Optimal. Leaf size=12 \[ \frac {F\left (\sin ^{-1}(x)|-\frac {1}{3}\right )}{\sqrt {3}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 12, 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 = {1095, 419} \[ \frac {F\left (\sin ^{-1}(x)|-\frac {1}{3}\right )}{\sqrt {3}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 419
Rule 1095
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {3-2 x^2-x^4}} \, dx &=2 \int \frac {1}{\sqrt {2-2 x^2} \sqrt {6+2 x^2}} \, dx\\ &=\frac {F\left (\sin ^{-1}(x)|-\frac {1}{3}\right )}{\sqrt {3}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.02, size = 18, normalized size = 1.50 \[ -i F\left (\left .i \sinh ^{-1}\left (\frac {x}{\sqrt {3}}\right )\right |-3\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.83, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-x^{4} - 2 \, x^{2} + 3}}{x^{4} + 2 \, x^{2} - 3}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-x^{4} - 2 \, x^{2} + 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.01, size = 43, normalized size = 3.58 \[ \frac {\sqrt {-x^{2}+1}\, \sqrt {3 x^{2}+9}\, \EllipticF \left (x , \frac {i \sqrt {3}}{3}\right )}{3 \sqrt {-x^{4}-2 x^{2}+3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-x^{4} - 2 \, x^{2} + 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.08 \[ \int \frac {1}{\sqrt {-x^4-2\,x^2+3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {- x^{4} - 2 x^{2} + 3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________