Integrand size = 12, antiderivative size = 38 \[ \int x^{3/2} \coth ^{-1}\left (\sqrt {x}\right ) \, dx=\frac {x}{5}+\frac {x^2}{10}+\frac {2}{5} x^{5/2} \coth ^{-1}\left (\sqrt {x}\right )+\frac {1}{5} \log (1-x) \]
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Time = 0.01 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6038, 45} \[ \int x^{3/2} \coth ^{-1}\left (\sqrt {x}\right ) \, dx=\frac {2}{5} x^{5/2} \coth ^{-1}\left (\sqrt {x}\right )+\frac {x^2}{10}+\frac {x}{5}+\frac {1}{5} \log (1-x) \]
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Rule 45
Rule 6038
Rubi steps \begin{align*} \text {integral}& = \frac {2}{5} x^{5/2} \coth ^{-1}\left (\sqrt {x}\right )-\frac {1}{5} \int \frac {x^2}{1-x} \, dx \\ & = \frac {2}{5} x^{5/2} \coth ^{-1}\left (\sqrt {x}\right )-\frac {1}{5} \int \left (-1+\frac {1}{1-x}-x\right ) \, dx \\ & = \frac {x}{5}+\frac {x^2}{10}+\frac {2}{5} x^{5/2} \coth ^{-1}\left (\sqrt {x}\right )+\frac {1}{5} \log (1-x) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.82 \[ \int x^{3/2} \coth ^{-1}\left (\sqrt {x}\right ) \, dx=\frac {1}{10} \left (x (2+x)+4 x^{5/2} \coth ^{-1}\left (\sqrt {x}\right )+2 \log (1-x)\right ) \]
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Time = 0.04 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.92
method | result | size |
derivativedivides | \(\frac {2 x^{\frac {5}{2}} \operatorname {arccoth}\left (\sqrt {x}\right )}{5}+\frac {x^{2}}{10}+\frac {x}{5}+\frac {\ln \left (\sqrt {x}-1\right )}{5}+\frac {\ln \left (\sqrt {x}+1\right )}{5}\) | \(35\) |
default | \(\frac {2 x^{\frac {5}{2}} \operatorname {arccoth}\left (\sqrt {x}\right )}{5}+\frac {x^{2}}{10}+\frac {x}{5}+\frac {\ln \left (\sqrt {x}-1\right )}{5}+\frac {\ln \left (\sqrt {x}+1\right )}{5}\) | \(35\) |
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Time = 0.28 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.92 \[ \int x^{3/2} \coth ^{-1}\left (\sqrt {x}\right ) \, dx=\frac {1}{5} \, x^{\frac {5}{2}} \log \left (\frac {x + 2 \, \sqrt {x} + 1}{x - 1}\right ) + \frac {1}{10} \, x^{2} + \frac {1}{5} \, x + \frac {1}{5} \, \log \left (x - 1\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 121 vs. \(2 (29) = 58\).
Time = 0.99 (sec) , antiderivative size = 121, normalized size of antiderivative = 3.18 \[ \int x^{3/2} \coth ^{-1}\left (\sqrt {x}\right ) \, dx=\frac {4 x^{\frac {7}{2}} \operatorname {acoth}{\left (\sqrt {x} \right )}}{10 x - 10} - \frac {4 x^{\frac {5}{2}} \operatorname {acoth}{\left (\sqrt {x} \right )}}{10 x - 10} + \frac {x^{3}}{10 x - 10} + \frac {x^{2}}{10 x - 10} + \frac {4 x \log {\left (\sqrt {x} + 1 \right )}}{10 x - 10} - \frac {4 x \operatorname {acoth}{\left (\sqrt {x} \right )}}{10 x - 10} - \frac {4 \log {\left (\sqrt {x} + 1 \right )}}{10 x - 10} + \frac {4 \operatorname {acoth}{\left (\sqrt {x} \right )}}{10 x - 10} - \frac {2}{10 x - 10} \]
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Time = 0.19 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.63 \[ \int x^{3/2} \coth ^{-1}\left (\sqrt {x}\right ) \, dx=\frac {2}{5} \, x^{\frac {5}{2}} \operatorname {arcoth}\left (\sqrt {x}\right ) + \frac {1}{10} \, x^{2} + \frac {1}{5} \, x + \frac {1}{5} \, \log \left (x - 1\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 168 vs. \(2 (26) = 52\).
Time = 0.27 (sec) , antiderivative size = 168, normalized size of antiderivative = 4.42 \[ \int x^{3/2} \coth ^{-1}\left (\sqrt {x}\right ) \, dx=\frac {8 \, {\left (\frac {{\left (\sqrt {x} + 1\right )}^{3}}{{\left (\sqrt {x} - 1\right )}^{3}} - \frac {{\left (\sqrt {x} + 1\right )}^{2}}{{\left (\sqrt {x} - 1\right )}^{2}} + \frac {\sqrt {x} + 1}{\sqrt {x} - 1}\right )}}{5 \, {\left (\frac {\sqrt {x} + 1}{\sqrt {x} - 1} - 1\right )}^{4}} + \frac {2 \, {\left (\frac {5 \, {\left (\sqrt {x} + 1\right )}^{4}}{{\left (\sqrt {x} - 1\right )}^{4}} + \frac {10 \, {\left (\sqrt {x} + 1\right )}^{2}}{{\left (\sqrt {x} - 1\right )}^{2}} + 1\right )} \log \left (\frac {\sqrt {x} + 1}{\sqrt {x} - 1}\right )}{5 \, {\left (\frac {\sqrt {x} + 1}{\sqrt {x} - 1} - 1\right )}^{5}} + \frac {2}{5} \, \log \left (\frac {\sqrt {x} + 1}{{\left | \sqrt {x} - 1 \right |}}\right ) - \frac {2}{5} \, \log \left ({\left | \frac {\sqrt {x} + 1}{\sqrt {x} - 1} - 1 \right |}\right ) \]
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Time = 4.56 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.63 \[ \int x^{3/2} \coth ^{-1}\left (\sqrt {x}\right ) \, dx=\frac {x}{5}+\frac {\ln \left (x-1\right )}{5}+\frac {2\,x^{5/2}\,\mathrm {acoth}\left (\sqrt {x}\right )}{5}+\frac {x^2}{10} \]
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