Optimal. Leaf size=45 \[ \frac{5-7 x^2}{8 \left (x^4+2 x^2+3\right )}+\frac{9 \tan ^{-1}\left (\frac{x^2+1}{\sqrt{2}}\right )}{8 \sqrt{2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0669286, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {1594, 1663, 1660, 12, 618, 204} \[ \frac{5-7 x^2}{8 \left (x^4+2 x^2+3\right )}+\frac{9 \tan ^{-1}\left (\frac{x^2+1}{\sqrt{2}}\right )}{8 \sqrt{2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1594
Rule 1663
Rule 1660
Rule 12
Rule 618
Rule 204
Rubi steps
\begin{align*} \int \frac{-x+2 x^3+4 x^5}{\left (3+2 x^2+x^4\right )^2} \, dx &=\int \frac{x \left (-1+2 x^2+4 x^4\right )}{\left (3+2 x^2+x^4\right )^2} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{-1+2 x+4 x^2}{\left (3+2 x+x^2\right )^2} \, dx,x,x^2\right )\\ &=\frac{5-7 x^2}{8 \left (3+2 x^2+x^4\right )}+\frac{1}{16} \operatorname{Subst}\left (\int \frac{18}{3+2 x+x^2} \, dx,x,x^2\right )\\ &=\frac{5-7 x^2}{8 \left (3+2 x^2+x^4\right )}+\frac{9}{8} \operatorname{Subst}\left (\int \frac{1}{3+2 x+x^2} \, dx,x,x^2\right )\\ &=\frac{5-7 x^2}{8 \left (3+2 x^2+x^4\right )}-\frac{9}{4} \operatorname{Subst}\left (\int \frac{1}{-8-x^2} \, dx,x,2 \left (1+x^2\right )\right )\\ &=\frac{5-7 x^2}{8 \left (3+2 x^2+x^4\right )}+\frac{9 \tan ^{-1}\left (\frac{1+x^2}{\sqrt{2}}\right )}{8 \sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.0254557, size = 45, normalized size = 1. \[ \frac{5-7 x^2}{8 \left (x^4+2 x^2+3\right )}+\frac{9 \tan ^{-1}\left (\frac{x^2+1}{\sqrt{2}}\right )}{8 \sqrt{2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.008, size = 41, normalized size = 0.9 \begin{align*}{\frac{1}{2\,{x}^{4}+4\,{x}^{2}+6} \left ( -{\frac{7\,{x}^{2}}{4}}+{\frac{5}{4}} \right ) }+{\frac{9\,\sqrt{2}}{16}\arctan \left ({\frac{ \left ( 2\,{x}^{2}+2 \right ) \sqrt{2}}{4}} \right ) } \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*} -\frac{7 \, x^{2} - 5}{8 \,{\left (x^{4} + 2 \, x^{2} + 3\right )}} + \frac{9}{4} \, \int \frac{x}{x^{4} + 2 \, x^{2} + 3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.48295, size = 132, normalized size = 2.93 \begin{align*} \frac{9 \, \sqrt{2}{\left (x^{4} + 2 \, x^{2} + 3\right )} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (x^{2} + 1\right )}\right ) - 14 \, x^{2} + 10}{16 \,{\left (x^{4} + 2 \, x^{2} + 3\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.147528, size = 44, normalized size = 0.98 \begin{align*} - \frac{7 x^{2} - 5}{8 x^{4} + 16 x^{2} + 24} + \frac{9 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x^{2}}{2} + \frac{\sqrt{2}}{2} \right )}}{16} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.11674, size = 51, normalized size = 1.13 \begin{align*} \frac{9}{16} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (x^{2} + 1\right )}\right ) - \frac{7 \, x^{2} - 5}{8 \,{\left (x^{4} + 2 \, x^{2} + 3\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]