Optimal. Leaf size=22 \[ -\log (x)+\log (x+1)-\frac{2 \cot ^{-1}\left (\sqrt{x}\right )}{\sqrt{x}} \]
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Rubi [A] time = 0.0076322, antiderivative size = 22, normalized size of antiderivative = 1., 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 = {5034, 36, 29, 31} \[ -\log (x)+\log (x+1)-\frac{2 \cot ^{-1}\left (\sqrt{x}\right )}{\sqrt{x}} \]
Antiderivative was successfully verified.
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Rule 5034
Rule 36
Rule 29
Rule 31
Rubi steps
\begin{align*} \int \frac{\cot ^{-1}\left (\sqrt{x}\right )}{x^{3/2}} \, dx &=-\frac{2 \cot ^{-1}\left (\sqrt{x}\right )}{\sqrt{x}}-\int \frac{1}{x (1+x)} \, dx\\ &=-\frac{2 \cot ^{-1}\left (\sqrt{x}\right )}{\sqrt{x}}-\int \frac{1}{x} \, dx+\int \frac{1}{1+x} \, dx\\ &=-\frac{2 \cot ^{-1}\left (\sqrt{x}\right )}{\sqrt{x}}-\log (x)+\log (1+x)\\ \end{align*}
Mathematica [A] time = 0.0118346, size = 22, normalized size = 1. \[ -\log (x)+\log (x+1)-\frac{2 \cot ^{-1}\left (\sqrt{x}\right )}{\sqrt{x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.027, size = 19, normalized size = 0.9 \begin{align*} -\ln \left ( x \right ) +\ln \left ( x+1 \right ) -2\,{\frac{{\rm arccot} \left (\sqrt{x}\right )}{\sqrt{x}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.978603, size = 24, normalized size = 1.09 \begin{align*} -\frac{2 \, \operatorname{arccot}\left (\sqrt{x}\right )}{\sqrt{x}} + \log \left (x + 1\right ) - \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.13551, size = 77, normalized size = 3.5 \begin{align*} \frac{x \log \left (x + 1\right ) - x \log \left (x\right ) - 2 \, \sqrt{x} \operatorname{arccot}\left (\sqrt{x}\right )}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.76621, size = 20, normalized size = 0.91 \begin{align*} - \log{\left (x \right )} + \log{\left (x + 1 \right )} - \frac{2 \operatorname{acot}{\left (\sqrt{x} \right )}}{\sqrt{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12818, size = 22, normalized size = 1. \begin{align*} -\frac{2 \, \arctan \left (\frac{1}{\sqrt{x}}\right )}{\sqrt{x}} + \log \left (\frac{1}{x} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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