Optimal. Leaf size=44 \[ \frac {F\left (\sin ^{-1}\left (\sqrt {\frac {2}{-2+\sqrt {10}}} x\right )|\frac {1}{3} \left (-7+2 \sqrt {10}\right )\right )}{\sqrt {2+\sqrt {10}}} \]
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Rubi [A]
time = 0.05, antiderivative size = 44, 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 = {1109, 430}
\begin {gather*} \frac {F\left (\text {ArcSin}\left (\sqrt {\frac {2}{-2+\sqrt {10}}} x\right )|\frac {1}{3} \left (-7+2 \sqrt {10}\right )\right )}{\sqrt {2+\sqrt {10}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 430
Rule 1109
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {3-4 x^2-2 x^4}} \, dx &=\left (2 \sqrt {2}\right ) \int \frac {1}{\sqrt {-4+2 \sqrt {10}-4 x^2} \sqrt {4+2 \sqrt {10}+4 x^2}} \, dx\\ &=\frac {F\left (\sin ^{-1}\left (\sqrt {\frac {2}{-2+\sqrt {10}}} x\right )|\frac {1}{3} \left (-7+2 \sqrt {10}\right )\right )}{\sqrt {2+\sqrt {10}}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 10.05, size = 51, normalized size = 1.16 \begin {gather*} -\frac {i 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}}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 83 vs. \(2 (34 ) = 68\).
time = 0.05, size = 84, normalized size = 1.91
method | result | size |
default | \(\frac {3 \sqrt {1-\left (\frac {2}{3}+\frac {\sqrt {10}}{3}\right ) x^{2}}\, \sqrt {1-\left (\frac {2}{3}-\frac {\sqrt {10}}{3}\right ) x^{2}}\, \EllipticF \left (\frac {x \sqrt {6+3 \sqrt {10}}}{3}, \frac {i \sqrt {15}}{3}-\frac {i \sqrt {6}}{3}\right )}{\sqrt {6+3 \sqrt {10}}\, \sqrt {-2 x^{4}-4 x^{2}+3}}\) | \(84\) |
elliptic | \(\frac {3 \sqrt {1-\left (\frac {2}{3}+\frac {\sqrt {10}}{3}\right ) x^{2}}\, \sqrt {1-\left (\frac {2}{3}-\frac {\sqrt {10}}{3}\right ) x^{2}}\, \EllipticF \left (\frac {x \sqrt {6+3 \sqrt {10}}}{3}, \frac {i \sqrt {15}}{3}-\frac {i \sqrt {6}}{3}\right )}{\sqrt {6+3 \sqrt {10}}\, \sqrt {-2 x^{4}-4 x^{2}+3}}\) | \(84\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.10, size = 35, normalized size = 0.80 \begin {gather*} \frac {1}{6} \, \sqrt {\sqrt {10} + 2} {\left (\sqrt {10} - 2\right )} {\rm ellipticF}\left (\frac {1}{3} \, \sqrt {3} x \sqrt {\sqrt {10} + 2}, \frac {2}{3} \, \sqrt {10} - \frac {7}{3}\right ) \end {gather*}
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 \begin {gather*} \int \frac {1}{\sqrt {- 2 x^{4} - 4 x^{2} + 3}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {1}{\sqrt {-2\,x^4-4\,x^2+3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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