Integrand size = 71, antiderivative size = 25 \[ \int \frac {-8+9 x-8 e^x x+8 x^2+\left (-18-32 x+e^x (16+16 x)\right ) \log \left (\frac {1}{2} \left (8-9 x+8 e^x x-8 x^2\right )\right )}{8-9 x+8 e^x x-8 x^2} \, dx=-x+\log ^2\left (4-\frac {x}{2}-4 x \left (1-e^x+x\right )\right ) \]
[Out]
\[ \int \frac {-8+9 x-8 e^x x+8 x^2+\left (-18-32 x+e^x (16+16 x)\right ) \log \left (\frac {1}{2} \left (8-9 x+8 e^x x-8 x^2\right )\right )}{8-9 x+8 e^x x-8 x^2} \, dx=\int \frac {-8+9 x-8 e^x x+8 x^2+\left (-18-32 x+e^x (16+16 x)\right ) \log \left (\frac {1}{2} \left (8-9 x+8 e^x x-8 x^2\right )\right )}{8-9 x+8 e^x x-8 x^2} \, dx \]
[In]
[Out]
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {2 \left (-8-8 x+x^2+8 x^3\right ) \log \left (\frac {1}{2} \left (8-9 x+8 e^x x-8 x^2\right )\right )}{x \left (8-9 x+8 e^x x-8 x^2\right )}+\frac {-x+2 \log \left (4+\left (-\frac {9}{2}+4 e^x\right ) x-4 x^2\right )+2 x \log \left (4+\left (-\frac {9}{2}+4 e^x\right ) x-4 x^2\right )}{x}\right ) \, dx \\ & = 2 \int \frac {\left (-8-8 x+x^2+8 x^3\right ) \log \left (\frac {1}{2} \left (8-9 x+8 e^x x-8 x^2\right )\right )}{x \left (8-9 x+8 e^x x-8 x^2\right )} \, dx+\int \frac {-x+2 \log \left (4+\left (-\frac {9}{2}+4 e^x\right ) x-4 x^2\right )+2 x \log \left (4+\left (-\frac {9}{2}+4 e^x\right ) x-4 x^2\right )}{x} \, dx \\ & = -\left (2 \int \frac {\left (9+16 x-8 e^x (1+x)\right ) \left (-8 \int \frac {1}{-8+9 x-8 e^x x+8 x^2} \, dx-8 \int \frac {1}{x \left (-8+9 x-8 e^x x+8 x^2\right )} \, dx+\int \frac {x}{-8+9 x-8 e^x x+8 x^2} \, dx+8 \int \frac {x^2}{-8+9 x-8 e^x x+8 x^2} \, dx\right )}{8+\left (-9+8 e^x\right ) x-8 x^2} \, dx\right )-\left (2 \log \left (\frac {1}{2} \left (8-9 x+8 e^x x-8 x^2\right )\right )\right ) \int \frac {x}{-8+9 x-8 e^x x+8 x^2} \, dx+\left (16 \log \left (\frac {1}{2} \left (8-9 x+8 e^x x-8 x^2\right )\right )\right ) \int \frac {1}{-8+9 x-8 e^x x+8 x^2} \, dx+\left (16 \log \left (\frac {1}{2} \left (8-9 x+8 e^x x-8 x^2\right )\right )\right ) \int \frac {1}{x \left (-8+9 x-8 e^x x+8 x^2\right )} \, dx-\left (16 \log \left (\frac {1}{2} \left (8-9 x+8 e^x x-8 x^2\right )\right )\right ) \int \frac {x^2}{-8+9 x-8 e^x x+8 x^2} \, dx+\int \left (-1+\frac {2 (1+x) \log \left (\frac {1}{2} \left (8-9 x+8 e^x x-8 x^2\right )\right )}{x}\right ) \, dx \\ & = -x+2 \int \frac {(1+x) \log \left (\frac {1}{2} \left (8-9 x+8 e^x x-8 x^2\right )\right )}{x} \, dx-2 \int \left (\frac {(1+x) \left (8 \int \frac {1}{-8+9 x-8 e^x x+8 x^2} \, dx+8 \int \frac {1}{x \left (-8+9 x-8 e^x x+8 x^2\right )} \, dx-\int \frac {x}{-8+9 x-8 e^x x+8 x^2} \, dx-8 \int \frac {x^2}{-8+9 x-8 e^x x+8 x^2} \, dx\right )}{x}-\frac {\left (-8-8 x+x^2+8 x^3\right ) \left (8 \int \frac {1}{-8+9 x-8 e^x x+8 x^2} \, dx+8 \int \frac {1}{x \left (-8+9 x-8 e^x x+8 x^2\right )} \, dx-\int \frac {x}{-8+9 x-8 e^x x+8 x^2} \, dx-8 \int \frac {x^2}{-8+9 x-8 e^x x+8 x^2} \, dx\right )}{x \left (-8+9 x-8 e^x x+8 x^2\right )}\right ) \, dx-\left (2 \log \left (\frac {1}{2} \left (8-9 x+8 e^x x-8 x^2\right )\right )\right ) \int \frac {x}{-8+9 x-8 e^x x+8 x^2} \, dx+\left (16 \log \left (\frac {1}{2} \left (8-9 x+8 e^x x-8 x^2\right )\right )\right ) \int \frac {1}{-8+9 x-8 e^x x+8 x^2} \, dx+\left (16 \log \left (\frac {1}{2} \left (8-9 x+8 e^x x-8 x^2\right )\right )\right ) \int \frac {1}{x \left (-8+9 x-8 e^x x+8 x^2\right )} \, dx-\left (16 \log \left (\frac {1}{2} \left (8-9 x+8 e^x x-8 x^2\right )\right )\right ) \int \frac {x^2}{-8+9 x-8 e^x x+8 x^2} \, dx \\ & = -x+2 \int \left (\log \left (\frac {1}{2} \left (8-9 x+8 e^x x-8 x^2\right )\right )+\frac {\log \left (\frac {1}{2} \left (8-9 x+8 e^x x-8 x^2\right )\right )}{x}\right ) \, dx-2 \int \frac {(1+x) \left (8 \int \frac {1}{-8+9 x-8 e^x x+8 x^2} \, dx+8 \int \frac {1}{x \left (-8+9 x-8 e^x x+8 x^2\right )} \, dx-\int \frac {x}{-8+9 x-8 e^x x+8 x^2} \, dx-8 \int \frac {x^2}{-8+9 x-8 e^x x+8 x^2} \, dx\right )}{x} \, dx+2 \int \frac {\left (-8-8 x+x^2+8 x^3\right ) \left (8 \int \frac {1}{-8+9 x-8 e^x x+8 x^2} \, dx+8 \int \frac {1}{x \left (-8+9 x-8 e^x x+8 x^2\right )} \, dx-\int \frac {x}{-8+9 x-8 e^x x+8 x^2} \, dx-8 \int \frac {x^2}{-8+9 x-8 e^x x+8 x^2} \, dx\right )}{x \left (-8+9 x-8 e^x x+8 x^2\right )} \, dx-\left (2 \log \left (\frac {1}{2} \left (8-9 x+8 e^x x-8 x^2\right )\right )\right ) \int \frac {x}{-8+9 x-8 e^x x+8 x^2} \, dx+\left (16 \log \left (\frac {1}{2} \left (8-9 x+8 e^x x-8 x^2\right )\right )\right ) \int \frac {1}{-8+9 x-8 e^x x+8 x^2} \, dx+\left (16 \log \left (\frac {1}{2} \left (8-9 x+8 e^x x-8 x^2\right )\right )\right ) \int \frac {1}{x \left (-8+9 x-8 e^x x+8 x^2\right )} \, dx-\left (16 \log \left (\frac {1}{2} \left (8-9 x+8 e^x x-8 x^2\right )\right )\right ) \int \frac {x^2}{-8+9 x-8 e^x x+8 x^2} \, dx \\ & = -x+2 \int \log \left (\frac {1}{2} \left (8-9 x+8 e^x x-8 x^2\right )\right ) \, dx+2 \int \frac {\log \left (\frac {1}{2} \left (8-9 x+8 e^x x-8 x^2\right )\right )}{x} \, dx+2 \int \left (-\frac {8 \left (8 \int \frac {1}{-8+9 x-8 e^x x+8 x^2} \, dx+8 \int \frac {1}{x \left (-8+9 x-8 e^x x+8 x^2\right )} \, dx-\int \frac {x}{-8+9 x-8 e^x x+8 x^2} \, dx-8 \int \frac {x^2}{-8+9 x-8 e^x x+8 x^2} \, dx\right )}{-8+9 x-8 e^x x+8 x^2}-\frac {8 \left (8 \int \frac {1}{-8+9 x-8 e^x x+8 x^2} \, dx+8 \int \frac {1}{x \left (-8+9 x-8 e^x x+8 x^2\right )} \, dx-\int \frac {x}{-8+9 x-8 e^x x+8 x^2} \, dx-8 \int \frac {x^2}{-8+9 x-8 e^x x+8 x^2} \, dx\right )}{x \left (-8+9 x-8 e^x x+8 x^2\right )}+\frac {x \left (8 \int \frac {1}{-8+9 x-8 e^x x+8 x^2} \, dx+8 \int \frac {1}{x \left (-8+9 x-8 e^x x+8 x^2\right )} \, dx-\int \frac {x}{-8+9 x-8 e^x x+8 x^2} \, dx-8 \int \frac {x^2}{-8+9 x-8 e^x x+8 x^2} \, dx\right )}{-8+9 x-8 e^x x+8 x^2}+\frac {8 x^2 \left (8 \int \frac {1}{-8+9 x-8 e^x x+8 x^2} \, dx+8 \int \frac {1}{x \left (-8+9 x-8 e^x x+8 x^2\right )} \, dx-\int \frac {x}{-8+9 x-8 e^x x+8 x^2} \, dx-8 \int \frac {x^2}{-8+9 x-8 e^x x+8 x^2} \, dx\right )}{-8+9 x-8 e^x x+8 x^2}\right ) \, dx-2 \int \left (\frac {(1+x) \left (8 \int \frac {1}{-8+9 x-8 e^x x+8 x^2} \, dx+8 \int \frac {1}{x \left (-8+9 x-8 e^x x+8 x^2\right )} \, dx-\int \frac {x}{-8+9 x-8 e^x x+8 x^2} \, dx\right )}{x}-\frac {8 (1+x) \int \frac {x^2}{-8+\left (9-8 e^x\right ) x+8 x^2} \, dx}{x}\right ) \, dx-\left (2 \log \left (\frac {1}{2} \left (8-9 x+8 e^x x-8 x^2\right )\right )\right ) \int \frac {x}{-8+9 x-8 e^x x+8 x^2} \, dx+\left (16 \log \left (\frac {1}{2} \left (8-9 x+8 e^x x-8 x^2\right )\right )\right ) \int \frac {1}{-8+9 x-8 e^x x+8 x^2} \, dx+\left (16 \log \left (\frac {1}{2} \left (8-9 x+8 e^x x-8 x^2\right )\right )\right ) \int \frac {1}{x \left (-8+9 x-8 e^x x+8 x^2\right )} \, dx-\left (16 \log \left (\frac {1}{2} \left (8-9 x+8 e^x x-8 x^2\right )\right )\right ) \int \frac {x^2}{-8+9 x-8 e^x x+8 x^2} \, dx \\ & = -x+2 x \log \left (\frac {1}{2} \left (8-9 x+8 e^x x-8 x^2\right )\right )-2 \int \frac {x \left (-9-16 x+8 e^x (1+x)\right )}{8-\left (9-8 e^x\right ) x-8 x^2} \, dx+2 \int \frac {\log \left (\frac {1}{2} \left (8-9 x+8 e^x x-8 x^2\right )\right )}{x} \, dx-2 \int \frac {(1+x) \left (8 \int \frac {1}{-8+9 x-8 e^x x+8 x^2} \, dx+8 \int \frac {1}{x \left (-8+9 x-8 e^x x+8 x^2\right )} \, dx-\int \frac {x}{-8+9 x-8 e^x x+8 x^2} \, dx\right )}{x} \, dx+2 \int \frac {x \left (8 \int \frac {1}{-8+9 x-8 e^x x+8 x^2} \, dx+8 \int \frac {1}{x \left (-8+9 x-8 e^x x+8 x^2\right )} \, dx-\int \frac {x}{-8+9 x-8 e^x x+8 x^2} \, dx-8 \int \frac {x^2}{-8+9 x-8 e^x x+8 x^2} \, dx\right )}{-8+9 x-8 e^x x+8 x^2} \, dx-16 \int \frac {8 \int \frac {1}{-8+9 x-8 e^x x+8 x^2} \, dx+8 \int \frac {1}{x \left (-8+9 x-8 e^x x+8 x^2\right )} \, dx-\int \frac {x}{-8+9 x-8 e^x x+8 x^2} \, dx-8 \int \frac {x^2}{-8+9 x-8 e^x x+8 x^2} \, dx}{-8+9 x-8 e^x x+8 x^2} \, dx-16 \int \frac {8 \int \frac {1}{-8+9 x-8 e^x x+8 x^2} \, dx+8 \int \frac {1}{x \left (-8+9 x-8 e^x x+8 x^2\right )} \, dx-\int \frac {x}{-8+9 x-8 e^x x+8 x^2} \, dx-8 \int \frac {x^2}{-8+9 x-8 e^x x+8 x^2} \, dx}{x \left (-8+9 x-8 e^x x+8 x^2\right )} \, dx+16 \int \frac {x^2 \left (8 \int \frac {1}{-8+9 x-8 e^x x+8 x^2} \, dx+8 \int \frac {1}{x \left (-8+9 x-8 e^x x+8 x^2\right )} \, dx-\int \frac {x}{-8+9 x-8 e^x x+8 x^2} \, dx-8 \int \frac {x^2}{-8+9 x-8 e^x x+8 x^2} \, dx\right )}{-8+9 x-8 e^x x+8 x^2} \, dx+16 \int \frac {(1+x) \int \frac {x^2}{-8+\left (9-8 e^x\right ) x+8 x^2} \, dx}{x} \, dx-\left (2 \log \left (\frac {1}{2} \left (8-9 x+8 e^x x-8 x^2\right )\right )\right ) \int \frac {x}{-8+9 x-8 e^x x+8 x^2} \, dx+\left (16 \log \left (\frac {1}{2} \left (8-9 x+8 e^x x-8 x^2\right )\right )\right ) \int \frac {1}{-8+9 x-8 e^x x+8 x^2} \, dx+\left (16 \log \left (\frac {1}{2} \left (8-9 x+8 e^x x-8 x^2\right )\right )\right ) \int \frac {1}{x \left (-8+9 x-8 e^x x+8 x^2\right )} \, dx-\left (16 \log \left (\frac {1}{2} \left (8-9 x+8 e^x x-8 x^2\right )\right )\right ) \int \frac {x^2}{-8+9 x-8 e^x x+8 x^2} \, dx \\ & = -x+2 x \log \left (\frac {1}{2} \left (8-9 x+8 e^x x-8 x^2\right )\right )-2 \int \left (1+x-\frac {-8-8 x+x^2+8 x^3}{-8+9 x-8 e^x x+8 x^2}\right ) \, dx+2 \int \frac {\log \left (\frac {1}{2} \left (8-9 x+8 e^x x-8 x^2\right )\right )}{x} \, dx-2 \int \left (\frac {8 (1+x) \left (\int \frac {1}{-8+9 x-8 e^x x+8 x^2} \, dx+\int \frac {1}{x \left (-8+9 x-8 e^x x+8 x^2\right )} \, dx\right )}{x}-\frac {(1+x) \int \frac {x}{-8+\left (9-8 e^x\right ) x+8 x^2} \, dx}{x}\right ) \, dx+2 \int \left (\frac {8 x \int \frac {1}{-8+\left (9-8 e^x\right ) x+8 x^2} \, dx}{-8+9 x-8 e^x x+8 x^2}+\frac {8 x \int \frac {1}{x \left (-8+\left (9-8 e^x\right ) x+8 x^2\right )} \, dx}{-8+9 x-8 e^x x+8 x^2}-\frac {x \int \frac {x}{-8+\left (9-8 e^x\right ) x+8 x^2} \, dx}{-8+9 x-8 e^x x+8 x^2}-\frac {8 x \int \frac {x^2}{-8+\left (9-8 e^x\right ) x+8 x^2} \, dx}{-8+9 x-8 e^x x+8 x^2}\right ) \, dx+16 \int \left (\int \frac {x^2}{-8+\left (9-8 e^x\right ) x+8 x^2} \, dx+\frac {\int \frac {x^2}{-8+\left (9-8 e^x\right ) x+8 x^2} \, dx}{x}\right ) \, dx-16 \int \left (\frac {8 \int \frac {1}{-8+\left (9-8 e^x\right ) x+8 x^2} \, dx}{-8+9 x-8 e^x x+8 x^2}+\frac {8 \int \frac {1}{x \left (-8+\left (9-8 e^x\right ) x+8 x^2\right )} \, dx}{-8+9 x-8 e^x x+8 x^2}-\frac {\int \frac {x}{-8+\left (9-8 e^x\right ) x+8 x^2} \, dx}{-8+9 x-8 e^x x+8 x^2}-\frac {8 \int \frac {x^2}{-8+\left (9-8 e^x\right ) x+8 x^2} \, dx}{-8+9 x-8 e^x x+8 x^2}\right ) \, dx-16 \int \left (\frac {8 \int \frac {1}{-8+\left (9-8 e^x\right ) x+8 x^2} \, dx}{x \left (-8+9 x-8 e^x x+8 x^2\right )}+\frac {8 \int \frac {1}{x \left (-8+\left (9-8 e^x\right ) x+8 x^2\right )} \, dx}{x \left (-8+9 x-8 e^x x+8 x^2\right )}-\frac {\int \frac {x}{-8+\left (9-8 e^x\right ) x+8 x^2} \, dx}{x \left (-8+9 x-8 e^x x+8 x^2\right )}-\frac {8 \int \frac {x^2}{-8+\left (9-8 e^x\right ) x+8 x^2} \, dx}{x \left (-8+9 x-8 e^x x+8 x^2\right )}\right ) \, dx+16 \int \left (\frac {8 x^2 \int \frac {1}{-8+\left (9-8 e^x\right ) x+8 x^2} \, dx}{-8+9 x-8 e^x x+8 x^2}+\frac {8 x^2 \int \frac {1}{x \left (-8+\left (9-8 e^x\right ) x+8 x^2\right )} \, dx}{-8+9 x-8 e^x x+8 x^2}-\frac {x^2 \int \frac {x}{-8+\left (9-8 e^x\right ) x+8 x^2} \, dx}{-8+9 x-8 e^x x+8 x^2}-\frac {8 x^2 \int \frac {x^2}{-8+\left (9-8 e^x\right ) x+8 x^2} \, dx}{-8+9 x-8 e^x x+8 x^2}\right ) \, dx-\left (2 \log \left (\frac {1}{2} \left (8-9 x+8 e^x x-8 x^2\right )\right )\right ) \int \frac {x}{-8+9 x-8 e^x x+8 x^2} \, dx+\left (16 \log \left (\frac {1}{2} \left (8-9 x+8 e^x x-8 x^2\right )\right )\right ) \int \frac {1}{-8+9 x-8 e^x x+8 x^2} \, dx+\left (16 \log \left (\frac {1}{2} \left (8-9 x+8 e^x x-8 x^2\right )\right )\right ) \int \frac {1}{x \left (-8+9 x-8 e^x x+8 x^2\right )} \, dx-\left (16 \log \left (\frac {1}{2} \left (8-9 x+8 e^x x-8 x^2\right )\right )\right ) \int \frac {x^2}{-8+9 x-8 e^x x+8 x^2} \, dx \\ & = -3 x-x^2+2 x \log \left (\frac {1}{2} \left (8-9 x+8 e^x x-8 x^2\right )\right )+2 \int \frac {-8-8 x+x^2+8 x^3}{-8+9 x-8 e^x x+8 x^2} \, dx+2 \int \frac {\log \left (\frac {1}{2} \left (8-9 x+8 e^x x-8 x^2\right )\right )}{x} \, dx+2 \int \frac {(1+x) \int \frac {x}{-8+\left (9-8 e^x\right ) x+8 x^2} \, dx}{x} \, dx-2 \int \frac {x \int \frac {x}{-8+\left (9-8 e^x\right ) x+8 x^2} \, dx}{-8+9 x-8 e^x x+8 x^2} \, dx-16 \int \frac {(1+x) \left (\int \frac {1}{-8+9 x-8 e^x x+8 x^2} \, dx+\int \frac {1}{x \left (-8+9 x-8 e^x x+8 x^2\right )} \, dx\right )}{x} \, dx+16 \int \frac {x \int \frac {1}{-8+\left (9-8 e^x\right ) x+8 x^2} \, dx}{-8+9 x-8 e^x x+8 x^2} \, dx+16 \int \frac {x \int \frac {1}{x \left (-8+\left (9-8 e^x\right ) x+8 x^2\right )} \, dx}{-8+9 x-8 e^x x+8 x^2} \, dx+16 \int \frac {\int \frac {x}{-8+\left (9-8 e^x\right ) x+8 x^2} \, dx}{-8+9 x-8 e^x x+8 x^2} \, dx+16 \int \frac {\int \frac {x}{-8+\left (9-8 e^x\right ) x+8 x^2} \, dx}{x \left (-8+9 x-8 e^x x+8 x^2\right )} \, dx-16 \int \frac {x^2 \int \frac {x}{-8+\left (9-8 e^x\right ) x+8 x^2} \, dx}{-8+9 x-8 e^x x+8 x^2} \, dx+16 \int \left (\int \frac {x^2}{-8+\left (9-8 e^x\right ) x+8 x^2} \, dx\right ) \, dx+16 \int \frac {\int \frac {x^2}{-8+\left (9-8 e^x\right ) x+8 x^2} \, dx}{x} \, dx-16 \int \frac {x \int \frac {x^2}{-8+\left (9-8 e^x\right ) x+8 x^2} \, dx}{-8+9 x-8 e^x x+8 x^2} \, dx-128 \int \frac {\int \frac {1}{-8+\left (9-8 e^x\right ) x+8 x^2} \, dx}{-8+9 x-8 e^x x+8 x^2} \, dx-128 \int \frac {\int \frac {1}{-8+\left (9-8 e^x\right ) x+8 x^2} \, dx}{x \left (-8+9 x-8 e^x x+8 x^2\right )} \, dx+128 \int \frac {x^2 \int \frac {1}{-8+\left (9-8 e^x\right ) x+8 x^2} \, dx}{-8+9 x-8 e^x x+8 x^2} \, dx-128 \int \frac {\int \frac {1}{x \left (-8+\left (9-8 e^x\right ) x+8 x^2\right )} \, dx}{-8+9 x-8 e^x x+8 x^2} \, dx-128 \int \frac {\int \frac {1}{x \left (-8+\left (9-8 e^x\right ) x+8 x^2\right )} \, dx}{x \left (-8+9 x-8 e^x x+8 x^2\right )} \, dx+128 \int \frac {x^2 \int \frac {1}{x \left (-8+\left (9-8 e^x\right ) x+8 x^2\right )} \, dx}{-8+9 x-8 e^x x+8 x^2} \, dx+128 \int \frac {\int \frac {x^2}{-8+\left (9-8 e^x\right ) x+8 x^2} \, dx}{-8+9 x-8 e^x x+8 x^2} \, dx+128 \int \frac {\int \frac {x^2}{-8+\left (9-8 e^x\right ) x+8 x^2} \, dx}{x \left (-8+9 x-8 e^x x+8 x^2\right )} \, dx-128 \int \frac {x^2 \int \frac {x^2}{-8+\left (9-8 e^x\right ) x+8 x^2} \, dx}{-8+9 x-8 e^x x+8 x^2} \, dx-\left (2 \log \left (\frac {1}{2} \left (8-9 x+8 e^x x-8 x^2\right )\right )\right ) \int \frac {x}{-8+9 x-8 e^x x+8 x^2} \, dx+\left (16 \log \left (\frac {1}{2} \left (8-9 x+8 e^x x-8 x^2\right )\right )\right ) \int \frac {1}{-8+9 x-8 e^x x+8 x^2} \, dx+\left (16 \log \left (\frac {1}{2} \left (8-9 x+8 e^x x-8 x^2\right )\right )\right ) \int \frac {1}{x \left (-8+9 x-8 e^x x+8 x^2\right )} \, dx-\left (16 \log \left (\frac {1}{2} \left (8-9 x+8 e^x x-8 x^2\right )\right )\right ) \int \frac {x^2}{-8+9 x-8 e^x x+8 x^2} \, dx \\ & = -3 x-x^2+2 x \log \left (\frac {1}{2} \left (8-9 x+8 e^x x-8 x^2\right )\right )+2 \int \left (-\frac {8}{-8+9 x-8 e^x x+8 x^2}-\frac {8 x}{-8+9 x-8 e^x x+8 x^2}+\frac {x^2}{-8+9 x-8 e^x x+8 x^2}+\frac {8 x^3}{-8+9 x-8 e^x x+8 x^2}\right ) \, dx+2 \int \frac {\log \left (\frac {1}{2} \left (8-9 x+8 e^x x-8 x^2\right )\right )}{x} \, dx-2 \int \frac {x \int \frac {x}{-8+\left (9-8 e^x\right ) x+8 x^2} \, dx}{-8+9 x-8 e^x x+8 x^2} \, dx+2 \int \left (\int \frac {x}{-8+\left (9-8 e^x\right ) x+8 x^2} \, dx+\frac {\int \frac {x}{-8+\left (9-8 e^x\right ) x+8 x^2} \, dx}{x}\right ) \, dx+16 \int \frac {x \int \frac {1}{-8+\left (9-8 e^x\right ) x+8 x^2} \, dx}{-8+9 x-8 e^x x+8 x^2} \, dx+16 \int \frac {x \int \frac {1}{x \left (-8+\left (9-8 e^x\right ) x+8 x^2\right )} \, dx}{-8+9 x-8 e^x x+8 x^2} \, dx-16 \int \left (\frac {(1+x) \int \frac {1}{-8+\left (9-8 e^x\right ) x+8 x^2} \, dx}{x}+\frac {(1+x) \int \frac {1}{x \left (-8+\left (9-8 e^x\right ) x+8 x^2\right )} \, dx}{x}\right ) \, dx+16 \int \frac {\int \frac {x}{-8+\left (9-8 e^x\right ) x+8 x^2} \, dx}{-8+9 x-8 e^x x+8 x^2} \, dx+16 \int \frac {\int \frac {x}{-8+\left (9-8 e^x\right ) x+8 x^2} \, dx}{x \left (-8+9 x-8 e^x x+8 x^2\right )} \, dx-16 \int \frac {x^2 \int \frac {x}{-8+\left (9-8 e^x\right ) x+8 x^2} \, dx}{-8+9 x-8 e^x x+8 x^2} \, dx+16 \int \left (\int \frac {x^2}{-8+\left (9-8 e^x\right ) x+8 x^2} \, dx\right ) \, dx+16 \int \frac {\int \frac {x^2}{-8+\left (9-8 e^x\right ) x+8 x^2} \, dx}{x} \, dx-16 \int \frac {x \int \frac {x^2}{-8+\left (9-8 e^x\right ) x+8 x^2} \, dx}{-8+9 x-8 e^x x+8 x^2} \, dx-128 \int \frac {\int \frac {1}{-8+\left (9-8 e^x\right ) x+8 x^2} \, dx}{-8+9 x-8 e^x x+8 x^2} \, dx-128 \int \frac {\int \frac {1}{-8+\left (9-8 e^x\right ) x+8 x^2} \, dx}{x \left (-8+9 x-8 e^x x+8 x^2\right )} \, dx+128 \int \frac {x^2 \int \frac {1}{-8+\left (9-8 e^x\right ) x+8 x^2} \, dx}{-8+9 x-8 e^x x+8 x^2} \, dx-128 \int \frac {\int \frac {1}{x \left (-8+\left (9-8 e^x\right ) x+8 x^2\right )} \, dx}{-8+9 x-8 e^x x+8 x^2} \, dx-128 \int \frac {\int \frac {1}{x \left (-8+\left (9-8 e^x\right ) x+8 x^2\right )} \, dx}{x \left (-8+9 x-8 e^x x+8 x^2\right )} \, dx+128 \int \frac {x^2 \int \frac {1}{x \left (-8+\left (9-8 e^x\right ) x+8 x^2\right )} \, dx}{-8+9 x-8 e^x x+8 x^2} \, dx+128 \int \frac {\int \frac {x^2}{-8+\left (9-8 e^x\right ) x+8 x^2} \, dx}{-8+9 x-8 e^x x+8 x^2} \, dx+128 \int \frac {\int \frac {x^2}{-8+\left (9-8 e^x\right ) x+8 x^2} \, dx}{x \left (-8+9 x-8 e^x x+8 x^2\right )} \, dx-128 \int \frac {x^2 \int \frac {x^2}{-8+\left (9-8 e^x\right ) x+8 x^2} \, dx}{-8+9 x-8 e^x x+8 x^2} \, dx-\left (2 \log \left (\frac {1}{2} \left (8-9 x+8 e^x x-8 x^2\right )\right )\right ) \int \frac {x}{-8+9 x-8 e^x x+8 x^2} \, dx+\left (16 \log \left (\frac {1}{2} \left (8-9 x+8 e^x x-8 x^2\right )\right )\right ) \int \frac {1}{-8+9 x-8 e^x x+8 x^2} \, dx+\left (16 \log \left (\frac {1}{2} \left (8-9 x+8 e^x x-8 x^2\right )\right )\right ) \int \frac {1}{x \left (-8+9 x-8 e^x x+8 x^2\right )} \, dx-\left (16 \log \left (\frac {1}{2} \left (8-9 x+8 e^x x-8 x^2\right )\right )\right ) \int \frac {x^2}{-8+9 x-8 e^x x+8 x^2} \, dx \\ & = \text {Too large to display} \\ \end{align*}
Time = 0.57 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {-8+9 x-8 e^x x+8 x^2+\left (-18-32 x+e^x (16+16 x)\right ) \log \left (\frac {1}{2} \left (8-9 x+8 e^x x-8 x^2\right )\right )}{8-9 x+8 e^x x-8 x^2} \, dx=-x+\log ^2\left (4+\left (-\frac {9}{2}+4 e^x\right ) x-4 x^2\right ) \]
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Time = 0.10 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92
method | result | size |
norman | \(\ln \left (4 \,{\mathrm e}^{x} x -4 x^{2}-\frac {9 x}{2}+4\right )^{2}-x\) | \(23\) |
risch | \(\ln \left (4 \,{\mathrm e}^{x} x -4 x^{2}-\frac {9 x}{2}+4\right )^{2}-x\) | \(23\) |
parallelrisch | \(\ln \left (4 \,{\mathrm e}^{x} x -4 x^{2}-\frac {9 x}{2}+4\right )^{2}-x\) | \(23\) |
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Time = 0.26 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88 \[ \int \frac {-8+9 x-8 e^x x+8 x^2+\left (-18-32 x+e^x (16+16 x)\right ) \log \left (\frac {1}{2} \left (8-9 x+8 e^x x-8 x^2\right )\right )}{8-9 x+8 e^x x-8 x^2} \, dx=\log \left (-4 \, x^{2} + 4 \, x e^{x} - \frac {9}{2} \, x + 4\right )^{2} - x \]
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Time = 0.18 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88 \[ \int \frac {-8+9 x-8 e^x x+8 x^2+\left (-18-32 x+e^x (16+16 x)\right ) \log \left (\frac {1}{2} \left (8-9 x+8 e^x x-8 x^2\right )\right )}{8-9 x+8 e^x x-8 x^2} \, dx=- x + \log {\left (- 4 x^{2} + 4 x e^{x} - \frac {9 x}{2} + 4 \right )}^{2} \]
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\[ \int \frac {-8+9 x-8 e^x x+8 x^2+\left (-18-32 x+e^x (16+16 x)\right ) \log \left (\frac {1}{2} \left (8-9 x+8 e^x x-8 x^2\right )\right )}{8-9 x+8 e^x x-8 x^2} \, dx=\int { -\frac {8 \, x^{2} - 8 \, x e^{x} + 2 \, {\left (8 \, {\left (x + 1\right )} e^{x} - 16 \, x - 9\right )} \log \left (-4 \, x^{2} + 4 \, x e^{x} - \frac {9}{2} \, x + 4\right ) + 9 \, x - 8}{8 \, x^{2} - 8 \, x e^{x} + 9 \, x - 8} \,d x } \]
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\[ \int \frac {-8+9 x-8 e^x x+8 x^2+\left (-18-32 x+e^x (16+16 x)\right ) \log \left (\frac {1}{2} \left (8-9 x+8 e^x x-8 x^2\right )\right )}{8-9 x+8 e^x x-8 x^2} \, dx=\int { -\frac {8 \, x^{2} - 8 \, x e^{x} + 2 \, {\left (8 \, {\left (x + 1\right )} e^{x} - 16 \, x - 9\right )} \log \left (-4 \, x^{2} + 4 \, x e^{x} - \frac {9}{2} \, x + 4\right ) + 9 \, x - 8}{8 \, x^{2} - 8 \, x e^{x} + 9 \, x - 8} \,d x } \]
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Time = 13.86 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88 \[ \int \frac {-8+9 x-8 e^x x+8 x^2+\left (-18-32 x+e^x (16+16 x)\right ) \log \left (\frac {1}{2} \left (8-9 x+8 e^x x-8 x^2\right )\right )}{8-9 x+8 e^x x-8 x^2} \, dx={\ln \left (4\,x\,{\mathrm {e}}^x-\frac {9\,x}{2}-4\,x^2+4\right )}^2-x \]
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