Optimal. Leaf size=49 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {1+\coth (x)}}{\sqrt {2}}\right )}{2 \sqrt {2}}-\frac {1}{3 (1+\coth (x))^{3/2}}-\frac {1}{2 \sqrt {1+\coth (x)}} \]
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Rubi [A]
time = 0.02, antiderivative size = 49, 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 = {3560, 3561, 212}
\begin {gather*} -\frac {1}{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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 3560
Rule 3561
Rubi steps
\begin {align*} \int \frac {1}{(1+\coth (x))^{3/2}} \, dx &=-\frac {1}{3 (1+\coth (x))^{3/2}}+\frac {1}{2} \int \frac {1}{\sqrt {1+\coth (x)}} \, dx\\ &=-\frac {1}{3 (1+\coth (x))^{3/2}}-\frac {1}{2 \sqrt {1+\coth (x)}}+\frac {1}{4} \int \sqrt {1+\coth (x)} \, dx\\ &=-\frac {1}{3 (1+\coth (x))^{3/2}}-\frac {1}{2 \sqrt {1+\coth (x)}}+\frac {1}{2} \text {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 {1}{2 \sqrt {1+\coth (x)}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.22, size = 86, normalized size = 1.76 \begin {gather*} \left (\frac {1}{4}+\frac {i}{4}\right ) \sqrt {1+\coth (x)} \left (-\frac {i \text {ArcTan}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {i (1+\coth (x))}\right )}{\sqrt {i (1+\coth (x))}}+\left (\frac {1}{6}-\frac {i}{6}\right ) (-4+5 \cosh (2 x)-\cosh (4 x)-5 \sinh (2 x)+\sinh (4 x))\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.67, size = 35, normalized size = 0.71
method | result | size |
derivativedivides | \(-\frac {1}{3 \left (1+\coth \left (x \right )\right )^{\frac {3}{2}}}+\frac {\arctanh \left (\frac {\sqrt {1+\coth \left (x \right )}\, \sqrt {2}}{2}\right ) \sqrt {2}}{4}-\frac {1}{2 \sqrt {1+\coth \left (x \right )}}\) | \(35\) |
default | \(-\frac {1}{3 \left (1+\coth \left (x \right )\right )^{\frac {3}{2}}}+\frac {\arctanh \left (\frac {\sqrt {1+\coth \left (x \right )}\, \sqrt {2}}{2}\right ) \sqrt {2}}{4}-\frac {1}{2 \sqrt {1+\coth \left (x \right )}}\) | \(35\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 168 vs.
\(2 (34) = 68\).
time = 0.39, size = 168, normalized size = 3.43 \begin {gather*} -\frac {2 \, \sqrt {2} {\left (4 \, \sqrt {2} \cosh \left (x\right )^{2} + 8 \, \sqrt {2} \cosh \left (x\right ) \sinh \left (x\right ) + 4 \, \sqrt {2} \sinh \left (x\right )^{2} - \sqrt {2}\right )} \sqrt {\frac {\sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}} - 3 \, {\left (\sqrt {2} \cosh \left (x\right )^{3} + 3 \, \sqrt {2} \cosh \left (x\right )^{2} \sinh \left (x\right ) + 3 \, \sqrt {2} \cosh \left (x\right ) \sinh \left (x\right )^{2} + \sqrt {2} \sinh \left (x\right )^{3}\right )} \log \left (2 \, \sqrt {2} \sqrt {\frac {\sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}} {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} + 2 \, \cosh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) \sinh \left (x\right ) + 2 \, \sinh \left (x\right )^{2} - 1\right )}{24 \, {\left (\cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right )^{2} \sinh \left (x\right ) + 3 \, \cosh \left (x\right ) \sinh \left (x\right )^{2} + \sinh \left (x\right )^{3}\right )}} \end {gather*}
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 \begin {gather*} \int \frac {1}{\left (\coth {\left (x \right )} + 1\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 113 vs.
\(2 (34) = 68\).
time = 0.43, size = 113, normalized size = 2.31 \begin {gather*} \frac {\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} + 3 \, \sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - 3 \, e^{\left (2 \, x\right )} + 1\right )}}{{\left (\sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )}\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 )}}{24 \, \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.20, size = 32, normalized size = 0.65 \begin {gather*} \frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {\mathrm {coth}\left (x\right )+1}}{2}\right )}{4}-\frac {\frac {\mathrm {coth}\left (x\right )}{2}+\frac {5}{6}}{{\left (\mathrm {coth}\left (x\right )+1\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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