Optimal. Leaf size=61 \[ -\frac {1}{4 \sqrt {\coth (x)+1}}-\frac {1}{6 (\coth (x)+1)^{3/2}}-\frac {1}{5 (\coth (x)+1)^{5/2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {\coth (x)+1}}{\sqrt {2}}\right )}{4 \sqrt {2}} \]
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Rubi [A] time = 0.04, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {3479, 3480, 206} \[ -\frac {1}{4 \sqrt {\coth (x)+1}}-\frac {1}{6 (\coth (x)+1)^{3/2}}-\frac {1}{5 (\coth (x)+1)^{5/2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {\coth (x)+1}}{\sqrt {2}}\right )}{4 \sqrt {2}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 3479
Rule 3480
Rubi steps
\begin {align*} \int \frac {1}{(1+\coth (x))^{5/2}} \, dx &=-\frac {1}{5 (1+\coth (x))^{5/2}}+\frac {1}{2} \int \frac {1}{(1+\coth (x))^{3/2}} \, dx\\ &=-\frac {1}{5 (1+\coth (x))^{5/2}}-\frac {1}{6 (1+\coth (x))^{3/2}}+\frac {1}{4} \int \frac {1}{\sqrt {1+\coth (x)}} \, dx\\ &=-\frac {1}{5 (1+\coth (x))^{5/2}}-\frac {1}{6 (1+\coth (x))^{3/2}}-\frac {1}{4 \sqrt {1+\coth (x)}}+\frac {1}{8} \int \sqrt {1+\coth (x)} \, dx\\ &=-\frac {1}{5 (1+\coth (x))^{5/2}}-\frac {1}{6 (1+\coth (x))^{3/2}}-\frac {1}{4 \sqrt {1+\coth (x)}}+\frac {1}{4} \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 )}{4 \sqrt {2}}-\frac {1}{5 (1+\coth (x))^{5/2}}-\frac {1}{6 (1+\coth (x))^{3/2}}-\frac {1}{4 \sqrt {1+\coth (x)}}\\ \end {align*}
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Mathematica [C] time = 0.80, size = 94, normalized size = 1.54 \[ -\frac {1}{60} \sqrt {\coth (x)+1} (\cosh (3 x)-\sinh (3 x)) (-24 \sinh (x)+13 \sinh (3 x)-10 \cosh (x)+10 \cosh (3 x))+\frac {\left (\frac {1}{8}+\frac {i}{8}\right ) (\coth (x)+1)^{3/2} \tan ^{-1}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {i (\coth (x)+1)}\right )}{(i (\coth (x)+1))^{3/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.40, size = 266, normalized size = 4.36 \[ -\frac {2 \, \sqrt {2} {\left (23 \, \sqrt {2} \cosh \relax (x)^{4} + 92 \, \sqrt {2} \cosh \relax (x) \sinh \relax (x)^{3} + 23 \, \sqrt {2} \sinh \relax (x)^{4} + {\left (138 \, \sqrt {2} \cosh \relax (x)^{2} - 11 \, \sqrt {2}\right )} \sinh \relax (x)^{2} - 11 \, \sqrt {2} \cosh \relax (x)^{2} + 2 \, {\left (46 \, \sqrt {2} \cosh \relax (x)^{3} - 11 \, \sqrt {2} \cosh \relax (x)\right )} \sinh \relax (x) + 3 \, \sqrt {2}\right )} \sqrt {\frac {\sinh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}} - 15 \, {\left (\sqrt {2} \cosh \relax (x)^{5} + 5 \, \sqrt {2} \cosh \relax (x)^{4} \sinh \relax (x) + 10 \, \sqrt {2} \cosh \relax (x)^{3} \sinh \relax (x)^{2} + 10 \, \sqrt {2} \cosh \relax (x)^{2} \sinh \relax (x)^{3} + 5 \, \sqrt {2} \cosh \relax (x) \sinh \relax (x)^{4} + \sqrt {2} \sinh \relax (x)^{5}\right )} \log \left (2 \, \sqrt {2} \sqrt {\frac {\sinh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}} {\left (\cosh \relax (x) + \sinh \relax (x)\right )} + 2 \, \cosh \relax (x)^{2} + 4 \, \cosh \relax (x) \sinh \relax (x) + 2 \, \sinh \relax (x)^{2} - 1\right )}{240 \, {\left (\cosh \relax (x)^{5} + 5 \, \cosh \relax (x)^{4} \sinh \relax (x) + 10 \, \cosh \relax (x)^{3} \sinh \relax (x)^{2} + 10 \, \cosh \relax (x)^{2} \sinh \relax (x)^{3} + 5 \, \cosh \relax (x) \sinh \relax (x)^{4} + \sinh \relax (x)^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.15, size = 179, normalized size = 2.93 \[ -\frac {1}{240} \, \sqrt {2} {\left (\frac {15 \, \log \left ({\left | 2 \, \sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - 2 \, e^{\left (2 \, x\right )} + 1 \right |}\right )}{\mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right )} - \frac {2 \, {\left (45 \, {\left (\sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )}\right )}^{4} + 45 \, {\left (\sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )}\right )}^{3} + 35 \, {\left (\sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )}\right )}^{2} + 15 \, \sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - 15 \, e^{\left (2 \, x\right )} + 3\right )}}{{\left (\sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )}\right )}^{5} \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right )} - 46 \, \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 43, normalized size = 0.70 \[ -\frac {1}{5 \left (1+\coth \relax (x )\right )^{\frac {5}{2}}}-\frac {1}{6 \left (1+\coth \relax (x )\right )^{\frac {3}{2}}}+\frac {\arctanh \left (\frac {\sqrt {1+\coth \relax (x )}\, \sqrt {2}}{2}\right ) \sqrt {2}}{8}-\frac {1}{4 \sqrt {1+\coth \relax (x )}} \]
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 \[ \int \frac {1}{{\left (\coth \relax (x) + 1\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.20, size = 40, normalized size = 0.66 \[ \frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {\mathrm {coth}\relax (x)+1}}{2}\right )}{8}-\frac {\frac {\mathrm {coth}\relax (x)}{6}+\frac {{\left (\mathrm {coth}\relax (x)+1\right )}^2}{4}+\frac {11}{30}}{{\left (\mathrm {coth}\relax (x)+1\right )}^{5/2}} \]
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 \[ \int \frac {1}{\left (\coth {\relax (x )} + 1\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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