Optimal. Leaf size=12 \[ \frac {F\left (\sin ^{-1}(x)|-\frac {1}{3}\right )}{\sqrt {3}} \]
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Rubi [A] time = 0.01, antiderivative size = 12, normalized size of antiderivative = 1.00, 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 419
Rule 1095
Rule 1988
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*}
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Mathematica [C] time = 0.02, size = 18, normalized size = 1.50 \[ -i F\left (\left .i \sinh ^{-1}\left (\frac {x}{\sqrt {3}}\right )\right |-3\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.88, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-x^{4} - 2 \, x^{2} + 3}}{x^{4} + 2 \, x^{2} - 3}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-{\left (x^{2} + 3\right )} {\left (x^{2} - 1\right )}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 43, normalized size = 3.58 \[ \frac {\sqrt {-x^{2}+1}\, \sqrt {3 x^{2}+9}\, \EllipticF \left (x , \frac {i \sqrt {3}}{3}\right )}{3 \sqrt {-x^{4}-2 x^{2}+3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-{\left (x^{2} + 3\right )} {\left (x^{2} - 1\right )}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.08 \[ \int \frac {1}{\sqrt {-\left (x^2-1\right )\,\left (x^2+3\right )}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {\left (1 - x^{2}\right ) \left (x^{2} + 3\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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