Optimal. Leaf size=63 \[ \frac {\sqrt {x^2-1} \sqrt {2 x^2+3} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {5}{3}} x}{\sqrt {x^2-1}}\right )|\frac {3}{5}\right )}{\sqrt {5} \sqrt {2 x^4+x^2-3}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {1097} \[ \frac {\sqrt {x^2-1} \sqrt {2 x^2+3} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {5}{3}} x}{\sqrt {x^2-1}}\right )|\frac {3}{5}\right )}{\sqrt {5} \sqrt {2 x^4+x^2-3}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1097
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-3+x^2+2 x^4}} \, dx &=\frac {\sqrt {-1+x^2} \sqrt {3+2 x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {5}{3}} x}{\sqrt {-1+x^2}}\right )|\frac {3}{5}\right )}{\sqrt {5} \sqrt {-3+x^2+2 x^4}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.02, size = 63, normalized size = 1.00 \[ -\frac {i \sqrt {1-x^2} \sqrt {2 x^2+3} F\left (i \sinh ^{-1}\left (\sqrt {\frac {2}{3}} x\right )|-\frac {3}{2}\right )}{\sqrt {2} \sqrt {2 x^4+x^2-3}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.81, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{\sqrt {2 \, x^{4} + x^{2} - 3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {2 \, x^{4} + x^{2} - 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.03, size = 51, normalized size = 0.81 \[ -\frac {i \sqrt {6}\, \sqrt {6 x^{2}+9}\, \sqrt {-x^{2}+1}\, \EllipticF \left (\frac {i \sqrt {6}\, x}{3}, \frac {i \sqrt {6}}{2}\right )}{6 \sqrt {2 x^{4}+x^{2}-3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {2 \, x^{4} + x^{2} - 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{\sqrt {2\,x^4+x^2-3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {2 x^{4} + x^{2} - 3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________