Integrand size = 12, antiderivative size = 31 \[ \int \sqrt {x} \coth ^{-1}\left (\sqrt {x}\right ) \, dx=\frac {x}{3}+\frac {2}{3} x^{3/2} \coth ^{-1}\left (\sqrt {x}\right )+\frac {1}{3} \log (1-x) \]
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Time = 0.01 (sec) , antiderivative size = 31, 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} \[ \int \sqrt {x} \coth ^{-1}\left (\sqrt {x}\right ) \, dx=\frac {2}{3} x^{3/2} \coth ^{-1}\left (\sqrt {x}\right )+\frac {x}{3}+\frac {1}{3} \log (1-x) \]
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Rule 45
Rule 6038
Rubi steps \begin{align*} \text {integral}& = \frac {2}{3} x^{3/2} \coth ^{-1}\left (\sqrt {x}\right )-\frac {1}{3} \int \frac {x}{1-x} \, dx \\ & = \frac {2}{3} x^{3/2} \coth ^{-1}\left (\sqrt {x}\right )-\frac {1}{3} \int \left (-1+\frac {1}{1-x}\right ) \, dx \\ & = \frac {x}{3}+\frac {2}{3} x^{3/2} \coth ^{-1}\left (\sqrt {x}\right )+\frac {1}{3} \log (1-x) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.81 \[ \int \sqrt {x} \coth ^{-1}\left (\sqrt {x}\right ) \, dx=\frac {1}{3} \left (x+2 x^{3/2} \coth ^{-1}\left (\sqrt {x}\right )+\log (1-x)\right ) \]
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Time = 0.04 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.97
method | result | size |
derivativedivides | \(\frac {2 x^{\frac {3}{2}} \operatorname {arccoth}\left (\sqrt {x}\right )}{3}+\frac {x}{3}+\frac {\ln \left (\sqrt {x}-1\right )}{3}+\frac {\ln \left (\sqrt {x}+1\right )}{3}\) | \(30\) |
default | \(\frac {2 x^{\frac {3}{2}} \operatorname {arccoth}\left (\sqrt {x}\right )}{3}+\frac {x}{3}+\frac {\ln \left (\sqrt {x}-1\right )}{3}+\frac {\ln \left (\sqrt {x}+1\right )}{3}\) | \(30\) |
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Time = 0.26 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.97 \[ \int \sqrt {x} \coth ^{-1}\left (\sqrt {x}\right ) \, dx=\frac {1}{3} \, x^{\frac {3}{2}} \log \left (\frac {x + 2 \, \sqrt {x} + 1}{x - 1}\right ) + \frac {1}{3} \, x + \frac {1}{3} \, \log \left (x - 1\right ) \]
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\[ \int \sqrt {x} \coth ^{-1}\left (\sqrt {x}\right ) \, dx=\int \sqrt {x} \operatorname {acoth}{\left (\sqrt {x} \right )}\, dx \]
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Time = 0.19 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.61 \[ \int \sqrt {x} \coth ^{-1}\left (\sqrt {x}\right ) \, dx=\frac {2}{3} \, x^{\frac {3}{2}} \operatorname {arcoth}\left (\sqrt {x}\right ) + \frac {1}{3} \, x + \frac {1}{3} \, \log \left (x - 1\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 119 vs. \(2 (21) = 42\).
Time = 0.28 (sec) , antiderivative size = 119, normalized size of antiderivative = 3.84 \[ \int \sqrt {x} \coth ^{-1}\left (\sqrt {x}\right ) \, dx=\frac {2 \, {\left (\frac {3 \, {\left (\sqrt {x} + 1\right )}^{2}}{{\left (\sqrt {x} - 1\right )}^{2}} + 1\right )} \log \left (\frac {\sqrt {x} + 1}{\sqrt {x} - 1}\right )}{3 \, {\left (\frac {\sqrt {x} + 1}{\sqrt {x} - 1} - 1\right )}^{3}} + \frac {4 \, {\left (\sqrt {x} + 1\right )}}{3 \, {\left (\sqrt {x} - 1\right )} {\left (\frac {\sqrt {x} + 1}{\sqrt {x} - 1} - 1\right )}^{2}} + \frac {2}{3} \, \log \left (\frac {\sqrt {x} + 1}{{\left | \sqrt {x} - 1 \right |}}\right ) - \frac {2}{3} \, \log \left ({\left | \frac {\sqrt {x} + 1}{\sqrt {x} - 1} - 1 \right |}\right ) \]
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Timed out. \[ \int \sqrt {x} \coth ^{-1}\left (\sqrt {x}\right ) \, dx=\int \sqrt {x}\,\mathrm {acoth}\left (\sqrt {x}\right ) \,d x \]
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