Optimal. Leaf size=27 \[ \sqrt{\frac{1}{1-x^2}} \sqrt{1-x^2} \sin ^{-1}(x) \]
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Rubi [A] time = 0.0206499, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {6688, 6720, 216} \[ \sqrt{\frac{1}{1-x^2}} \sqrt{1-x^2} \sin ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 6688
Rule 6720
Rule 216
Rubi steps
\begin{align*} \int \sqrt{\frac{1+x^2}{1-x^4}} \, dx &=\int \sqrt{\frac{1}{1-x^2}} \, dx\\ &=\left (\sqrt{\frac{1}{1-x^2}} \sqrt{1-x^2}\right ) \int \frac{1}{\sqrt{1-x^2}} \, dx\\ &=\sqrt{\frac{1}{1-x^2}} \sqrt{1-x^2} \sin ^{-1}(x)\\ \end{align*}
Mathematica [A] time = 0.0049208, size = 27, normalized size = 1. \[ \sqrt{\frac{1}{1-x^2}} \sqrt{1-x^2} \sin ^{-1}(x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 30, normalized size = 1.1 \begin{align*} \sqrt{- \left ({x}^{2}-1 \right ) ^{-1}}\sqrt{{x}^{2}-1}\ln \left ( x+\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 \sqrt{-\frac{x^{2} + 1}{x^{4} - 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.47535, size = 65, normalized size = 2.41 \begin{align*} 2 \, \arctan \left (\frac{{\left (x^{2} - 1\right )} \sqrt{-\frac{1}{x^{2} - 1}} + 1}{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 \sqrt{\frac{x^{2} + 1}{1 - x^{4}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13012, size = 14, normalized size = 0.52 \begin{align*} -\arcsin \left (x\right ) \mathrm{sgn}\left (x^{2} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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