Optimal. Leaf size=72 \[ \frac {\left (\sqrt {6} x^2+3\right ) \sqrt {\frac {2 x^4+3}{\left (\sqrt {6} x^2+3\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{\frac {2}{3}} x\right )|\frac {1}{2}\right )}{2 \sqrt [4]{6} \sqrt {2 x^4+3}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {220} \[ \frac {\left (\sqrt {6} x^2+3\right ) \sqrt {\frac {2 x^4+3}{\left (\sqrt {6} x^2+3\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{\frac {2}{3}} x\right )|\frac {1}{2}\right )}{2 \sqrt [4]{6} \sqrt {2 x^4+3}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 220
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {3+2 x^4}} \, dx &=\frac {\left (3+\sqrt {6} x^2\right ) \sqrt {\frac {3+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}\right )}{2 \sqrt [4]{6} \sqrt {3+2 x^4}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.03, size = 25, normalized size = 0.35 \[ -\sqrt [4]{-\frac {1}{6}} F\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-\frac {2}{3}} x\right )\right |-1\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.81, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{\sqrt {2 \, x^{4} + 3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {2 \, x^{4} + 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.05, size = 66, normalized size = 0.92 \[ \frac {\sqrt {3}\, \sqrt {-3 i \sqrt {6}\, x^{2}+9}\, \sqrt {3 i \sqrt {6}\, x^{2}+9}\, \EllipticF \left (\frac {\sqrt {3}\, \sqrt {i \sqrt {6}}\, x}{3}, i\right )}{9 \sqrt {i \sqrt {6}}\, \sqrt {2 x^{4}+3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {2 \, x^{4} + 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.09, size = 16, normalized size = 0.22 \[ \frac {\sqrt {3}\,x\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{4},\frac {1}{2};\ \frac {5}{4};\ -\frac {2\,x^4}{3}\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [C] time = 1.19, size = 36, normalized size = 0.50 \[ \frac {\sqrt {3} x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {2 x^{4} e^{i \pi }}{3}} \right )}}{12 \Gamma \left (\frac {5}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________