Integrand size = 239, antiderivative size = 28 \[ \int \frac {2 e^{4 x} x+8 x^3+e^{2 x} \left (-2 x+12 x^2\right )+e^{x^2} \left (4 e^{4 x} x^2+16 e^{2 x} x^3+16 x^4\right )+\left (12 x^2+8 x^3+e^{4 x} (4+2 x)+e^{2 x} \left (14 x+8 x^2\right )+e^{x^2} \left (2 e^{4 x}+8 e^{2 x} x+8 x^2\right )\right ) \log \left (\frac {3 x+2 x^2+e^{2 x} (2+x)+e^{x^2} \left (e^{2 x}+2 x\right )}{e^{2 x}+2 x}\right )}{30 x^2+20 x^3+e^{4 x} (10+5 x)+e^{2 x} \left (35 x+20 x^2\right )+e^{x^2} \left (5 e^{4 x}+20 e^{2 x} x+20 x^2\right )} \, dx=\frac {2}{5} x \log \left (2+e^{x^2}+x-\frac {x}{e^{2 x}+2 x}\right ) \] Output:
2/5*ln(x-x/(exp(x)^2+2*x)+2+exp(x^2))*x
Time = 0.16 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.79 \[ \int \frac {2 e^{4 x} x+8 x^3+e^{2 x} \left (-2 x+12 x^2\right )+e^{x^2} \left (4 e^{4 x} x^2+16 e^{2 x} x^3+16 x^4\right )+\left (12 x^2+8 x^3+e^{4 x} (4+2 x)+e^{2 x} \left (14 x+8 x^2\right )+e^{x^2} \left (2 e^{4 x}+8 e^{2 x} x+8 x^2\right )\right ) \log \left (\frac {3 x+2 x^2+e^{2 x} (2+x)+e^{x^2} \left (e^{2 x}+2 x\right )}{e^{2 x}+2 x}\right )}{30 x^2+20 x^3+e^{4 x} (10+5 x)+e^{2 x} \left (35 x+20 x^2\right )+e^{x^2} \left (5 e^{4 x}+20 e^{2 x} x+20 x^2\right )} \, dx=\frac {2}{5} x \log \left (\frac {e^{x (2+x)}+2 e^{x^2} x+e^{2 x} (2+x)+x (3+2 x)}{e^{2 x}+2 x}\right ) \] Input:
Integrate[(2*E^(4*x)*x + 8*x^3 + E^(2*x)*(-2*x + 12*x^2) + E^x^2*(4*E^(4*x )*x^2 + 16*E^(2*x)*x^3 + 16*x^4) + (12*x^2 + 8*x^3 + E^(4*x)*(4 + 2*x) + E ^(2*x)*(14*x + 8*x^2) + E^x^2*(2*E^(4*x) + 8*E^(2*x)*x + 8*x^2))*Log[(3*x + 2*x^2 + E^(2*x)*(2 + x) + E^x^2*(E^(2*x) + 2*x))/(E^(2*x) + 2*x)])/(30*x ^2 + 20*x^3 + E^(4*x)*(10 + 5*x) + E^(2*x)*(35*x + 20*x^2) + E^x^2*(5*E^(4 *x) + 20*E^(2*x)*x + 20*x^2)),x]
Output:
(2*x*Log[(E^(x*(2 + x)) + 2*E^x^2*x + E^(2*x)*(2 + x) + x*(3 + 2*x))/(E^(2 *x) + 2*x)])/5
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.
| \(\displaystyle \int \frac {8 x^3+e^{2 x} \left (12 x^2-2 x\right )+\left (8 x^3+12 x^2+e^{2 x} \left (8 x^2+14 x\right )+e^{x^2} \left (8 x^2+8 e^{2 x} x+2 e^{4 x}\right )+e^{4 x} (2 x+4)\right ) \log \left (\frac {2 x^2+e^{x^2} \left (2 x+e^{2 x}\right )+3 x+e^{2 x} (x+2)}{2 x+e^{2 x}}\right )+e^{x^2} \left (16 x^4+16 e^{2 x} x^3+4 e^{4 x} x^2\right )+2 e^{4 x} x}{20 x^3+30 x^2+e^{2 x} \left (20 x^2+35 x\right )+e^{x^2} \left (20 x^2+20 e^{2 x} x+5 e^{4 x}\right )+e^{4 x} (5 x+10)} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {8 x^3+e^{2 x} \left (12 x^2-2 x\right )+\left (8 x^3+12 x^2+e^{2 x} \left (8 x^2+14 x\right )+e^{x^2} \left (8 x^2+8 e^{2 x} x+2 e^{4 x}\right )+e^{4 x} (2 x+4)\right ) \log \left (\frac {2 x^2+e^{x^2} \left (2 x+e^{2 x}\right )+3 x+e^{2 x} (x+2)}{2 x+e^{2 x}}\right )+e^{x^2} \left (16 x^4+16 e^{2 x} x^3+4 e^{4 x} x^2\right )+2 e^{4 x} x}{5 \left (2 x+e^{2 x}\right ) \left (2 x^2+2 e^{x^2} x+e^{2 x} x+3 x+2 e^{2 x}+e^{x (x+2)}\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} \int \frac {2 \left (4 x^3+e^{4 x} x-e^{2 x} \left (x-6 x^2\right )+2 e^{x^2} \left (4 x^4+4 e^{2 x} x^3+e^{4 x} x^2\right )+\left (4 x^3+6 x^2+e^{4 x} (x+2)+e^{2 x} \left (4 x^2+7 x\right )+e^{x^2} \left (4 x^2+4 e^{2 x} x+e^{4 x}\right )\right ) \log \left (\frac {2 x^2+3 x+e^{2 x} (x+2)+e^{x^2} \left (2 x+e^{2 x}\right )}{2 x+e^{2 x}}\right )\right )}{\left (2 x+e^{2 x}\right ) \left (2 x^2+2 e^{x^2} x+e^{2 x} x+3 x+2 e^{2 x}+e^{x (x+2)}\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{5} \int \frac {4 x^3+e^{4 x} x-e^{2 x} \left (x-6 x^2\right )+2 e^{x^2} \left (4 x^4+4 e^{2 x} x^3+e^{4 x} x^2\right )+\left (4 x^3+6 x^2+e^{4 x} (x+2)+e^{2 x} \left (4 x^2+7 x\right )+e^{x^2} \left (4 x^2+4 e^{2 x} x+e^{4 x}\right )\right ) \log \left (\frac {2 x^2+3 x+e^{2 x} (x+2)+e^{x^2} \left (2 x+e^{2 x}\right )}{2 x+e^{2 x}}\right )}{\left (2 x+e^{2 x}\right ) \left (2 x^2+2 e^{x^2} x+e^{2 x} x+3 x+2 e^{2 x}+e^{x (x+2)}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {2}{5} \int \left (\frac {8 e^{x^2} x^4}{\left (2 x+e^{2 x}\right ) \left (2 x^2+2 e^{x^2} x+e^{2 x} x+3 x+2 e^{2 x}+e^{x (x+2)}\right )}+\frac {4 \log \left (\frac {2 e^{x^2} x+(2 x+3) x+e^{x (x+2)}+e^{2 x} (x+2)}{2 x+e^{2 x}}\right ) x^3}{\left (2 x+e^{2 x}\right ) \left (2 x^2+2 e^{x^2} x+e^{2 x} x+3 x+2 e^{2 x}+e^{x (x+2)}\right )}+\frac {8 e^{x^2+2 x} x^3}{\left (2 x+e^{2 x}\right ) \left (2 x^2+2 e^{x^2} x+e^{2 x} x+3 x+2 e^{2 x}+e^{x (x+2)}\right )}+\frac {4 x^3}{\left (2 x+e^{2 x}\right ) \left (2 x^2+2 e^{x^2} x+e^{2 x} x+3 x+2 e^{2 x}+e^{x (x+2)}\right )}+\frac {4 e^{x^2} \log \left (\frac {2 e^{x^2} x+(2 x+3) x+e^{x (x+2)}+e^{2 x} (x+2)}{2 x+e^{2 x}}\right ) x^2}{\left (2 x+e^{2 x}\right ) \left (2 x^2+2 e^{x^2} x+e^{2 x} x+3 x+2 e^{2 x}+e^{x (x+2)}\right )}+\frac {4 e^{2 x} \log \left (\frac {2 e^{x^2} x+(2 x+3) x+e^{x (x+2)}+e^{2 x} (x+2)}{2 x+e^{2 x}}\right ) x^2}{\left (2 x+e^{2 x}\right ) \left (2 x^2+2 e^{x^2} x+e^{2 x} x+3 x+2 e^{2 x}+e^{x (x+2)}\right )}+\frac {6 \log \left (\frac {2 e^{x^2} x+(2 x+3) x+e^{x (x+2)}+e^{2 x} (x+2)}{2 x+e^{2 x}}\right ) x^2}{\left (2 x+e^{2 x}\right ) \left (2 x^2+2 e^{x^2} x+e^{2 x} x+3 x+2 e^{2 x}+e^{x (x+2)}\right )}+\frac {6 e^{2 x} x^2}{\left (2 x+e^{2 x}\right ) \left (2 x^2+2 e^{x^2} x+e^{2 x} x+3 x+2 e^{2 x}+e^{x (x+2)}\right )}+\frac {2 e^{x^2+4 x} x^2}{\left (2 x+e^{2 x}\right ) \left (2 x^2+2 e^{x^2} x+e^{2 x} x+3 x+2 e^{2 x}+e^{x (x+2)}\right )}+\frac {7 e^{2 x} \log \left (\frac {2 e^{x^2} x+(2 x+3) x+e^{x (x+2)}+e^{2 x} (x+2)}{2 x+e^{2 x}}\right ) x}{\left (2 x+e^{2 x}\right ) \left (2 x^2+2 e^{x^2} x+e^{2 x} x+3 x+2 e^{2 x}+e^{x (x+2)}\right )}+\frac {e^{4 x} \log \left (\frac {2 e^{x^2} x+(2 x+3) x+e^{x (x+2)}+e^{2 x} (x+2)}{2 x+e^{2 x}}\right ) x}{\left (2 x+e^{2 x}\right ) \left (2 x^2+2 e^{x^2} x+e^{2 x} x+3 x+2 e^{2 x}+e^{x (x+2)}\right )}+\frac {4 e^{x^2+2 x} \log \left (\frac {2 e^{x^2} x+(2 x+3) x+e^{x (x+2)}+e^{2 x} (x+2)}{2 x+e^{2 x}}\right ) x}{\left (2 x+e^{2 x}\right ) \left (2 x^2+2 e^{x^2} x+e^{2 x} x+3 x+2 e^{2 x}+e^{x (x+2)}\right )}-\frac {e^{2 x} x}{\left (2 x+e^{2 x}\right ) \left (2 x^2+2 e^{x^2} x+e^{2 x} x+3 x+2 e^{2 x}+e^{x (x+2)}\right )}+\frac {e^{4 x} x}{\left (2 x+e^{2 x}\right ) \left (2 x^2+2 e^{x^2} x+e^{2 x} x+3 x+2 e^{2 x}+e^{x (x+2)}\right )}+\frac {2 e^{4 x} \log \left (\frac {2 e^{x^2} x+(2 x+3) x+e^{x (x+2)}+e^{2 x} (x+2)}{2 x+e^{2 x}}\right )}{\left (2 x+e^{2 x}\right ) \left (2 x^2+2 e^{x^2} x+e^{2 x} x+3 x+2 e^{2 x}+e^{x (x+2)}\right )}+\frac {e^{x^2+4 x} \log \left (\frac {2 e^{x^2} x+(2 x+3) x+e^{x (x+2)}+e^{2 x} (x+2)}{2 x+e^{2 x}}\right )}{\left (2 x+e^{2 x}\right ) \left (2 x^2+2 e^{x^2} x+e^{2 x} x+3 x+2 e^{2 x}+e^{x (x+2)}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {2}{5} \int \left (\frac {x \left (8 e^{x^2} x^3+8 e^{x (x+2)} x^2+4 x^2+2 e^{x (x+4)} x+e^{4 x}+e^{2 x} (6 x-1)\right )}{\left (2 x+e^{2 x}\right ) \left (2 e^{x^2} x+(2 x+3) x+e^{x (x+2)}+e^{2 x} (x+2)\right )}+\log \left (\frac {2 e^{x^2} x+(2 x+3) x+e^{x (x+2)}+e^{2 x} (x+2)}{2 x+e^{2 x}}\right )\right )dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {2}{5} \int \left (\frac {x \left (8 e^{x^2} x^3+8 e^{x (x+2)} x^2+4 x^2+6 e^{2 x} x+2 e^{x (x+4)} x-e^{2 x}+e^{4 x}\right )}{\left (2 x+e^{2 x}\right ) \left (2 x^2+2 e^{x^2} x+e^{2 x} x+3 x+2 e^{2 x}+e^{x (x+2)}\right )}+\log \left (\frac {2 e^{x^2} x+(2 x+3) x+e^{x (x+2)}+e^{2 x} (x+2)}{2 x+e^{2 x}}\right )\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \frac {2}{5} \int \left (\frac {x \left (8 e^{x^2} x^3+8 e^{x (x+2)} x^2+4 x^2+6 e^{2 x} x+2 e^{x (x+4)} x-e^{2 x}+e^{4 x}\right )}{\left (2 x+e^{2 x}\right ) \left (2 x^2+2 e^{x^2} x+e^{2 x} x+3 x+2 e^{2 x}+e^{x (x+2)}\right )}+\log \left (\frac {2 e^{x^2} x+(2 x+3) x+e^{x (x+2)}+e^{2 x} (x+2)}{2 x+e^{2 x}}\right )\right )dx\) |
Input:
Int[(2*E^(4*x)*x + 8*x^3 + E^(2*x)*(-2*x + 12*x^2) + E^x^2*(4*E^(4*x)*x^2 + 16*E^(2*x)*x^3 + 16*x^4) + (12*x^2 + 8*x^3 + E^(4*x)*(4 + 2*x) + E^(2*x) *(14*x + 8*x^2) + E^x^2*(2*E^(4*x) + 8*E^(2*x)*x + 8*x^2))*Log[(3*x + 2*x^ 2 + E^(2*x)*(2 + x) + E^x^2*(E^(2*x) + 2*x))/(E^(2*x) + 2*x)])/(30*x^2 + 2 0*x^3 + E^(4*x)*(10 + 5*x) + E^(2*x)*(35*x + 20*x^2) + E^x^2*(5*E^(4*x) + 20*E^(2*x)*x + 20*x^2)),x]
Output:
$Aborted
Time = 108.69 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.64
| method | result | size |
| parallelrisch | \(\frac {2 x \ln \left (\frac {\left (2 x +{\mathrm e}^{2 x}\right ) {\mathrm e}^{x^{2}}+\left (2+x \right ) {\mathrm e}^{2 x}+2 x^{2}+3 x}{2 x +{\mathrm e}^{2 x}}\right )}{5}\) | \(46\) |
| risch | \(\frac {2 x \ln \left (x^{2}+\left (\frac {{\mathrm e}^{2 x}}{2}+{\mathrm e}^{x^{2}}+\frac {3}{2}\right ) x +\left (\frac {{\mathrm e}^{x^{2}}}{2}+1\right ) {\mathrm e}^{2 x}\right )}{5}-\frac {2 x \ln \left (\frac {{\mathrm e}^{2 x}}{2}+x \right )}{5}-\frac {i \pi x \,\operatorname {csgn}\left (\frac {i}{\frac {{\mathrm e}^{2 x}}{2}+x}\right ) \operatorname {csgn}\left (i \left (x^{2}+\left (\frac {{\mathrm e}^{2 x}}{2}+{\mathrm e}^{x^{2}}+\frac {3}{2}\right ) x +\left (\frac {{\mathrm e}^{x^{2}}}{2}+1\right ) {\mathrm e}^{2 x}\right )\right ) \operatorname {csgn}\left (\frac {i \left (x^{2}+\left (\frac {{\mathrm e}^{2 x}}{2}+{\mathrm e}^{x^{2}}+\frac {3}{2}\right ) x +\left (\frac {{\mathrm e}^{x^{2}}}{2}+1\right ) {\mathrm e}^{2 x}\right )}{\frac {{\mathrm e}^{2 x}}{2}+x}\right )}{5}+\frac {i \pi x \,\operatorname {csgn}\left (\frac {i}{\frac {{\mathrm e}^{2 x}}{2}+x}\right ) {\operatorname {csgn}\left (\frac {i \left (x^{2}+\left (\frac {{\mathrm e}^{2 x}}{2}+{\mathrm e}^{x^{2}}+\frac {3}{2}\right ) x +\left (\frac {{\mathrm e}^{x^{2}}}{2}+1\right ) {\mathrm e}^{2 x}\right )}{\frac {{\mathrm e}^{2 x}}{2}+x}\right )}^{2}}{5}+\frac {i \pi x \,\operatorname {csgn}\left (i \left (x^{2}+\left (\frac {{\mathrm e}^{2 x}}{2}+{\mathrm e}^{x^{2}}+\frac {3}{2}\right ) x +\left (\frac {{\mathrm e}^{x^{2}}}{2}+1\right ) {\mathrm e}^{2 x}\right )\right ) {\operatorname {csgn}\left (\frac {i \left (x^{2}+\left (\frac {{\mathrm e}^{2 x}}{2}+{\mathrm e}^{x^{2}}+\frac {3}{2}\right ) x +\left (\frac {{\mathrm e}^{x^{2}}}{2}+1\right ) {\mathrm e}^{2 x}\right )}{\frac {{\mathrm e}^{2 x}}{2}+x}\right )}^{2}}{5}-\frac {i \pi x {\operatorname {csgn}\left (\frac {i \left (x^{2}+\left (\frac {{\mathrm e}^{2 x}}{2}+{\mathrm e}^{x^{2}}+\frac {3}{2}\right ) x +\left (\frac {{\mathrm e}^{x^{2}}}{2}+1\right ) {\mathrm e}^{2 x}\right )}{\frac {{\mathrm e}^{2 x}}{2}+x}\right )}^{3}}{5}\) | \(353\) |
Input:
int((((2*exp(x)^4+8*x*exp(x)^2+8*x^2)*exp(x^2)+(4+2*x)*exp(x)^4+(8*x^2+14* x)*exp(x)^2+8*x^3+12*x^2)*ln(((exp(x)^2+2*x)*exp(x^2)+(2+x)*exp(x)^2+2*x^2 +3*x)/(exp(x)^2+2*x))+(4*x^2*exp(x)^4+16*exp(x)^2*x^3+16*x^4)*exp(x^2)+2*x *exp(x)^4+(12*x^2-2*x)*exp(x)^2+8*x^3)/((5*exp(x)^4+20*x*exp(x)^2+20*x^2)* exp(x^2)+(5*x+10)*exp(x)^4+(20*x^2+35*x)*exp(x)^2+20*x^3+30*x^2),x,method= _RETURNVERBOSE)
Output:
2/5*x*ln(((exp(x)^2+2*x)*exp(x^2)+(2+x)*exp(x)^2+2*x^2+3*x)/(exp(x)^2+2*x) )
Time = 0.08 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.61 \[ \int \frac {2 e^{4 x} x+8 x^3+e^{2 x} \left (-2 x+12 x^2\right )+e^{x^2} \left (4 e^{4 x} x^2+16 e^{2 x} x^3+16 x^4\right )+\left (12 x^2+8 x^3+e^{4 x} (4+2 x)+e^{2 x} \left (14 x+8 x^2\right )+e^{x^2} \left (2 e^{4 x}+8 e^{2 x} x+8 x^2\right )\right ) \log \left (\frac {3 x+2 x^2+e^{2 x} (2+x)+e^{x^2} \left (e^{2 x}+2 x\right )}{e^{2 x}+2 x}\right )}{30 x^2+20 x^3+e^{4 x} (10+5 x)+e^{2 x} \left (35 x+20 x^2\right )+e^{x^2} \left (5 e^{4 x}+20 e^{2 x} x+20 x^2\right )} \, dx=\frac {2}{5} \, x \log \left (\frac {2 \, x^{2} + {\left (2 \, x + e^{\left (2 \, x\right )}\right )} e^{\left (x^{2}\right )} + {\left (x + 2\right )} e^{\left (2 \, x\right )} + 3 \, x}{2 \, x + e^{\left (2 \, x\right )}}\right ) \] Input:
integrate((((2*exp(x)^4+8*x*exp(x)^2+8*x^2)*exp(x^2)+(4+2*x)*exp(x)^4+(8*x ^2+14*x)*exp(x)^2+8*x^3+12*x^2)*log(((exp(x)^2+2*x)*exp(x^2)+(2+x)*exp(x)^ 2+2*x^2+3*x)/(exp(x)^2+2*x))+(4*x^2*exp(x)^4+16*exp(x)^2*x^3+16*x^4)*exp(x ^2)+2*x*exp(x)^4+(12*x^2-2*x)*exp(x)^2+8*x^3)/((5*exp(x)^4+20*x*exp(x)^2+2 0*x^2)*exp(x^2)+(5*x+10)*exp(x)^4+(20*x^2+35*x)*exp(x)^2+20*x^3+30*x^2),x, algorithm="fricas")
Output:
2/5*x*log((2*x^2 + (2*x + e^(2*x))*e^(x^2) + (x + 2)*e^(2*x) + 3*x)/(2*x + e^(2*x)))
Timed out. \[ \int \frac {2 e^{4 x} x+8 x^3+e^{2 x} \left (-2 x+12 x^2\right )+e^{x^2} \left (4 e^{4 x} x^2+16 e^{2 x} x^3+16 x^4\right )+\left (12 x^2+8 x^3+e^{4 x} (4+2 x)+e^{2 x} \left (14 x+8 x^2\right )+e^{x^2} \left (2 e^{4 x}+8 e^{2 x} x+8 x^2\right )\right ) \log \left (\frac {3 x+2 x^2+e^{2 x} (2+x)+e^{x^2} \left (e^{2 x}+2 x\right )}{e^{2 x}+2 x}\right )}{30 x^2+20 x^3+e^{4 x} (10+5 x)+e^{2 x} \left (35 x+20 x^2\right )+e^{x^2} \left (5 e^{4 x}+20 e^{2 x} x+20 x^2\right )} \, dx=\text {Timed out} \] Input:
integrate((((2*exp(x)**4+8*x*exp(x)**2+8*x**2)*exp(x**2)+(4+2*x)*exp(x)**4 +(8*x**2+14*x)*exp(x)**2+8*x**3+12*x**2)*ln(((exp(x)**2+2*x)*exp(x**2)+(2+ x)*exp(x)**2+2*x**2+3*x)/(exp(x)**2+2*x))+(4*x**2*exp(x)**4+16*exp(x)**2*x **3+16*x**4)*exp(x**2)+2*x*exp(x)**4+(12*x**2-2*x)*exp(x)**2+8*x**3)/((5*e xp(x)**4+20*x*exp(x)**2+20*x**2)*exp(x**2)+(5*x+10)*exp(x)**4+(20*x**2+35* x)*exp(x)**2+20*x**3+30*x**2),x)
Output:
Timed out
Time = 0.12 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.68 \[ \int \frac {2 e^{4 x} x+8 x^3+e^{2 x} \left (-2 x+12 x^2\right )+e^{x^2} \left (4 e^{4 x} x^2+16 e^{2 x} x^3+16 x^4\right )+\left (12 x^2+8 x^3+e^{4 x} (4+2 x)+e^{2 x} \left (14 x+8 x^2\right )+e^{x^2} \left (2 e^{4 x}+8 e^{2 x} x+8 x^2\right )\right ) \log \left (\frac {3 x+2 x^2+e^{2 x} (2+x)+e^{x^2} \left (e^{2 x}+2 x\right )}{e^{2 x}+2 x}\right )}{30 x^2+20 x^3+e^{4 x} (10+5 x)+e^{2 x} \left (35 x+20 x^2\right )+e^{x^2} \left (5 e^{4 x}+20 e^{2 x} x+20 x^2\right )} \, dx=\frac {2}{5} \, x \log \left (2 \, x^{2} + {\left (2 \, x + e^{\left (2 \, x\right )}\right )} e^{\left (x^{2}\right )} + {\left (x + 2\right )} e^{\left (2 \, x\right )} + 3 \, x\right ) - \frac {2}{5} \, x \log \left (2 \, x + e^{\left (2 \, x\right )}\right ) \] Input:
integrate((((2*exp(x)^4+8*x*exp(x)^2+8*x^2)*exp(x^2)+(4+2*x)*exp(x)^4+(8*x ^2+14*x)*exp(x)^2+8*x^3+12*x^2)*log(((exp(x)^2+2*x)*exp(x^2)+(2+x)*exp(x)^ 2+2*x^2+3*x)/(exp(x)^2+2*x))+(4*x^2*exp(x)^4+16*exp(x)^2*x^3+16*x^4)*exp(x ^2)+2*x*exp(x)^4+(12*x^2-2*x)*exp(x)^2+8*x^3)/((5*exp(x)^4+20*x*exp(x)^2+2 0*x^2)*exp(x^2)+(5*x+10)*exp(x)^4+(20*x^2+35*x)*exp(x)^2+20*x^3+30*x^2),x, algorithm="maxima")
Output:
2/5*x*log(2*x^2 + (2*x + e^(2*x))*e^(x^2) + (x + 2)*e^(2*x) + 3*x) - 2/5*x *log(2*x + e^(2*x))
Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (24) = 48\).
Time = 0.38 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.82 \[ \int \frac {2 e^{4 x} x+8 x^3+e^{2 x} \left (-2 x+12 x^2\right )+e^{x^2} \left (4 e^{4 x} x^2+16 e^{2 x} x^3+16 x^4\right )+\left (12 x^2+8 x^3+e^{4 x} (4+2 x)+e^{2 x} \left (14 x+8 x^2\right )+e^{x^2} \left (2 e^{4 x}+8 e^{2 x} x+8 x^2\right )\right ) \log \left (\frac {3 x+2 x^2+e^{2 x} (2+x)+e^{x^2} \left (e^{2 x}+2 x\right )}{e^{2 x}+2 x}\right )}{30 x^2+20 x^3+e^{4 x} (10+5 x)+e^{2 x} \left (35 x+20 x^2\right )+e^{x^2} \left (5 e^{4 x}+20 e^{2 x} x+20 x^2\right )} \, dx=\frac {2}{5} \, x \log \left (\frac {2 \, x^{2} + 2 \, x e^{\left (x^{2}\right )} + x e^{\left (2 \, x\right )} + 3 \, x + e^{\left (x^{2} + 2 \, x\right )} + 2 \, e^{\left (2 \, x\right )}}{2 \, x + e^{\left (2 \, x\right )}}\right ) \] Input:
integrate((((2*exp(x)^4+8*x*exp(x)^2+8*x^2)*exp(x^2)+(4+2*x)*exp(x)^4+(8*x ^2+14*x)*exp(x)^2+8*x^3+12*x^2)*log(((exp(x)^2+2*x)*exp(x^2)+(2+x)*exp(x)^ 2+2*x^2+3*x)/(exp(x)^2+2*x))+(4*x^2*exp(x)^4+16*exp(x)^2*x^3+16*x^4)*exp(x ^2)+2*x*exp(x)^4+(12*x^2-2*x)*exp(x)^2+8*x^3)/((5*exp(x)^4+20*x*exp(x)^2+2 0*x^2)*exp(x^2)+(5*x+10)*exp(x)^4+(20*x^2+35*x)*exp(x)^2+20*x^3+30*x^2),x, algorithm="giac")
Output:
2/5*x*log((2*x^2 + 2*x*e^(x^2) + x*e^(2*x) + 3*x + e^(x^2 + 2*x) + 2*e^(2* x))/(2*x + e^(2*x)))
Time = 1.77 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.61 \[ \int \frac {2 e^{4 x} x+8 x^3+e^{2 x} \left (-2 x+12 x^2\right )+e^{x^2} \left (4 e^{4 x} x^2+16 e^{2 x} x^3+16 x^4\right )+\left (12 x^2+8 x^3+e^{4 x} (4+2 x)+e^{2 x} \left (14 x+8 x^2\right )+e^{x^2} \left (2 e^{4 x}+8 e^{2 x} x+8 x^2\right )\right ) \log \left (\frac {3 x+2 x^2+e^{2 x} (2+x)+e^{x^2} \left (e^{2 x}+2 x\right )}{e^{2 x}+2 x}\right )}{30 x^2+20 x^3+e^{4 x} (10+5 x)+e^{2 x} \left (35 x+20 x^2\right )+e^{x^2} \left (5 e^{4 x}+20 e^{2 x} x+20 x^2\right )} \, dx=\frac {2\,x\,\ln \left (\frac {3\,x+{\mathrm {e}}^{x^2}\,\left (2\,x+{\mathrm {e}}^{2\,x}\right )+{\mathrm {e}}^{2\,x}\,\left (x+2\right )+2\,x^2}{2\,x+{\mathrm {e}}^{2\,x}}\right )}{5} \] Input:
int((2*x*exp(4*x) - exp(2*x)*(2*x - 12*x^2) + exp(x^2)*(16*x^3*exp(2*x) + 4*x^2*exp(4*x) + 16*x^4) + log((3*x + exp(x^2)*(2*x + exp(2*x)) + exp(2*x) *(x + 2) + 2*x^2)/(2*x + exp(2*x)))*(exp(2*x)*(14*x + 8*x^2) + exp(4*x)*(2 *x + 4) + 12*x^2 + 8*x^3 + exp(x^2)*(2*exp(4*x) + 8*x*exp(2*x) + 8*x^2)) + 8*x^3)/(exp(2*x)*(35*x + 20*x^2) + exp(4*x)*(5*x + 10) + 30*x^2 + 20*x^3 + exp(x^2)*(5*exp(4*x) + 20*x*exp(2*x) + 20*x^2)),x)
Output:
(2*x*log((3*x + exp(x^2)*(2*x + exp(2*x)) + exp(2*x)*(x + 2) + 2*x^2)/(2*x + exp(2*x))))/5
\[ \int \frac {2 e^{4 x} x+8 x^3+e^{2 x} \left (-2 x+12 x^2\right )+e^{x^2} \left (4 e^{4 x} x^2+16 e^{2 x} x^3+16 x^4\right )+\left (12 x^2+8 x^3+e^{4 x} (4+2 x)+e^{2 x} \left (14 x+8 x^2\right )+e^{x^2} \left (2 e^{4 x}+8 e^{2 x} x+8 x^2\right )\right ) \log \left (\frac {3 x+2 x^2+e^{2 x} (2+x)+e^{x^2} \left (e^{2 x}+2 x\right )}{e^{2 x}+2 x}\right )}{30 x^2+20 x^3+e^{4 x} (10+5 x)+e^{2 x} \left (35 x+20 x^2\right )+e^{x^2} \left (5 e^{4 x}+20 e^{2 x} x+20 x^2\right )} \, dx=\int \frac {\left (\left (2 \left ({\mathrm e}^{x}\right )^{4}+8 x \left ({\mathrm e}^{x}\right )^{2}+8 x^{2}\right ) {\mathrm e}^{x^{2}}+\left (4+2 x \right ) \left ({\mathrm e}^{x}\right )^{4}+\left (8 x^{2}+14 x \right ) \left ({\mathrm e}^{x}\right )^{2}+8 x^{3}+12 x^{2}\right ) \mathrm {log}\left (\frac {\left (\left ({\mathrm e}^{x}\right )^{2}+2 x \right ) {\mathrm e}^{x^{2}}+\left (x +2\right ) \left ({\mathrm e}^{x}\right )^{2}+2 x^{2}+3 x}{\left ({\mathrm e}^{x}\right )^{2}+2 x}\right )+\left (4 x^{2} \left ({\mathrm e}^{x}\right )^{4}+16 \left ({\mathrm e}^{x}\right )^{2} x^{3}+16 x^{4}\right ) {\mathrm e}^{x^{2}}+2 x \left ({\mathrm e}^{x}\right )^{4}+\left (12 x^{2}-2 x \right ) \left ({\mathrm e}^{x}\right )^{2}+8 x^{3}}{\left (5 \left ({\mathrm e}^{x}\right )^{4}+20 x \left ({\mathrm e}^{x}\right )^{2}+20 x^{2}\right ) {\mathrm e}^{x^{2}}+\left (5 x +10\right ) \left ({\mathrm e}^{x}\right )^{4}+\left (20 x^{2}+35 x \right ) \left ({\mathrm e}^{x}\right )^{2}+20 x^{3}+30 x^{2}}d x \] Input:
int((((2*exp(x)^4+8*x*exp(x)^2+8*x^2)*exp(x^2)+(4+2*x)*exp(x)^4+(8*x^2+14* x)*exp(x)^2+8*x^3+12*x^2)*log(((exp(x)^2+2*x)*exp(x^2)+(2+x)*exp(x)^2+2*x^ 2+3*x)/(exp(x)^2+2*x))+(4*x^2*exp(x)^4+16*exp(x)^2*x^3+16*x^4)*exp(x^2)+2* x*exp(x)^4+(12*x^2-2*x)*exp(x)^2+8*x^3)/((5*exp(x)^4+20*x*exp(x)^2+20*x^2) *exp(x^2)+(5*x+10)*exp(x)^4+(20*x^2+35*x)*exp(x)^2+20*x^3+30*x^2),x)
Output:
int((((2*exp(x)^4+8*x*exp(x)^2+8*x^2)*exp(x^2)+(4+2*x)*exp(x)^4+(8*x^2+14* x)*exp(x)^2+8*x^3+12*x^2)*log(((exp(x)^2+2*x)*exp(x^2)+(2+x)*exp(x)^2+2*x^ 2+3*x)/(exp(x)^2+2*x))+(4*x^2*exp(x)^4+16*exp(x)^2*x^3+16*x^4)*exp(x^2)+2* x*exp(x)^4+(12*x^2-2*x)*exp(x)^2+8*x^3)/((5*exp(x)^4+20*x*exp(x)^2+20*x^2) *exp(x^2)+(5*x+10)*exp(x)^4+(20*x^2+35*x)*exp(x)^2+20*x^3+30*x^2),x)