Optimal. Leaf size=45 \[ \sqrt {\frac {2}{\sqrt {41}-5}} F\left (\sin ^{-1}\left (\frac {2 x}{\sqrt {5+\sqrt {41}}}\right )|\frac {1}{8} \left (-33-5 \sqrt {41}\right )\right ) \]
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Rubi [A] time = 0.06, 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} \[ \sqrt {\frac {2}{\sqrt {41}-5}} F\left (\sin ^{-1}\left (\frac {2 x}{\sqrt {5+\sqrt {41}}}\right )|\frac {1}{8} \left (-33-5 \sqrt {41}\right )\right ) \]
Antiderivative was successfully verified.
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Rule 419
Rule 1095
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {2+5 x^2-2 x^4}} \, dx &=\left (2 \sqrt {2}\right ) \int \frac {1}{\sqrt {5+\sqrt {41}-4 x^2} \sqrt {-5+\sqrt {41}+4 x^2}} \, dx\\ &=\sqrt {\frac {2}{-5+\sqrt {41}}} F\left (\sin ^{-1}\left (\frac {2 x}{\sqrt {5+\sqrt {41}}}\right )|\frac {1}{8} \left (-33-5 \sqrt {41}\right )\right )\\ \end {align*}
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Mathematica [C] time = 0.05, size = 52, normalized size = 1.16 \[ -i \sqrt {\frac {2}{5+\sqrt {41}}} F\left (i \sinh ^{-1}\left (\frac {2 x}{\sqrt {-5+\sqrt {41}}}\right )|-\frac {33}{8}+\frac {5 \sqrt {41}}{8}\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.84, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-2 \, x^{4} + 5 \, x^{2} + 2}}{2 \, x^{4} - 5 \, x^{2} - 2}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {-2 \, x^{4} + 5 \, x^{2} + 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.09, size = 76, normalized size = 1.69 \[ \frac {2 \sqrt {-\left (-\frac {5}{4}+\frac {\sqrt {41}}{4}\right ) x^{2}+1}\, \sqrt {-\left (-\frac {5}{4}-\frac {\sqrt {41}}{4}\right ) x^{2}+1}\, \EllipticF \left (\frac {\sqrt {-5+\sqrt {41}}\, x}{2}, \frac {5 i}{4}+\frac {i \sqrt {41}}{4}\right )}{\sqrt {-5+\sqrt {41}}\, \sqrt {-2 x^{4}+5 x^{2}+2}} \]
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 \[ \int \frac {1}{\sqrt {-2 \, x^{4} + 5 \, x^{2} + 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{\sqrt {-2\,x^4+5\,x^2+2}} \,d x \]
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 \[ \int \frac {1}{\sqrt {- 2 x^{4} + 5 x^{2} + 2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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