Optimal. Leaf size=23 \[ 4 x-\frac {1}{2} (\coth (x)+1)^2-2 \coth (x)+4 \log (\sinh (x)) \]
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Rubi [A] time = 0.02, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3478, 3477, 3475} \[ 4 x-\frac {1}{2} (\coth (x)+1)^2-2 \coth (x)+4 \log (\sinh (x)) \]
Antiderivative was successfully verified.
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Rule 3475
Rule 3477
Rule 3478
Rubi steps
\begin {align*} \int (1+\coth (x))^3 \, dx &=-\frac {1}{2} (1+\coth (x))^2+2 \int (1+\coth (x))^2 \, dx\\ &=4 x-2 \coth (x)-\frac {1}{2} (1+\coth (x))^2+4 \int \coth (x) \, dx\\ &=4 x-2 \coth (x)-\frac {1}{2} (1+\coth (x))^2+4 \log (\sinh (x))\\ \end {align*}
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Mathematica [C] time = 0.16, size = 61, normalized size = 2.65 \[ \frac {1}{4} \text {csch}^2(x) \left (-6 \sinh (2 x) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\tanh ^2(x)\right )-2 x-8 \log (\tanh (x))-8 \log (\cosh (x))+\cosh (2 x) (2 x+8 \log (\tanh (x))+8 \log (\cosh (x))-1)-1\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.39, size = 142, normalized size = 6.17 \[ -\frac {2 \, {\left (4 \, \cosh \relax (x)^{2} - 2 \, {\left (\cosh \relax (x)^{4} + 4 \, \cosh \relax (x) \sinh \relax (x)^{3} + \sinh \relax (x)^{4} + 2 \, {\left (3 \, \cosh \relax (x)^{2} - 1\right )} \sinh \relax (x)^{2} - 2 \, \cosh \relax (x)^{2} + 4 \, {\left (\cosh \relax (x)^{3} - \cosh \relax (x)\right )} \sinh \relax (x) + 1\right )} \log \left (\frac {2 \, \sinh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}\right ) + 8 \, \cosh \relax (x) \sinh \relax (x) + 4 \, \sinh \relax (x)^{2} - 3\right )}}{\cosh \relax (x)^{4} + 4 \, \cosh \relax (x) \sinh \relax (x)^{3} + \sinh \relax (x)^{4} + 2 \, {\left (3 \, \cosh \relax (x)^{2} - 1\right )} \sinh \relax (x)^{2} - 2 \, \cosh \relax (x)^{2} + 4 \, {\left (\cosh \relax (x)^{3} - \cosh \relax (x)\right )} \sinh \relax (x) + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.11, size = 29, normalized size = 1.26 \[ -\frac {2 \, {\left (4 \, e^{\left (2 \, x\right )} - 3\right )}}{{\left (e^{\left (2 \, x\right )} - 1\right )}^{2}} + 4 \, \log \left ({\left | e^{\left (2 \, x\right )} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 19, normalized size = 0.83 \[ -\frac {\left (\coth ^{2}\relax (x )\right )}{2}-3 \coth \relax (x )-4 \ln \left (\coth \relax (x )-1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.32, size = 55, normalized size = 2.39 \[ 5 \, x + \frac {2 \, e^{\left (-2 \, x\right )}}{2 \, e^{\left (-2 \, x\right )} - e^{\left (-4 \, x\right )} - 1} + \frac {6}{e^{\left (-2 \, x\right )} - 1} + \log \left (e^{\left (-x\right )} + 1\right ) + \log \left (e^{\left (-x\right )} - 1\right ) + 3 \, \log \left (\sinh \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 36, normalized size = 1.57 \[ 4\,\ln \left ({\mathrm {e}}^{2\,x}-1\right )-\frac {2}{{\mathrm {e}}^{4\,x}-2\,{\mathrm {e}}^{2\,x}+1}-\frac {8}{{\mathrm {e}}^{2\,x}-1} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.68, size = 31, normalized size = 1.35 \[ 8 x - 4 \log {\left (\tanh {\relax (x )} + 1 \right )} + 4 \log {\left (\tanh {\relax (x )} \right )} - \frac {3}{\tanh {\relax (x )}} - \frac {1}{2 \tanh ^{2}{\relax (x )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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