Optimal. Leaf size=38 \[ \frac {\tan ^{-1}\left (\sqrt {\frac {3}{2}} x^2\right )}{8 \sqrt {6}}+\frac {x^2}{8 \left (3 x^4+2\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {275, 199, 203} \[ \frac {x^2}{8 \left (3 x^4+2\right )}+\frac {\tan ^{-1}\left (\sqrt {\frac {3}{2}} x^2\right )}{8 \sqrt {6}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 199
Rule 203
Rule 275
Rubi steps
\begin {align*} \int \frac {x}{\left (2+3 x^4\right )^2} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\left (2+3 x^2\right )^2} \, dx,x,x^2\right )\\ &=\frac {x^2}{8 \left (2+3 x^4\right )}+\frac {1}{8} \operatorname {Subst}\left (\int \frac {1}{2+3 x^2} \, dx,x,x^2\right )\\ &=\frac {x^2}{8 \left (2+3 x^4\right )}+\frac {\tan ^{-1}\left (\sqrt {\frac {3}{2}} x^2\right )}{8 \sqrt {6}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 38, normalized size = 1.00 \[ \frac {\tan ^{-1}\left (\sqrt {\frac {3}{2}} x^2\right )}{8 \sqrt {6}}+\frac {x^2}{8 \left (3 x^4+2\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.68, size = 37, normalized size = 0.97 \[ \frac {\sqrt {6} {\left (3 \, x^{4} + 2\right )} \arctan \left (\frac {1}{2} \, \sqrt {6} x^{2}\right ) + 6 \, x^{2}}{48 \, {\left (3 \, x^{4} + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.16, size = 29, normalized size = 0.76 \[ \frac {1}{48} \, \sqrt {6} \arctan \left (\frac {1}{2} \, \sqrt {6} x^{2}\right ) + \frac {x^{2}}{8 \, {\left (3 \, x^{4} + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 30, normalized size = 0.79 \[ \frac {x^{2}}{24 x^{4}+16}+\frac {\sqrt {6}\, \arctan \left (\frac {\sqrt {6}\, x^{2}}{2}\right )}{48} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 2.91, size = 29, normalized size = 0.76 \[ \frac {1}{48} \, \sqrt {6} \arctan \left (\frac {1}{2} \, \sqrt {6} x^{2}\right ) + \frac {x^{2}}{8 \, {\left (3 \, x^{4} + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.04, size = 28, normalized size = 0.74 \[ \frac {\sqrt {6}\,\mathrm {atan}\left (\frac {\sqrt {6}\,x^2}{2}\right )}{48}+\frac {x^2}{24\,\left (x^4+\frac {2}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.17, size = 27, normalized size = 0.71 \[ \frac {x^{2}}{24 x^{4} + 16} + \frac {\sqrt {6} \operatorname {atan}{\left (\frac {\sqrt {6} x^{2}}{2} \right )}}{48} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________