Optimal. Leaf size=146 \[ \frac {\sqrt {\frac {4-\left (3-\sqrt {33}\right ) x^2}{4-\left (3+\sqrt {33}\right ) x^2}} \sqrt {-4+\left (3+\sqrt {33}\right ) x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{33} x}{\sqrt {-4+\left (3+\sqrt {33}\right ) x^2}}\right )|\frac {1}{22} \left (11+\sqrt {33}\right )\right )}{2 \sqrt {2} \sqrt [4]{33} \sqrt {\frac {1}{4-\left (3+\sqrt {33}\right ) x^2}} \sqrt {-2+3 x^2+3 x^4}} \]
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Rubi [A]
time = 0.03, antiderivative size = 146, 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 {\frac {4-\left (3-\sqrt {33}\right ) x^2}{4-\left (3+\sqrt {33}\right ) x^2}} \sqrt {\left (3+\sqrt {33}\right ) x^2-4} F\left (\text {ArcSin}\left (\frac {\sqrt {2} \sqrt [4]{33} x}{\sqrt {\left (3+\sqrt {33}\right ) x^2-4}}\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 {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 {\frac {4-\left (3-\sqrt {33}\right ) x^2}{4-\left (3+\sqrt {33}\right ) x^2}} \sqrt {-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 = 83, normalized size = 0.57 \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 {7}{4}-\frac {\sqrt {33}}{4}\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 = 0.58
method | result | size |
default | \(\frac {2 \sqrt {1-\left (\frac {3}{4}-\frac {\sqrt {33}}{4}\right ) x^{2}}\, \sqrt {1-\left (\frac {\sqrt {33}}{4}+\frac {3}{4}\right ) x^{2}}\, \EllipticF \left (\frac {x \sqrt {3-\sqrt {33}}}{2}, \frac {i \sqrt {6}}{4}+\frac {i \sqrt {22}}{4}\right )}{\sqrt {3-\sqrt {33}}\, \sqrt {3 x^{4}+3 x^{2}-2}}\) | \(84\) |
elliptic | \(\frac {2 \sqrt {1-\left (\frac {3}{4}-\frac {\sqrt {33}}{4}\right ) x^{2}}\, \sqrt {1-\left (\frac {\sqrt {33}}{4}+\frac {3}{4}\right ) x^{2}}\, \EllipticF \left (\frac {x \sqrt {3-\sqrt {33}}}{2}, \frac {i \sqrt {6}}{4}+\frac {i \sqrt {22}}{4}\right )}{\sqrt {3-\sqrt {33}}\, \sqrt {3 x^{4}+3 x^{2}-2}}\) | \(84\) |
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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