Optimal. Leaf size=43 \[ \sqrt{\frac{2}{3+\sqrt{33}}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{2 x}{\sqrt{\sqrt{33}-3}}\right ),\frac{1}{4} \left (\sqrt{33}-7\right )\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0314177, antiderivative size = 43, normalized size of antiderivative = 1., 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} \[ \sqrt{\frac{2}{3+\sqrt{33}}} F\left (\sin ^{-1}\left (\frac{2 x}{\sqrt{-3+\sqrt{33}}}\right )|\frac{1}{4} \left (-7+\sqrt{33}\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1095
Rule 419
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] time = 0.0444487, size = 52, normalized size = 1.21 \[ -i \sqrt{\frac{2}{\sqrt{33}-3}} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{2 x}{\sqrt{3+\sqrt{33}}}\right ),-\frac{7}{4}-\frac{\sqrt{33}}{4}\right ) \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.211, size = 84, normalized size = 2. \begin{align*} 6\,{\frac{\sqrt{1- \left ( 1/6\,\sqrt{33}+1/2 \right ){x}^{2}}\sqrt{1- \left ( -1/6\,\sqrt{33}+1/2 \right ){x}^{2}}{\it EllipticF} \left ( 1/6\,x\sqrt{18+6\,\sqrt{33}},i/4\sqrt{22}-i/4\sqrt{6} \right ) }{\sqrt{18+6\,\sqrt{33}}\sqrt{-2\,{x}^{4}-3\,{x}^{2}+3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-2 \, x^{4} - 3 \, x^{2} + 3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-2 \, x^{4} - 3 \, x^{2} + 3}}{2 \, x^{4} + 3 \, x^{2} - 3}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{- 2 x^{4} - 3 x^{2} + 3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-2 \, x^{4} - 3 \, x^{2} + 3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]