Integrand size = 43, antiderivative size = 59 \[ \int \frac {9-40 x-18 x^2+174 x^4+24 x^5+26 x^6-39 x^8}{\left (3+2 x^2+x^4\right )^3} \, dx=\frac {2 \left (1-2 x^2\right )}{\left (3+2 x^2+x^4\right )^2}-\frac {2 x \left (18+13 x^2\right )}{\left (3+2 x^2+x^4\right )^2}+\frac {13 x}{3+2 x^2+x^4} \]
[Out]
Time = 0.06 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.140, Rules used = {1687, 1692, 1602, 1677, 1674, 8} \[ \int \frac {9-40 x-18 x^2+174 x^4+24 x^5+26 x^6-39 x^8}{\left (3+2 x^2+x^4\right )^3} \, dx=\frac {13 x}{x^4+2 x^2+3}-\frac {2 \left (13 x^2+18\right ) x}{\left (x^4+2 x^2+3\right )^2}+\frac {2 \left (1-2 x^2\right )}{\left (x^4+2 x^2+3\right )^2} \]
[In]
[Out]
Rule 8
Rule 1602
Rule 1674
Rule 1677
Rule 1687
Rule 1692
Rubi steps \begin{align*} \text {integral}& = \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} \text {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} \text {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*}
Time = 0.01 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.47 \[ \int \frac {9-40 x-18 x^2+174 x^4+24 x^5+26 x^6-39 x^8}{\left (3+2 x^2+x^4\right )^3} \, dx=\frac {2+3 x-4 x^2+13 x^5}{\left (3+2 x^2+x^4\right )^2} \]
[In]
[Out]
Time = 0.05 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.49
method | result | size |
gosper | \(\frac {13 x^{5}-4 x^{2}+3 x +2}{\left (x^{4}+2 x^{2}+3\right )^{2}}\) | \(29\) |
norman | \(\frac {13 x^{5}-4 x^{2}+3 x +2}{\left (x^{4}+2 x^{2}+3\right )^{2}}\) | \(29\) |
risch | \(\frac {13 x^{5}-4 x^{2}+3 x +2}{\left (x^{4}+2 x^{2}+3\right )^{2}}\) | \(29\) |
parallelrisch | \(\frac {13 x^{5}-4 x^{2}+3 x +2}{\left (x^{4}+2 x^{2}+3\right )^{2}}\) | \(29\) |
default | \(-\frac {-13 x^{5}+4 x^{2}-3 x -2}{\left (x^{4}+2 x^{2}+3\right )^{2}}\) | \(30\) |
[In]
[Out]
none
Time = 0.32 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.64 \[ \int \frac {9-40 x-18 x^2+174 x^4+24 x^5+26 x^6-39 x^8}{\left (3+2 x^2+x^4\right )^3} \, dx=\frac {13 \, x^{5} - 4 \, x^{2} + 3 \, x + 2}{x^{8} + 4 \, x^{6} + 10 \, x^{4} + 12 \, x^{2} + 9} \]
[In]
[Out]
Time = 0.06 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.61 \[ \int \frac {9-40 x-18 x^2+174 x^4+24 x^5+26 x^6-39 x^8}{\left (3+2 x^2+x^4\right )^3} \, dx=- \frac {- 13 x^{5} + 4 x^{2} - 3 x - 2}{x^{8} + 4 x^{6} + 10 x^{4} + 12 x^{2} + 9} \]
[In]
[Out]
none
Time = 0.22 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.64 \[ \int \frac {9-40 x-18 x^2+174 x^4+24 x^5+26 x^6-39 x^8}{\left (3+2 x^2+x^4\right )^3} \, dx=\frac {13 \, x^{5} - 4 \, x^{2} + 3 \, x + 2}{x^{8} + 4 \, x^{6} + 10 \, x^{4} + 12 \, x^{2} + 9} \]
[In]
[Out]
none
Time = 0.57 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.47 \[ \int \frac {9-40 x-18 x^2+174 x^4+24 x^5+26 x^6-39 x^8}{\left (3+2 x^2+x^4\right )^3} \, dx=\frac {13 \, x^{5} - 4 \, x^{2} + 3 \, x + 2}{{\left (x^{4} + 2 \, x^{2} + 3\right )}^{2}} \]
[In]
[Out]
Time = 0.03 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.47 \[ \int \frac {9-40 x-18 x^2+174 x^4+24 x^5+26 x^6-39 x^8}{\left (3+2 x^2+x^4\right )^3} \, dx=\frac {13\,x^5-4\,x^2+3\,x+2}{{\left (x^4+2\,x^2+3\right )}^2} \]
[In]
[Out]