Optimal. Leaf size=10 \[ \frac{\text{EllipticF}\left (\sin ^{-1}(x),-1\right )}{\sqrt{2}} \]
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Rubi [A] time = 0.0047369, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {248, 221} \[ \frac{F\left (\left .\sin ^{-1}(x)\right |-1\right )}{\sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 248
Rule 221
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{2-2 x^2} \sqrt{1+x^2}} \, dx &=\int \frac{1}{\sqrt{2-2 x^4}} \, dx\\ &=\frac{F\left (\left .\sin ^{-1}(x)\right |-1\right )}{\sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.0118819, size = 10, normalized size = 1. \[ \frac{\text{EllipticF}\left (\sin ^{-1}(x),-1\right )}{\sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 10, normalized size = 1. \begin{align*}{\frac{{\it EllipticF} \left ( x,i \right ) \sqrt{2}}{2}} \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}{\sqrt{x^{2} + 1} \sqrt{-2 \, x^{2} + 2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{x^{2} + 1} \sqrt{-2 \, x^{2} + 2}}{2 \,{\left (x^{4} - 1\right )}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 7.75348, size = 76, normalized size = 7.6 \begin{align*} \frac{\sqrt{2} i{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{1}{2}, 1, 1 & \frac{3}{4}, \frac{3}{4}, \frac{5}{4} \\\frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1, \frac{5}{4} & 0 \end{matrix} \middle |{\frac{1}{x^{4}}} \right )}}{16 \pi ^{\frac{3}{2}}} - \frac{\sqrt{2} i{G_{6, 6}^{3, 5}\left (\begin{matrix} - \frac{1}{4}, 0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4} & 1 \\0, \frac{1}{2}, 0 & - \frac{1}{4}, \frac{1}{4}, \frac{1}{4} \end{matrix} \middle |{\frac{e^{- 2 i \pi }}{x^{4}}} \right )}}{16 \pi ^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{x^{2} + 1} \sqrt{-2 \, x^{2} + 2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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