Integrand size = 8, antiderivative size = 49 \[ \int \frac {1}{(1+\coth (x))^{3/2}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {1+\coth (x)}}{\sqrt {2}}\right )}{2 \sqrt {2}}-\frac {1}{3 (1+\coth (x))^{3/2}}-\frac {1}{2 \sqrt {1+\coth (x)}} \]
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Time = 0.03 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {3560, 3561, 212} \[ \int \frac {1}{(1+\coth (x))^{3/2}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {\coth (x)+1}}{\sqrt {2}}\right )}{2 \sqrt {2}}-\frac {1}{2 \sqrt {\coth (x)+1}}-\frac {1}{3 (\coth (x)+1)^{3/2}} \]
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Rule 212
Rule 3560
Rule 3561
Rubi steps \begin{align*} \text {integral}& = -\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 {\text {arctanh}\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*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.56 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.57 \[ \int \frac {1}{(1+\coth (x))^{3/2}} \, dx=-\frac {\operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},\frac {1}{2} (1+\coth (x))\right )}{3 (1+\coth (x))^{3/2}} \]
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Time = 0.08 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.71
method | result | size |
derivativedivides | \(-\frac {1}{3 \left (1+\coth \left (x \right )\right )^{\frac {3}{2}}}+\frac {\operatorname {arctanh}\left (\frac {\sqrt {1+\coth \left (x \right )}\, \sqrt {2}}{2}\right ) \sqrt {2}}{4}-\frac {1}{2 \sqrt {1+\coth \left (x \right )}}\) | \(35\) |
default | \(-\frac {1}{3 \left (1+\coth \left (x \right )\right )^{\frac {3}{2}}}+\frac {\operatorname {arctanh}\left (\frac {\sqrt {1+\coth \left (x \right )}\, \sqrt {2}}{2}\right ) \sqrt {2}}{4}-\frac {1}{2 \sqrt {1+\coth \left (x \right )}}\) | \(35\) |
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Leaf count of result is larger than twice the leaf count of optimal. 168 vs. \(2 (34) = 68\).
Time = 0.27 (sec) , antiderivative size = 168, normalized size of antiderivative = 3.43 \[ \int \frac {1}{(1+\coth (x))^{3/2}} \, dx=-\frac {2 \, \sqrt {2} {\left (4 \, \sqrt {2} \cosh \left (x\right )^{2} + 8 \, \sqrt {2} \cosh \left (x\right ) \sinh \left (x\right ) + 4 \, \sqrt {2} \sinh \left (x\right )^{2} - \sqrt {2}\right )} \sqrt {\frac {\sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}} - 3 \, {\left (\sqrt {2} \cosh \left (x\right )^{3} + 3 \, \sqrt {2} \cosh \left (x\right )^{2} \sinh \left (x\right ) + 3 \, \sqrt {2} \cosh \left (x\right ) \sinh \left (x\right )^{2} + \sqrt {2} \sinh \left (x\right )^{3}\right )} \log \left (2 \, \sqrt {2} \sqrt {\frac {\sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}} {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} + 2 \, \cosh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) \sinh \left (x\right ) + 2 \, \sinh \left (x\right )^{2} - 1\right )}{24 \, {\left (\cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right )^{2} \sinh \left (x\right ) + 3 \, \cosh \left (x\right ) \sinh \left (x\right )^{2} + \sinh \left (x\right )^{3}\right )}} \]
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\[ \int \frac {1}{(1+\coth (x))^{3/2}} \, dx=\int \frac {1}{\left (\coth {\left (x \right )} + 1\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {1}{(1+\coth (x))^{3/2}} \, dx=\int { \frac {1}{{\left (\coth \left (x\right ) + 1\right )}^{\frac {3}{2}}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 113 vs. \(2 (34) = 68\).
Time = 0.28 (sec) , antiderivative size = 113, normalized size of antiderivative = 2.31 \[ \int \frac {1}{(1+\coth (x))^{3/2}} \, dx=\frac {\sqrt {2} {\left (\frac {2 \, {\left (6 \, {\left (\sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )}\right )}^{2} + 3 \, \sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - 3 \, e^{\left (2 \, x\right )} + 1\right )}}{{\left (\sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )}\right )}^{3}} - 3 \, \log \left ({\left | 2 \, \sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - 2 \, e^{\left (2 \, x\right )} + 1 \right |}\right )\right )}}{24 \, \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right )} \]
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Time = 1.91 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.65 \[ \int \frac {1}{(1+\coth (x))^{3/2}} \, dx=\frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {\mathrm {coth}\left (x\right )+1}}{2}\right )}{4}-\frac {\frac {\mathrm {coth}\left (x\right )}{2}+\frac {5}{6}}{{\left (\mathrm {coth}\left (x\right )+1\right )}^{3/2}} \]
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