Integrand size = 219, antiderivative size = 33 \[ \int \frac {-32+8 x+e^{2 x} (-2+2 x)+e^x \left (16-10 x+2 x^2\right )+e^{4+8 x-2 x^2+(-4+x) \log (x)} \left (96-344 x+280 x^2-82 x^3+8 x^4+e^{2 x} \left (6-16 x+8 x^2\right )+e^x \left (-48+150 x-98 x^2+16 x^3\right )+\left (-32 x-2 e^{2 x} x+16 x^2-2 x^3+e^x \left (16 x-4 x^2\right )\right ) \log (x)\right )}{x^3+3 e^{4+8 x-2 x^2+(-4+x) \log (x)} x^3+3 e^{8+16 x-4 x^2+2 (-4+x) \log (x)} x^3+e^{12+24 x-6 x^2+3 (-4+x) \log (x)} x^3} \, dx=\frac {\left (-4+e^x+x\right )^2}{\left (x+e^{4+(4-x) (2 x-\log (x))} x\right )^2} \]
Time = 0.21 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.33 \[ \int \frac {-32+8 x+e^{2 x} (-2+2 x)+e^x \left (16-10 x+2 x^2\right )+e^{4+8 x-2 x^2+(-4+x) \log (x)} \left (96-344 x+280 x^2-82 x^3+8 x^4+e^{2 x} \left (6-16 x+8 x^2\right )+e^x \left (-48+150 x-98 x^2+16 x^3\right )+\left (-32 x-2 e^{2 x} x+16 x^2-2 x^3+e^x \left (16 x-4 x^2\right )\right ) \log (x)\right )}{x^3+3 e^{4+8 x-2 x^2+(-4+x) \log (x)} x^3+3 e^{8+16 x-4 x^2+2 (-4+x) \log (x)} x^3+e^{12+24 x-6 x^2+3 (-4+x) \log (x)} x^3} \, dx=\frac {e^{4 x^2} x^6 \left (-4+e^x+x\right )^2}{\left (e^{2 x^2} x^4+e^{4+8 x} x^x\right )^2} \]
Integrate[(-32 + 8*x + E^(2*x)*(-2 + 2*x) + E^x*(16 - 10*x + 2*x^2) + E^(4 + 8*x - 2*x^2 + (-4 + x)*Log[x])*(96 - 344*x + 280*x^2 - 82*x^3 + 8*x^4 + E^(2*x)*(6 - 16*x + 8*x^2) + E^x*(-48 + 150*x - 98*x^2 + 16*x^3) + (-32*x - 2*E^(2*x)*x + 16*x^2 - 2*x^3 + E^x*(16*x - 4*x^2))*Log[x]))/(x^3 + 3*E^ (4 + 8*x - 2*x^2 + (-4 + x)*Log[x])*x^3 + 3*E^(8 + 16*x - 4*x^2 + 2*(-4 + x)*Log[x])*x^3 + E^(12 + 24*x - 6*x^2 + 3*(-4 + x)*Log[x])*x^3),x]
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {e^x \left (2 x^2-10 x+16\right )+e^{-2 x^2+8 x+(x-4) \log (x)+4} \left (8 x^4-82 x^3+280 x^2+e^{2 x} \left (8 x^2-16 x+6\right )+e^x \left (16 x^3-98 x^2+150 x-48\right )+\left (-2 x^3+16 x^2+e^x \left (16 x-4 x^2\right )-2 e^{2 x} x-32 x\right ) \log (x)-344 x+96\right )+8 x+e^{2 x} (2 x-2)-32}{x^3+3 x^3 e^{-2 x^2+8 x+(x-4) \log (x)+4}+3 x^3 e^{-4 x^2+16 x+2 (x-4) \log (x)+8}+x^3 e^{-6 x^2+24 x+3 (x-4) \log (x)+12}} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {e^{6 x^2} x^9 \left (e^x \left (2 x^2-10 x+16\right )+e^{-2 x^2+8 x+(x-4) \log (x)+4} \left (8 x^4-82 x^3+280 x^2+e^{2 x} \left (8 x^2-16 x+6\right )+e^x \left (16 x^3-98 x^2+150 x-48\right )+\left (-2 x^3+16 x^2+e^x \left (16 x-4 x^2\right )-2 e^{2 x} x-32 x\right ) \log (x)-344 x+96\right )+8 x+e^{2 x} (2 x-2)-32\right )}{\left (e^{8 x+4} x^x+e^{2 x^2} x^4\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {96 e^{4 x^2+8 x+4} x^{x+5}}{\left (e^{8 x+4} x^x+e^{2 x^2} x^4\right )^3}-\frac {48 e^{4 x^2+9 x+4} x^{x+5}}{\left (e^{8 x+4} x^x+e^{2 x^2} x^4\right )^3}+\frac {6 e^{4 x^2+10 x+4} x^{x+5}}{\left (e^{8 x+4} x^x+e^{2 x^2} x^4\right )^3}-\frac {32 e^{4 x^2+8 x+4} \log (x) x^{x+6}}{\left (e^{8 x+4} x^x+e^{2 x^2} x^4\right )^3}+\frac {16 e^{4 x^2+9 x+4} \log (x) x^{x+6}}{\left (e^{8 x+4} x^x+e^{2 x^2} x^4\right )^3}-\frac {2 e^{4 x^2+10 x+4} \log (x) x^{x+6}}{\left (e^{8 x+4} x^x+e^{2 x^2} x^4\right )^3}-\frac {344 e^{4 x^2+8 x+4} x^{x+6}}{\left (e^{8 x+4} x^x+e^{2 x^2} x^4\right )^3}+\frac {150 e^{4 x^2+9 x+4} x^{x+6}}{\left (e^{8 x+4} x^x+e^{2 x^2} x^4\right )^3}-\frac {16 e^{4 x^2+10 x+4} x^{x+6}}{\left (e^{8 x+4} x^x+e^{2 x^2} x^4\right )^3}+\frac {16 e^{4 x^2+8 x+4} \log (x) x^{x+7}}{\left (e^{8 x+4} x^x+e^{2 x^2} x^4\right )^3}-\frac {4 e^{4 x^2+9 x+4} \log (x) x^{x+7}}{\left (e^{8 x+4} x^x+e^{2 x^2} x^4\right )^3}+\frac {280 e^{4 x^2+8 x+4} x^{x+7}}{\left (e^{8 x+4} x^x+e^{2 x^2} x^4\right )^3}-\frac {98 e^{4 x^2+9 x+4} x^{x+7}}{\left (e^{8 x+4} x^x+e^{2 x^2} x^4\right )^3}+\frac {8 e^{4 x^2+10 x+4} x^{x+7}}{\left (e^{8 x+4} x^x+e^{2 x^2} x^4\right )^3}-\frac {2 e^{4 x^2+8 x+4} \log (x) x^{x+8}}{\left (e^{8 x+4} x^x+e^{2 x^2} x^4\right )^3}-\frac {82 e^{4 x^2+8 x+4} x^{x+8}}{\left (e^{8 x+4} x^x+e^{2 x^2} x^4\right )^3}+\frac {16 e^{4 x^2+9 x+4} x^{x+8}}{\left (e^{8 x+4} x^x+e^{2 x^2} x^4\right )^3}+\frac {8 e^{4 x^2+8 x+4} x^{x+9}}{\left (e^{8 x+4} x^x+e^{2 x^2} x^4\right )^3}+\frac {2 e^{6 x^2+x} x^{11}}{\left (e^{8 x+4} x^x+e^{2 x^2} x^4\right )^3}+\frac {8 e^{6 x^2} x^{10}}{\left (e^{8 x+4} x^x+e^{2 x^2} x^4\right )^3}-\frac {10 e^{6 x^2+x} x^{10}}{\left (e^{8 x+4} x^x+e^{2 x^2} x^4\right )^3}+\frac {2 e^{6 x^2+2 x} x^{10}}{\left (e^{8 x+4} x^x+e^{2 x^2} x^4\right )^3}-\frac {32 e^{6 x^2} x^9}{\left (e^{8 x+4} x^x+e^{2 x^2} x^4\right )^3}+\frac {16 e^{6 x^2+x} x^9}{\left (e^{8 x+4} x^x+e^{2 x^2} x^4\right )^3}-\frac {2 e^{6 x^2+2 x} x^9}{\left (e^{8 x+4} x^x+e^{2 x^2} x^4\right )^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -32 \int \frac {e^{6 x^2} x^9}{\left (e^{8 x+4} x^x+e^{2 x^2} x^4\right )^3}dx+16 \int \frac {e^{6 x^2+x} x^9}{\left (e^{8 x+4} x^x+e^{2 x^2} x^4\right )^3}dx-2 \int \frac {e^{6 x^2+2 x} x^9}{\left (e^{8 x+4} x^x+e^{2 x^2} x^4\right )^3}dx+8 \int \frac {e^{6 x^2} x^{10}}{\left (e^{8 x+4} x^x+e^{2 x^2} x^4\right )^3}dx-10 \int \frac {e^{6 x^2+x} x^{10}}{\left (e^{8 x+4} x^x+e^{2 x^2} x^4\right )^3}dx+2 \int \frac {e^{6 x^2+2 x} x^{10}}{\left (e^{8 x+4} x^x+e^{2 x^2} x^4\right )^3}dx+2 \int \frac {e^{6 x^2+x} x^{11}}{\left (e^{8 x+4} x^x+e^{2 x^2} x^4\right )^3}dx+96 \int \frac {e^{4 x^2+8 x+4} x^{x+5}}{\left (e^{8 x+4} x^x+e^{2 x^2} x^4\right )^3}dx-48 \int \frac {e^{4 x^2+9 x+4} x^{x+5}}{\left (e^{8 x+4} x^x+e^{2 x^2} x^4\right )^3}dx+6 \int \frac {e^{4 x^2+10 x+4} x^{x+5}}{\left (e^{8 x+4} x^x+e^{2 x^2} x^4\right )^3}dx-32 \log (x) \int \frac {e^{4 x^2+8 x+4} x^{x+6}}{\left (e^{8 x+4} x^x+e^{2 x^2} x^4\right )^3}dx-344 \int \frac {e^{4 x^2+8 x+4} x^{x+6}}{\left (e^{8 x+4} x^x+e^{2 x^2} x^4\right )^3}dx+16 \log (x) \int \frac {e^{4 x^2+9 x+4} x^{x+6}}{\left (e^{8 x+4} x^x+e^{2 x^2} x^4\right )^3}dx+150 \int \frac {e^{4 x^2+9 x+4} x^{x+6}}{\left (e^{8 x+4} x^x+e^{2 x^2} x^4\right )^3}dx-2 \log (x) \int \frac {e^{4 x^2+10 x+4} x^{x+6}}{\left (e^{8 x+4} x^x+e^{2 x^2} x^4\right )^3}dx-16 \int \frac {e^{4 x^2+10 x+4} x^{x+6}}{\left (e^{8 x+4} x^x+e^{2 x^2} x^4\right )^3}dx+16 \log (x) \int \frac {e^{4 x^2+8 x+4} x^{x+7}}{\left (e^{8 x+4} x^x+e^{2 x^2} x^4\right )^3}dx+280 \int \frac {e^{4 x^2+8 x+4} x^{x+7}}{\left (e^{8 x+4} x^x+e^{2 x^2} x^4\right )^3}dx-4 \log (x) \int \frac {e^{4 x^2+9 x+4} x^{x+7}}{\left (e^{8 x+4} x^x+e^{2 x^2} x^4\right )^3}dx-98 \int \frac {e^{4 x^2+9 x+4} x^{x+7}}{\left (e^{8 x+4} x^x+e^{2 x^2} x^4\right )^3}dx+8 \int \frac {e^{4 x^2+10 x+4} x^{x+7}}{\left (e^{8 x+4} x^x+e^{2 x^2} x^4\right )^3}dx-2 \log (x) \int \frac {e^{4 x^2+8 x+4} x^{x+8}}{\left (e^{8 x+4} x^x+e^{2 x^2} x^4\right )^3}dx-82 \int \frac {e^{4 x^2+8 x+4} x^{x+8}}{\left (e^{8 x+4} x^x+e^{2 x^2} x^4\right )^3}dx+16 \int \frac {e^{4 x^2+9 x+4} x^{x+8}}{\left (e^{8 x+4} x^x+e^{2 x^2} x^4\right )^3}dx+8 \int \frac {e^{4 x^2+8 x+4} x^{x+9}}{\left (e^{8 x+4} x^x+e^{2 x^2} x^4\right )^3}dx+32 \int \frac {\int \frac {e^{4 (x+1)^2} x^{x+6}}{\left (e^{8 x+4} x^x+e^{2 x^2} x^4\right )^3}dx}{x}dx-16 \int \frac {\int \frac {e^{4 x^2+9 x+4} x^{x+6}}{\left (e^{8 x+4} x^x+e^{2 x^2} x^4\right )^3}dx}{x}dx+2 \int \frac {\int \frac {e^{4 x^2+10 x+4} x^{x+6}}{\left (e^{8 x+4} x^x+e^{2 x^2} x^4\right )^3}dx}{x}dx-16 \int \frac {\int \frac {e^{4 (x+1)^2} x^{x+7}}{\left (e^{8 x+4} x^x+e^{2 x^2} x^4\right )^3}dx}{x}dx+4 \int \frac {\int \frac {e^{4 x^2+9 x+4} x^{x+7}}{\left (e^{8 x+4} x^x+e^{2 x^2} x^4\right )^3}dx}{x}dx+2 \int \frac {\int \frac {e^{4 (x+1)^2} x^{x+8}}{\left (e^{8 x+4} x^x+e^{2 x^2} x^4\right )^3}dx}{x}dx\) |
Int[(-32 + 8*x + E^(2*x)*(-2 + 2*x) + E^x*(16 - 10*x + 2*x^2) + E^(4 + 8*x - 2*x^2 + (-4 + x)*Log[x])*(96 - 344*x + 280*x^2 - 82*x^3 + 8*x^4 + E^(2* x)*(6 - 16*x + 8*x^2) + E^x*(-48 + 150*x - 98*x^2 + 16*x^3) + (-32*x - 2*E ^(2*x)*x + 16*x^2 - 2*x^3 + E^x*(16*x - 4*x^2))*Log[x]))/(x^3 + 3*E^(4 + 8 *x - 2*x^2 + (-4 + x)*Log[x])*x^3 + 3*E^(8 + 16*x - 4*x^2 + 2*(-4 + x)*Log [x])*x^3 + E^(12 + 24*x - 6*x^2 + 3*(-4 + x)*Log[x])*x^3),x]
3.4.29.3.1 Defintions of rubi rules used
Time = 4.12 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.42
| method | result | size |
| risch | \(\frac {x^{2}+2 \,{\mathrm e}^{x} x +{\mathrm e}^{2 x}-8 x -8 \,{\mathrm e}^{x}+16}{x^{2} \left (x^{x -4} {\mathrm e}^{-2 x^{2}+8 x +4}+1\right )^{2}}\) | \(47\) |
| parallelrisch | \(-\frac {-32-2 x^{2}-4 \,{\mathrm e}^{x} x -2 \,{\mathrm e}^{2 x}+16 x +16 \,{\mathrm e}^{x}}{2 x^{2} \left ({\mathrm e}^{\left (2 x -8\right ) \ln \left (x \right )-4 x^{2}+16 x +8}+2 \,{\mathrm e}^{\left (x -4\right ) \ln \left (x \right )-2 x^{2}+8 x +4}+1\right )}\) | \(73\) |
int((((-2*x*exp(x)^2+(-4*x^2+16*x)*exp(x)-2*x^3+16*x^2-32*x)*ln(x)+(8*x^2- 16*x+6)*exp(x)^2+(16*x^3-98*x^2+150*x-48)*exp(x)+8*x^4-82*x^3+280*x^2-344* x+96)*exp((x-4)*ln(x)-2*x^2+8*x+4)+(-2+2*x)*exp(x)^2+(2*x^2-10*x+16)*exp(x )+8*x-32)/(x^3*exp((x-4)*ln(x)-2*x^2+8*x+4)^3+3*x^3*exp((x-4)*ln(x)-2*x^2+ 8*x+4)^2+3*x^3*exp((x-4)*ln(x)-2*x^2+8*x+4)+x^3),x,method=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (30) = 60\).
Time = 0.24 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.12 \[ \int \frac {-32+8 x+e^{2 x} (-2+2 x)+e^x \left (16-10 x+2 x^2\right )+e^{4+8 x-2 x^2+(-4+x) \log (x)} \left (96-344 x+280 x^2-82 x^3+8 x^4+e^{2 x} \left (6-16 x+8 x^2\right )+e^x \left (-48+150 x-98 x^2+16 x^3\right )+\left (-32 x-2 e^{2 x} x+16 x^2-2 x^3+e^x \left (16 x-4 x^2\right )\right ) \log (x)\right )}{x^3+3 e^{4+8 x-2 x^2+(-4+x) \log (x)} x^3+3 e^{8+16 x-4 x^2+2 (-4+x) \log (x)} x^3+e^{12+24 x-6 x^2+3 (-4+x) \log (x)} x^3} \, dx=\frac {x^{2} + 2 \, {\left (x - 4\right )} e^{x} - 8 \, x + e^{\left (2 \, x\right )} + 16}{2 \, x^{2} e^{\left (-2 \, x^{2} + {\left (x - 4\right )} \log \left (x\right ) + 8 \, x + 4\right )} + x^{2} e^{\left (-4 \, x^{2} + 2 \, {\left (x - 4\right )} \log \left (x\right ) + 16 \, x + 8\right )} + x^{2}} \]
integrate((((-2*x*exp(x)^2+(-4*x^2+16*x)*exp(x)-2*x^3+16*x^2-32*x)*log(x)+ (8*x^2-16*x+6)*exp(x)^2+(16*x^3-98*x^2+150*x-48)*exp(x)+8*x^4-82*x^3+280*x ^2-344*x+96)*exp((x-4)*log(x)-2*x^2+8*x+4)+(-2+2*x)*exp(x)^2+(2*x^2-10*x+1 6)*exp(x)+8*x-32)/(x^3*exp((x-4)*log(x)-2*x^2+8*x+4)^3+3*x^3*exp((x-4)*log (x)-2*x^2+8*x+4)^2+3*x^3*exp((x-4)*log(x)-2*x^2+8*x+4)+x^3),x, algorithm=\
(x^2 + 2*(x - 4)*e^x - 8*x + e^(2*x) + 16)/(2*x^2*e^(-2*x^2 + (x - 4)*log( x) + 8*x + 4) + x^2*e^(-4*x^2 + 2*(x - 4)*log(x) + 16*x + 8) + x^2)
Leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (26) = 52\).
Time = 0.31 (sec) , antiderivative size = 75, normalized size of antiderivative = 2.27 \[ \int \frac {-32+8 x+e^{2 x} (-2+2 x)+e^x \left (16-10 x+2 x^2\right )+e^{4+8 x-2 x^2+(-4+x) \log (x)} \left (96-344 x+280 x^2-82 x^3+8 x^4+e^{2 x} \left (6-16 x+8 x^2\right )+e^x \left (-48+150 x-98 x^2+16 x^3\right )+\left (-32 x-2 e^{2 x} x+16 x^2-2 x^3+e^x \left (16 x-4 x^2\right )\right ) \log (x)\right )}{x^3+3 e^{4+8 x-2 x^2+(-4+x) \log (x)} x^3+3 e^{8+16 x-4 x^2+2 (-4+x) \log (x)} x^3+e^{12+24 x-6 x^2+3 (-4+x) \log (x)} x^3} \, dx=\frac {x^{2} + 2 x e^{x} - 8 x + e^{2 x} - 8 e^{x} + 16}{x^{2} e^{- 4 x^{2} + 16 x + 2 \left (x - 4\right ) \log {\left (x \right )} + 8} + 2 x^{2} e^{- 2 x^{2} + 8 x + \left (x - 4\right ) \log {\left (x \right )} + 4} + x^{2}} \]
integrate((((-2*x*exp(x)**2+(-4*x**2+16*x)*exp(x)-2*x**3+16*x**2-32*x)*ln( x)+(8*x**2-16*x+6)*exp(x)**2+(16*x**3-98*x**2+150*x-48)*exp(x)+8*x**4-82*x **3+280*x**2-344*x+96)*exp((x-4)*ln(x)-2*x**2+8*x+4)+(-2+2*x)*exp(x)**2+(2 *x**2-10*x+16)*exp(x)+8*x-32)/(x**3*exp((x-4)*ln(x)-2*x**2+8*x+4)**3+3*x** 3*exp((x-4)*ln(x)-2*x**2+8*x+4)**2+3*x**3*exp((x-4)*ln(x)-2*x**2+8*x+4)+x* *3),x)
(x**2 + 2*x*exp(x) - 8*x + exp(2*x) - 8*exp(x) + 16)/(x**2*exp(-4*x**2 + 1 6*x + 2*(x - 4)*log(x) + 8) + 2*x**2*exp(-2*x**2 + 8*x + (x - 4)*log(x) + 4) + x**2)
Leaf count of result is larger than twice the leaf count of optimal. 86 vs. \(2 (30) = 60\).
Time = 0.30 (sec) , antiderivative size = 86, normalized size of antiderivative = 2.61 \[ \int \frac {-32+8 x+e^{2 x} (-2+2 x)+e^x \left (16-10 x+2 x^2\right )+e^{4+8 x-2 x^2+(-4+x) \log (x)} \left (96-344 x+280 x^2-82 x^3+8 x^4+e^{2 x} \left (6-16 x+8 x^2\right )+e^x \left (-48+150 x-98 x^2+16 x^3\right )+\left (-32 x-2 e^{2 x} x+16 x^2-2 x^3+e^x \left (16 x-4 x^2\right )\right ) \log (x)\right )}{x^3+3 e^{4+8 x-2 x^2+(-4+x) \log (x)} x^3+3 e^{8+16 x-4 x^2+2 (-4+x) \log (x)} x^3+e^{12+24 x-6 x^2+3 (-4+x) \log (x)} x^3} \, dx=\frac {{\left (x^{8} - 8 \, x^{7} + x^{6} e^{\left (2 \, x\right )} + 16 \, x^{6} + 2 \, {\left (x^{7} - 4 \, x^{6}\right )} e^{x}\right )} e^{\left (4 \, x^{2}\right )}}{x^{8} e^{\left (4 \, x^{2}\right )} + 2 \, x^{4} e^{\left (2 \, x^{2} + x \log \left (x\right ) + 8 \, x + 4\right )} + e^{\left (2 \, x \log \left (x\right ) + 16 \, x + 8\right )}} \]
integrate((((-2*x*exp(x)^2+(-4*x^2+16*x)*exp(x)-2*x^3+16*x^2-32*x)*log(x)+ (8*x^2-16*x+6)*exp(x)^2+(16*x^3-98*x^2+150*x-48)*exp(x)+8*x^4-82*x^3+280*x ^2-344*x+96)*exp((x-4)*log(x)-2*x^2+8*x+4)+(-2+2*x)*exp(x)^2+(2*x^2-10*x+1 6)*exp(x)+8*x-32)/(x^3*exp((x-4)*log(x)-2*x^2+8*x+4)^3+3*x^3*exp((x-4)*log (x)-2*x^2+8*x+4)^2+3*x^3*exp((x-4)*log(x)-2*x^2+8*x+4)+x^3),x, algorithm=\
(x^8 - 8*x^7 + x^6*e^(2*x) + 16*x^6 + 2*(x^7 - 4*x^6)*e^x)*e^(4*x^2)/(x^8* e^(4*x^2) + 2*x^4*e^(2*x^2 + x*log(x) + 8*x + 4) + e^(2*x*log(x) + 16*x + 8))
Leaf count of result is larger than twice the leaf count of optimal. 4807 vs. \(2 (30) = 60\).
Time = 0.55 (sec) , antiderivative size = 4807, normalized size of antiderivative = 145.67 \[ \int \frac {-32+8 x+e^{2 x} (-2+2 x)+e^x \left (16-10 x+2 x^2\right )+e^{4+8 x-2 x^2+(-4+x) \log (x)} \left (96-344 x+280 x^2-82 x^3+8 x^4+e^{2 x} \left (6-16 x+8 x^2\right )+e^x \left (-48+150 x-98 x^2+16 x^3\right )+\left (-32 x-2 e^{2 x} x+16 x^2-2 x^3+e^x \left (16 x-4 x^2\right )\right ) \log (x)\right )}{x^3+3 e^{4+8 x-2 x^2+(-4+x) \log (x)} x^3+3 e^{8+16 x-4 x^2+2 (-4+x) \log (x)} x^3+e^{12+24 x-6 x^2+3 (-4+x) \log (x)} x^3} \, dx=\text {Too large to display} \]
integrate((((-2*x*exp(x)^2+(-4*x^2+16*x)*exp(x)-2*x^3+16*x^2-32*x)*log(x)+ (8*x^2-16*x+6)*exp(x)^2+(16*x^3-98*x^2+150*x-48)*exp(x)+8*x^4-82*x^3+280*x ^2-344*x+96)*exp((x-4)*log(x)-2*x^2+8*x+4)+(-2+2*x)*exp(x)^2+(2*x^2-10*x+1 6)*exp(x)+8*x-32)/(x^3*exp((x-4)*log(x)-2*x^2+8*x+4)^3+3*x^3*exp((x-4)*log (x)-2*x^2+8*x+4)^2+3*x^3*exp((x-4)*log(x)-2*x^2+8*x+4)+x^3),x, algorithm=\
(64*x^24*e^(4*x^2) - 48*x^23*e^(4*x^2)*log(x) + 12*x^22*e^(4*x^2)*log(x)^2 - x^21*e^(4*x^2)*log(x)^3 - 944*x^23*e^(4*x^2) + 128*x^23*e^(4*x^2 + x) + 600*x^22*e^(4*x^2)*log(x) - 96*x^22*e^(4*x^2 + x)*log(x) - 123*x^21*e^(4* x^2)*log(x)^2 + 24*x^21*e^(4*x^2 + x)*log(x)^2 + 8*x^20*e^(4*x^2)*log(x)^3 - 2*x^20*e^(4*x^2 + x)*log(x)^3 + 5644*x^22*e^(4*x^2) + 64*x^22*e^(4*x^2 + 2*x) - 1376*x^22*e^(4*x^2 + x) - 2835*x^21*e^(4*x^2)*log(x) - 48*x^21*e^ (4*x^2 + 2*x)*log(x) + 816*x^21*e^(4*x^2 + x)*log(x) + 420*x^20*e^(4*x^2)* log(x)^2 + 12*x^20*e^(4*x^2 + 2*x)*log(x)^2 - 150*x^20*e^(4*x^2 + x)*log(x )^2 - 16*x^19*e^(4*x^2)*log(x)^3 - x^19*e^(4*x^2 + 2*x)*log(x)^3 + 8*x^19* e^(4*x^2 + x)*log(x)^3 - 17817*x^21*e^(4*x^2) - 432*x^21*e^(4*x^2 + 2*x) + 5784*x^21*e^(4*x^2 + x) + 6384*x^20*e^(4*x^2)*log(x) + 216*x^20*e^(4*x^2 + 2*x)*log(x) - 2406*x^20*e^(4*x^2 + x)*log(x) - 528*x^19*e^(4*x^2)*log(x) ^2 - 27*x^19*e^(4*x^2 + 2*x)*log(x)^2 + 240*x^19*e^(4*x^2 + x)*log(x)^2 + 32532*x^20*e^(4*x^2) + 1164*x^20*e^(4*x^2 + 2*x) - 12498*x^20*e^(4*x^2 + x ) + 128*x^20*e^(2*x^2 + x*log(x) + 8*x + 4) - 7200*x^19*e^(4*x^2)*log(x) - 339*x^19*e^(4*x^2 + 2*x)*log(x) + 3144*x^19*e^(4*x^2 + x)*log(x) - 96*x^1 9*e^(2*x^2 + x*log(x) + 8*x + 4)*log(x) + 192*x^18*e^(4*x^2)*log(x)^2 + 12 *x^18*e^(4*x^2 + 2*x)*log(x)^2 - 96*x^18*e^(4*x^2 + x)*log(x)^2 + 24*x^18* e^(2*x^2 + x*log(x) + 8*x + 4)*log(x)^2 - 2*x^17*e^(2*x^2 + x*log(x) + 8*x + 4)*log(x)^3 - 35232*x^19*e^(4*x^2) - 1593*x^19*e^(4*x^2 + 2*x) + 150...
Time = 9.28 (sec) , antiderivative size = 168, normalized size of antiderivative = 5.09 \[ \int \frac {-32+8 x+e^{2 x} (-2+2 x)+e^x \left (16-10 x+2 x^2\right )+e^{4+8 x-2 x^2+(-4+x) \log (x)} \left (96-344 x+280 x^2-82 x^3+8 x^4+e^{2 x} \left (6-16 x+8 x^2\right )+e^x \left (-48+150 x-98 x^2+16 x^3\right )+\left (-32 x-2 e^{2 x} x+16 x^2-2 x^3+e^x \left (16 x-4 x^2\right )\right ) \log (x)\right )}{x^3+3 e^{4+8 x-2 x^2+(-4+x) \log (x)} x^3+3 e^{8+16 x-4 x^2+2 (-4+x) \log (x)} x^3+e^{12+24 x-6 x^2+3 (-4+x) \log (x)} x^3} \, dx=\frac {176\,x-4\,{\mathrm {e}}^{2\,x}+32\,{\mathrm {e}}^x+9\,x\,{\mathrm {e}}^{2\,x}+50\,x^2\,{\mathrm {e}}^x-8\,x^3\,{\mathrm {e}}^x-8\,x^2\,\ln \left (x\right )+x^3\,\ln \left (x\right )-4\,x^2\,{\mathrm {e}}^{2\,x}-80\,x\,{\mathrm {e}}^x+16\,x\,\ln \left (x\right )-140\,x^2+41\,x^3-4\,x^4-8\,x\,{\mathrm {e}}^x\,\ln \left (x\right )+x\,{\mathrm {e}}^{2\,x}\,\ln \left (x\right )+2\,x^2\,{\mathrm {e}}^x\,\ln \left (x\right )-64}{x^2\,\left (x^{2\,x-8}\,{\mathrm {e}}^{-4\,x^2+16\,x+8}+2\,x^{x-4}\,{\mathrm {e}}^{-2\,x^2+8\,x+4}+1\right )\,\left (9\,x+x\,\ln \left (x\right )-4\,x^2-4\right )} \]
int((8*x + exp(x)*(2*x^2 - 10*x + 16) + exp(8*x + log(x)*(x - 4) - 2*x^2 + 4)*(exp(2*x)*(8*x^2 - 16*x + 6) - 344*x - log(x)*(32*x + 2*x*exp(2*x) - e xp(x)*(16*x - 4*x^2) - 16*x^2 + 2*x^3) + 280*x^2 - 82*x^3 + 8*x^4 + exp(x) *(150*x - 98*x^2 + 16*x^3 - 48) + 96) + exp(2*x)*(2*x - 2) - 32)/(3*x^3*ex p(8*x + log(x)*(x - 4) - 2*x^2 + 4) + 3*x^3*exp(16*x + 2*log(x)*(x - 4) - 4*x^2 + 8) + x^3*exp(24*x + 3*log(x)*(x - 4) - 6*x^2 + 12) + x^3),x)
(176*x - 4*exp(2*x) + 32*exp(x) + 9*x*exp(2*x) + 50*x^2*exp(x) - 8*x^3*exp (x) - 8*x^2*log(x) + x^3*log(x) - 4*x^2*exp(2*x) - 80*x*exp(x) + 16*x*log( x) - 140*x^2 + 41*x^3 - 4*x^4 - 8*x*exp(x)*log(x) + x*exp(2*x)*log(x) + 2* x^2*exp(x)*log(x) - 64)/(x^2*(x^(2*x - 8)*exp(16*x - 4*x^2 + 8) + 2*x^(x - 4)*exp(8*x - 2*x^2 + 4) + 1)*(9*x + x*log(x) - 4*x^2 - 4))