Optimal. Leaf size=148 \[ \frac {\sqrt {-2-\left (1-\sqrt {7}\right ) x^2} \sqrt {\frac {2+\left (1+\sqrt {7}\right ) x^2}{2+\left (1-\sqrt {7}\right ) x^2}} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{7} x}{\sqrt {-2-\left (1-\sqrt {7}\right ) x^2}}\right )|\frac {1}{14} \left (7-\sqrt {7}\right )\right )}{2 \sqrt [4]{7} \sqrt {\frac {1}{2+\left (1-\sqrt {7}\right ) x^2}} \sqrt {-2-2 x^2+3 x^4}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 148, 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 (1-\sqrt {7}\right ) x^2\right )-2} \sqrt {\frac {\left (1+\sqrt {7}\right ) x^2+2}{\left (1-\sqrt {7}\right ) x^2+2}} F\left (\text {ArcSin}\left (\frac {\sqrt {2} \sqrt [4]{7} x}{\sqrt {-\left (\left (1-\sqrt {7}\right ) x^2\right )-2}}\right )|\frac {1}{14} \left (7-\sqrt {7}\right )\right )}{2 \sqrt [4]{7} \sqrt {\frac {1}{\left (1-\sqrt {7}\right ) x^2+2}} \sqrt {3 x^4-2 x^2-2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 1112
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-2-2 x^2+3 x^4}} \, dx &=\frac {\sqrt {-2-\left (1-\sqrt {7}\right ) x^2} \sqrt {\frac {2+\left (1+\sqrt {7}\right ) x^2}{2+\left (1-\sqrt {7}\right ) x^2}} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{7} x}{\sqrt {-2-\left (1-\sqrt {7}\right ) x^2}}\right )|\frac {1}{14} \left (7-\sqrt {7}\right )\right )}{2 \sqrt [4]{7} \sqrt {\frac {1}{2+\left (1-\sqrt {7}\right ) x^2}} \sqrt {-2-2 x^2+3 x^4}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains complex when optimal does not.
time = 10.04, size = 81, normalized size = 0.55 \begin {gather*} -\frac {i \sqrt {2+2 x^2-3 x^4} F\left (i \sinh ^{-1}\left (\sqrt {\frac {3}{-1+\sqrt {7}}} x\right )|\frac {1}{3} \left (-4+\sqrt {7}\right )\right )}{\sqrt {1+\sqrt {7}} \sqrt {-2-2 x^2+3 x^4}} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [C] Result contains complex when optimal does not.
time = 0.04, size = 84, normalized size = 0.57
method | result | size |
default | \(\frac {2 \sqrt {1-\left (-\frac {1}{2}-\frac {\sqrt {7}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}+\frac {\sqrt {7}}{2}\right ) x^{2}}\, \EllipticF \left (\frac {\sqrt {-2-2 \sqrt {7}}\, x}{2}, \frac {i \sqrt {42}}{6}-\frac {i \sqrt {6}}{6}\right )}{\sqrt {-2-2 \sqrt {7}}\, \sqrt {3 x^{4}-2 x^{2}-2}}\) | \(84\) |
elliptic | \(\frac {2 \sqrt {1-\left (-\frac {1}{2}-\frac {\sqrt {7}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}+\frac {\sqrt {7}}{2}\right ) x^{2}}\, \EllipticF \left (\frac {\sqrt {-2-2 \sqrt {7}}\, x}{2}, \frac {i \sqrt {42}}{6}-\frac {i \sqrt {6}}{6}\right )}{\sqrt {-2-2 \sqrt {7}}\, \sqrt {3 x^{4}-2 x^{2}-2}}\) | \(84\) |
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 [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]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {3 x^{4} - 2 x^{2} - 2}}\, 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 {3\,x^4-2\,x^2-2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________