Optimal. Leaf size=32 \[ \frac{\sqrt{1-x^2} \text{EllipticF}\left (\sin ^{-1}(x),\frac{3}{2}\right )}{\sqrt{2} \sqrt{x^2-1}} \]
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Rubi [A] time = 0.0147927, antiderivative size = 32, 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 = {421, 419} \[ \frac{\sqrt{1-x^2} F\left (\sin ^{-1}(x)|\frac{3}{2}\right )}{\sqrt{2} \sqrt{x^2-1}} \]
Antiderivative was successfully verified.
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Rule 421
Rule 419
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{2-3 x^2} \sqrt{-1+x^2}} \, dx &=\frac{\sqrt{1-x^2} \int \frac{1}{\sqrt{2-3 x^2} \sqrt{1-x^2}} \, dx}{\sqrt{-1+x^2}}\\ &=\frac{\sqrt{1-x^2} F\left (\sin ^{-1}(x)|\frac{3}{2}\right )}{\sqrt{2} \sqrt{-1+x^2}}\\ \end{align*}
Mathematica [A] time = 0.0232915, size = 40, normalized size = 1.25 \[ \frac{\sqrt{1-x^2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right ),\frac{2}{3}\right )}{\sqrt{3} \sqrt{x^2-1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.022, size = 29, normalized size = 0.9 \begin{align*}{\frac{\sqrt{2}}{2}{\it EllipticF} \left ( x,{\frac{\sqrt{6}}{2}} \right ) \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{-3 \, 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{-3 \, x^{2} + 2}}{3 \, x^{4} - 5 \, x^{2} + 2}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 4.80676, size = 37, normalized size = 1.16 \begin{align*} \begin{cases} - \frac{\sqrt{3} i F\left (\operatorname{asin}{\left (\frac{\sqrt{6} x}{2} \right )}\middle | \frac{2}{3}\right )}{3} & \text{for}\: x > - \frac{\sqrt{6}}{3} \wedge x < \frac{\sqrt{6}}{3} \end{cases} \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{-3 \, x^{2} + 2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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