Optimal. Leaf size=22 \[ \log (x)-\log (x+1)-\frac{2 \tan ^{-1}\left (\sqrt{x}\right )}{\sqrt{x}} \]
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Rubi [A] time = 0.0087961, 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 = {5033, 36, 29, 31} \[ \log (x)-\log (x+1)-\frac{2 \tan ^{-1}\left (\sqrt{x}\right )}{\sqrt{x}} \]
Antiderivative was successfully verified.
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Rule 5033
Rule 36
Rule 29
Rule 31
Rubi steps
\begin{align*} \int \frac{\tan ^{-1}\left (\sqrt{x}\right )}{x^{3/2}} \, dx &=-\frac{2 \tan ^{-1}\left (\sqrt{x}\right )}{\sqrt{x}}+\int \frac{1}{x (1+x)} \, dx\\ &=-\frac{2 \tan ^{-1}\left (\sqrt{x}\right )}{\sqrt{x}}+\int \frac{1}{x} \, dx-\int \frac{1}{1+x} \, dx\\ &=-\frac{2 \tan ^{-1}\left (\sqrt{x}\right )}{\sqrt{x}}+\log (x)-\log (1+x)\\ \end{align*}
Mathematica [A] time = 0.0109185, size = 22, normalized size = 1. \[ \log (x)-\log (x+1)-\frac{2 \tan ^{-1}\left (\sqrt{x}\right )}{\sqrt{x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.026, size = 19, normalized size = 0.9 \begin{align*} \ln \left ( x \right ) -\ln \left ( x+1 \right ) -2\,{\frac{\arctan \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 = 1.14953, size = 24, normalized size = 1.09 \begin{align*} -\frac{2 \, \arctan \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 = 2.18452, size = 78, normalized size = 3.55 \begin{align*} -\frac{x \log \left (x + 1\right ) - x \log \left (x\right ) + 2 \, \sqrt{x} \arctan \left (\sqrt{x}\right )}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.91852, size = 20, normalized size = 0.91 \begin{align*} \log{\left (x \right )} - \log{\left (x + 1 \right )} - \frac{2 \operatorname{atan}{\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 = 1.11018, size = 24, normalized size = 1.09 \begin{align*} -\frac{2 \, \arctan \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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