Integrand size = 153, antiderivative size = 36 \[ \int \frac {48+e^{2 e^{\frac {x^2}{2}}} \left (4+e^{3+x} (-1+x)\right )+e^{3+x} (-12+12 x)+e^{e^{\frac {x^2}{2}}} \left (-28+e^{3+x} (7-7 x)+e^{\frac {x^2}{2}} \left (4 x^2-e^{3+x} x^2\right )\right )}{-48 x+12 e^{3+x} x+e^{e^{\frac {x^2}{2}}} \left (28 x-7 e^{3+x} x\right )+e^{2 e^{\frac {x^2}{2}}} \left (-4 x+e^{3+x} x\right )} \, dx=-2+\log \left (\frac {4-e^{3+x}}{x+\frac {x}{3-e^{e^{\frac {x^2}{2}}}}}\right ) \] Output:
-2+ln((4-exp(3+x))/(x/(3-exp(exp(1/2*x^2)))+x))
Time = 0.09 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.36 \[ \int \frac {48+e^{2 e^{\frac {x^2}{2}}} \left (4+e^{3+x} (-1+x)\right )+e^{3+x} (-12+12 x)+e^{e^{\frac {x^2}{2}}} \left (-28+e^{3+x} (7-7 x)+e^{\frac {x^2}{2}} \left (4 x^2-e^{3+x} x^2\right )\right )}{-48 x+12 e^{3+x} x+e^{e^{\frac {x^2}{2}}} \left (28 x-7 e^{3+x} x\right )+e^{2 e^{\frac {x^2}{2}}} \left (-4 x+e^{3+x} x\right )} \, dx=\log \left (3-e^{e^{\frac {x^2}{2}}}\right )-\log \left (4-e^{e^{\frac {x^2}{2}}}\right )+\log \left (4-e^{3+x}\right )-\log (x) \] Input:
Integrate[(48 + E^(2*E^(x^2/2))*(4 + E^(3 + x)*(-1 + x)) + E^(3 + x)*(-12 + 12*x) + E^E^(x^2/2)*(-28 + E^(3 + x)*(7 - 7*x) + E^(x^2/2)*(4*x^2 - E^(3 + x)*x^2)))/(-48*x + 12*E^(3 + x)*x + E^E^(x^2/2)*(28*x - 7*E^(3 + x)*x) + E^(2*E^(x^2/2))*(-4*x + E^(3 + x)*x)),x]
Output:
Log[3 - E^E^(x^2/2)] - Log[4 - E^E^(x^2/2)] + Log[4 - E^(3 + x)] - Log[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.
| \(\displaystyle \int \frac {e^{2 e^{\frac {x^2}{2}}} \left (e^{x+3} (x-1)+4\right )+e^{e^{\frac {x^2}{2}}} \left (e^{\frac {x^2}{2}} \left (4 x^2-e^{x+3} x^2\right )+e^{x+3} (7-7 x)-28\right )+e^{x+3} (12 x-12)+48}{e^{e^{\frac {x^2}{2}}} \left (28 x-7 e^{x+3} x\right )+e^{2 e^{\frac {x^2}{2}}} \left (e^{x+3} x-4 x\right )+12 e^{x+3} x-48 x} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {-e^{2 e^{\frac {x^2}{2}}} \left (e^{x+3} (x-1)+4\right )-e^{e^{\frac {x^2}{2}}} \left (e^{\frac {x^2}{2}} \left (4 x^2-e^{x+3} x^2\right )+e^{x+3} (7-7 x)-28\right )-e^{x+3} (12 x-12)-48}{\left (-7 e^{e^{\frac {x^2}{2}}}+e^{2 e^{\frac {x^2}{2}}}+12\right ) \left (4-e^{x+3}\right ) x}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {12 e^{x+3} (x-1)}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}-\frac {7 e^{e^{\frac {x^2}{2}}+x+3} (x-1)}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}+\frac {e^{\frac {1}{2} \left (x^2+2 e^{\frac {x^2}{2}}\right )} x}{7 e^{e^{\frac {x^2}{2}}}-e^{2 e^{\frac {x^2}{2}}}-12}+\frac {e^{2 e^{\frac {x^2}{2}}} \left (e^{x+3} x-e^{x+3}+4\right )}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}-\frac {28 e^{e^{\frac {x^2}{2}}}}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}+\frac {48}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {-4 e^{\frac {x^2}{2}+e^{\frac {x^2}{2}}} x^2+e^{\frac {x^2}{2}+e^{\frac {x^2}{2}}+x+3} x^2+28 e^{e^{\frac {x^2}{2}}}-4 e^{2 e^{\frac {x^2}{2}}}+7 e^{e^{\frac {x^2}{2}}+x+3} (x-1)-e^{2 e^{\frac {x^2}{2}}+x+3} (x-1)-12 e^{x+3} (x-1)-48}{\left (-7 e^{e^{\frac {x^2}{2}}}+e^{2 e^{\frac {x^2}{2}}}+12\right ) \left (4-e^{x+3}\right ) x}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {12 e^{x+3} (x-1)}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}-\frac {7 e^{e^{\frac {x^2}{2}}+x+3} (x-1)}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}+\frac {e^{2 e^{\frac {x^2}{2}}+x+3} (x-1)}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}+\frac {e^{\frac {1}{2} \left (x^2+2 e^{\frac {x^2}{2}}\right )} x}{7 e^{e^{\frac {x^2}{2}}}-e^{2 e^{\frac {x^2}{2}}}-12}-\frac {28 e^{e^{\frac {x^2}{2}}}}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}+\frac {4 e^{2 e^{\frac {x^2}{2}}}}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}+\frac {48}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {-4 e^{\frac {x^2}{2}+e^{\frac {x^2}{2}}} x^2+e^{\frac {x^2}{2}+e^{\frac {x^2}{2}}+x+3} x^2+28 e^{e^{\frac {x^2}{2}}}-4 e^{2 e^{\frac {x^2}{2}}}+7 e^{e^{\frac {x^2}{2}}+x+3} (x-1)-e^{2 e^{\frac {x^2}{2}}+x+3} (x-1)-12 e^{x+3} (x-1)-48}{\left (-7 e^{e^{\frac {x^2}{2}}}+e^{2 e^{\frac {x^2}{2}}}+12\right ) \left (4-e^{x+3}\right ) x}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {12 e^{x+3} (x-1)}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}-\frac {7 e^{e^{\frac {x^2}{2}}+x+3} (x-1)}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}+\frac {e^{2 e^{\frac {x^2}{2}}+x+3} (x-1)}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}+\frac {e^{\frac {1}{2} \left (x^2+2 e^{\frac {x^2}{2}}\right )} x}{7 e^{e^{\frac {x^2}{2}}}-e^{2 e^{\frac {x^2}{2}}}-12}-\frac {28 e^{e^{\frac {x^2}{2}}}}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}+\frac {4 e^{2 e^{\frac {x^2}{2}}}}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}+\frac {48}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {-4 e^{\frac {x^2}{2}+e^{\frac {x^2}{2}}} x^2+e^{\frac {x^2}{2}+e^{\frac {x^2}{2}}+x+3} x^2+28 e^{e^{\frac {x^2}{2}}}-4 e^{2 e^{\frac {x^2}{2}}}+7 e^{e^{\frac {x^2}{2}}+x+3} (x-1)-e^{2 e^{\frac {x^2}{2}}+x+3} (x-1)-12 e^{x+3} (x-1)-48}{\left (-7 e^{e^{\frac {x^2}{2}}}+e^{2 e^{\frac {x^2}{2}}}+12\right ) \left (4-e^{x+3}\right ) x}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {12 e^{x+3} (x-1)}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}-\frac {7 e^{e^{\frac {x^2}{2}}+x+3} (x-1)}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}+\frac {e^{2 e^{\frac {x^2}{2}}+x+3} (x-1)}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}+\frac {e^{\frac {1}{2} \left (x^2+2 e^{\frac {x^2}{2}}\right )} x}{7 e^{e^{\frac {x^2}{2}}}-e^{2 e^{\frac {x^2}{2}}}-12}-\frac {28 e^{e^{\frac {x^2}{2}}}}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}+\frac {4 e^{2 e^{\frac {x^2}{2}}}}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}+\frac {48}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {-4 e^{\frac {x^2}{2}+e^{\frac {x^2}{2}}} x^2+e^{\frac {x^2}{2}+e^{\frac {x^2}{2}}+x+3} x^2+28 e^{e^{\frac {x^2}{2}}}-4 e^{2 e^{\frac {x^2}{2}}}+7 e^{e^{\frac {x^2}{2}}+x+3} (x-1)-e^{2 e^{\frac {x^2}{2}}+x+3} (x-1)-12 e^{x+3} (x-1)-48}{\left (-7 e^{e^{\frac {x^2}{2}}}+e^{2 e^{\frac {x^2}{2}}}+12\right ) \left (4-e^{x+3}\right ) x}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {12 e^{x+3} (x-1)}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}-\frac {7 e^{e^{\frac {x^2}{2}}+x+3} (x-1)}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}+\frac {e^{2 e^{\frac {x^2}{2}}+x+3} (x-1)}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}+\frac {e^{\frac {1}{2} \left (x^2+2 e^{\frac {x^2}{2}}\right )} x}{7 e^{e^{\frac {x^2}{2}}}-e^{2 e^{\frac {x^2}{2}}}-12}-\frac {28 e^{e^{\frac {x^2}{2}}}}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}+\frac {4 e^{2 e^{\frac {x^2}{2}}}}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}+\frac {48}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {-4 e^{\frac {x^2}{2}+e^{\frac {x^2}{2}}} x^2+e^{\frac {x^2}{2}+e^{\frac {x^2}{2}}+x+3} x^2+28 e^{e^{\frac {x^2}{2}}}-4 e^{2 e^{\frac {x^2}{2}}}+7 e^{e^{\frac {x^2}{2}}+x+3} (x-1)-e^{2 e^{\frac {x^2}{2}}+x+3} (x-1)-12 e^{x+3} (x-1)-48}{\left (-7 e^{e^{\frac {x^2}{2}}}+e^{2 e^{\frac {x^2}{2}}}+12\right ) \left (4-e^{x+3}\right ) x}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {12 e^{x+3} (x-1)}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}-\frac {7 e^{e^{\frac {x^2}{2}}+x+3} (x-1)}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}+\frac {e^{2 e^{\frac {x^2}{2}}+x+3} (x-1)}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}+\frac {e^{\frac {1}{2} \left (x^2+2 e^{\frac {x^2}{2}}\right )} x}{7 e^{e^{\frac {x^2}{2}}}-e^{2 e^{\frac {x^2}{2}}}-12}-\frac {28 e^{e^{\frac {x^2}{2}}}}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}+\frac {4 e^{2 e^{\frac {x^2}{2}}}}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}+\frac {48}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {-4 e^{\frac {x^2}{2}+e^{\frac {x^2}{2}}} x^2+e^{\frac {x^2}{2}+e^{\frac {x^2}{2}}+x+3} x^2+28 e^{e^{\frac {x^2}{2}}}-4 e^{2 e^{\frac {x^2}{2}}}+7 e^{e^{\frac {x^2}{2}}+x+3} (x-1)-e^{2 e^{\frac {x^2}{2}}+x+3} (x-1)-12 e^{x+3} (x-1)-48}{\left (-7 e^{e^{\frac {x^2}{2}}}+e^{2 e^{\frac {x^2}{2}}}+12\right ) \left (4-e^{x+3}\right ) x}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {12 e^{x+3} (x-1)}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}-\frac {7 e^{e^{\frac {x^2}{2}}+x+3} (x-1)}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}+\frac {e^{2 e^{\frac {x^2}{2}}+x+3} (x-1)}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}+\frac {e^{\frac {1}{2} \left (x^2+2 e^{\frac {x^2}{2}}\right )} x}{7 e^{e^{\frac {x^2}{2}}}-e^{2 e^{\frac {x^2}{2}}}-12}-\frac {28 e^{e^{\frac {x^2}{2}}}}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}+\frac {4 e^{2 e^{\frac {x^2}{2}}}}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}+\frac {48}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {-4 e^{\frac {x^2}{2}+e^{\frac {x^2}{2}}} x^2+e^{\frac {x^2}{2}+e^{\frac {x^2}{2}}+x+3} x^2+28 e^{e^{\frac {x^2}{2}}}-4 e^{2 e^{\frac {x^2}{2}}}+7 e^{e^{\frac {x^2}{2}}+x+3} (x-1)-e^{2 e^{\frac {x^2}{2}}+x+3} (x-1)-12 e^{x+3} (x-1)-48}{\left (-7 e^{e^{\frac {x^2}{2}}}+e^{2 e^{\frac {x^2}{2}}}+12\right ) \left (4-e^{x+3}\right ) x}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {12 e^{x+3} (x-1)}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}-\frac {7 e^{e^{\frac {x^2}{2}}+x+3} (x-1)}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}+\frac {e^{2 e^{\frac {x^2}{2}}+x+3} (x-1)}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}+\frac {e^{\frac {1}{2} \left (x^2+2 e^{\frac {x^2}{2}}\right )} x}{7 e^{e^{\frac {x^2}{2}}}-e^{2 e^{\frac {x^2}{2}}}-12}-\frac {28 e^{e^{\frac {x^2}{2}}}}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}+\frac {4 e^{2 e^{\frac {x^2}{2}}}}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}+\frac {48}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {-4 e^{\frac {x^2}{2}+e^{\frac {x^2}{2}}} x^2+e^{\frac {x^2}{2}+e^{\frac {x^2}{2}}+x+3} x^2+28 e^{e^{\frac {x^2}{2}}}-4 e^{2 e^{\frac {x^2}{2}}}+7 e^{e^{\frac {x^2}{2}}+x+3} (x-1)-e^{2 e^{\frac {x^2}{2}}+x+3} (x-1)-12 e^{x+3} (x-1)-48}{\left (-7 e^{e^{\frac {x^2}{2}}}+e^{2 e^{\frac {x^2}{2}}}+12\right ) \left (4-e^{x+3}\right ) x}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {12 e^{x+3} (x-1)}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}-\frac {7 e^{e^{\frac {x^2}{2}}+x+3} (x-1)}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}+\frac {e^{2 e^{\frac {x^2}{2}}+x+3} (x-1)}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}+\frac {e^{\frac {1}{2} \left (x^2+2 e^{\frac {x^2}{2}}\right )} x}{7 e^{e^{\frac {x^2}{2}}}-e^{2 e^{\frac {x^2}{2}}}-12}-\frac {28 e^{e^{\frac {x^2}{2}}}}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}+\frac {4 e^{2 e^{\frac {x^2}{2}}}}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}+\frac {48}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {-4 e^{\frac {x^2}{2}+e^{\frac {x^2}{2}}} x^2+e^{\frac {x^2}{2}+e^{\frac {x^2}{2}}+x+3} x^2+28 e^{e^{\frac {x^2}{2}}}-4 e^{2 e^{\frac {x^2}{2}}}+7 e^{e^{\frac {x^2}{2}}+x+3} (x-1)-e^{2 e^{\frac {x^2}{2}}+x+3} (x-1)-12 e^{x+3} (x-1)-48}{\left (-7 e^{e^{\frac {x^2}{2}}}+e^{2 e^{\frac {x^2}{2}}}+12\right ) \left (4-e^{x+3}\right ) x}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {12 e^{x+3} (x-1)}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}-\frac {7 e^{e^{\frac {x^2}{2}}+x+3} (x-1)}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}+\frac {e^{2 e^{\frac {x^2}{2}}+x+3} (x-1)}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}+\frac {e^{\frac {1}{2} \left (x^2+2 e^{\frac {x^2}{2}}\right )} x}{7 e^{e^{\frac {x^2}{2}}}-e^{2 e^{\frac {x^2}{2}}}-12}-\frac {28 e^{e^{\frac {x^2}{2}}}}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}+\frac {4 e^{2 e^{\frac {x^2}{2}}}}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}+\frac {48}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {-4 e^{\frac {x^2}{2}+e^{\frac {x^2}{2}}} x^2+e^{\frac {x^2}{2}+e^{\frac {x^2}{2}}+x+3} x^2+28 e^{e^{\frac {x^2}{2}}}-4 e^{2 e^{\frac {x^2}{2}}}+7 e^{e^{\frac {x^2}{2}}+x+3} (x-1)-e^{2 e^{\frac {x^2}{2}}+x+3} (x-1)-12 e^{x+3} (x-1)-48}{\left (-7 e^{e^{\frac {x^2}{2}}}+e^{2 e^{\frac {x^2}{2}}}+12\right ) \left (4-e^{x+3}\right ) x}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {12 e^{x+3} (x-1)}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}-\frac {7 e^{e^{\frac {x^2}{2}}+x+3} (x-1)}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}+\frac {e^{2 e^{\frac {x^2}{2}}+x+3} (x-1)}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}+\frac {e^{\frac {1}{2} \left (x^2+2 e^{\frac {x^2}{2}}\right )} x}{7 e^{e^{\frac {x^2}{2}}}-e^{2 e^{\frac {x^2}{2}}}-12}-\frac {28 e^{e^{\frac {x^2}{2}}}}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}+\frac {4 e^{2 e^{\frac {x^2}{2}}}}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}+\frac {48}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {-4 e^{\frac {x^2}{2}+e^{\frac {x^2}{2}}} x^2+e^{\frac {x^2}{2}+e^{\frac {x^2}{2}}+x+3} x^2+28 e^{e^{\frac {x^2}{2}}}-4 e^{2 e^{\frac {x^2}{2}}}+7 e^{e^{\frac {x^2}{2}}+x+3} (x-1)-e^{2 e^{\frac {x^2}{2}}+x+3} (x-1)-12 e^{x+3} (x-1)-48}{\left (-7 e^{e^{\frac {x^2}{2}}}+e^{2 e^{\frac {x^2}{2}}}+12\right ) \left (4-e^{x+3}\right ) x}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {12 e^{x+3} (x-1)}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}-\frac {7 e^{e^{\frac {x^2}{2}}+x+3} (x-1)}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}+\frac {e^{2 e^{\frac {x^2}{2}}+x+3} (x-1)}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}+\frac {e^{\frac {1}{2} \left (x^2+2 e^{\frac {x^2}{2}}\right )} x}{7 e^{e^{\frac {x^2}{2}}}-e^{2 e^{\frac {x^2}{2}}}-12}-\frac {28 e^{e^{\frac {x^2}{2}}}}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}+\frac {4 e^{2 e^{\frac {x^2}{2}}}}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}+\frac {48}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {-4 e^{\frac {x^2}{2}+e^{\frac {x^2}{2}}} x^2+e^{\frac {x^2}{2}+e^{\frac {x^2}{2}}+x+3} x^2+28 e^{e^{\frac {x^2}{2}}}-4 e^{2 e^{\frac {x^2}{2}}}+7 e^{e^{\frac {x^2}{2}}+x+3} (x-1)-e^{2 e^{\frac {x^2}{2}}+x+3} (x-1)-12 e^{x+3} (x-1)-48}{\left (-7 e^{e^{\frac {x^2}{2}}}+e^{2 e^{\frac {x^2}{2}}}+12\right ) \left (4-e^{x+3}\right ) x}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {12 e^{x+3} (x-1)}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}-\frac {7 e^{e^{\frac {x^2}{2}}+x+3} (x-1)}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}+\frac {e^{2 e^{\frac {x^2}{2}}+x+3} (x-1)}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}+\frac {e^{\frac {1}{2} \left (x^2+2 e^{\frac {x^2}{2}}\right )} x}{7 e^{e^{\frac {x^2}{2}}}-e^{2 e^{\frac {x^2}{2}}}-12}-\frac {28 e^{e^{\frac {x^2}{2}}}}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}+\frac {4 e^{2 e^{\frac {x^2}{2}}}}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}+\frac {48}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {-4 e^{\frac {x^2}{2}+e^{\frac {x^2}{2}}} x^2+e^{\frac {x^2}{2}+e^{\frac {x^2}{2}}+x+3} x^2+28 e^{e^{\frac {x^2}{2}}}-4 e^{2 e^{\frac {x^2}{2}}}+7 e^{e^{\frac {x^2}{2}}+x+3} (x-1)-e^{2 e^{\frac {x^2}{2}}+x+3} (x-1)-12 e^{x+3} (x-1)-48}{\left (-7 e^{e^{\frac {x^2}{2}}}+e^{2 e^{\frac {x^2}{2}}}+12\right ) \left (4-e^{x+3}\right ) x}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {12 e^{x+3} (x-1)}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}-\frac {7 e^{e^{\frac {x^2}{2}}+x+3} (x-1)}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}+\frac {e^{2 e^{\frac {x^2}{2}}+x+3} (x-1)}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}+\frac {e^{\frac {1}{2} \left (x^2+2 e^{\frac {x^2}{2}}\right )} x}{7 e^{e^{\frac {x^2}{2}}}-e^{2 e^{\frac {x^2}{2}}}-12}-\frac {28 e^{e^{\frac {x^2}{2}}}}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}+\frac {4 e^{2 e^{\frac {x^2}{2}}}}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}+\frac {48}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {-4 e^{\frac {x^2}{2}+e^{\frac {x^2}{2}}} x^2+e^{\frac {x^2}{2}+e^{\frac {x^2}{2}}+x+3} x^2+28 e^{e^{\frac {x^2}{2}}}-4 e^{2 e^{\frac {x^2}{2}}}+7 e^{e^{\frac {x^2}{2}}+x+3} (x-1)-e^{2 e^{\frac {x^2}{2}}+x+3} (x-1)-12 e^{x+3} (x-1)-48}{\left (-7 e^{e^{\frac {x^2}{2}}}+e^{2 e^{\frac {x^2}{2}}}+12\right ) \left (4-e^{x+3}\right ) x}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {12 e^{x+3} (x-1)}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}-\frac {7 e^{e^{\frac {x^2}{2}}+x+3} (x-1)}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}+\frac {e^{2 e^{\frac {x^2}{2}}+x+3} (x-1)}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}+\frac {e^{\frac {1}{2} \left (x^2+2 e^{\frac {x^2}{2}}\right )} x}{7 e^{e^{\frac {x^2}{2}}}-e^{2 e^{\frac {x^2}{2}}}-12}-\frac {28 e^{e^{\frac {x^2}{2}}}}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}+\frac {4 e^{2 e^{\frac {x^2}{2}}}}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}+\frac {48}{\left (e^{e^{\frac {x^2}{2}}}-4\right ) \left (e^{e^{\frac {x^2}{2}}}-3\right ) \left (e^{x+3}-4\right ) x}\right )dx\) |
Input:
Int[(48 + E^(2*E^(x^2/2))*(4 + E^(3 + x)*(-1 + x)) + E^(3 + x)*(-12 + 12*x ) + E^E^(x^2/2)*(-28 + E^(3 + x)*(7 - 7*x) + E^(x^2/2)*(4*x^2 - E^(3 + x)* x^2)))/(-48*x + 12*E^(3 + x)*x + E^E^(x^2/2)*(28*x - 7*E^(3 + x)*x) + E^(2 *E^(x^2/2))*(-4*x + E^(3 + x)*x)),x]
Output:
$Aborted
Time = 1.18 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.97
| method | result | size |
| parallelrisch | \(-\ln \left (x \right )+\ln \left ({\mathrm e}^{3+x}-4\right )-\ln \left ({\mathrm e}^{{\mathrm e}^{\frac {x^{2}}{2}}}-4\right )+\ln \left ({\mathrm e}^{{\mathrm e}^{\frac {x^{2}}{2}}}-3\right )\) | \(35\) |
| risch | \(-\ln \left (x \right )-3+\ln \left ({\mathrm e}^{3+x}-4\right )+\ln \left ({\mathrm e}^{{\mathrm e}^{\frac {x^{2}}{2}}}-3\right )-\ln \left ({\mathrm e}^{{\mathrm e}^{\frac {x^{2}}{2}}}-4\right )\) | \(36\) |
Input:
int((((-1+x)*exp(3+x)+4)*exp(exp(1/2*x^2))^2+((-x^2*exp(3+x)+4*x^2)*exp(1/ 2*x^2)+(-7*x+7)*exp(3+x)-28)*exp(exp(1/2*x^2))+(12*x-12)*exp(3+x)+48)/((ex p(3+x)*x-4*x)*exp(exp(1/2*x^2))^2+(-7*exp(3+x)*x+28*x)*exp(exp(1/2*x^2))+1 2*exp(3+x)*x-48*x),x,method=_RETURNVERBOSE)
Output:
-ln(x)+ln(exp(3+x)-4)-ln(exp(exp(1/2*x^2))-4)+ln(exp(exp(1/2*x^2))-3)
Time = 0.10 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.94 \[ \int \frac {48+e^{2 e^{\frac {x^2}{2}}} \left (4+e^{3+x} (-1+x)\right )+e^{3+x} (-12+12 x)+e^{e^{\frac {x^2}{2}}} \left (-28+e^{3+x} (7-7 x)+e^{\frac {x^2}{2}} \left (4 x^2-e^{3+x} x^2\right )\right )}{-48 x+12 e^{3+x} x+e^{e^{\frac {x^2}{2}}} \left (28 x-7 e^{3+x} x\right )+e^{2 e^{\frac {x^2}{2}}} \left (-4 x+e^{3+x} x\right )} \, dx=-\log \left (x\right ) + \log \left (e^{\left (x + 3\right )} - 4\right ) + \log \left (e^{\left (e^{\left (\frac {1}{2} \, x^{2}\right )}\right )} - 3\right ) - \log \left (e^{\left (e^{\left (\frac {1}{2} \, x^{2}\right )}\right )} - 4\right ) \] Input:
integrate((((-1+x)*exp(3+x)+4)*exp(exp(1/2*x^2))^2+((-x^2*exp(3+x)+4*x^2)* exp(1/2*x^2)+(-7*x+7)*exp(3+x)-28)*exp(exp(1/2*x^2))+(12*x-12)*exp(3+x)+48 )/((exp(3+x)*x-4*x)*exp(exp(1/2*x^2))^2+(-7*exp(3+x)*x+28*x)*exp(exp(1/2*x ^2))+12*exp(3+x)*x-48*x),x, algorithm="fricas")
Output:
-log(x) + log(e^(x + 3) - 4) + log(e^(e^(1/2*x^2)) - 3) - log(e^(e^(1/2*x^ 2)) - 4)
Time = 0.18 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.94 \[ \int \frac {48+e^{2 e^{\frac {x^2}{2}}} \left (4+e^{3+x} (-1+x)\right )+e^{3+x} (-12+12 x)+e^{e^{\frac {x^2}{2}}} \left (-28+e^{3+x} (7-7 x)+e^{\frac {x^2}{2}} \left (4 x^2-e^{3+x} x^2\right )\right )}{-48 x+12 e^{3+x} x+e^{e^{\frac {x^2}{2}}} \left (28 x-7 e^{3+x} x\right )+e^{2 e^{\frac {x^2}{2}}} \left (-4 x+e^{3+x} x\right )} \, dx=- \log {\left (x \right )} + \log {\left (e^{x + 3} - 4 \right )} - \log {\left (e^{e^{\frac {x^{2}}{2}}} - 4 \right )} + \log {\left (e^{e^{\frac {x^{2}}{2}}} - 3 \right )} \] Input:
integrate((((-1+x)*exp(3+x)+4)*exp(exp(1/2*x**2))**2+((-x**2*exp(3+x)+4*x* *2)*exp(1/2*x**2)+(-7*x+7)*exp(3+x)-28)*exp(exp(1/2*x**2))+(12*x-12)*exp(3 +x)+48)/((exp(3+x)*x-4*x)*exp(exp(1/2*x**2))**2+(-7*exp(3+x)*x+28*x)*exp(e xp(1/2*x**2))+12*exp(3+x)*x-48*x),x)
Output:
-log(x) + log(exp(x + 3) - 4) - log(exp(exp(x**2/2)) - 4) + log(exp(exp(x* *2/2)) - 3)
Time = 0.10 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.03 \[ \int \frac {48+e^{2 e^{\frac {x^2}{2}}} \left (4+e^{3+x} (-1+x)\right )+e^{3+x} (-12+12 x)+e^{e^{\frac {x^2}{2}}} \left (-28+e^{3+x} (7-7 x)+e^{\frac {x^2}{2}} \left (4 x^2-e^{3+x} x^2\right )\right )}{-48 x+12 e^{3+x} x+e^{e^{\frac {x^2}{2}}} \left (28 x-7 e^{3+x} x\right )+e^{2 e^{\frac {x^2}{2}}} \left (-4 x+e^{3+x} x\right )} \, dx=\log \left ({\left (e^{\left (x + 3\right )} - 4\right )} e^{\left (-3\right )}\right ) - \log \left (x\right ) + \log \left (e^{\left (e^{\left (\frac {1}{2} \, x^{2}\right )}\right )} - 3\right ) - \log \left (e^{\left (e^{\left (\frac {1}{2} \, x^{2}\right )}\right )} - 4\right ) \] Input:
integrate((((-1+x)*exp(3+x)+4)*exp(exp(1/2*x^2))^2+((-x^2*exp(3+x)+4*x^2)* exp(1/2*x^2)+(-7*x+7)*exp(3+x)-28)*exp(exp(1/2*x^2))+(12*x-12)*exp(3+x)+48 )/((exp(3+x)*x-4*x)*exp(exp(1/2*x^2))^2+(-7*exp(3+x)*x+28*x)*exp(exp(1/2*x ^2))+12*exp(3+x)*x-48*x),x, algorithm="maxima")
Output:
log((e^(x + 3) - 4)*e^(-3)) - log(x) + log(e^(e^(1/2*x^2)) - 3) - log(e^(e ^(1/2*x^2)) - 4)
Time = 1.37 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.94 \[ \int \frac {48+e^{2 e^{\frac {x^2}{2}}} \left (4+e^{3+x} (-1+x)\right )+e^{3+x} (-12+12 x)+e^{e^{\frac {x^2}{2}}} \left (-28+e^{3+x} (7-7 x)+e^{\frac {x^2}{2}} \left (4 x^2-e^{3+x} x^2\right )\right )}{-48 x+12 e^{3+x} x+e^{e^{\frac {x^2}{2}}} \left (28 x-7 e^{3+x} x\right )+e^{2 e^{\frac {x^2}{2}}} \left (-4 x+e^{3+x} x\right )} \, dx=-\log \left (x\right ) + \log \left (e^{\left (x + 3\right )} - 4\right ) + \log \left (e^{\left (e^{\left (\frac {1}{2} \, x^{2}\right )}\right )} - 3\right ) - \log \left (e^{\left (e^{\left (\frac {1}{2} \, x^{2}\right )}\right )} - 4\right ) \] Input:
integrate((((-1+x)*exp(3+x)+4)*exp(exp(1/2*x^2))^2+((-x^2*exp(3+x)+4*x^2)* exp(1/2*x^2)+(-7*x+7)*exp(3+x)-28)*exp(exp(1/2*x^2))+(12*x-12)*exp(3+x)+48 )/((exp(3+x)*x-4*x)*exp(exp(1/2*x^2))^2+(-7*exp(3+x)*x+28*x)*exp(exp(1/2*x ^2))+12*exp(3+x)*x-48*x),x, algorithm="giac")
Output:
-log(x) + log(e^(x + 3) - 4) + log(e^(e^(1/2*x^2)) - 3) - log(e^(e^(1/2*x^ 2)) - 4)
Time = 2.74 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.92 \[ \int \frac {48+e^{2 e^{\frac {x^2}{2}}} \left (4+e^{3+x} (-1+x)\right )+e^{3+x} (-12+12 x)+e^{e^{\frac {x^2}{2}}} \left (-28+e^{3+x} (7-7 x)+e^{\frac {x^2}{2}} \left (4 x^2-e^{3+x} x^2\right )\right )}{-48 x+12 e^{3+x} x+e^{e^{\frac {x^2}{2}}} \left (28 x-7 e^{3+x} x\right )+e^{2 e^{\frac {x^2}{2}}} \left (-4 x+e^{3+x} x\right )} \, dx=\ln \left ({\mathrm {e}}^3\,{\mathrm {e}}^x-4\right )-\ln \left (24\,x\,\sqrt {{\mathrm {e}}^{x^2}}-6\,x\,\sqrt {{\mathrm {e}}^{x^2}}\,{\mathrm {e}}^{\sqrt {{\mathrm {e}}^{x^2}}}\right )+\ln \left (8\,x\,\sqrt {{\mathrm {e}}^{x^2}}\,{\mathrm {e}}^{\sqrt {{\mathrm {e}}^{x^2}}}-24\,x\,\sqrt {{\mathrm {e}}^{x^2}}\right )-\ln \left (x\right ) \] Input:
int(-(exp(2*exp(x^2/2))*(exp(x + 3)*(x - 1) + 4) - exp(exp(x^2/2))*(exp(x^ 2/2)*(x^2*exp(x + 3) - 4*x^2) + exp(x + 3)*(7*x - 7) + 28) + exp(x + 3)*(1 2*x - 12) + 48)/(48*x + exp(2*exp(x^2/2))*(4*x - x*exp(x + 3)) - 12*x*exp( x + 3) - exp(exp(x^2/2))*(28*x - 7*x*exp(x + 3))),x)
Output:
log(exp(3)*exp(x) - 4) - log(24*x*exp(x^2)^(1/2) - 6*x*exp(x^2)^(1/2)*exp( exp(x^2)^(1/2))) + log(8*x*exp(x^2)^(1/2)*exp(exp(x^2)^(1/2)) - 24*x*exp(x ^2)^(1/2)) - log(x)
Time = 0.23 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.14 \[ \int \frac {48+e^{2 e^{\frac {x^2}{2}}} \left (4+e^{3+x} (-1+x)\right )+e^{3+x} (-12+12 x)+e^{e^{\frac {x^2}{2}}} \left (-28+e^{3+x} (7-7 x)+e^{\frac {x^2}{2}} \left (4 x^2-e^{3+x} x^2\right )\right )}{-48 x+12 e^{3+x} x+e^{e^{\frac {x^2}{2}}} \left (28 x-7 e^{3+x} x\right )+e^{2 e^{\frac {x^2}{2}}} \left (-4 x+e^{3+x} x\right )} \, dx=-\mathrm {log}\left (e^{e^{\frac {x^{2}}{2}}}-4\right )+\mathrm {log}\left (e^{e^{\frac {x^{2}}{2}}}-3\right )+\mathrm {log}\left (e^{x} e^{3}-4\right )-\mathrm {log}\left (x \right ) \] Input:
int((((-1+x)*exp(3+x)+4)*exp(exp(1/2*x^2))^2+((-x^2*exp(3+x)+4*x^2)*exp(1/ 2*x^2)+(-7*x+7)*exp(3+x)-28)*exp(exp(1/2*x^2))+(12*x-12)*exp(3+x)+48)/((ex p(3+x)*x-4*x)*exp(exp(1/2*x^2))^2+(-7*exp(3+x)*x+28*x)*exp(exp(1/2*x^2))+1 2*exp(3+x)*x-48*x),x)
Output:
- log(e**(e**(x**2/2)) - 4) + log(e**(e**(x**2/2)) - 3) + log(e**x*e**3 - 4) - log(x)