1.08 \[ -\frac {7}{2} \, x + \frac {{\left (e^{\left (4 \, x\right )} - 10 \, e^{\left (2 \, x\right )} + 1\right )} e^{\left (-2 \, x\right )}}{4 \, {\left (e^{\left (2 \, x\right )} - 1\right )}^{2}} + 2 \, \log \left ({\left | e^{\left (2 \, x\right )} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 32, normalized size = 0.86 \[ -\frac {\left (\coth ^{2}\relax (x )\right )}{2}+\coth \relax (x )-\frac {\ln \left (\coth \relax (x )-1\right )}{4}-\frac {1}{2 \left (1+\coth \relax (x )\right )}-\frac {7 \ln \left (1+\coth \relax (x )\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 54, normalized size = 1.46 \[ \frac {1}{2} \, x + \frac {2 \, {\left (2 \, e^{\left (-2 \, x\right )} - 1\right )}}{2 \, e^{\left (-2 \, x\right )} - e^{\left (-4 \, x\right )} - 1} + \frac {1}{4} \, e^{\left (-2 \, x\right )} + 2 \, \log \left (e^{\left (-x\right )} + 1\right ) + 2 \, \log \left (e^{\left (-x\right )} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 29, normalized size = 0.78 \[ \frac {x}{2}-2\,\ln \left (\mathrm {coth}\relax (x)+1\right )+\mathrm {coth}\relax (x)-\frac {{\mathrm {coth}\relax (x)}^2}{2}-\frac {1}{2\,\left (\mathrm {coth}\relax (x)+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.18, size = 197, normalized size = 5.32 \[ \frac {x \tanh ^{3}{\relax (x )}}{2 \tanh ^{3}{\relax (x )} + 2 \tanh ^{2}{\relax (x )}} + \frac {x \tanh ^{2}{\relax (x )}}{2 \tanh ^{3}{\relax (x )} + 2 \tanh ^{2}{\relax (x )}} - \frac {4 \log {\left (\tanh {\relax (x )} + 1 \right )} \tanh ^{3}{\relax (x )}}{2 \tanh ^{3}{\relax (x )} + 2 \tanh ^{2}{\relax (x )}} - \frac {4 \log {\left (\tanh {\relax (x )} + 1 \right )} \tanh ^{2}{\relax (x )}}{2 \tanh ^{3}{\relax (x )} + 2 \tanh ^{2}{\relax (x )}} + \frac {4 \log {\left (\tanh {\relax (x )} \right )} \tanh ^{3}{\relax (x )}}{2 \tanh ^{3}{\relax (x )} + 2 \tanh ^{2}{\relax (x )}} + \frac {4 \log {\left (\tanh {\relax (x )} \right )} \tanh ^{2}{\relax (x )}}{2 \tanh ^{3}{\relax (x )} + 2 \tanh ^{2}{\relax (x )}} + \frac {3 \tanh ^{2}{\relax (x )}}{2 \tanh ^{3}{\relax (x )} + 2 \tanh ^{2}{\relax (x )}} + \frac {\tanh {\relax (x )}}{2 \tanh ^{3}{\relax (x )} + 2 \tanh ^{2}{\relax (x )}} - \frac {1}{2 \tanh ^{3}{\relax (x )} + 2 \tanh ^{2}{\relax (x )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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