Optimal. Leaf size=27 \[ -\frac{1}{\sqrt{x}}-\frac{\tan ^{-1}\left (\sqrt{x}\right )}{x}-\tan ^{-1}\left (\sqrt{x}\right ) \]
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Rubi [A] time = 0.011758, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {5033, 51, 63, 203} \[ -\frac{1}{\sqrt{x}}-\frac{\tan ^{-1}\left (\sqrt{x}\right )}{x}-\tan ^{-1}\left (\sqrt{x}\right ) \]
Antiderivative was successfully verified.
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Rule 5033
Rule 51
Rule 63
Rule 203
Rubi steps
\begin{align*} \int \frac{\tan ^{-1}\left (\sqrt{x}\right )}{x^2} \, dx &=-\frac{\tan ^{-1}\left (\sqrt{x}\right )}{x}+\frac{1}{2} \int \frac{1}{x^{3/2} (1+x)} \, dx\\ &=-\frac{1}{\sqrt{x}}-\frac{\tan ^{-1}\left (\sqrt{x}\right )}{x}-\frac{1}{2} \int \frac{1}{\sqrt{x} (1+x)} \, dx\\ &=-\frac{1}{\sqrt{x}}-\frac{\tan ^{-1}\left (\sqrt{x}\right )}{x}-\operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sqrt{x}\right )\\ &=-\frac{1}{\sqrt{x}}-\tan ^{-1}\left (\sqrt{x}\right )-\frac{\tan ^{-1}\left (\sqrt{x}\right )}{x}\\ \end{align*}
Mathematica [C] time = 0.0095062, size = 30, normalized size = 1.11 \[ -\frac{\text{Hypergeometric2F1}\left (-\frac{1}{2},1,\frac{1}{2},-x\right )}{\sqrt{x}}-\frac{\tan ^{-1}\left (\sqrt{x}\right )}{x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.026, size = 22, normalized size = 0.8 \begin{align*} -\arctan \left ( \sqrt{x} \right ) -{\frac{1}{x}\arctan \left ( \sqrt{x} \right ) }-{\frac{1}{\sqrt{x}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.46563, size = 28, normalized size = 1.04 \begin{align*} -\frac{\arctan \left (\sqrt{x}\right )}{x} - \frac{1}{\sqrt{x}} - \arctan \left (\sqrt{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.13151, size = 54, normalized size = 2. \begin{align*} -\frac{{\left (x + 1\right )} \arctan \left (\sqrt{x}\right ) + \sqrt{x}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 3.14029, size = 94, normalized size = 3.48 \begin{align*} - \frac{x^{\frac{5}{2}} \operatorname{atan}{\left (\sqrt{x} \right )}}{x^{\frac{5}{2}} + x^{\frac{3}{2}}} - \frac{2 x^{\frac{3}{2}} \operatorname{atan}{\left (\sqrt{x} \right )}}{x^{\frac{5}{2}} + x^{\frac{3}{2}}} - \frac{\sqrt{x} \operatorname{atan}{\left (\sqrt{x} \right )}}{x^{\frac{5}{2}} + x^{\frac{3}{2}}} - \frac{x^{2}}{x^{\frac{5}{2}} + x^{\frac{3}{2}}} - \frac{x}{x^{\frac{5}{2}} + x^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.34476, size = 28, normalized size = 1.04 \begin{align*} -\frac{\arctan \left (\sqrt{x}\right )}{x} - \frac{1}{\sqrt{x}} - \arctan \left (\sqrt{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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