Optimal. Leaf size=42 \[ -\frac{1}{6 x^{3/2}}-\frac{\tan ^{-1}\left (\sqrt{x}\right )}{2 x^2}+\frac{1}{2 \sqrt{x}}+\frac{1}{2} \tan ^{-1}\left (\sqrt{x}\right ) \]
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Rubi [A] time = 0.0138852, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {5033, 51, 63, 203} \[ -\frac{1}{6 x^{3/2}}-\frac{\tan ^{-1}\left (\sqrt{x}\right )}{2 x^2}+\frac{1}{2 \sqrt{x}}+\frac{1}{2} \tan ^{-1}\left (\sqrt{x}\right ) \]
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^3} \, dx &=-\frac{\tan ^{-1}\left (\sqrt{x}\right )}{2 x^2}+\frac{1}{4} \int \frac{1}{x^{5/2} (1+x)} \, dx\\ &=-\frac{1}{6 x^{3/2}}-\frac{\tan ^{-1}\left (\sqrt{x}\right )}{2 x^2}-\frac{1}{4} \int \frac{1}{x^{3/2} (1+x)} \, dx\\ &=-\frac{1}{6 x^{3/2}}+\frac{1}{2 \sqrt{x}}-\frac{\tan ^{-1}\left (\sqrt{x}\right )}{2 x^2}+\frac{1}{4} \int \frac{1}{\sqrt{x} (1+x)} \, dx\\ &=-\frac{1}{6 x^{3/2}}+\frac{1}{2 \sqrt{x}}-\frac{\tan ^{-1}\left (\sqrt{x}\right )}{2 x^2}+\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sqrt{x}\right )\\ &=-\frac{1}{6 x^{3/2}}+\frac{1}{2 \sqrt{x}}+\frac{1}{2} \tan ^{-1}\left (\sqrt{x}\right )-\frac{\tan ^{-1}\left (\sqrt{x}\right )}{2 x^2}\\ \end{align*}
Mathematica [C] time = 0.0110679, size = 34, normalized size = 0.81 \[ -\frac{\text{Hypergeometric2F1}\left (-\frac{3}{2},1,-\frac{1}{2},-x\right )}{6 x^{3/2}}-\frac{\tan ^{-1}\left (\sqrt{x}\right )}{2 x^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.027, size = 27, normalized size = 0.6 \begin{align*} -{\frac{1}{6}{x}^{-{\frac{3}{2}}}}+{\frac{1}{2}\arctan \left ( \sqrt{x} \right ) }-{\frac{1}{2\,{x}^{2}}\arctan \left ( \sqrt{x} \right ) }+{\frac{1}{2}{\frac{1}{\sqrt{x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.45289, size = 35, normalized size = 0.83 \begin{align*} \frac{3 \, x - 1}{6 \, x^{\frac{3}{2}}} - \frac{\arctan \left (\sqrt{x}\right )}{2 \, x^{2}} + \frac{1}{2} \, \arctan \left (\sqrt{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.19196, size = 80, normalized size = 1.9 \begin{align*} \frac{3 \,{\left (x^{2} - 1\right )} \arctan \left (\sqrt{x}\right ) +{\left (3 \, x - 1\right )} \sqrt{x}}{6 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 8.43446, size = 160, normalized size = 3.81 \begin{align*} \frac{3 x^{\frac{7}{2}} \operatorname{atan}{\left (\sqrt{x} \right )}}{6 x^{\frac{7}{2}} + 6 x^{\frac{5}{2}}} + \frac{3 x^{\frac{5}{2}} \operatorname{atan}{\left (\sqrt{x} \right )}}{6 x^{\frac{7}{2}} + 6 x^{\frac{5}{2}}} - \frac{3 x^{\frac{3}{2}} \operatorname{atan}{\left (\sqrt{x} \right )}}{6 x^{\frac{7}{2}} + 6 x^{\frac{5}{2}}} - \frac{3 \sqrt{x} \operatorname{atan}{\left (\sqrt{x} \right )}}{6 x^{\frac{7}{2}} + 6 x^{\frac{5}{2}}} + \frac{3 x^{3}}{6 x^{\frac{7}{2}} + 6 x^{\frac{5}{2}}} + \frac{2 x^{2}}{6 x^{\frac{7}{2}} + 6 x^{\frac{5}{2}}} - \frac{x}{6 x^{\frac{7}{2}} + 6 x^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.24529, size = 35, normalized size = 0.83 \begin{align*} \frac{3 \, x - 1}{6 \, x^{\frac{3}{2}}} - \frac{\arctan \left (\sqrt{x}\right )}{2 \, x^{2}} + \frac{1}{2} \, \arctan \left (\sqrt{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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