Integrand size = 8, antiderivative size = 42 \[ \int x \coth ^{-1}\left (\sqrt {x}\right ) \, dx=\frac {\sqrt {x}}{2}+\frac {x^{3/2}}{6}+\frac {1}{2} x^2 \coth ^{-1}\left (\sqrt {x}\right )-\frac {\text {arctanh}\left (\sqrt {x}\right )}{2} \]
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Time = 0.01 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6038, 52, 65, 212} \[ \int x \coth ^{-1}\left (\sqrt {x}\right ) \, dx=-\frac {\text {arctanh}\left (\sqrt {x}\right )}{2}+\frac {x^{3/2}}{6}+\frac {1}{2} x^2 \coth ^{-1}\left (\sqrt {x}\right )+\frac {\sqrt {x}}{2} \]
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Rule 52
Rule 65
Rule 212
Rule 6038
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} x^2 \coth ^{-1}\left (\sqrt {x}\right )-\frac {1}{4} \int \frac {x^{3/2}}{1-x} \, dx \\ & = \frac {x^{3/2}}{6}+\frac {1}{2} x^2 \coth ^{-1}\left (\sqrt {x}\right )-\frac {1}{4} \int \frac {\sqrt {x}}{1-x} \, dx \\ & = \frac {\sqrt {x}}{2}+\frac {x^{3/2}}{6}+\frac {1}{2} x^2 \coth ^{-1}\left (\sqrt {x}\right )-\frac {1}{4} \int \frac {1}{(1-x) \sqrt {x}} \, dx \\ & = \frac {\sqrt {x}}{2}+\frac {x^{3/2}}{6}+\frac {1}{2} x^2 \coth ^{-1}\left (\sqrt {x}\right )-\frac {1}{2} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {x}\right ) \\ & = \frac {\sqrt {x}}{2}+\frac {x^{3/2}}{6}+\frac {1}{2} x^2 \coth ^{-1}\left (\sqrt {x}\right )-\frac {\text {arctanh}\left (\sqrt {x}\right )}{2} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.24 \[ \int x \coth ^{-1}\left (\sqrt {x}\right ) \, dx=\frac {1}{12} \left (6 \sqrt {x}+2 x^{3/2}+6 x^2 \coth ^{-1}\left (\sqrt {x}\right )+3 \log \left (1-\sqrt {x}\right )-3 \log \left (1+\sqrt {x}\right )\right ) \]
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Time = 0.04 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.88
method | result | size |
derivativedivides | \(\frac {x^{2} \operatorname {arccoth}\left (\sqrt {x}\right )}{2}+\frac {x^{\frac {3}{2}}}{6}+\frac {\sqrt {x}}{2}+\frac {\ln \left (\sqrt {x}-1\right )}{4}-\frac {\ln \left (\sqrt {x}+1\right )}{4}\) | \(37\) |
default | \(\frac {x^{2} \operatorname {arccoth}\left (\sqrt {x}\right )}{2}+\frac {x^{\frac {3}{2}}}{6}+\frac {\sqrt {x}}{2}+\frac {\ln \left (\sqrt {x}-1\right )}{4}-\frac {\ln \left (\sqrt {x}+1\right )}{4}\) | \(37\) |
parts | \(\frac {x^{2} \operatorname {arccoth}\left (\sqrt {x}\right )}{2}+\frac {x^{\frac {3}{2}}}{6}+\frac {\sqrt {x}}{2}+\frac {\ln \left (\sqrt {x}-1\right )}{4}-\frac {\ln \left (\sqrt {x}+1\right )}{4}\) | \(37\) |
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Time = 0.25 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.74 \[ \int x \coth ^{-1}\left (\sqrt {x}\right ) \, dx=\frac {1}{4} \, {\left (x^{2} - 1\right )} \log \left (\frac {x + 2 \, \sqrt {x} + 1}{x - 1}\right ) + \frac {1}{6} \, {\left (x + 3\right )} \sqrt {x} \]
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\[ \int x \coth ^{-1}\left (\sqrt {x}\right ) \, dx=\int x \operatorname {acoth}{\left (\sqrt {x} \right )}\, dx \]
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Time = 0.20 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.86 \[ \int x \coth ^{-1}\left (\sqrt {x}\right ) \, dx=\frac {1}{2} \, x^{2} \operatorname {arcoth}\left (\sqrt {x}\right ) + \frac {1}{6} \, x^{\frac {3}{2}} + \frac {1}{2} \, \sqrt {x} - \frac {1}{4} \, \log \left (\sqrt {x} + 1\right ) + \frac {1}{4} \, \log \left (\sqrt {x} - 1\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 114 vs. \(2 (26) = 52\).
Time = 0.28 (sec) , antiderivative size = 114, normalized size of antiderivative = 2.71 \[ \int x \coth ^{-1}\left (\sqrt {x}\right ) \, dx=\frac {2 \, {\left (\frac {3 \, {\left (\sqrt {x} + 1\right )}^{2}}{{\left (\sqrt {x} - 1\right )}^{2}} - \frac {3 \, {\left (\sqrt {x} + 1\right )}}{\sqrt {x} - 1} + 2\right )}}{3 \, {\left (\frac {\sqrt {x} + 1}{\sqrt {x} - 1} - 1\right )}^{3}} + \frac {2 \, {\left (\frac {{\left (\sqrt {x} + 1\right )}^{3}}{{\left (\sqrt {x} - 1\right )}^{3}} + \frac {\sqrt {x} + 1}{\sqrt {x} - 1}\right )} \log \left (\frac {\sqrt {x} + 1}{\sqrt {x} - 1}\right )}{{\left (\frac {\sqrt {x} + 1}{\sqrt {x} - 1} - 1\right )}^{4}} \]
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Time = 4.25 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.62 \[ \int x \coth ^{-1}\left (\sqrt {x}\right ) \, dx=\frac {x^2\,\mathrm {acoth}\left (\sqrt {x}\right )}{2}-\frac {\mathrm {acoth}\left (\sqrt {x}\right )}{2}+\frac {\sqrt {x}}{2}+\frac {x^{3/2}}{6} \]
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