Optimal. Leaf size=25 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{1-x^2}}\right )}{\sqrt{2}} \]
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Rubi [A] time = 0.0076408, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {377, 203} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{1-x^2}}\right )}{\sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 377
Rule 203
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{1-x^2} \left (1+x^2\right )} \, dx &=\operatorname{Subst}\left (\int \frac{1}{1+2 x^2} \, dx,x,\frac{x}{\sqrt{1-x^2}}\right )\\ &=\frac{\tan ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{1-x^2}}\right )}{\sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.0050939, size = 25, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{1-x^2}}\right )}{\sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 28, normalized size = 1.1 \begin{align*} -{\frac{\sqrt{2}}{2}\arctan \left ({\frac{x\sqrt{2}}{{x}^{2}-1}\sqrt{-{x}^{2}+1}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (x^{2} + 1\right )} \sqrt{-x^{2} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.50801, size = 69, normalized size = 2.76 \begin{align*} -\frac{1}{2} \, \sqrt{2} \arctan \left (\frac{\sqrt{2} \sqrt{-x^{2} + 1}}{2 \, x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{- \left (x - 1\right ) \left (x + 1\right )} \left (x^{2} + 1\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.14642, size = 69, normalized size = 2.76 \begin{align*} \frac{1}{4} \, \sqrt{2}{\left (\pi \mathrm{sgn}\left (x\right ) + 2 \, \arctan \left (-\frac{\sqrt{2} x{\left (\frac{{\left (\sqrt{-x^{2} + 1} - 1\right )}^{2}}{x^{2}} - 1\right )}}{4 \,{\left (\sqrt{-x^{2} + 1} - 1\right )}}\right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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