Optimal. Leaf size=148 \[ \frac{\sqrt{\frac{3-\left (2-\sqrt{10}\right ) x^2}{3-\left (2+\sqrt{10}\right ) x^2}} \sqrt{\left (2+\sqrt{10}\right ) x^2-3} \text{EllipticF}\left (\sin ^{-1}\left (\frac{2^{3/4} \sqrt [4]{5} x}{\sqrt{\left (2+\sqrt{10}\right ) x^2-3}}\right ),\frac{1}{10} \left (5+\sqrt{10}\right )\right )}{2^{3/4} \sqrt{3} \sqrt [4]{5} \sqrt{\frac{1}{3-\left (2+\sqrt{10}\right ) x^2}} \sqrt{2 x^4+4 x^2-3}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0271006, antiderivative size = 148, 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 = {1098} \[ \frac{\sqrt{\frac{3-\left (2-\sqrt{10}\right ) x^2}{3-\left (2+\sqrt{10}\right ) x^2}} \sqrt{\left (2+\sqrt{10}\right ) x^2-3} F\left (\sin ^{-1}\left (\frac{2^{3/4} \sqrt [4]{5} x}{\sqrt{\left (2+\sqrt{10}\right ) x^2-3}}\right )|\frac{1}{10} \left (5+\sqrt{10}\right )\right )}{2^{3/4} \sqrt{3} \sqrt [4]{5} \sqrt{\frac{1}{3-\left (2+\sqrt{10}\right ) x^2}} \sqrt{2 x^4+4 x^2-3}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1098
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{-3+4 x^2+2 x^4}} \, dx &=\frac{\sqrt{\frac{3-\left (2-\sqrt{10}\right ) x^2}{3-\left (2+\sqrt{10}\right ) x^2}} \sqrt{-3+\left (2+\sqrt{10}\right ) x^2} F\left (\sin ^{-1}\left (\frac{2^{3/4} \sqrt [4]{5} x}{\sqrt{-3+\left (2+\sqrt{10}\right ) x^2}}\right )|\frac{1}{10} \left (5+\sqrt{10}\right )\right )}{2^{3/4} \sqrt{3} \sqrt [4]{5} \sqrt{\frac{1}{3-\left (2+\sqrt{10}\right ) x^2}} \sqrt{-3+4 x^2+2 x^4}}\\ \end{align*}
Mathematica [C] time = 0.0687126, size = 83, normalized size = 0.56 \[ -\frac{i \sqrt{-2 x^4-4 x^2+3} \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{\frac{2}{2+\sqrt{10}}} x\right ),-\frac{7}{3}-\frac{2 \sqrt{10}}{3}\right )}{\sqrt{\sqrt{10}-2} \sqrt{2 x^4+4 x^2-3}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.181, size = 84, normalized size = 0.6 \begin{align*} 3\,{\frac{\sqrt{1- \left ( 2/3-1/3\,\sqrt{10} \right ){x}^{2}}\sqrt{1- \left ( 2/3+1/3\,\sqrt{10} \right ){x}^{2}}{\it EllipticF} \left ( 1/3\,\sqrt{6-3\,\sqrt{10}}x,i/3\sqrt{6}+i/3\sqrt{15} \right ) }{\sqrt{6-3\,\sqrt{10}}\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]