Optimal. Leaf size=90 \[ \frac{\left (\sqrt{6} x^2+3\right ) \sqrt{\frac{2 x^4+4 x^2+3}{\left (\sqrt{6} x^2+3\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\sqrt [4]{\frac{2}{3}} x\right ),\frac{1}{2}-\frac{1}{\sqrt{6}}\right )}{2 \sqrt [4]{6} \sqrt{2 x^4+4 x^2+3}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0163867, antiderivative size = 90, normalized size of antiderivative = 1., 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 = {1103} \[ \frac{\left (\sqrt{6} x^2+3\right ) \sqrt{\frac{2 x^4+4 x^2+3}{\left (\sqrt{6} x^2+3\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{\frac{2}{3}} x\right )|\frac{1}{2}-\frac{1}{\sqrt{6}}\right )}{2 \sqrt [4]{6} \sqrt{2 x^4+4 x^2+3}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1103
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] time = 0.0949952, size = 144, normalized size = 1.6 \[ -\frac{i \sqrt{1-\frac{2 x^2}{-2-i \sqrt{2}}} \sqrt{1-\frac{2 x^2}{-2+i \sqrt{2}}} \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{-\frac{2}{-2-i \sqrt{2}}} x\right ),\frac{-2-i \sqrt{2}}{-2+i \sqrt{2}}\right )}{\sqrt{2} \sqrt{-\frac{1}{-2-i \sqrt{2}}} \sqrt{2 x^4+4 x^2+3}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.737, size = 87, normalized size = 1. \begin{align*} 3\,{\frac{\sqrt{1- \left ( -2/3+i/3\sqrt{2} \right ){x}^{2}}\sqrt{1- \left ( -2/3-i/3\sqrt{2} \right ){x}^{2}}{\it EllipticF} \left ( 1/3\,x\sqrt{-6+3\,i\sqrt{2}},1/3\,\sqrt{3+6\,i\sqrt{2}} \right ) }{\sqrt{-6+3\,i\sqrt{2}}\sqrt{2\,{x}^{4}+4\,{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} + 4 \, 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{1}{\sqrt{2 \, x^{4} + 4 \, 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} + 4 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} + 4 \, x^{2} + 3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]