Optimal. Leaf size=45 \[ -\frac {2}{3} (\coth (x)+1)^{3/2}-4 \sqrt {\coth (x)+1}+4 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {\coth (x)+1}}{\sqrt {2}}\right ) \]
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Rubi [A] time = 0.03, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {3478, 3480, 206} \[ -\frac {2}{3} (\coth (x)+1)^{3/2}-4 \sqrt {\coth (x)+1}+4 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {\coth (x)+1}}{\sqrt {2}}\right ) \]
Antiderivative was successfully verified.
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Rule 206
Rule 3478
Rule 3480
Rubi steps
\begin {align*} \int (1+\coth (x))^{5/2} \, dx &=-\frac {2}{3} (1+\coth (x))^{3/2}+2 \int (1+\coth (x))^{3/2} \, dx\\ &=-4 \sqrt {1+\coth (x)}-\frac {2}{3} (1+\coth (x))^{3/2}+4 \int \sqrt {1+\coth (x)} \, dx\\ &=-4 \sqrt {1+\coth (x)}-\frac {2}{3} (1+\coth (x))^{3/2}+8 \operatorname {Subst}\left (\int \frac {1}{2-x^2} \, dx,x,\sqrt {1+\coth (x)}\right )\\ &=4 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {1+\coth (x)}}{\sqrt {2}}\right )-4 \sqrt {1+\coth (x)}-\frac {2}{3} (1+\coth (x))^{3/2}\\ \end {align*}
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Mathematica [C] time = 0.16, size = 92, normalized size = 2.04 \[ -\frac {2 \sinh (x) (\coth (x)+1)^{5/2} \left (\cosh (x) \sqrt {i (\coth (x)+1)}+\sinh (x) \left (7 \sqrt {i (\coth (x)+1)}-(6-6 i) \tan ^{-1}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {i (\coth (x)+1)}\right )\right )\right )}{3 \sqrt {i (\coth (x)+1)} (\sinh (x)+\cosh (x))^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.41, size = 259, normalized size = 5.76 \[ -\frac {2 \, {\left (2 \, \sqrt {2} {\left (4 \, \sqrt {2} \cosh \relax (x)^{3} + 12 \, \sqrt {2} \cosh \relax (x) \sinh \relax (x)^{2} + 4 \, \sqrt {2} \sinh \relax (x)^{3} + 3 \, {\left (4 \, \sqrt {2} \cosh \relax (x)^{2} - \sqrt {2}\right )} \sinh \relax (x) - 3 \, \sqrt {2} \cosh \relax (x)\right )} \sqrt {\frac {\sinh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}} - 3 \, {\left (\sqrt {2} \cosh \relax (x)^{4} + 4 \, \sqrt {2} \cosh \relax (x) \sinh \relax (x)^{3} + \sqrt {2} \sinh \relax (x)^{4} + 2 \, {\left (3 \, \sqrt {2} \cosh \relax (x)^{2} - \sqrt {2}\right )} \sinh \relax (x)^{2} - 2 \, \sqrt {2} \cosh \relax (x)^{2} + 4 \, {\left (\sqrt {2} \cosh \relax (x)^{3} - \sqrt {2} \cosh \relax (x)\right )} \sinh \relax (x) + \sqrt {2}\right )} \log \left (2 \, \sqrt {2} \sqrt {\frac {\sinh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}} {\left (\cosh \relax (x) + \sinh \relax (x)\right )} + 2 \, \cosh \relax (x)^{2} + 4 \, \cosh \relax (x) \sinh \relax (x) + 2 \, \sinh \relax (x)^{2} - 1\right )\right )}}{3 \, {\left (\cosh \relax (x)^{4} + 4 \, \cosh \relax (x) \sinh \relax (x)^{3} + \sinh \relax (x)^{4} + 2 \, {\left (3 \, \cosh \relax (x)^{2} - 1\right )} \sinh \relax (x)^{2} - 2 \, \cosh \relax (x)^{2} + 4 \, {\left (\cosh \relax (x)^{3} - \cosh \relax (x)\right )} \sinh \relax (x) + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.15, size = 112, normalized size = 2.49 \[ -\frac {2}{3} \, \sqrt {2} {\left (\frac {2 \, {\left (6 \, {\left (\sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )}\right )}^{2} + 9 \, \sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - 9 \, e^{\left (2 \, x\right )} + 4\right )}}{{\left (\sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )} + 1\right )}^{3}} + 3 \, \log \left ({\left | 2 \, \sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - 2 \, e^{\left (2 \, x\right )} + 1 \right |}\right )\right )} \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 35, normalized size = 0.78 \[ -\frac {2 \left (1+\coth \relax (x )\right )^{\frac {3}{2}}}{3}+4 \arctanh \left (\frac {\sqrt {1+\coth \relax (x )}\, \sqrt {2}}{2}\right ) \sqrt {2}-4 \sqrt {1+\coth \relax (x )} \]
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 \[ \int {\left (\coth \relax (x) + 1\right )}^{\frac {5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.19, size = 54, normalized size = 1.20 \[ \sqrt {8}\,\ln \left (-2\,\sqrt {8}\,\sqrt {\mathrm {coth}\relax (x)+1}-8\right )-\frac {2\,{\left (\mathrm {coth}\relax (x)+1\right )}^{3/2}}{3}-2\,\sqrt {2}\,\ln \left (4\,\sqrt {2}\,\sqrt {\mathrm {coth}\relax (x)+1}-8\right )-4\,\sqrt {\mathrm {coth}\relax (x)+1} \]
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 \[ \int \left (\coth {\relax (x )} + 1\right )^{\frac {5}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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