Optimal. Leaf size=49 \[ \frac {3}{2 \sqrt {\coth (x)+1}}-\frac {1}{3 (\coth (x)+1)^{3/2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {\coth (x)+1}}{\sqrt {2}}\right )}{2 \sqrt {2}} \]
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Rubi [A] time = 0.08, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {3540, 3526, 3480, 206} \[ \frac {3}{2 \sqrt {\coth (x)+1}}-\frac {1}{3 (\coth (x)+1)^{3/2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {\coth (x)+1}}{\sqrt {2}}\right )}{2 \sqrt {2}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 3480
Rule 3526
Rule 3540
Rubi steps
\begin {align*} \int \frac {\coth ^2(x)}{(1+\coth (x))^{3/2}} \, dx &=-\frac {1}{3 (1+\coth (x))^{3/2}}-\frac {1}{2} \int \frac {1-2 \coth (x)}{\sqrt {1+\coth (x)}} \, dx\\ &=-\frac {1}{3 (1+\coth (x))^{3/2}}+\frac {3}{2 \sqrt {1+\coth (x)}}+\frac {1}{4} \int \sqrt {1+\coth (x)} \, dx\\ &=-\frac {1}{3 (1+\coth (x))^{3/2}}+\frac {3}{2 \sqrt {1+\coth (x)}}+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{2-x^2} \, dx,x,\sqrt {1+\coth (x)}\right )\\ &=\frac {\tanh ^{-1}\left (\frac {\sqrt {1+\coth (x)}}{\sqrt {2}}\right )}{2 \sqrt {2}}-\frac {1}{3 (1+\coth (x))^{3/2}}+\frac {3}{2 \sqrt {1+\coth (x)}}\\ \end {align*}
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Mathematica [C] time = 0.37, size = 86, normalized size = 1.76 \[ \left (\frac {1}{4}+\frac {i}{4}\right ) \sqrt {\coth (x)+1} \left (-\left (\frac {1}{6}-\frac {i}{6}\right ) (-7 \sinh (2 x)-\sinh (4 x)+7 \cosh (2 x)+\cosh (4 x)-8)-\frac {i \tan ^{-1}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {i (\coth (x)+1)}\right )}{\sqrt {i (\coth (x)+1)}}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.41, size = 166, normalized size = 3.39 \[ \frac {2 \, \sqrt {2} {\left (8 \, \sqrt {2} \cosh \relax (x)^{2} + 16 \, \sqrt {2} \cosh \relax (x) \sinh \relax (x) + 8 \, \sqrt {2} \sinh \relax (x)^{2} + \sqrt {2}\right )} \sqrt {\frac {\sinh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}} + 3 \, {\left (\sqrt {2} \cosh \relax (x)^{3} + 3 \, \sqrt {2} \cosh \relax (x)^{2} \sinh \relax (x) + 3 \, \sqrt {2} \cosh \relax (x) \sinh \relax (x)^{2} + \sqrt {2} \sinh \relax (x)^{3}\right )} \log \left (2 \, \sqrt {2} \sqrt {\frac {\sinh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}} {\left (\cosh \relax (x) + \sinh \relax (x)\right )} + 2 \, \cosh \relax (x)^{2} + 4 \, \cosh \relax (x) \sinh \relax (x) + 2 \, \sinh \relax (x)^{2} - 1\right )}{24 \, {\left (\cosh \relax (x)^{3} + 3 \, \cosh \relax (x)^{2} \sinh \relax (x) + 3 \, \cosh \relax (x) \sinh \relax (x)^{2} + \sinh \relax (x)^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.19, size = 135, normalized size = 2.76 \[ -\frac {2}{3} \, \sqrt {2} \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right ) - \frac {\sqrt {2} \log \left ({\left | 2 \, \sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - 2 \, e^{\left (2 \, x\right )} + 1 \right |}\right )}{8 \, \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right )} - \frac {\sqrt {2} {\left (6 \, {\left (\sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )}\right )}^{2} - 3 \, \sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} + 3 \, e^{\left (2 \, x\right )} - 1\right )}}{12 \, {\left (\sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )}\right )}^{3} \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 35, normalized size = 0.71 \[ -\frac {1}{3 \left (1+\coth \relax (x )\right )^{\frac {3}{2}}}+\frac {\arctanh \left (\frac {\sqrt {1+\coth \relax (x )}\, \sqrt {2}}{2}\right ) \sqrt {2}}{4}+\frac {3}{2 \sqrt {1+\coth \relax (x )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\coth \relax (x)^{2}}{{\left (\coth \relax (x) + 1\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.23, size = 31, normalized size = 0.63 \[ \frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {\mathrm {coth}\relax (x)+1}}{2}\right )}{4}+\frac {\frac {3\,\mathrm {coth}\relax (x)}{2}+\frac {7}{6}}{{\left (\mathrm {coth}\relax (x)+1\right )}^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\coth ^{2}{\relax (x )}}{\left (\coth {\relax (x )} + 1\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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