Optimal. Leaf size=88 \[ \frac {\left (3+\sqrt {6} x^2\right ) \sqrt {\frac {3-4 x^2+2 x^4}{\left (3+\sqrt {6} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{\frac {2}{3}} x\right )|\frac {1}{2}+\frac {1}{\sqrt {6}}\right )}{2 \sqrt [4]{6} \sqrt {-3+4 x^2-2 x^4}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 88, 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 = {1117}
\begin {gather*} \frac {\left (\sqrt {6} x^2+3\right ) \sqrt {\frac {2 x^4-4 x^2+3}{\left (\sqrt {6} x^2+3\right )^2}} F\left (2 \text {ArcTan}\left (\sqrt [4]{\frac {2}{3}} x\right )|\frac {1}{2}+\frac {1}{\sqrt {6}}\right )}{2 \sqrt [4]{6} \sqrt {-2 x^4+4 x^2-3}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 1117
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-3+4 x^2-2 x^4}} \, dx &=\frac {\left (3+\sqrt {6} x^2\right ) \sqrt {\frac {3-4 x^2+2 x^4}{\left (3+\sqrt {6} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{\frac {2}{3}} x\right )|\frac {1}{2}+\frac {1}{\sqrt {6}}\right )}{2 \sqrt [4]{6} \sqrt {-3+4 x^2-2 x^4}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains complex when optimal does not.
time = 10.04, size = 144, normalized size = 1.64 \begin {gather*} -\frac {i \sqrt {1-\frac {2 x^2}{2-i \sqrt {2}}} \sqrt {1-\frac {2 x^2}{2+i \sqrt {2}}} F\left (i \sinh ^{-1}\left (\sqrt {-\frac {2}{2-i \sqrt {2}}} x\right )|\frac {2-i \sqrt {2}}{2+i \sqrt {2}}\right )}{\sqrt {2} \sqrt {-\frac {1}{2-i \sqrt {2}}} \sqrt {-3+4 x^2-2 x^4}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] Result contains complex when optimal does not.
time = 0.04, size = 87, normalized size = 0.99
method | result | size |
default | \(\frac {3 \sqrt {1-\left (\frac {2}{3}-\frac {i \sqrt {2}}{3}\right ) x^{2}}\, \sqrt {1-\left (\frac {2}{3}+\frac {i \sqrt {2}}{3}\right ) x^{2}}\, \EllipticF \left (\frac {\sqrt {6-3 i \sqrt {2}}\, x}{3}, \frac {\sqrt {3+6 i \sqrt {2}}}{3}\right )}{\sqrt {6-3 i \sqrt {2}}\, \sqrt {-2 x^{4}+4 x^{2}-3}}\) | \(87\) |
elliptic | \(\frac {3 \sqrt {1-\left (\frac {2}{3}-\frac {i \sqrt {2}}{3}\right ) x^{2}}\, \sqrt {1-\left (\frac {2}{3}+\frac {i \sqrt {2}}{3}\right ) x^{2}}\, \EllipticF \left (\frac {\sqrt {6-3 i \sqrt {2}}\, x}{3}, \frac {\sqrt {3+6 i \sqrt {2}}}{3}\right )}{\sqrt {6-3 i \sqrt {2}}\, \sqrt {-2 x^{4}+4 x^{2}-3}}\) | \(87\) |
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 = 0.09, size = 46, normalized size = 0.52 \begin {gather*} \frac {1}{18} \, \sqrt {3} {\left (\sqrt {-2} \sqrt {-3} - 2 \, \sqrt {-3}\right )} \sqrt {\sqrt {-2} + 2} {\rm ellipticF}\left (\frac {1}{3} \, \sqrt {3} x \sqrt {\sqrt {-2} + 2}, -\frac {2}{3} \, \sqrt {-2} + \frac {1}{3}\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} + 4 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 = -0.01 \begin {gather*} \int \frac {1}{\sqrt {-2\,x^4+4\,x^2-3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________