Optimal. Leaf size=37 \[ -\frac {2 \cot ^{-1}\left (\sqrt {x}\right )}{3 x^{3/2}}+\frac {1}{3 x}+\frac {\log (x)}{3}-\frac {1}{3} \log (x+1) \]
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Rubi [A] time = 0.01, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {5034, 44} \[ -\frac {2 \cot ^{-1}\left (\sqrt {x}\right )}{3 x^{3/2}}+\frac {1}{3 x}+\frac {\log (x)}{3}-\frac {1}{3} \log (x+1) \]
Antiderivative was successfully verified.
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Rule 44
Rule 5034
Rubi steps
\begin {align*} \int \frac {\cot ^{-1}\left (\sqrt {x}\right )}{x^{5/2}} \, dx &=-\frac {2 \cot ^{-1}\left (\sqrt {x}\right )}{3 x^{3/2}}-\frac {1}{3} \int \frac {1}{x^2 (1+x)} \, dx\\ &=-\frac {2 \cot ^{-1}\left (\sqrt {x}\right )}{3 x^{3/2}}-\frac {1}{3} \int \left (\frac {1}{x^2}-\frac {1}{x}+\frac {1}{1+x}\right ) \, dx\\ &=\frac {1}{3 x}-\frac {2 \cot ^{-1}\left (\sqrt {x}\right )}{3 x^{3/2}}+\frac {\log (x)}{3}-\frac {1}{3} \log (1+x)\\ \end {align*}
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Mathematica [A] time = 0.02, size = 29, normalized size = 0.78 \[ \frac {1}{3} \left (-\frac {2 \cot ^{-1}\left (\sqrt {x}\right )}{x^{3/2}}+\frac {1}{x}+\log (x)-\log (x+1)\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 33, normalized size = 0.89 \[ -\frac {x^{2} \log \left (x + 1\right ) - x^{2} \log \relax (x) + 2 \, \sqrt {x} \operatorname {arccot}\left (\sqrt {x}\right ) - x}{3 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 23, normalized size = 0.62 \[ -\frac {2 \, \arctan \left (\frac {1}{\sqrt {x}}\right )}{3 \, x^{\frac {3}{2}}} + \frac {1}{3 \, x} - \frac {1}{3} \, \log \left (\frac {1}{x} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 26, normalized size = 0.70 \[ \frac {1}{3 x}-\frac {2 \,\mathrm {arccot}\left (\sqrt {x}\right )}{3 x^{\frac {3}{2}}}+\frac {\ln \relax (x )}{3}-\frac {\ln \left (x +1\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 25, normalized size = 0.68 \[ -\frac {2 \, \operatorname {arccot}\left (\sqrt {x}\right )}{3 \, x^{\frac {3}{2}}} + \frac {1}{3 \, x} - \frac {1}{3} \, \log \left (x + 1\right ) + \frac {1}{3} \, \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.65, size = 27, normalized size = 0.73 \[ \frac {2\,\ln \left (\sqrt {x}\right )}{3}-\frac {\ln \left (x+1\right )}{3}-\frac {2\,\mathrm {acot}\left (\sqrt {x}\right )}{3\,x^{3/2}}+\frac {1}{3\,x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 4.92, size = 143, normalized size = 3.86 \[ - \frac {2 x^{\frac {3}{2}} \operatorname {acot}{\left (\sqrt {x} \right )}}{3 x^{3} + 3 x^{2}} - \frac {2 \sqrt {x} \operatorname {acot}{\left (\sqrt {x} \right )}}{3 x^{3} + 3 x^{2}} + \frac {x^{3} \log {\relax (x )}}{3 x^{3} + 3 x^{2}} - \frac {x^{3} \log {\left (x + 1 \right )}}{3 x^{3} + 3 x^{2}} + \frac {x^{2} \log {\relax (x )}}{3 x^{3} + 3 x^{2}} - \frac {x^{2} \log {\left (x + 1 \right )}}{3 x^{3} + 3 x^{2}} + \frac {x^{2}}{3 x^{3} + 3 x^{2}} + \frac {x}{3 x^{3} + 3 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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