Optimal. Leaf size=24 \[ -\log (1-x)+\log (x)-\frac {2 \coth ^{-1}\left (\sqrt {x}\right )}{\sqrt {x}} \]
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Rubi [A] time = 0.01, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6098, 36, 31, 29} \[ -\log (1-x)+\log (x)-\frac {2 \coth ^{-1}\left (\sqrt {x}\right )}{\sqrt {x}} \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 6098
Rubi steps
\begin {align*} \int \frac {\coth ^{-1}\left (\sqrt {x}\right )}{x^{3/2}} \, dx &=-\frac {2 \coth ^{-1}\left (\sqrt {x}\right )}{\sqrt {x}}+\int \frac {1}{(1-x) x} \, dx\\ &=-\frac {2 \coth ^{-1}\left (\sqrt {x}\right )}{\sqrt {x}}+\int \frac {1}{1-x} \, dx+\int \frac {1}{x} \, dx\\ &=-\frac {2 \coth ^{-1}\left (\sqrt {x}\right )}{\sqrt {x}}-\log (1-x)+\log (x)\\ \end {align*}
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Mathematica [A] time = 0.02, size = 24, normalized size = 1.00 \[ -\log (1-x)+\log (x)-\frac {2 \coth ^{-1}\left (\sqrt {x}\right )}{\sqrt {x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 36, normalized size = 1.50 \[ -\frac {x \log \left (x - 1\right ) - x \log \relax (x) + \sqrt {x} \log \left (\frac {x + 2 \, \sqrt {x} + 1}{x - 1}\right )}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {arcoth}\left (\sqrt {x}\right )}{x^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 29, normalized size = 1.21 \[ -\frac {2 \,\mathrm {arccoth}\left (\sqrt {x}\right )}{\sqrt {x}}+\ln \relax (x )-\ln \left (-1+\sqrt {x}\right )-\ln \left (1+\sqrt {x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 18, normalized size = 0.75 \[ -\frac {2 \, \operatorname {arcoth}\left (\sqrt {x}\right )}{\sqrt {x}} - \log \left (x - 1\right ) + \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.25, size = 22, normalized size = 0.92 \[ 2\,\ln \left (\sqrt {x}\right )-\ln \left (x-1\right )-\frac {2\,\mathrm {acoth}\left (\sqrt {x}\right )}{\sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.38, size = 126, normalized size = 5.25 \[ - \frac {2 x^{\frac {3}{2}} \operatorname {acoth}{\left (\sqrt {x} \right )}}{x^{2} - x} + \frac {2 \sqrt {x} \operatorname {acoth}{\left (\sqrt {x} \right )}}{x^{2} - x} + \frac {x^{2} \log {\relax (x )}}{x^{2} - x} - \frac {2 x^{2} \log {\left (\sqrt {x} + 1 \right )}}{x^{2} - x} + \frac {2 x^{2} \operatorname {acoth}{\left (\sqrt {x} \right )}}{x^{2} - x} - \frac {x \log {\relax (x )}}{x^{2} - x} + \frac {2 x \log {\left (\sqrt {x} + 1 \right )}}{x^{2} - x} - \frac {2 x \operatorname {acoth}{\left (\sqrt {x} \right )}}{x^{2} - x} \]
Verification of antiderivative is not currently implemented for this CAS.
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