Optimal. Leaf size=12 \[ \frac{F\left (\sin ^{-1}(x)|-\frac{1}{3}\right )}{\sqrt{3}} \]
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Rubi [A] time = 0.0131315, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {1988, 1095, 419} \[ \frac{F\left (\sin ^{-1}(x)|-\frac{1}{3}\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 1988
Rule 1095
Rule 419
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{\left (1-x^2\right ) \left (3+x^2\right )}} \, dx &=\int \frac{1}{\sqrt{3-2 x^2-x^4}} \, dx\\ &=2 \int \frac{1}{\sqrt{2-2 x^2} \sqrt{6+2 x^2}} \, dx\\ &=\frac{F\left (\sin ^{-1}(x)|-\frac{1}{3}\right )}{\sqrt{3}}\\ \end{align*}
Mathematica [C] time = 0.0157029, size = 18, normalized size = 1.5 \[ -i F\left (\left .i \sinh ^{-1}\left (\frac{x}{\sqrt{3}}\right )\right |-3\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.007, size = 43, normalized size = 3.6 \begin{align*}{\frac{{\it EllipticF} \left ( x,{\frac{i}{3}}\sqrt{3} \right ) }{3}\sqrt{-{x}^{2}+1}\sqrt{3\,{x}^{2}+9}{\frac{1}{\sqrt{-{x}^{4}-2\,{x}^{2}+3}}}} \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{-{\left (x^{2} + 3\right )}{\left (x^{2} - 1\right )}}}\,{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^{4} - 2 \, x^{2} + 3}}{x^{4} + 2 \, x^{2} - 3}, 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{\left (1 - x^{2}\right ) \left (x^{2} + 3\right )}}\, 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{-{\left (x^{2} + 3\right )}{\left (x^{2} - 1\right )}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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