Optimal. Leaf size=59 \[ \frac{13 x}{x^4+2 x^2+3}-\frac{2 \left (13 x^2+18\right ) x}{\left (x^4+2 x^2+3\right )^2}+\frac{2 \left (1-2 x^2\right )}{\left (x^4+2 x^2+3\right )^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0906975, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.14, Rules used = {1673, 1678, 1588, 1663, 1660, 8} \[ \frac{13 x}{x^4+2 x^2+3}-\frac{2 \left (13 x^2+18\right ) x}{\left (x^4+2 x^2+3\right )^2}+\frac{2 \left (1-2 x^2\right )}{\left (x^4+2 x^2+3\right )^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1673
Rule 1678
Rule 1588
Rule 1663
Rule 1660
Rule 8
Rubi steps
\begin{align*} \int \frac{9-40 x-18 x^2+174 x^4+24 x^5+26 x^6-39 x^8}{\left (3+2 x^2+x^4\right )^3} \, dx &=\int \frac{x \left (-40+24 x^4\right )}{\left (3+2 x^2+x^4\right )^3} \, dx+\int \frac{9-18 x^2+174 x^4+26 x^6-39 x^8}{\left (3+2 x^2+x^4\right )^3} \, dx\\ &=-\frac{2 x \left (18+13 x^2\right )}{\left (3+2 x^2+x^4\right )^2}+\frac{1}{96} \int \frac{3744-2496 x^2-3744 x^4}{\left (3+2 x^2+x^4\right )^2} \, dx+\frac{1}{2} \operatorname{Subst}\left (\int \frac{-40+24 x^2}{\left (3+2 x+x^2\right )^3} \, dx,x,x^2\right )\\ &=\frac{2 \left (1-2 x^2\right )}{\left (3+2 x^2+x^4\right )^2}-\frac{2 x \left (18+13 x^2\right )}{\left (3+2 x^2+x^4\right )^2}+\frac{13 x}{3+2 x^2+x^4}+\frac{1}{32} \operatorname{Subst}\left (\int 0 \, dx,x,x^2\right )\\ &=\frac{2 \left (1-2 x^2\right )}{\left (3+2 x^2+x^4\right )^2}-\frac{2 x \left (18+13 x^2\right )}{\left (3+2 x^2+x^4\right )^2}+\frac{13 x}{3+2 x^2+x^4}\\ \end{align*}
Mathematica [A] time = 0.0118335, size = 28, normalized size = 0.47 \[ \frac{13 x^5-4 x^2+3 x+2}{\left (x^4+2 x^2+3\right )^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.006, size = 30, normalized size = 0.5 \begin{align*} -{\frac{-13\,{x}^{5}+4\,{x}^{2}-3\,x-2}{ \left ({x}^{4}+2\,{x}^{2}+3 \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.01136, size = 51, normalized size = 0.86 \begin{align*} \frac{13 \, x^{5} - 4 \, x^{2} + 3 \, x + 2}{x^{8} + 4 \, x^{6} + 10 \, x^{4} + 12 \, x^{2} + 9} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.20708, size = 86, normalized size = 1.46 \begin{align*} \frac{13 \, x^{5} - 4 \, x^{2} + 3 \, x + 2}{x^{8} + 4 \, x^{6} + 10 \, x^{4} + 12 \, x^{2} + 9} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.181473, size = 34, normalized size = 0.58 \begin{align*} \frac{13 x^{5} - 4 x^{2} + 3 x + 2}{x^{8} + 4 x^{6} + 10 x^{4} + 12 x^{2} + 9} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.19465, size = 38, normalized size = 0.64 \begin{align*} \frac{13 \, x^{5} - 4 \, x^{2} + 3 \, x + 2}{{\left (x^{4} + 2 \, x^{2} + 3\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]