Optimal. Leaf size=52 \[ \frac{\sqrt{1-2 x^2} \sqrt{1-x^2} \text{EllipticF}\left (\sin ^{-1}(x),2\right )}{\sqrt{x-1} \sqrt{x+1} \sqrt{2 x^2-1}} \]
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Rubi [A] time = 0.0463509, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {519, 421, 419} \[ \frac{\sqrt{1-2 x^2} \sqrt{1-x^2} F\left (\left .\sin ^{-1}(x)\right |2\right )}{\sqrt{x-1} \sqrt{x+1} \sqrt{2 x^2-1}} \]
Antiderivative was successfully verified.
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Rule 519
Rule 421
Rule 419
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{-1+x} \sqrt{1+x} \sqrt{-1+2 x^2}} \, dx &=\frac{\sqrt{-1+x^2} \int \frac{1}{\sqrt{-1+x^2} \sqrt{-1+2 x^2}} \, dx}{\sqrt{-1+x} \sqrt{1+x}}\\ &=\frac{\left (\sqrt{1-2 x^2} \sqrt{-1+x^2}\right ) \int \frac{1}{\sqrt{1-2 x^2} \sqrt{-1+x^2}} \, dx}{\sqrt{-1+x} \sqrt{1+x} \sqrt{-1+2 x^2}}\\ &=\frac{\left (\sqrt{1-2 x^2} \sqrt{1-x^2}\right ) \int \frac{1}{\sqrt{1-2 x^2} \sqrt{1-x^2}} \, dx}{\sqrt{-1+x} \sqrt{1+x} \sqrt{-1+2 x^2}}\\ &=\frac{\sqrt{1-2 x^2} \sqrt{1-x^2} F\left (\left .\sin ^{-1}(x)\right |2\right )}{\sqrt{-1+x} \sqrt{1+x} \sqrt{-1+2 x^2}}\\ \end{align*}
Mathematica [B] time = 0.238413, size = 107, normalized size = 2.06 \[ -\frac{2 (x-1)^{3/2} \sqrt{\frac{x+1}{1-x}} \sqrt{\frac{1-2 x^2}{(x-1)^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{1}{x-1}+\sqrt{2}+2}}{2^{3/4}}\right ),4 \left (3 \sqrt{2}-4\right )\right )}{\sqrt{3+2 \sqrt{2}} \sqrt{x+1} \sqrt{2 x^2-1}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.115, size = 58, normalized size = 1.1 \begin{align*}{\frac{{\it EllipticF} \left ( x,\sqrt{2} \right ) }{2\,{x}^{4}-3\,{x}^{2}+1}\sqrt{-1+x}\sqrt{1+x}\sqrt{2\,{x}^{2}-1}\sqrt{-{x}^{2}+1}\sqrt{-2\,{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{2 \, x^{2} - 1} \sqrt{x + 1} \sqrt{x - 1}}\,{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{2 \, x^{2} - 1} \sqrt{x + 1} \sqrt{x - 1}}{2 \, x^{4} - 3 \, x^{2} + 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 \frac{1}{\sqrt{x - 1} \sqrt{x + 1} \sqrt{2 x^{2} - 1}}\, dx \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{2 \, x^{2} - 1} \sqrt{x + 1} \sqrt{x - 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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