Optimal. Leaf size=153 \[ \frac {\sqrt {-4-\left (3-\sqrt {33}\right ) x^2} \sqrt {\frac {4+\left (3+\sqrt {33}\right ) x^2}{4+\left (3-\sqrt {33}\right ) x^2}} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{33} x}{\sqrt {-4-\left (3-\sqrt {33}\right ) x^2}}\right )|\frac {1}{22} \left (11-\sqrt {33}\right )\right )}{2 \sqrt {2} \sqrt [4]{33} \sqrt {\frac {1}{4+\left (3-\sqrt {33}\right ) x^2}} \sqrt {-2-3 x^2+3 x^4}} \]
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Rubi [A]
time = 0.02, antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {1112}
\begin {gather*} \frac {\sqrt {-\left (\left (3-\sqrt {33}\right ) x^2\right )-4} \sqrt {\frac {\left (3+\sqrt {33}\right ) x^2+4}{\left (3-\sqrt {33}\right ) x^2+4}} F\left (\text {ArcSin}\left (\frac {\sqrt {2} \sqrt [4]{33} x}{\sqrt {-\left (\left (3-\sqrt {33}\right ) x^2\right )-4}}\right )|\frac {1}{22} \left (11-\sqrt {33}\right )\right )}{2 \sqrt {2} \sqrt [4]{33} \sqrt {\frac {1}{\left (3-\sqrt {33}\right ) x^2+4}} \sqrt {3 x^4-3 x^2-2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 1112
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-2-3 x^2+3 x^4}} \, dx &=\frac {\sqrt {-4-\left (3-\sqrt {33}\right ) x^2} \sqrt {\frac {4+\left (3+\sqrt {33}\right ) x^2}{4+\left (3-\sqrt {33}\right ) x^2}} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{33} x}{\sqrt {-4-\left (3-\sqrt {33}\right ) x^2}}\right )|\frac {1}{22} \left (11-\sqrt {33}\right )\right )}{2 \sqrt {2} \sqrt [4]{33} \sqrt {\frac {1}{4+\left (3-\sqrt {33}\right ) x^2}} \sqrt {-2-3 x^2+3 x^4}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 10.05, size = 81, normalized size = 0.53 \begin {gather*} -\frac {i \sqrt {4+6 x^2-6 x^4} F\left (i \sinh ^{-1}\left (\sqrt {\frac {6}{-3+\sqrt {33}}} x\right )|\frac {1}{4} \left (-7+\sqrt {33}\right )\right )}{\sqrt {3+\sqrt {33}} \sqrt {-2-3 x^2+3 x^4}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [C] Result contains complex when optimal does not.
time = 0.04, size = 84, normalized size = 0.55
method | result | size |
default | \(\frac {2 \sqrt {1-\left (-\frac {3}{4}-\frac {\sqrt {33}}{4}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{4}+\frac {\sqrt {33}}{4}\right ) x^{2}}\, \EllipticF \left (\frac {\sqrt {-3-\sqrt {33}}\, x}{2}, \frac {i \sqrt {22}}{4}-\frac {i \sqrt {6}}{4}\right )}{\sqrt {-3-\sqrt {33}}\, \sqrt {3 x^{4}-3 x^{2}-2}}\) | \(84\) |
elliptic | \(\frac {2 \sqrt {1-\left (-\frac {3}{4}-\frac {\sqrt {33}}{4}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{4}+\frac {\sqrt {33}}{4}\right ) x^{2}}\, \EllipticF \left (\frac {\sqrt {-3-\sqrt {33}}\, x}{2}, \frac {i \sqrt {22}}{4}-\frac {i \sqrt {6}}{4}\right )}{\sqrt {-3-\sqrt {33}}\, \sqrt {3 x^{4}-3 x^{2}-2}}\) | \(84\) |
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 [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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {3 x^{4} - 3 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 = -0.01 \begin {gather*} \int \frac {1}{\sqrt {3\,x^4-3\,x^2-2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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