Optimal. Leaf size=45 \[ \sqrt {\frac {2}{-5+\sqrt {89}}} F\left (\sin ^{-1}\left (\frac {4 x}{\sqrt {5+\sqrt {89}}}\right )|\frac {1}{32} \left (-57-5 \sqrt {89}\right )\right ) \]
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Rubi [A]
time = 0.05, 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 {89}-5}} F\left (\text {ArcSin}\left (\frac {4 x}{\sqrt {5+\sqrt {89}}}\right )|\frac {1}{32} \left (-57-5 \sqrt {89}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 430
Rule 1109
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {2+5 x^2-8 x^4}} \, dx &=\left (4 \sqrt {2}\right ) \int \frac {1}{\sqrt {5+\sqrt {89}-16 x^2} \sqrt {-5+\sqrt {89}+16 x^2}} \, dx\\ &=\sqrt {\frac {2}{-5+\sqrt {89}}} F\left (\sin ^{-1}\left (\frac {4 x}{\sqrt {5+\sqrt {89}}}\right )|\frac {1}{32} \left (-57-5 \sqrt {89}\right )\right )\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 10.05, size = 52, normalized size = 1.16 \begin {gather*} -i \sqrt {\frac {2}{5+\sqrt {89}}} F\left (i \sinh ^{-1}\left (\frac {4 x}{\sqrt {-5+\sqrt {89}}}\right )|\frac {1}{32} \left (-57+5 \sqrt {89}\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. 75 vs. \(2 (31 ) = 62\).
time = 0.08, size = 76, normalized size = 1.69
method | result | size |
default | \(\frac {2 \sqrt {1-\left (-\frac {5}{4}+\frac {\sqrt {89}}{4}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{4}-\frac {\sqrt {89}}{4}\right ) x^{2}}\, \EllipticF \left (\frac {x \sqrt {-5+\sqrt {89}}}{2}, \frac {5 i}{8}+\frac {i \sqrt {89}}{8}\right )}{\sqrt {-5+\sqrt {89}}\, \sqrt {-8 x^{4}+5 x^{2}+2}}\) | \(76\) |
elliptic | \(\frac {2 \sqrt {1-\left (-\frac {5}{4}+\frac {\sqrt {89}}{4}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{4}-\frac {\sqrt {89}}{4}\right ) x^{2}}\, \EllipticF \left (\frac {x \sqrt {-5+\sqrt {89}}}{2}, \frac {5 i}{8}+\frac {i \sqrt {89}}{8}\right )}{\sqrt {-5+\sqrt {89}}\, \sqrt {-8 x^{4}+5 x^{2}+2}}\) | \(76\) |
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 = 40, normalized size = 0.89 \begin {gather*} \frac {1}{64} \, {\left (\sqrt {89} \sqrt {2} + 5 \, \sqrt {2}\right )} \sqrt {\sqrt {89} - 5} {\rm ellipticF}\left (\frac {1}{2} \, x \sqrt {\sqrt {89} - 5}, -\frac {5}{32} \, \sqrt {89} - \frac {57}{32}\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 {- 8 x^{4} + 5 x^{2} + 2}}\, 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 {-8\,x^4+5\,x^2+2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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