Optimal. Leaf size=143 \[ \frac {\sqrt {\frac {3-\left (1-\sqrt {7}\right ) x^2}{3-\left (1+\sqrt {7}\right ) x^2}} \sqrt {-3+\left (1+\sqrt {7}\right ) x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{7} x}{\sqrt {-3+\left (1+\sqrt {7}\right ) x^2}}\right )|\frac {1}{14} \left (7+\sqrt {7}\right )\right )}{\sqrt {6} \sqrt [4]{7} \sqrt {\frac {1}{3-\left (1+\sqrt {7}\right ) x^2}} \sqrt {-3+2 x^2+2 x^4}} \]
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Rubi [A]
time = 0.02, antiderivative size = 143, 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 {3-\left (1-\sqrt {7}\right ) x^2}{3-\left (1+\sqrt {7}\right ) x^2}} \sqrt {\left (1+\sqrt {7}\right ) x^2-3} F\left (\text {ArcSin}\left (\frac {\sqrt {2} \sqrt [4]{7} x}{\sqrt {\left (1+\sqrt {7}\right ) x^2-3}}\right )|\frac {1}{14} \left (7+\sqrt {7}\right )\right )}{\sqrt {6} \sqrt [4]{7} \sqrt {\frac {1}{3-\left (1+\sqrt {7}\right ) x^2}} \sqrt {2 x^4+2 x^2-3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 1112
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-3+2 x^2+2 x^4}} \, dx &=\frac {\sqrt {\frac {3-\left (1-\sqrt {7}\right ) x^2}{3-\left (1+\sqrt {7}\right ) x^2}} \sqrt {-3+\left (1+\sqrt {7}\right ) x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{7} x}{\sqrt {-3+\left (1+\sqrt {7}\right ) x^2}}\right )|\frac {1}{14} \left (7+\sqrt {7}\right )\right )}{\sqrt {6} \sqrt [4]{7} \sqrt {\frac {1}{3-\left (1+\sqrt {7}\right ) x^2}} \sqrt {-3+2 x^2+2 x^4}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 10.05, size = 83, normalized size = 0.58 \begin {gather*} -\frac {i \sqrt {3-2 x^2-2 x^4} F\left (i \sinh ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {7}}} x\right )|-\frac {4}{3}-\frac {\sqrt {7}}{3}\right )}{\sqrt {-1+\sqrt {7}} \sqrt {-3+2 x^2+2 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 = 0.59
method | result | size |
default | \(\frac {3 \sqrt {1-\left (\frac {1}{3}-\frac {\sqrt {7}}{3}\right ) x^{2}}\, \sqrt {1-\left (\frac {\sqrt {7}}{3}+\frac {1}{3}\right ) x^{2}}\, \EllipticF \left (\frac {\sqrt {3-3 \sqrt {7}}\, x}{3}, \frac {i \sqrt {6}}{6}+\frac {i \sqrt {42}}{6}\right )}{\sqrt {3-3 \sqrt {7}}\, \sqrt {2 x^{4}+2 x^{2}-3}}\) | \(84\) |
elliptic | \(\frac {3 \sqrt {1-\left (\frac {1}{3}-\frac {\sqrt {7}}{3}\right ) x^{2}}\, \sqrt {1-\left (\frac {\sqrt {7}}{3}+\frac {1}{3}\right ) x^{2}}\, \EllipticF \left (\frac {\sqrt {3-3 \sqrt {7}}\, x}{3}, \frac {i \sqrt {6}}{6}+\frac {i \sqrt {42}}{6}\right )}{\sqrt {3-3 \sqrt {7}}\, \sqrt {2 x^{4}+2 x^{2}-3}}\) | \(84\) |
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 [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 {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 = -0.01 \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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