Optimal. Leaf size=44 \[ \frac {F\left (\sin ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {7}}} x\right )|\frac {1}{3} \left (-4-\sqrt {7}\right )\right )}{\sqrt {-1+\sqrt {7}}} \]
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Rubi [A]
time = 0.04, 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}{1+\sqrt {7}}} x\right )|\frac {1}{3} \left (-4-\sqrt {7}\right )\right )}{\sqrt {\sqrt {7}-1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 430
Rule 1109
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {3+2 x^2-2 x^4}} \, dx &=\left (2 \sqrt {2}\right ) \int \frac {1}{\sqrt {2+2 \sqrt {7}-4 x^2} \sqrt {-2+2 \sqrt {7}+4 x^2}} \, dx\\ &=\frac {F\left (\sin ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {7}}} x\right )|\frac {1}{3} \left (-4-\sqrt {7}\right )\right )}{\sqrt {-1+\sqrt {7}}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 10.03, size = 49, normalized size = 1.11 \begin {gather*} -\frac {i F\left (i \sinh ^{-1}\left (\sqrt {\frac {2}{-1+\sqrt {7}}} x\right )|\frac {1}{3} \left (-4+\sqrt {7}\right )\right )}{\sqrt {1+\sqrt {7}}} \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 = 0.07, size = 84, normalized size = 1.91
method | result | size |
default | \(\frac {3 \sqrt {1-\left (-\frac {1}{3}+\frac {\sqrt {7}}{3}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{3}-\frac {\sqrt {7}}{3}\right ) x^{2}}\, \EllipticF \left (\frac {x \sqrt {-3+3 \sqrt {7}}}{3}, \frac {i \sqrt {6}}{6}+\frac {i \sqrt {42}}{6}\right )}{\sqrt {-3+3 \sqrt {7}}\, \sqrt {-2 x^{4}+2 x^{2}+3}}\) | \(84\) |
elliptic | \(\frac {3 \sqrt {1-\left (-\frac {1}{3}+\frac {\sqrt {7}}{3}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{3}-\frac {\sqrt {7}}{3}\right ) x^{2}}\, \EllipticF \left (\frac {x \sqrt {-3+3 \sqrt {7}}}{3}, \frac {i \sqrt {6}}{6}+\frac {i \sqrt {42}}{6}\right )}{\sqrt {-3+3 \sqrt {7}}\, \sqrt {-2 x^{4}+2 x^{2}+3}}\) | \(84\) |
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 = 0.08, size = 35, normalized size = 0.80 \begin {gather*} \frac {1}{6} \, {\left (\sqrt {7} + 1\right )} \sqrt {\sqrt {7} - 1} {\rm ellipticF}\left (\frac {1}{3} \, \sqrt {3} x \sqrt {\sqrt {7} - 1}, -\frac {1}{3} \, \sqrt {7} - \frac {4}{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} + 2 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+2\,x^2+3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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