Integrand size = 135, antiderivative size = 30 \[ \int \frac {-16-56 x+55 x^2+146 x^3-28 x^4+x^5+e^3 \left (8 x^2+9 x^3\right )+\left (8 x^2+9 x^3\right ) \log (x)}{112 x+488 x^2+439 x^3-199 x^4+25 x^5-x^6+e^3 \left (16 x+72 x^2+73 x^3-18 x^4+x^5\right )+\left (16 x+72 x^2+73 x^3-18 x^4+x^5\right ) \log (x)} \, dx=\frac {x}{9+\frac {4}{x}-x}-\log \left (-7-e^3+x-\log (x)\right ) \]
[Out]
Time = 0.57 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.07, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.030, Rules used = {6820, 6874, 789, 6816} \[ \int \frac {-16-56 x+55 x^2+146 x^3-28 x^4+x^5+e^3 \left (8 x^2+9 x^3\right )+\left (8 x^2+9 x^3\right ) \log (x)}{112 x+488 x^2+439 x^3-199 x^4+25 x^5-x^6+e^3 \left (16 x+72 x^2+73 x^3-18 x^4+x^5\right )+\left (16 x+72 x^2+73 x^3-18 x^4+x^5\right ) \log (x)} \, dx=\frac {9 x+4}{-x^2+9 x+4}-\log \left (-x+\log (x)+e^3+7\right ) \]
[In]
[Out]
Rule 789
Rule 6816
Rule 6820
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \frac {-16-56 x+\left (55+8 e^3\right ) x^2+\left (146+9 e^3\right ) x^3-28 x^4+x^5+x^2 (8+9 x) \log (x)}{x \left (4+9 x-x^2\right )^2 \left (7 \left (1+\frac {e^3}{7}\right )-x+\log (x)\right )} \, dx \\ & = \int \left (\frac {x (8+9 x)}{\left (-4-9 x+x^2\right )^2}+\frac {-1+x}{x \left (7 \left (1+\frac {e^3}{7}\right )-x+\log (x)\right )}\right ) \, dx \\ & = \int \frac {x (8+9 x)}{\left (-4-9 x+x^2\right )^2} \, dx+\int \frac {-1+x}{x \left (7 \left (1+\frac {e^3}{7}\right )-x+\log (x)\right )} \, dx \\ & = \frac {4+9 x}{4+9 x-x^2}-\log \left (7+e^3-x+\log (x)\right ) \\ \end{align*}
Time = 0.36 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.03 \[ \int \frac {-16-56 x+55 x^2+146 x^3-28 x^4+x^5+e^3 \left (8 x^2+9 x^3\right )+\left (8 x^2+9 x^3\right ) \log (x)}{112 x+488 x^2+439 x^3-199 x^4+25 x^5-x^6+e^3 \left (16 x+72 x^2+73 x^3-18 x^4+x^5\right )+\left (16 x+72 x^2+73 x^3-18 x^4+x^5\right ) \log (x)} \, dx=-\frac {4+9 x}{-4-9 x+x^2}-\log \left (7+e^3-x+\log (x)\right ) \]
[In]
[Out]
Time = 0.85 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.97
method | result | size |
default | \(-\frac {x^{2}}{x^{2}-9 x -4}-\ln \left ({\mathrm e}^{3}+\ln \left (x \right )-x +7\right )\) | \(29\) |
norman | \(-\frac {x^{2}}{x^{2}-9 x -4}-\ln \left ({\mathrm e}^{3}+\ln \left (x \right )-x +7\right )\) | \(29\) |
risch | \(-\frac {9 x +4}{x^{2}-9 x -4}-\ln \left ({\mathrm e}^{3}+\ln \left (x \right )-x +7\right )\) | \(31\) |
parallelrisch | \(\frac {-\ln \left (-7-{\mathrm e}^{3}+x -\ln \left (x \right )\right ) x^{2}-4+9 \ln \left (-7-{\mathrm e}^{3}+x -\ln \left (x \right )\right ) x +4 \ln \left (-7-{\mathrm e}^{3}+x -\ln \left (x \right )\right )-9 x}{x^{2}-9 x -4}\) | \(63\) |
[In]
[Out]
none
Time = 0.25 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.20 \[ \int \frac {-16-56 x+55 x^2+146 x^3-28 x^4+x^5+e^3 \left (8 x^2+9 x^3\right )+\left (8 x^2+9 x^3\right ) \log (x)}{112 x+488 x^2+439 x^3-199 x^4+25 x^5-x^6+e^3 \left (16 x+72 x^2+73 x^3-18 x^4+x^5\right )+\left (16 x+72 x^2+73 x^3-18 x^4+x^5\right ) \log (x)} \, dx=-\frac {{\left (x^{2} - 9 \, x - 4\right )} \log \left (-x + e^{3} + \log \left (x\right ) + 7\right ) + 9 \, x + 4}{x^{2} - 9 \, x - 4} \]
[In]
[Out]
Time = 0.11 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.87 \[ \int \frac {-16-56 x+55 x^2+146 x^3-28 x^4+x^5+e^3 \left (8 x^2+9 x^3\right )+\left (8 x^2+9 x^3\right ) \log (x)}{112 x+488 x^2+439 x^3-199 x^4+25 x^5-x^6+e^3 \left (16 x+72 x^2+73 x^3-18 x^4+x^5\right )+\left (16 x+72 x^2+73 x^3-18 x^4+x^5\right ) \log (x)} \, dx=\frac {- 9 x - 4}{x^{2} - 9 x - 4} - \log {\left (- x + \log {\left (x \right )} + 7 + e^{3} \right )} \]
[In]
[Out]
none
Time = 0.23 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {-16-56 x+55 x^2+146 x^3-28 x^4+x^5+e^3 \left (8 x^2+9 x^3\right )+\left (8 x^2+9 x^3\right ) \log (x)}{112 x+488 x^2+439 x^3-199 x^4+25 x^5-x^6+e^3 \left (16 x+72 x^2+73 x^3-18 x^4+x^5\right )+\left (16 x+72 x^2+73 x^3-18 x^4+x^5\right ) \log (x)} \, dx=-\frac {9 \, x + 4}{x^{2} - 9 \, x - 4} - \log \left (-x + e^{3} + \log \left (x\right ) + 7\right ) \]
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.87 \[ \int \frac {-16-56 x+55 x^2+146 x^3-28 x^4+x^5+e^3 \left (8 x^2+9 x^3\right )+\left (8 x^2+9 x^3\right ) \log (x)}{112 x+488 x^2+439 x^3-199 x^4+25 x^5-x^6+e^3 \left (16 x+72 x^2+73 x^3-18 x^4+x^5\right )+\left (16 x+72 x^2+73 x^3-18 x^4+x^5\right ) \log (x)} \, dx=-\frac {x^{2} \log \left (-x + e^{3} + \log \left (x\right ) + 7\right ) - 9 \, x \log \left (-x + e^{3} + \log \left (x\right ) + 7\right ) + 9 \, x - 4 \, \log \left (-x + e^{3} + \log \left (x\right ) + 7\right ) + 4}{x^{2} - 9 \, x - 4} \]
[In]
[Out]
Time = 10.00 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.03 \[ \int \frac {-16-56 x+55 x^2+146 x^3-28 x^4+x^5+e^3 \left (8 x^2+9 x^3\right )+\left (8 x^2+9 x^3\right ) \log (x)}{112 x+488 x^2+439 x^3-199 x^4+25 x^5-x^6+e^3 \left (16 x+72 x^2+73 x^3-18 x^4+x^5\right )+\left (16 x+72 x^2+73 x^3-18 x^4+x^5\right ) \log (x)} \, dx=\frac {9\,x+4}{-x^2+9\,x+4}-\ln \left ({\mathrm {e}}^3-x+\ln \left (x\right )+7\right ) \]
[In]
[Out]