Optimal. Leaf size=13 \[ \tan ^{-1}\left (\sqrt{x}\right )+\tanh ^{-1}\left (\sqrt{x}\right ) \]
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Rubi [A] time = 0.0065155, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {329, 212, 206, 203} \[ \tan ^{-1}\left (\sqrt{x}\right )+\tanh ^{-1}\left (\sqrt{x}\right ) \]
Antiderivative was successfully verified.
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Rule 329
Rule 212
Rule 206
Rule 203
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{x} \left (1-x^2\right )} \, dx &=2 \operatorname{Subst}\left (\int \frac{1}{1-x^4} \, dx,x,\sqrt{x}\right )\\ &=\operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\sqrt{x}\right )+\operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sqrt{x}\right )\\ &=\tan ^{-1}\left (\sqrt{x}\right )+\tanh ^{-1}\left (\sqrt{x}\right )\\ \end{align*}
Mathematica [A] time = 0.0036707, size = 13, normalized size = 1. \[ \tan ^{-1}\left (\sqrt{x}\right )+\tanh ^{-1}\left (\sqrt{x}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 10, normalized size = 0.8 \begin{align*} \arctan \left ( \sqrt{x} \right ) +{\it Artanh} \left ( \sqrt{x} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.66338, size = 28, normalized size = 2.15 \begin{align*} \arctan \left (\sqrt{x}\right ) + \frac{1}{2} \, \log \left (\sqrt{x} + 1\right ) - \frac{1}{2} \, \log \left (\sqrt{x} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.49295, size = 85, normalized size = 6.54 \begin{align*} \arctan \left (\sqrt{x}\right ) + \frac{1}{2} \, \log \left (\sqrt{x} + 1\right ) - \frac{1}{2} \, \log \left (\sqrt{x} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.394427, size = 26, normalized size = 2. \begin{align*} - \frac{\log{\left (\sqrt{x} - 1 \right )}}{2} + \frac{\log{\left (\sqrt{x} + 1 \right )}}{2} + \operatorname{atan}{\left (\sqrt{x} \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.13098, size = 30, normalized size = 2.31 \begin{align*} \arctan \left (\sqrt{x}\right ) + \frac{1}{2} \, \log \left (\sqrt{x} + 1\right ) - \frac{1}{2} \, \log \left ({\left | \sqrt{x} - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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