Optimal. Leaf size=141 \[ \frac {\sqrt {\frac {2-\left (2-\sqrt {10}\right ) x^2}{2-\left (2+\sqrt {10}\right ) x^2}} \sqrt {\left (2+\sqrt {10}\right ) x^2-2} F\left (\sin ^{-1}\left (\frac {2^{3/4} \sqrt [4]{5} x}{\sqrt {\left (2+\sqrt {10}\right ) x^2-2}}\right )|\frac {1}{10} \left (5+\sqrt {10}\right )\right )}{2 \sqrt [4]{10} \sqrt {\frac {1}{2-\left (2+\sqrt {10}\right ) x^2}} \sqrt {3 x^4+4 x^2-2}} \]
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Rubi [A] time = 0.03, antiderivative size = 141, 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} \[ \frac {\sqrt {\frac {2-\left (2-\sqrt {10}\right ) x^2}{2-\left (2+\sqrt {10}\right ) x^2}} \sqrt {\left (2+\sqrt {10}\right ) x^2-2} F\left (\sin ^{-1}\left (\frac {2^{3/4} \sqrt [4]{5} x}{\sqrt {\left (2+\sqrt {10}\right ) x^2-2}}\right )|\frac {1}{10} \left (5+\sqrt {10}\right )\right )}{2 \sqrt [4]{10} \sqrt {\frac {1}{2-\left (2+\sqrt {10}\right ) x^2}} \sqrt {3 x^4+4 x^2-2}} \]
Antiderivative was successfully verified.
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Rule 1098
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-2+4 x^2+3 x^4}} \, dx &=\frac {\sqrt {\frac {2-\left (2-\sqrt {10}\right ) x^2}{2-\left (2+\sqrt {10}\right ) x^2}} \sqrt {-2+\left (2+\sqrt {10}\right ) x^2} F\left (\sin ^{-1}\left (\frac {2^{3/4} \sqrt [4]{5} x}{\sqrt {-2+\left (2+\sqrt {10}\right ) x^2}}\right )|\frac {1}{10} \left (5+\sqrt {10}\right )\right )}{2 \sqrt [4]{10} \sqrt {\frac {1}{2-\left (2+\sqrt {10}\right ) x^2}} \sqrt {-2+4 x^2+3 x^4}}\\ \end {align*}
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Mathematica [C] time = 0.05, size = 81, normalized size = 0.57 \[ -\frac {i \sqrt {-3 x^4-4 x^2+2} F\left (i \sinh ^{-1}\left (\sqrt {-1+\sqrt {\frac {5}{2}}} x\right )|\frac {1}{3} \left (-7-2 \sqrt {10}\right )\right )}{\sqrt {\sqrt {10}-2} \sqrt {3 x^4+4 x^2-2}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.86, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{\sqrt {3 \, x^{4} + 4 \, x^{2} - 2}}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {3 \, x^{4} + 4 \, x^{2} - 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.03, size = 84, normalized size = 0.60 \[ \frac {2 \sqrt {-\left (-\frac {\sqrt {10}}{2}+1\right ) x^{2}+1}\, \sqrt {-\left (1+\frac {\sqrt {10}}{2}\right ) x^{2}+1}\, \EllipticF \left (\frac {\sqrt {4-2 \sqrt {10}}\, x}{2}, \frac {i \sqrt {6}}{3}+\frac {i \sqrt {15}}{3}\right )}{\sqrt {4-2 \sqrt {10}}\, \sqrt {3 x^{4}+4 x^{2}-2}} \]
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 \[ \int \frac {1}{\sqrt {3 \, x^{4} + 4 \, x^{2} - 2}}\,{d x} \]
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 \[ \int \frac {1}{\sqrt {3\,x^4+4\,x^2-2}} \,d x \]
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 \[ \int \frac {1}{\sqrt {3 x^{4} + 4 x^{2} - 2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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