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 = 0.0120186, antiderivative size = 37, normalized size of antiderivative = 1., 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 5034
Rule 44
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*}
Mathematica [A] time = 0.018484, 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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Maple [A] time = 0.029, size = 26, normalized size = 0.7 \begin{align*}{\frac{1}{3\,x}}-{\frac{2}{3}{\rm arccot} \left (\sqrt{x}\right ){x}^{-{\frac{3}{2}}}}+{\frac{\ln \left ( x \right ) }{3}}-{\frac{\ln \left ( x+1 \right ) }{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00591, size = 34, normalized size = 0.92 \begin{align*} -\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 \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.26169, size = 97, normalized size = 2.62 \begin{align*} -\frac{x^{2} \log \left (x + 1\right ) - x^{2} \log \left (x\right ) + 2 \, \sqrt{x} \operatorname{arccot}\left (\sqrt{x}\right ) - x}{3 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 8.51655, size = 143, normalized size = 3.86 \begin{align*} - \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{\left (x \right )}}{3 x^{3} + 3 x^{2}} - \frac{x^{3} \log{\left (x + 1 \right )}}{3 x^{3} + 3 x^{2}} - \frac{x^{3}}{3 x^{3} + 3 x^{2}} + \frac{x^{2} \log{\left (x \right )}}{3 x^{3} + 3 x^{2}} - \frac{x^{2} \log{\left (x + 1 \right )}}{3 x^{3} + 3 x^{2}} + \frac{x}{3 x^{3} + 3 x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12866, size = 38, normalized size = 1.03 \begin{align*} -\frac{x - 1}{3 \, x} - \frac{2 \, \arctan \left (\frac{1}{\sqrt{x}}\right )}{3 \, x^{\frac{3}{2}}} - \frac{1}{3} \, \log \left (x + 1\right ) + \frac{1}{3} \, \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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