Integrand size = 251, antiderivative size = 27 \[ \int \frac {27 x^2-27 x^3-45 x^4-17 x^5-2 x^6+e^6 \left (x^2-2 x^3\right )+e^4 \left (9 x^2-15 x^3-6 x^4\right )+e^2 \left (27 x^2-36 x^3-33 x^4-6 x^5\right )+e^{\frac {-125-15 e^4+e^2 (-90-30 x)-90 x-15 x^2}{9 x+e^4 x+6 x^2+x^3+e^2 \left (6 x+2 x^2\right )}} \left (375+15 e^6+375 x+135 x^2+15 x^3+e^4 (135+45 x)+e^2 \left (395+270 x+45 x^2\right )\right )}{27 x^2+e^6 x^2+27 x^3+9 x^4+x^5+e^4 \left (9 x^2+3 x^3\right )+e^2 \left (27 x^2+18 x^3+3 x^4\right )} \, dx=9+e^{\frac {5 \left (-3+\frac {2}{\left (3+e^2+x\right )^2}\right )}{x}}+x-x^2 \] Output:
exp(5*(2/(3+exp(2)+x)^2-3)/x)-x^2+9+x
Time = 0.20 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.67 \[ \int \frac {27 x^2-27 x^3-45 x^4-17 x^5-2 x^6+e^6 \left (x^2-2 x^3\right )+e^4 \left (9 x^2-15 x^3-6 x^4\right )+e^2 \left (27 x^2-36 x^3-33 x^4-6 x^5\right )+e^{\frac {-125-15 e^4+e^2 (-90-30 x)-90 x-15 x^2}{9 x+e^4 x+6 x^2+x^3+e^2 \left (6 x+2 x^2\right )}} \left (375+15 e^6+375 x+135 x^2+15 x^3+e^4 (135+45 x)+e^2 \left (395+270 x+45 x^2\right )\right )}{27 x^2+e^6 x^2+27 x^3+9 x^4+x^5+e^4 \left (9 x^2+3 x^3\right )+e^2 \left (27 x^2+18 x^3+3 x^4\right )} \, dx=e^{-\frac {5 \left (25+3 e^4+18 x+3 x^2+6 e^2 (3+x)\right )}{x \left (3+e^2+x\right )^2}}+x-x^2 \] Input:
Integrate[(27*x^2 - 27*x^3 - 45*x^4 - 17*x^5 - 2*x^6 + E^6*(x^2 - 2*x^3) + E^4*(9*x^2 - 15*x^3 - 6*x^4) + E^2*(27*x^2 - 36*x^3 - 33*x^4 - 6*x^5) + E ^((-125 - 15*E^4 + E^2*(-90 - 30*x) - 90*x - 15*x^2)/(9*x + E^4*x + 6*x^2 + x^3 + E^2*(6*x + 2*x^2)))*(375 + 15*E^6 + 375*x + 135*x^2 + 15*x^3 + E^4 *(135 + 45*x) + E^2*(395 + 270*x + 45*x^2)))/(27*x^2 + E^6*x^2 + 27*x^3 + 9*x^4 + x^5 + E^4*(9*x^2 + 3*x^3) + E^2*(27*x^2 + 18*x^3 + 3*x^4)),x]
Output:
E^((-5*(25 + 3*E^4 + 18*x + 3*x^2 + 6*E^2*(3 + x)))/(x*(3 + E^2 + x)^2)) + x - x^2
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.
| \(\displaystyle \int \frac {\left (15 x^3+135 x^2+e^2 \left (45 x^2+270 x+395\right )+375 x+e^4 (45 x+135)+15 e^6+375\right ) \exp \left (\frac {-15 x^2-90 x+e^2 (-30 x-90)-15 e^4-125}{x^3+6 x^2+e^2 \left (2 x^2+6 x\right )+e^4 x+9 x}\right )-2 x^6-17 x^5-45 x^4-27 x^3+27 x^2+e^6 \left (x^2-2 x^3\right )+e^4 \left (-6 x^4-15 x^3+9 x^2\right )+e^2 \left (-6 x^5-33 x^4-36 x^3+27 x^2\right )}{x^5+9 x^4+27 x^3+e^6 x^2+27 x^2+e^4 \left (3 x^3+9 x^2\right )+e^2 \left (3 x^4+18 x^3+27 x^2\right )} \, dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {\left (15 x^3+135 x^2+e^2 \left (45 x^2+270 x+395\right )+375 x+e^4 (45 x+135)+15 e^6+375\right ) \exp \left (\frac {-15 x^2-90 x+e^2 (-30 x-90)-15 e^4-125}{x^3+6 x^2+e^2 \left (2 x^2+6 x\right )+e^4 x+9 x}\right )-2 x^6-17 x^5-45 x^4-27 x^3+27 x^2+e^6 \left (x^2-2 x^3\right )+e^4 \left (-6 x^4-15 x^3+9 x^2\right )+e^2 \left (-6 x^5-33 x^4-36 x^3+27 x^2\right )}{x^5+9 x^4+27 x^3+\left (27+e^6\right ) x^2+e^4 \left (3 x^3+9 x^2\right )+e^2 \left (3 x^4+18 x^3+27 x^2\right )}dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {\left (15 x^3+135 x^2+e^2 \left (45 x^2+270 x+395\right )+375 x+e^4 (45 x+135)+15 e^6+375\right ) \exp \left (\frac {-15 x^2-90 x+e^2 (-30 x-90)-15 e^4-125}{x^3+6 x^2+e^2 \left (2 x^2+6 x\right )+e^4 x+9 x}\right )-2 x^6-17 x^5-45 x^4-27 x^3+27 x^2+e^6 \left (x^2-2 x^3\right )+e^4 \left (-6 x^4-15 x^3+9 x^2\right )+e^2 \left (-6 x^5-33 x^4-36 x^3+27 x^2\right )}{x^2 \left (x^3+3 \left (3+e^2\right ) x^2+3 \left (3+e^2\right )^2 x+\left (3+e^2\right )^3\right )}dx\) |
| \(\Big \downarrow \) 2007 |
| \(\displaystyle \int \frac {\left (15 x^3+135 x^2+e^2 \left (45 x^2+270 x+395\right )+375 x+e^4 (45 x+135)+15 e^6+375\right ) \exp \left (\frac {-15 x^2-90 x+e^2 (-30 x-90)-15 e^4-125}{x^3+6 x^2+e^2 \left (2 x^2+6 x\right )+e^4 x+9 x}\right )-2 x^6-17 x^5-45 x^4-27 x^3+27 x^2+e^6 \left (x^2-2 x^3\right )+e^4 \left (-6 x^4-15 x^3+9 x^2\right )+e^2 \left (-6 x^5-33 x^4-36 x^3+27 x^2\right )}{x^2 \left (x+e^2+3\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {5 \left (3 x^3+9 \left (3+e^2\right ) x^2+3 \left (25+18 e^2+3 e^4\right ) x+3 e^6+27 e^4+79 e^2+75\right ) \exp \left (-\frac {5 \left (3 x^2+6 \left (3+e^2\right ) x+3 e^4+18 e^2+25\right )}{x \left (x+e^2+3\right )^2}\right )}{\left (x+e^2+3\right )^3 x^2}-\frac {2 x^4}{\left (x+e^2+3\right )^3}-\frac {17 x^3}{\left (x+e^2+3\right )^3}-\frac {45 x^2}{\left (x+e^2+3\right )^3}-\frac {27 x}{\left (x+e^2+3\right )^3}-\frac {3 e^2 (x+3)^2 (2 x-1)}{\left (x+e^2+3\right )^3}-\frac {3 e^4 (x+3) (2 x-1)}{\left (x+e^2+3\right )^3}-\frac {e^6 (2 x-1)}{\left (x+e^2+3\right )^3}+\frac {27}{\left (x+e^2+3\right )^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {5 \left (25+18 e^2+3 e^4\right ) \int \frac {\exp \left (-\frac {5 \left (3 x^2+6 \left (3+e^2\right ) x+3 e^4+18 e^2+25\right )}{x \left (x+e^2+3\right )^2}\right )}{x^2}dx}{\left (3+e^2\right )^2}+\frac {20 \int \frac {\exp \left (-\frac {5 \left (3 x^2+6 \left (3+e^2\right ) x+3 e^4+18 e^2+25\right )}{x \left (x+e^2+3\right )^2}\right )}{\left (x+e^2+3\right )^3}dx}{3+e^2}+\frac {10 \int \frac {\exp \left (-\frac {5 \left (3 x^2+6 \left (3+e^2\right ) x+3 e^4+18 e^2+25\right )}{x \left (x+e^2+3\right )^2}\right )}{\left (x+e^2+3\right )^2}dx}{\left (3+e^2\right )^2}-x^2-\frac {27 x^2}{2 \left (3+e^2\right ) \left (x+e^2+3\right )^2}-\frac {e^6 (1-2 x)^2}{2 \left (7+2 e^2\right ) \left (x+e^2+3\right )^2}+6 \left (3+e^2\right ) x-6 e^2 x-17 x-\frac {3 e^4 \left (7+4 e^2\right )}{x+e^2+3}+\frac {6 e^4 \left (7+3 e^2\right )}{x+e^2+3}-\frac {8 \left (3+e^2\right )^3}{x+e^2+3}+\frac {51 \left (3+e^2\right )^2}{x+e^2+3}-\frac {90 \left (3+e^2\right )}{x+e^2+3}+\frac {\left (3+e^2\right )^4}{\left (x+e^2+3\right )^2}-\frac {17 \left (3+e^2\right )^3}{2 \left (x+e^2+3\right )^2}+\frac {45 \left (3+e^2\right )^2}{2 \left (x+e^2+3\right )^2}-\frac {27}{2 \left (x+e^2+3\right )^2}+3 e^2 \left (7+6 e^2\right ) \log \left (x+e^2+3\right )-12 \left (3+e^2\right )^2 \log \left (x+e^2+3\right )+51 \left (3+e^2\right ) \log \left (x+e^2+3\right )-6 e^4 \log \left (x+e^2+3\right )-45 \log \left (x+e^2+3\right )\) |
Input:
Int[(27*x^2 - 27*x^3 - 45*x^4 - 17*x^5 - 2*x^6 + E^6*(x^2 - 2*x^3) + E^4*( 9*x^2 - 15*x^3 - 6*x^4) + E^2*(27*x^2 - 36*x^3 - 33*x^4 - 6*x^5) + E^((-12 5 - 15*E^4 + E^2*(-90 - 30*x) - 90*x - 15*x^2)/(9*x + E^4*x + 6*x^2 + x^3 + E^2*(6*x + 2*x^2)))*(375 + 15*E^6 + 375*x + 135*x^2 + 15*x^3 + E^4*(135 + 45*x) + E^2*(395 + 270*x + 45*x^2)))/(27*x^2 + E^6*x^2 + 27*x^3 + 9*x^4 + x^5 + E^4*(9*x^2 + 3*x^3) + E^2*(27*x^2 + 18*x^3 + 3*x^4)),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(273\) vs. \(2(25)=50\).
Time = 0.74 (sec) , antiderivative size = 274, normalized size of antiderivative = 10.15
\[\frac {x^{3} {\mathrm e}^{\frac {-15 \,{\mathrm e}^{4}+\left (-30 x -90\right ) {\mathrm e}^{2}-15 x^{2}-90 x -125}{x \,{\mathrm e}^{4}+\left (2 x^{2}+6 x \right ) {\mathrm e}^{2}+x^{3}+6 x^{2}+9 x}}+\left (-5-2 \,{\mathrm e}^{2}\right ) x^{4}+\left (2 \,{\mathrm e}^{6}+15 \,{\mathrm e}^{4}+36 \,{\mathrm e}^{2}+27\right ) x^{2}+\left ({\mathrm e}^{8}+10 \,{\mathrm e}^{6}+36 \,{\mathrm e}^{4}+54 \,{\mathrm e}^{2}+27\right ) x +\left (2 \,{\mathrm e}^{2}+6\right ) x^{2} {\mathrm e}^{\frac {-15 \,{\mathrm e}^{4}+\left (-30 x -90\right ) {\mathrm e}^{2}-15 x^{2}-90 x -125}{x \,{\mathrm e}^{4}+\left (2 x^{2}+6 x \right ) {\mathrm e}^{2}+x^{3}+6 x^{2}+9 x}}+\left (6 \,{\mathrm e}^{2}+{\mathrm e}^{4}+9\right ) x \,{\mathrm e}^{\frac {-15 \,{\mathrm e}^{4}+\left (-30 x -90\right ) {\mathrm e}^{2}-15 x^{2}-90 x -125}{x \,{\mathrm e}^{4}+\left (2 x^{2}+6 x \right ) {\mathrm e}^{2}+x^{3}+6 x^{2}+9 x}}-x^{5}}{x \left (3+{\mathrm e}^{2}+x \right )^{2}}\]
Input:
int(((15*exp(2)^3+(45*x+135)*exp(2)^2+(45*x^2+270*x+395)*exp(2)+15*x^3+135 *x^2+375*x+375)*exp((-15*exp(2)^2+(-30*x-90)*exp(2)-15*x^2-90*x-125)/(x*ex p(2)^2+(2*x^2+6*x)*exp(2)+x^3+6*x^2+9*x))+(-2*x^3+x^2)*exp(2)^3+(-6*x^4-15 *x^3+9*x^2)*exp(2)^2+(-6*x^5-33*x^4-36*x^3+27*x^2)*exp(2)-2*x^6-17*x^5-45* x^4-27*x^3+27*x^2)/(x^2*exp(2)^3+(3*x^3+9*x^2)*exp(2)^2+(3*x^4+18*x^3+27*x ^2)*exp(2)+x^5+9*x^4+27*x^3+27*x^2),x)
Output:
(x^3*exp((-15*exp(2)^2+(-30*x-90)*exp(2)-15*x^2-90*x-125)/(x*exp(2)^2+(2*x ^2+6*x)*exp(2)+x^3+6*x^2+9*x))+(-5-2*exp(2))*x^4+(2*exp(2)^3+15*exp(2)^2+3 6*exp(2)+27)*x^2+(exp(2)^4+10*exp(2)^3+36*exp(2)^2+54*exp(2)+27)*x+(2*exp( 2)+6)*x^2*exp((-15*exp(2)^2+(-30*x-90)*exp(2)-15*x^2-90*x-125)/(x*exp(2)^2 +(2*x^2+6*x)*exp(2)+x^3+6*x^2+9*x))+(exp(2)^2+6*exp(2)+9)*x*exp((-15*exp(2 )^2+(-30*x-90)*exp(2)-15*x^2-90*x-125)/(x*exp(2)^2+(2*x^2+6*x)*exp(2)+x^3+ 6*x^2+9*x))-x^5)/x/(3+exp(2)+x)^2
Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (25) = 50\).
Time = 0.08 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.22 \[ \int \frac {27 x^2-27 x^3-45 x^4-17 x^5-2 x^6+e^6 \left (x^2-2 x^3\right )+e^4 \left (9 x^2-15 x^3-6 x^4\right )+e^2 \left (27 x^2-36 x^3-33 x^4-6 x^5\right )+e^{\frac {-125-15 e^4+e^2 (-90-30 x)-90 x-15 x^2}{9 x+e^4 x+6 x^2+x^3+e^2 \left (6 x+2 x^2\right )}} \left (375+15 e^6+375 x+135 x^2+15 x^3+e^4 (135+45 x)+e^2 \left (395+270 x+45 x^2\right )\right )}{27 x^2+e^6 x^2+27 x^3+9 x^4+x^5+e^4 \left (9 x^2+3 x^3\right )+e^2 \left (27 x^2+18 x^3+3 x^4\right )} \, dx=-x^{2} + x + e^{\left (-\frac {5 \, {\left (3 \, x^{2} + 6 \, {\left (x + 3\right )} e^{2} + 18 \, x + 3 \, e^{4} + 25\right )}}{x^{3} + 6 \, x^{2} + x e^{4} + 2 \, {\left (x^{2} + 3 \, x\right )} e^{2} + 9 \, x}\right )} \] Input:
integrate(((15*exp(2)^3+(45*x+135)*exp(2)^2+(45*x^2+270*x+395)*exp(2)+15*x ^3+135*x^2+375*x+375)*exp((-15*exp(2)^2+(-30*x-90)*exp(2)-15*x^2-90*x-125) /(x*exp(2)^2+(2*x^2+6*x)*exp(2)+x^3+6*x^2+9*x))+(-2*x^3+x^2)*exp(2)^3+(-6* x^4-15*x^3+9*x^2)*exp(2)^2+(-6*x^5-33*x^4-36*x^3+27*x^2)*exp(2)-2*x^6-17*x ^5-45*x^4-27*x^3+27*x^2)/(x^2*exp(2)^3+(3*x^3+9*x^2)*exp(2)^2+(3*x^4+18*x^ 3+27*x^2)*exp(2)+x^5+9*x^4+27*x^3+27*x^2),x, algorithm="fricas")
Output:
-x^2 + x + e^(-5*(3*x^2 + 6*(x + 3)*e^2 + 18*x + 3*e^4 + 25)/(x^3 + 6*x^2 + x*e^4 + 2*(x^2 + 3*x)*e^2 + 9*x))
Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (20) = 40\).
Time = 0.71 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.22 \[ \int \frac {27 x^2-27 x^3-45 x^4-17 x^5-2 x^6+e^6 \left (x^2-2 x^3\right )+e^4 \left (9 x^2-15 x^3-6 x^4\right )+e^2 \left (27 x^2-36 x^3-33 x^4-6 x^5\right )+e^{\frac {-125-15 e^4+e^2 (-90-30 x)-90 x-15 x^2}{9 x+e^4 x+6 x^2+x^3+e^2 \left (6 x+2 x^2\right )}} \left (375+15 e^6+375 x+135 x^2+15 x^3+e^4 (135+45 x)+e^2 \left (395+270 x+45 x^2\right )\right )}{27 x^2+e^6 x^2+27 x^3+9 x^4+x^5+e^4 \left (9 x^2+3 x^3\right )+e^2 \left (27 x^2+18 x^3+3 x^4\right )} \, dx=- x^{2} + x + e^{\frac {- 15 x^{2} - 90 x + \left (- 30 x - 90\right ) e^{2} - 15 e^{4} - 125}{x^{3} + 6 x^{2} + 9 x + x e^{4} + \left (2 x^{2} + 6 x\right ) e^{2}}} \] Input:
integrate(((15*exp(2)**3+(45*x+135)*exp(2)**2+(45*x**2+270*x+395)*exp(2)+1 5*x**3+135*x**2+375*x+375)*exp((-15*exp(2)**2+(-30*x-90)*exp(2)-15*x**2-90 *x-125)/(x*exp(2)**2+(2*x**2+6*x)*exp(2)+x**3+6*x**2+9*x))+(-2*x**3+x**2)* exp(2)**3+(-6*x**4-15*x**3+9*x**2)*exp(2)**2+(-6*x**5-33*x**4-36*x**3+27*x **2)*exp(2)-2*x**6-17*x**5-45*x**4-27*x**3+27*x**2)/(x**2*exp(2)**3+(3*x** 3+9*x**2)*exp(2)**2+(3*x**4+18*x**3+27*x**2)*exp(2)+x**5+9*x**4+27*x**3+27 *x**2),x)
Output:
-x**2 + x + exp((-15*x**2 - 90*x + (-30*x - 90)*exp(2) - 15*exp(4) - 125)/ (x**3 + 6*x**2 + 9*x + x*exp(4) + (2*x**2 + 6*x)*exp(2)))
Leaf count of result is larger than twice the leaf count of optimal. 873 vs. \(2 (25) = 50\).
Time = 0.79 (sec) , antiderivative size = 873, normalized size of antiderivative = 32.33 \[ \int \frac {27 x^2-27 x^3-45 x^4-17 x^5-2 x^6+e^6 \left (x^2-2 x^3\right )+e^4 \left (9 x^2-15 x^3-6 x^4\right )+e^2 \left (27 x^2-36 x^3-33 x^4-6 x^5\right )+e^{\frac {-125-15 e^4+e^2 (-90-30 x)-90 x-15 x^2}{9 x+e^4 x+6 x^2+x^3+e^2 \left (6 x+2 x^2\right )}} \left (375+15 e^6+375 x+135 x^2+15 x^3+e^4 (135+45 x)+e^2 \left (395+270 x+45 x^2\right )\right )}{27 x^2+e^6 x^2+27 x^3+9 x^4+x^5+e^4 \left (9 x^2+3 x^3\right )+e^2 \left (27 x^2+18 x^3+3 x^4\right )} \, dx=\text {Too large to display} \] Input:
integrate(((15*exp(2)^3+(45*x+135)*exp(2)^2+(45*x^2+270*x+395)*exp(2)+15*x ^3+135*x^2+375*x+375)*exp((-15*exp(2)^2+(-30*x-90)*exp(2)-15*x^2-90*x-125) /(x*exp(2)^2+(2*x^2+6*x)*exp(2)+x^3+6*x^2+9*x))+(-2*x^3+x^2)*exp(2)^3+(-6* x^4-15*x^3+9*x^2)*exp(2)^2+(-6*x^5-33*x^4-36*x^3+27*x^2)*exp(2)-2*x^6-17*x ^5-45*x^4-27*x^3+27*x^2)/(x^2*exp(2)^3+(3*x^3+9*x^2)*exp(2)^2+(3*x^4+18*x^ 3+27*x^2)*exp(2)+x^5+9*x^4+27*x^3+27*x^2),x, algorithm="maxima")
Output:
-x^2 + 6*x*(e^2 + 3) - 3*((4*x*(e^2 + 3) + 3*e^4 + 18*e^2 + 27)/(x^2 + 2*x *(e^2 + 3) + e^4 + 6*e^2 + 9) + 2*log(x + e^2 + 3))*e^4 + 3*(6*(e^2 + 3)*l og(x + e^2 + 3) - 2*x + (6*x*(e^4 + 6*e^2 + 9) + 5*e^6 + 45*e^4 + 135*e^2 + 135)/(x^2 + 2*x*(e^2 + 3) + e^4 + 6*e^2 + 9))*e^2 - 33/2*((4*x*(e^2 + 3) + 3*e^4 + 18*e^2 + 27)/(x^2 + 2*x*(e^2 + 3) + e^4 + 6*e^2 + 9) + 2*log(x + e^2 + 3))*e^2 - 12*(e^4 + 6*e^2 + 9)*log(x + e^2 + 3) + 51*(e^2 + 3)*log (x + e^2 + 3) - 17*x + (2*x + e^2 + 3)*e^6/(x^2 + 2*x*(e^2 + 3) + e^4 + 6* e^2 + 9) + 15/2*(2*x + e^2 + 3)*e^4/(x^2 + 2*x*(e^2 + 3) + e^4 + 6*e^2 + 9 ) + 18*(2*x + e^2 + 3)*e^2/(x^2 + 2*x*(e^2 + 3) + e^4 + 6*e^2 + 9) - (8*x* (e^6 + 9*e^4 + 27*e^2 + 27) + 7*e^8 + 84*e^6 + 378*e^4 + 756*e^2 + 567)/(x ^2 + 2*x*(e^2 + 3) + e^4 + 6*e^2 + 9) + 17/2*(6*x*(e^4 + 6*e^2 + 9) + 5*e^ 6 + 45*e^4 + 135*e^2 + 135)/(x^2 + 2*x*(e^2 + 3) + e^4 + 6*e^2 + 9) - 45/2 *(4*x*(e^2 + 3) + 3*e^4 + 18*e^2 + 27)/(x^2 + 2*x*(e^2 + 3) + e^4 + 6*e^2 + 9) + 27/2*(2*x + e^2 + 3)/(x^2 + 2*x*(e^2 + 3) + e^4 + 6*e^2 + 9) - 1/2* e^6/(x^2 + 2*x*(e^2 + 3) + e^4 + 6*e^2 + 9) - 9/2*e^4/(x^2 + 2*x*(e^2 + 3) + e^4 + 6*e^2 + 9) - 27/2*e^2/(x^2 + 2*x*(e^2 + 3) + e^4 + 6*e^2 + 9) - 2 7/2/(x^2 + 2*x*(e^2 + 3) + e^4 + 6*e^2 + 9) + e^(15*e^4/(x^2*(e^2 + 3) + 2 *x*(e^4 + 6*e^2 + 9) + e^6 + 9*e^4 + 27*e^2 + 27) + 15*e^4/(x*(e^4 + 6*e^2 + 9) + e^6 + 9*e^4 + 27*e^2 + 27) + 90*e^2/(x^2*(e^2 + 3) + 2*x*(e^4 + 6* e^2 + 9) + e^6 + 9*e^4 + 27*e^2 + 27) - 15*e^2/(x^2 + 2*x*(e^2 + 3) + e...
\[ \int \frac {27 x^2-27 x^3-45 x^4-17 x^5-2 x^6+e^6 \left (x^2-2 x^3\right )+e^4 \left (9 x^2-15 x^3-6 x^4\right )+e^2 \left (27 x^2-36 x^3-33 x^4-6 x^5\right )+e^{\frac {-125-15 e^4+e^2 (-90-30 x)-90 x-15 x^2}{9 x+e^4 x+6 x^2+x^3+e^2 \left (6 x+2 x^2\right )}} \left (375+15 e^6+375 x+135 x^2+15 x^3+e^4 (135+45 x)+e^2 \left (395+270 x+45 x^2\right )\right )}{27 x^2+e^6 x^2+27 x^3+9 x^4+x^5+e^4 \left (9 x^2+3 x^3\right )+e^2 \left (27 x^2+18 x^3+3 x^4\right )} \, dx=\int { -\frac {2 \, x^{6} + 17 \, x^{5} + 45 \, x^{4} + 27 \, x^{3} - 27 \, x^{2} + {\left (2 \, x^{3} - x^{2}\right )} e^{6} + 3 \, {\left (2 \, x^{4} + 5 \, x^{3} - 3 \, x^{2}\right )} e^{4} + 3 \, {\left (2 \, x^{5} + 11 \, x^{4} + 12 \, x^{3} - 9 \, x^{2}\right )} e^{2} - 5 \, {\left (3 \, x^{3} + 27 \, x^{2} + 9 \, {\left (x + 3\right )} e^{4} + {\left (9 \, x^{2} + 54 \, x + 79\right )} e^{2} + 75 \, x + 3 \, e^{6} + 75\right )} e^{\left (-\frac {5 \, {\left (3 \, x^{2} + 6 \, {\left (x + 3\right )} e^{2} + 18 \, x + 3 \, e^{4} + 25\right )}}{x^{3} + 6 \, x^{2} + x e^{4} + 2 \, {\left (x^{2} + 3 \, x\right )} e^{2} + 9 \, x}\right )}}{x^{5} + 9 \, x^{4} + 27 \, x^{3} + x^{2} e^{6} + 27 \, x^{2} + 3 \, {\left (x^{3} + 3 \, x^{2}\right )} e^{4} + 3 \, {\left (x^{4} + 6 \, x^{3} + 9 \, x^{2}\right )} e^{2}} \,d x } \] Input:
integrate(((15*exp(2)^3+(45*x+135)*exp(2)^2+(45*x^2+270*x+395)*exp(2)+15*x ^3+135*x^2+375*x+375)*exp((-15*exp(2)^2+(-30*x-90)*exp(2)-15*x^2-90*x-125) /(x*exp(2)^2+(2*x^2+6*x)*exp(2)+x^3+6*x^2+9*x))+(-2*x^3+x^2)*exp(2)^3+(-6* x^4-15*x^3+9*x^2)*exp(2)^2+(-6*x^5-33*x^4-36*x^3+27*x^2)*exp(2)-2*x^6-17*x ^5-45*x^4-27*x^3+27*x^2)/(x^2*exp(2)^3+(3*x^3+9*x^2)*exp(2)^2+(3*x^4+18*x^ 3+27*x^2)*exp(2)+x^5+9*x^4+27*x^3+27*x^2),x, algorithm="giac")
Output:
integrate(-(2*x^6 + 17*x^5 + 45*x^4 + 27*x^3 - 27*x^2 + (2*x^3 - x^2)*e^6 + 3*(2*x^4 + 5*x^3 - 3*x^2)*e^4 + 3*(2*x^5 + 11*x^4 + 12*x^3 - 9*x^2)*e^2 - 5*(3*x^3 + 27*x^2 + 9*(x + 3)*e^4 + (9*x^2 + 54*x + 79)*e^2 + 75*x + 3*e ^6 + 75)*e^(-5*(3*x^2 + 6*(x + 3)*e^2 + 18*x + 3*e^4 + 25)/(x^3 + 6*x^2 + x*e^4 + 2*(x^2 + 3*x)*e^2 + 9*x)))/(x^5 + 9*x^4 + 27*x^3 + x^2*e^6 + 27*x^ 2 + 3*(x^3 + 3*x^2)*e^4 + 3*(x^4 + 6*x^3 + 9*x^2)*e^2), x)
Time = 2.59 (sec) , antiderivative size = 217, normalized size of antiderivative = 8.04 \[ \int \frac {27 x^2-27 x^3-45 x^4-17 x^5-2 x^6+e^6 \left (x^2-2 x^3\right )+e^4 \left (9 x^2-15 x^3-6 x^4\right )+e^2 \left (27 x^2-36 x^3-33 x^4-6 x^5\right )+e^{\frac {-125-15 e^4+e^2 (-90-30 x)-90 x-15 x^2}{9 x+e^4 x+6 x^2+x^3+e^2 \left (6 x+2 x^2\right )}} \left (375+15 e^6+375 x+135 x^2+15 x^3+e^4 (135+45 x)+e^2 \left (395+270 x+45 x^2\right )\right )}{27 x^2+e^6 x^2+27 x^3+9 x^4+x^5+e^4 \left (9 x^2+3 x^3\right )+e^2 \left (27 x^2+18 x^3+3 x^4\right )} \, dx=x-x^2+{\mathrm {e}}^{-\frac {15\,x^2}{9\,x+6\,x\,{\mathrm {e}}^2+x\,{\mathrm {e}}^4+2\,x^2\,{\mathrm {e}}^2+6\,x^2+x^3}}\,{\mathrm {e}}^{-\frac {30\,x\,{\mathrm {e}}^2}{9\,x+6\,x\,{\mathrm {e}}^2+x\,{\mathrm {e}}^4+2\,x^2\,{\mathrm {e}}^2+6\,x^2+x^3}}\,{\mathrm {e}}^{-\frac {125}{9\,x+6\,x\,{\mathrm {e}}^2+x\,{\mathrm {e}}^4+2\,x^2\,{\mathrm {e}}^2+6\,x^2+x^3}}\,{\mathrm {e}}^{-\frac {15\,{\mathrm {e}}^4}{9\,x+6\,x\,{\mathrm {e}}^2+x\,{\mathrm {e}}^4+2\,x^2\,{\mathrm {e}}^2+6\,x^2+x^3}}\,{\mathrm {e}}^{-\frac {90\,{\mathrm {e}}^2}{9\,x+6\,x\,{\mathrm {e}}^2+x\,{\mathrm {e}}^4+2\,x^2\,{\mathrm {e}}^2+6\,x^2+x^3}}\,{\mathrm {e}}^{-\frac {90\,x}{9\,x+6\,x\,{\mathrm {e}}^2+x\,{\mathrm {e}}^4+2\,x^2\,{\mathrm {e}}^2+6\,x^2+x^3}} \] Input:
int(-(exp(4)*(15*x^3 - 9*x^2 + 6*x^4) - exp(6)*(x^2 - 2*x^3) - 27*x^2 + 27 *x^3 + 45*x^4 + 17*x^5 + 2*x^6 - exp(-(90*x + 15*exp(4) + 15*x^2 + exp(2)* (30*x + 90) + 125)/(9*x + exp(2)*(6*x + 2*x^2) + x*exp(4) + 6*x^2 + x^3))* (375*x + 15*exp(6) + exp(2)*(270*x + 45*x^2 + 395) + 135*x^2 + 15*x^3 + ex p(4)*(45*x + 135) + 375) + exp(2)*(36*x^3 - 27*x^2 + 33*x^4 + 6*x^5))/(exp (4)*(9*x^2 + 3*x^3) + x^2*exp(6) + exp(2)*(27*x^2 + 18*x^3 + 3*x^4) + 27*x ^2 + 27*x^3 + 9*x^4 + x^5),x)
Output:
x - x^2 + exp(-(15*x^2)/(9*x + 6*x*exp(2) + x*exp(4) + 2*x^2*exp(2) + 6*x^ 2 + x^3))*exp(-(30*x*exp(2))/(9*x + 6*x*exp(2) + x*exp(4) + 2*x^2*exp(2) + 6*x^2 + x^3))*exp(-125/(9*x + 6*x*exp(2) + x*exp(4) + 2*x^2*exp(2) + 6*x^ 2 + x^3))*exp(-(15*exp(4))/(9*x + 6*x*exp(2) + x*exp(4) + 2*x^2*exp(2) + 6 *x^2 + x^3))*exp(-(90*exp(2))/(9*x + 6*x*exp(2) + x*exp(4) + 2*x^2*exp(2) + 6*x^2 + x^3))*exp(-(90*x)/(9*x + 6*x*exp(2) + x*exp(4) + 2*x^2*exp(2) + 6*x^2 + x^3))
Time = 0.24 (sec) , antiderivative size = 198, normalized size of antiderivative = 7.33 \[ \int \frac {27 x^2-27 x^3-45 x^4-17 x^5-2 x^6+e^6 \left (x^2-2 x^3\right )+e^4 \left (9 x^2-15 x^3-6 x^4\right )+e^2 \left (27 x^2-36 x^3-33 x^4-6 x^5\right )+e^{\frac {-125-15 e^4+e^2 (-90-30 x)-90 x-15 x^2}{9 x+e^4 x+6 x^2+x^3+e^2 \left (6 x+2 x^2\right )}} \left (375+15 e^6+375 x+135 x^2+15 x^3+e^4 (135+45 x)+e^2 \left (395+270 x+45 x^2\right )\right )}{27 x^2+e^6 x^2+27 x^3+9 x^4+x^5+e^4 \left (9 x^2+3 x^3\right )+e^2 \left (27 x^2+18 x^3+3 x^4\right )} \, dx=\frac {-e^{\frac {15 e^{4}+30 e^{2} x +90 e^{2}+15 x^{2}+90 x +125}{e^{4} x +2 e^{2} x^{2}+6 e^{2} x +x^{3}+6 x^{2}+9 x}} x^{2}+e^{\frac {15 e^{4}+30 e^{2} x +90 e^{2}+15 x^{2}+90 x +125}{e^{4} x +2 e^{2} x^{2}+6 e^{2} x +x^{3}+6 x^{2}+9 x}} x +1}{e^{\frac {15 e^{4}+30 e^{2} x +90 e^{2}+15 x^{2}+90 x +125}{e^{4} x +2 e^{2} x^{2}+6 e^{2} x +x^{3}+6 x^{2}+9 x}}} \] Input:
int(((15*exp(2)^3+(45*x+135)*exp(2)^2+(45*x^2+270*x+395)*exp(2)+15*x^3+135 *x^2+375*x+375)*exp((-15*exp(2)^2+(-30*x-90)*exp(2)-15*x^2-90*x-125)/(x*ex p(2)^2+(2*x^2+6*x)*exp(2)+x^3+6*x^2+9*x))+(-2*x^3+x^2)*exp(2)^3+(-6*x^4-15 *x^3+9*x^2)*exp(2)^2+(-6*x^5-33*x^4-36*x^3+27*x^2)*exp(2)-2*x^6-17*x^5-45* x^4-27*x^3+27*x^2)/(x^2*exp(2)^3+(3*x^3+9*x^2)*exp(2)^2+(3*x^4+18*x^3+27*x ^2)*exp(2)+x^5+9*x^4+27*x^3+27*x^2),x)
Output:
( - e**((15*e**4 + 30*e**2*x + 90*e**2 + 15*x**2 + 90*x + 125)/(e**4*x + 2 *e**2*x**2 + 6*e**2*x + x**3 + 6*x**2 + 9*x))*x**2 + e**((15*e**4 + 30*e** 2*x + 90*e**2 + 15*x**2 + 90*x + 125)/(e**4*x + 2*e**2*x**2 + 6*e**2*x + x **3 + 6*x**2 + 9*x))*x + 1)/e**((15*e**4 + 30*e**2*x + 90*e**2 + 15*x**2 + 90*x + 125)/(e**4*x + 2*e**2*x**2 + 6*e**2*x + x**3 + 6*x**2 + 9*x))