Optimal. Leaf size=45 \[ \sqrt {\frac {2}{7+\sqrt {73}}} F\left (\sin ^{-1}\left (\frac {2 x}{\sqrt {-7+\sqrt {73}}}\right )|\frac {1}{12} \left (-61+7 \sqrt {73}\right )\right ) \]
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Rubi [A] time = 0.05, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1095, 419} \[ \sqrt {\frac {2}{7+\sqrt {73}}} F\left (\sin ^{-1}\left (\frac {2 x}{\sqrt {-7+\sqrt {73}}}\right )|\frac {1}{12} \left (-61+7 \sqrt {73}\right )\right ) \]
Antiderivative was successfully verified.
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Rule 419
Rule 1095
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {3-7 x^2-2 x^4}} \, dx &=\left (2 \sqrt {2}\right ) \int \frac {1}{\sqrt {-7+\sqrt {73}-4 x^2} \sqrt {7+\sqrt {73}+4 x^2}} \, dx\\ &=\sqrt {\frac {2}{7+\sqrt {73}}} F\left (\sin ^{-1}\left (\frac {2 x}{\sqrt {-7+\sqrt {73}}}\right )|\frac {1}{12} \left (-61+7 \sqrt {73}\right )\right )\\ \end {align*}
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Mathematica [C] time = 0.05, size = 52, normalized size = 1.16 \[ -i \sqrt {\frac {2}{\sqrt {73}-7}} F\left (i \sinh ^{-1}\left (\frac {2 x}{\sqrt {7+\sqrt {73}}}\right )|\frac {1}{12} \left (-61-7 \sqrt {73}\right )\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.63, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-2 \, x^{4} - 7 \, x^{2} + 3}}{2 \, x^{4} + 7 \, x^{2} - 3}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {-2 \, x^{4} - 7 \, x^{2} + 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.09, size = 84, normalized size = 1.87 \[ \frac {6 \sqrt {-\left (\frac {\sqrt {73}}{6}+\frac {7}{6}\right ) x^{2}+1}\, \sqrt {-\left (\frac {7}{6}-\frac {\sqrt {73}}{6}\right ) x^{2}+1}\, \EllipticF \left (\frac {\sqrt {42+6 \sqrt {73}}\, x}{6}, \frac {i \sqrt {438}}{12}-\frac {7 i \sqrt {6}}{12}\right )}{\sqrt {42+6 \sqrt {73}}\, \sqrt {-2 x^{4}-7 x^{2}+3}} \]
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 \[ \int \frac {1}{\sqrt {-2 \, x^{4} - 7 \, x^{2} + 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{\sqrt {-2\,x^4-7\,x^2+3}} \,d x \]
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 \[ \int \frac {1}{\sqrt {- 2 x^{4} - 7 x^{2} + 3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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