Optimal. Leaf size=155 \[ \frac {\sqrt {-\left (\left (2-\sqrt {10}\right ) x^2\right )-3} \sqrt {\frac {\left (2+\sqrt {10}\right ) x^2+3}{\left (2-\sqrt {10}\right ) x^2+3}} F\left (\sin ^{-1}\left (\frac {2^{3/4} \sqrt [4]{5} x}{\sqrt {-\left (\left (2-\sqrt {10}\right ) x^2\right )-3}}\right )|\frac {1}{10} \left (5-\sqrt {10}\right )\right )}{2^{3/4} \sqrt {3} \sqrt [4]{5} \sqrt {\frac {1}{\left (2-\sqrt {10}\right ) x^2+3}} \sqrt {2 x^4-4 x^2-3}} \]
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Rubi [A] time = 0.02, antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {1098} \[ \frac {\sqrt {-\left (2-\sqrt {10}\right ) x^2-3} \sqrt {\frac {\left (2+\sqrt {10}\right ) x^2+3}{\left (2-\sqrt {10}\right ) x^2+3}} F\left (\sin ^{-1}\left (\frac {2^{3/4} \sqrt [4]{5} x}{\sqrt {-\left (2-\sqrt {10}\right ) x^2-3}}\right )|\frac {1}{10} \left (5-\sqrt {10}\right )\right )}{2^{3/4} \sqrt {3} \sqrt [4]{5} \sqrt {\frac {1}{\left (2-\sqrt {10}\right ) x^2+3}} \sqrt {2 x^4-4 x^2-3}} \]
Antiderivative was successfully verified.
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Rule 1098
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-3-4 x^2+2 x^4}} \, dx &=\frac {\sqrt {-3-\left (2-\sqrt {10}\right ) x^2} \sqrt {\frac {3+\left (2+\sqrt {10}\right ) x^2}{3+\left (2-\sqrt {10}\right ) x^2}} F\left (\sin ^{-1}\left (\frac {2^{3/4} \sqrt [4]{5} x}{\sqrt {-3-\left (2-\sqrt {10}\right ) x^2}}\right )|\frac {1}{10} \left (5-\sqrt {10}\right )\right )}{2^{3/4} \sqrt {3} \sqrt [4]{5} \sqrt {\frac {1}{3+\left (2-\sqrt {10}\right ) x^2}} \sqrt {-3-4 x^2+2 x^4}}\\ \end {align*}
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Mathematica [C] time = 0.06, size = 83, normalized size = 0.54 \[ -\frac {i \sqrt {-2 x^4+4 x^2+3} F\left (i \sinh ^{-1}\left (\sqrt {\frac {2}{-2+\sqrt {10}}} x\right )|-\frac {7}{3}+\frac {2 \sqrt {10}}{3}\right )}{\sqrt {2+\sqrt {10}} \sqrt {2 x^4-4 x^2-3}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.79, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{\sqrt {2 \, x^{4} - 4 \, x^{2} - 3}}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {2 \, x^{4} - 4 \, x^{2} - 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.03, size = 84, normalized size = 0.54 \[ \frac {3 \sqrt {-\left (-\frac {2}{3}-\frac {\sqrt {10}}{3}\right ) x^{2}+1}\, \sqrt {-\left (-\frac {2}{3}+\frac {\sqrt {10}}{3}\right ) x^{2}+1}\, \EllipticF \left (\frac {\sqrt {-6-3 \sqrt {10}}\, x}{3}, \frac {i \sqrt {15}}{3}-\frac {i \sqrt {6}}{3}\right )}{\sqrt {-6-3 \sqrt {10}}\, \sqrt {2 x^{4}-4 x^{2}-3}} \]
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 \[ \int \frac {1}{\sqrt {2 \, x^{4} - 4 \, x^{2} - 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{\sqrt {2\,x^4-4\,x^2-3}} \,d x \]
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 \[ \int \frac {1}{\sqrt {2 x^{4} - 4 x^{2} - 3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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