Optimal. Leaf size=42 \[ -\frac{\sqrt{1-\frac{1}{x^4}} x^2 \text{EllipticF}\left (\csc ^{-1}(x),-1\right )}{\sqrt{2-2 x^2} \sqrt{-x^2-1}} \]
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Rubi [A] time = 0.0122466, antiderivative size = 65, normalized size of antiderivative = 1.55, number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {253, 222} \[ \frac{\sqrt{x^2-1} \sqrt{x^2+1} F\left (\sin ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{x^2-1}}\right )|\frac{1}{2}\right )}{2 \sqrt{-x^2-1} \sqrt{1-x^2}} \]
Antiderivative was successfully verified.
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Rule 253
Rule 222
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{2-2 x^2} \sqrt{-1-x^2}} \, dx &=\frac{\sqrt{-2+2 x^4} \int \frac{1}{\sqrt{-2+2 x^4}} \, dx}{\sqrt{2-2 x^2} \sqrt{-1-x^2}}\\ &=\frac{\sqrt{-1+x^2} \sqrt{1+x^2} F\left (\sin ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{-1+x^2}}\right )|\frac{1}{2}\right )}{2 \sqrt{-1-x^2} \sqrt{1-x^2}}\\ \end{align*}
Mathematica [C] time = 0.0132972, size = 48, normalized size = 1.14 \[ \frac{x \sqrt{1-x^4} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};x^4\right )}{\sqrt{2-2 x^2} \sqrt{-x^2-1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 30, normalized size = 0.7 \begin{align*}{{\frac{i}{2}}{\it EllipticF} \left ( ix,i \right ) \sqrt{2}\sqrt{-{x}^{2}-1}{\frac{1}{\sqrt{{x}^{2}+1}}}} \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 [A] time = 7.92956, size = 73, normalized size = 1.74 \begin{align*} \frac{\sqrt{2}{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}{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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