Optimal. Leaf size=45 \[ \sqrt {\frac {2}{-7+\sqrt {73}}} F\left (\sin ^{-1}\left (\frac {2 x}{\sqrt {7+\sqrt {73}}}\right )|\frac {1}{12} \left (-61-7 \sqrt {73}\right )\right ) \]
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Rubi [A]
time = 0.04, 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 = {1109, 430}
\begin {gather*} \sqrt {\frac {2}{\sqrt {73}-7}} F\left (\text {ArcSin}\left (\frac {2 x}{\sqrt {7+\sqrt {73}}}\right )|\frac {1}{12} \left (-61-7 \sqrt {73}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 430
Rule 1109
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {3+7 x^2-2 x^4}} \, dx &=\left (2 \sqrt {2}\right ) \int \frac {1}{\sqrt {7+\sqrt {73}-4 x^2} \sqrt {-7+\sqrt {73}+4 x^2}} \, dx\\ &=\sqrt {\frac {2}{-7+\sqrt {73}}} F\left (\sin ^{-1}\left (\frac {2 x}{\sqrt {7+\sqrt {73}}}\right )|\frac {1}{12} \left (-61-7 \sqrt {73}\right )\right )\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 10.04, size = 52, normalized size = 1.16 \begin {gather*} -i \sqrt {\frac {2}{7+\sqrt {73}}} F\left (i \sinh ^{-1}\left (\frac {2 x}{\sqrt {-7+\sqrt {73}}}\right )|\frac {1}{12} \left (-61+7 \sqrt {73}\right )\right ) \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. \(2 (35 ) = 70\).
time = 0.07, size = 84, normalized size = 1.87
method | result | size |
default | \(\frac {6 \sqrt {1-\left (-\frac {7}{6}+\frac {\sqrt {73}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {7}{6}-\frac {\sqrt {73}}{6}\right ) x^{2}}\, \EllipticF \left (\frac {x \sqrt {-42+6 \sqrt {73}}}{6}, \frac {7 i \sqrt {6}}{12}+\frac {i \sqrt {438}}{12}\right )}{\sqrt {-42+6 \sqrt {73}}\, \sqrt {-2 x^{4}+7 x^{2}+3}}\) | \(84\) |
elliptic | \(\frac {6 \sqrt {1-\left (-\frac {7}{6}+\frac {\sqrt {73}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {7}{6}-\frac {\sqrt {73}}{6}\right ) x^{2}}\, \EllipticF \left (\frac {x \sqrt {-42+6 \sqrt {73}}}{6}, \frac {7 i \sqrt {6}}{12}+\frac {i \sqrt {438}}{12}\right )}{\sqrt {-42+6 \sqrt {73}}\, \sqrt {-2 x^{4}+7 x^{2}+3}}\) | \(84\) |
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 = 0.09, size = 49, normalized size = 1.09 \begin {gather*} \frac {1}{72} \, {\left (\sqrt {73} \sqrt {6} \sqrt {3} + 7 \, \sqrt {6} \sqrt {3}\right )} \sqrt {\sqrt {73} - 7} {\rm ellipticF}\left (\frac {1}{6} \, \sqrt {6} x \sqrt {\sqrt {73} - 7}, -\frac {7}{12} \, \sqrt {73} - \frac {61}{12}\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} + 7 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+7\,x^2+3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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