Optimal. Leaf size=38 \[ \frac {x}{2}-\frac {1}{6 (\coth (x)+1)}-\frac {2 \tan ^{-1}\left (\frac {1-2 \coth (x)}{\sqrt {3}}\right )}{3 \sqrt {3}} \]
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Rubi [A] time = 0.07, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {3661, 2074, 207, 618, 204} \[ \frac {x}{2}-\frac {1}{6 (\coth (x)+1)}-\frac {2 \tan ^{-1}\left (\frac {1-2 \coth (x)}{\sqrt {3}}\right )}{3 \sqrt {3}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 207
Rule 618
Rule 2074
Rule 3661
Rubi steps
\begin {align*} \int \frac {1}{1+\coth ^3(x)} \, dx &=\operatorname {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \left (1+x^3\right )} \, dx,x,\coth (x)\right )\\ &=\operatorname {Subst}\left (\int \left (\frac {1}{6 (1+x)^2}-\frac {1}{2 \left (-1+x^2\right )}+\frac {1}{3 \left (1-x+x^2\right )}\right ) \, dx,x,\coth (x)\right )\\ &=-\frac {1}{6 (1+\coth (x))}+\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\coth (x)\right )-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\coth (x)\right )\\ &=\frac {x}{2}-\frac {1}{6 (1+\coth (x))}-\frac {2}{3} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 \coth (x)\right )\\ &=\frac {x}{2}-\frac {2 \tan ^{-1}\left (\frac {1-2 \coth (x)}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {1}{6 (1+\coth (x))}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 40, normalized size = 1.05 \[ \frac {1}{2} \tanh ^{-1}(\tanh (x))+\frac {1}{6 (\tanh (x)+1)}+\frac {2 \tan ^{-1}\left (\frac {1-2 \tanh (x)}{\sqrt {3}}\right )}{3 \sqrt {3}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 95, normalized size = 2.50 \[ \frac {18 \, x \cosh \relax (x)^{2} + 36 \, x \cosh \relax (x) \sinh \relax (x) + 18 \, x \sinh \relax (x)^{2} + 8 \, {\left (\sqrt {3} \cosh \relax (x)^{2} + 2 \, \sqrt {3} \cosh \relax (x) \sinh \relax (x) + \sqrt {3} \sinh \relax (x)^{2}\right )} \arctan \left (-\frac {\sqrt {3} \cosh \relax (x) + \sqrt {3} \sinh \relax (x)}{3 \, {\left (\cosh \relax (x) - \sinh \relax (x)\right )}}\right ) + 3}{36 \, {\left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 25, normalized size = 0.66 \[ -\frac {2}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} e^{\left (2 \, x\right )}\right ) + \frac {1}{2} \, x + \frac {1}{12} \, e^{\left (-2 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 41, normalized size = 1.08 \[ -\frac {\ln \left (\coth \relax (x )-1\right )}{4}-\frac {1}{6 \left (1+\coth \relax (x )\right )}+\frac {\ln \left (1+\coth \relax (x )\right )}{4}+\frac {2 \sqrt {3}\, \arctan \left (\frac {\left (2 \coth \relax (x )-1\right ) \sqrt {3}}{3}\right )}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.41, size = 73, normalized size = 1.92 \[ -\frac {2}{9} \, \sqrt {3} \arctan \left (\frac {1}{6} \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (2 \, \sqrt {3} e^{\left (-x\right )} + 3^{\frac {1}{4}} \sqrt {2}\right )}\right ) + \frac {2}{9} \, \sqrt {3} \arctan \left (\frac {1}{6} \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (2 \, \sqrt {3} e^{\left (-x\right )} - 3^{\frac {1}{4}} \sqrt {2}\right )}\right ) + \frac {1}{2} \, x + \frac {1}{12} \, e^{\left (-2 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 38, normalized size = 1.00 \[ \frac {\frac {x}{2}+\frac {\mathrm {coth}\relax (x)}{6}+\frac {x\,\mathrm {coth}\relax (x)}{2}}{\mathrm {coth}\relax (x)+1}+\frac {2\,\sqrt {3}\,\mathrm {atan}\left (\frac {\sqrt {3}\,\left (2\,\mathrm {coth}\relax (x)-1\right )}{3}\right )}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.98, size = 102, normalized size = 2.68 \[ \frac {9 x \tanh {\relax (x )}}{18 \tanh {\relax (x )} + 18} + \frac {9 x}{18 \tanh {\relax (x )} + 18} - \frac {4 \sqrt {3} \tanh {\relax (x )} \operatorname {atan}{\left (\frac {2 \sqrt {3} \tanh {\relax (x )}}{3} - \frac {\sqrt {3}}{3} \right )}}{18 \tanh {\relax (x )} + 18} - \frac {4 \sqrt {3} \operatorname {atan}{\left (\frac {2 \sqrt {3} \tanh {\relax (x )}}{3} - \frac {\sqrt {3}}{3} \right )}}{18 \tanh {\relax (x )} + 18} + \frac {3}{18 \tanh {\relax (x )} + 18} \]
Verification of antiderivative is not currently implemented for this CAS.
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