Optimal. Leaf size=43 \[ \sqrt {\frac {2}{3+\sqrt {33}}} F\left (\sin ^{-1}\left (\frac {2 x}{\sqrt {-3+\sqrt {33}}}\right )|\frac {1}{4} \left (-7+\sqrt {33}\right )\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 43, 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}{3+\sqrt {33}}} F\left (\text {ArcSin}\left (\frac {2 x}{\sqrt {-3+\sqrt {33}}}\right )|\frac {1}{4} \left (-7+\sqrt {33}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 430
Rule 1109
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {3-3 x^2-2 x^4}} \, dx &=\left (2 \sqrt {2}\right ) \int \frac {1}{\sqrt {-3+\sqrt {33}-4 x^2} \sqrt {3+\sqrt {33}+4 x^2}} \, dx\\ &=\sqrt {\frac {2}{3+\sqrt {33}}} F\left (\sin ^{-1}\left (\frac {2 x}{\sqrt {-3+\sqrt {33}}}\right )|\frac {1}{4} \left (-7+\sqrt {33}\right )\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains complex when optimal does not.
time = 10.04, size = 52, normalized size = 1.21 \begin {gather*} -i \sqrt {\frac {2}{-3+\sqrt {33}}} F\left (i \sinh ^{-1}\left (\frac {2 x}{\sqrt {3+\sqrt {33}}}\right )|-\frac {7}{4}-\frac {\sqrt {33}}{4}\right ) \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
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 = 0.05, size = 84, normalized size = 1.95
method | result | size |
default | \(\frac {6 \sqrt {1-\left (\frac {\sqrt {33}}{6}+\frac {1}{2}\right ) x^{2}}\, \sqrt {1-\left (\frac {1}{2}-\frac {\sqrt {33}}{6}\right ) x^{2}}\, \EllipticF \left (\frac {x \sqrt {18+6 \sqrt {33}}}{6}, \frac {i \sqrt {22}}{4}-\frac {i \sqrt {6}}{4}\right )}{\sqrt {18+6 \sqrt {33}}\, \sqrt {-2 x^{4}-3 x^{2}+3}}\) | \(84\) |
elliptic | \(\frac {6 \sqrt {1-\left (\frac {\sqrt {33}}{6}+\frac {1}{2}\right ) x^{2}}\, \sqrt {1-\left (\frac {1}{2}-\frac {\sqrt {33}}{6}\right ) x^{2}}\, \EllipticF \left (\frac {x \sqrt {18+6 \sqrt {33}}}{6}, \frac {i \sqrt {22}}{4}-\frac {i \sqrt {6}}{4}\right )}{\sqrt {18+6 \sqrt {33}}\, \sqrt {-2 x^{4}-3 x^{2}+3}}\) | \(84\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.08, size = 57, normalized size = 1.33 \begin {gather*} \frac {1}{24} \, {\left (\sqrt {11} \sqrt {6} - \sqrt {6} \sqrt {3}\right )} \sqrt {\sqrt {11} \sqrt {3} + 3} {\rm ellipticF}\left (\frac {1}{6} \, \sqrt {6} \sqrt {\sqrt {11} \sqrt {3} + 3} x, \frac {1}{4} \, \sqrt {11} \sqrt {3} - \frac {7}{4}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {- 2 x^{4} - 3 x^{2} + 3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{\sqrt {-2\,x^4-3\,x^2+3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________