Integrand size = 338, antiderivative size = 31 \[ \int \frac {e^{16} \left (54+3 e^2-54 x-18 x^2-12 e x^2-18 x^3\right )+e^{16-\frac {x}{2}} \left (36+e^2 (6-3 x)-36 x-12 x^2-15 e x^2-12 x^3\right )+e^{16+\frac {x}{2}} \left (36-36 x-12 x^2-3 e x^2-12 x^3\right )+e^{16+x} \left (9-9 x-3 x^2-3 x^3\right )+e^{16-x} \left (9+e^2 (3-3 x)-9 x-3 x^2-6 e x^2-3 x^3\right )}{9+e^4+4 e^3 x+6 x^2+x^4+e^2 \left (6+6 x^2\right )+e \left (12 x+4 x^3\right )+e^{2 x} \left (9+6 x^2+x^4\right )+e^{3 x/2} \left (36+24 x^2+4 x^4+e \left (12 x+4 x^3\right )\right )+e^x \left (54+36 x^2+6 x^4+e^2 \left (6+6 x^2\right )+e \left (36 x+12 x^3\right )\right )+e^{x/2} \left (36+4 e^3 x+24 x^2+4 x^4+e^2 \left (12+12 x^2\right )+e \left (36 x+12 x^3\right )\right )} \, dx=\frac {3 e^{16-x} x}{3+\left (\frac {e}{1+e^{x/2}}+x\right )^2} \] Output:
3*exp(16-x)*x/(3+(x+exp(1)/(1+exp(1/2*x)))^2)
Leaf count is larger than twice the leaf count of optimal. \(70\) vs. \(2(31)=62\).
Time = 7.56 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.26 \[ \int \frac {e^{16} \left (54+3 e^2-54 x-18 x^2-12 e x^2-18 x^3\right )+e^{16-\frac {x}{2}} \left (36+e^2 (6-3 x)-36 x-12 x^2-15 e x^2-12 x^3\right )+e^{16+\frac {x}{2}} \left (36-36 x-12 x^2-3 e x^2-12 x^3\right )+e^{16+x} \left (9-9 x-3 x^2-3 x^3\right )+e^{16-x} \left (9+e^2 (3-3 x)-9 x-3 x^2-6 e x^2-3 x^3\right )}{9+e^4+4 e^3 x+6 x^2+x^4+e^2 \left (6+6 x^2\right )+e \left (12 x+4 x^3\right )+e^{2 x} \left (9+6 x^2+x^4\right )+e^{3 x/2} \left (36+24 x^2+4 x^4+e \left (12 x+4 x^3\right )\right )+e^x \left (54+36 x^2+6 x^4+e^2 \left (6+6 x^2\right )+e \left (36 x+12 x^3\right )\right )+e^{x/2} \left (36+4 e^3 x+24 x^2+4 x^4+e^2 \left (12+12 x^2\right )+e \left (36 x+12 x^3\right )\right )} \, dx=\frac {3 e^{16-x} \left (1+e^{x/2}\right )^2 x}{3+e^2+2 e x+2 e^{1+\frac {x}{2}} x+x^2+2 e^{x/2} \left (3+x^2\right )+e^x \left (3+x^2\right )} \] Input:
Integrate[(E^16*(54 + 3*E^2 - 54*x - 18*x^2 - 12*E*x^2 - 18*x^3) + E^(16 - x/2)*(36 + E^2*(6 - 3*x) - 36*x - 12*x^2 - 15*E*x^2 - 12*x^3) + E^(16 + x /2)*(36 - 36*x - 12*x^2 - 3*E*x^2 - 12*x^3) + E^(16 + x)*(9 - 9*x - 3*x^2 - 3*x^3) + E^(16 - x)*(9 + E^2*(3 - 3*x) - 9*x - 3*x^2 - 6*E*x^2 - 3*x^3)) /(9 + E^4 + 4*E^3*x + 6*x^2 + x^4 + E^2*(6 + 6*x^2) + E*(12*x + 4*x^3) + E ^(2*x)*(9 + 6*x^2 + x^4) + E^((3*x)/2)*(36 + 24*x^2 + 4*x^4 + E*(12*x + 4* x^3)) + E^x*(54 + 36*x^2 + 6*x^4 + E^2*(6 + 6*x^2) + E*(36*x + 12*x^3)) + E^(x/2)*(36 + 4*E^3*x + 24*x^2 + 4*x^4 + E^2*(12 + 12*x^2) + E*(36*x + 12* x^3))),x]
Output:
(3*E^(16 - x)*(1 + E^(x/2))^2*x)/(3 + E^2 + 2*E*x + 2*E^(1 + x/2)*x + x^2 + 2*E^(x/2)*(3 + x^2) + E^x*(3 + 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.
| \(\displaystyle \int \frac {e^{16} \left (-18 x^3-12 e x^2-18 x^2-54 x+3 e^2+54\right )+e^{16-\frac {x}{2}} \left (-12 x^3-15 e x^2-12 x^2-36 x+e^2 (6-3 x)+36\right )+e^{\frac {x}{2}+16} \left (-12 x^3-3 e x^2-12 x^2-36 x+36\right )+e^{x+16} \left (-3 x^3-3 x^2-9 x+9\right )+e^{16-x} \left (-3 x^3-6 e x^2-3 x^2-9 x+e^2 (3-3 x)+9\right )}{x^4+e \left (4 x^3+12 x\right )+6 x^2+e^2 \left (6 x^2+6\right )+e^{2 x} \left (x^4+6 x^2+9\right )+e^{3 x/2} \left (4 x^4+e \left (4 x^3+12 x\right )+24 x^2+36\right )+e^x \left (6 x^4+e \left (12 x^3+36 x\right )+36 x^2+e^2 \left (6 x^2+6\right )+54\right )+e^{x/2} \left (4 x^4+e \left (12 x^3+36 x\right )+24 x^2+e^2 \left (12 x^2+12\right )+4 e^3 x+36\right )+4 e^3 x+e^4+9} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {3 e^{16-x} \left (e^{x/2}+1\right ) \left (-x^3-3 e^{\frac {x}{2}+1} x^2-e^{x+1} x^2-(1+2 e) x^2-3 e^{x/2} \left (x^3+x^2+3 x-3\right )-3 e^x \left (x^3+x^2+3 x-3\right )-e^{3 x/2} \left (x^3+x^2+3 x-3\right )-3 x+e^{\frac {x}{2}+2}-e^2 (x-1)+3\right )}{\left (x^2+2 e^{x/2} \left (x^2+3\right )+e^x \left (x^2+3\right )+2 e^{\frac {x}{2}+1} x+2 e x+3 \left (1+\frac {e^2}{3}\right )\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 3 \int \frac {e^{16-x} \left (1+e^{x/2}\right ) \left (-x^3-3 e^{\frac {x}{2}+1} x^2-e^{x+1} x^2-(1+2 e) x^2-3 x+e^{\frac {x}{2}+2}+e^2 (1-x)+3 e^{x/2} \left (-x^3-x^2-3 x+3\right )+3 e^x \left (-x^3-x^2-3 x+3\right )+e^{3 x/2} \left (-x^3-x^2-3 x+3\right )+3\right )}{\left (x^2+2 e^{\frac {x}{2}+1} x+2 e x+2 e^{x/2} \left (x^2+3\right )+e^x \left (x^2+3\right )+e^2+3\right )^2}dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle 3 \int \frac {e^{16-x} \left (1+e^{x/2}\right ) \left (-x^3-3 e^{\frac {x}{2}+1} x^2-e^{x+1} x^2-(1+2 e) x^2-3 x+e^{\frac {x}{2}+2}+e^2 (1-x)+3 e^{x/2} \left (-x^3-x^2-3 x+3\right )+3 e^x \left (-x^3-x^2-3 x+3\right )+e^{3 x/2} \left (-x^3-x^2-3 x+3\right )+3\right )}{\left (x^2+2 e^{\frac {x}{2}+1} x+2 e x+2 e^{x/2} \left (x^2+3\right )+e^x \left (x^2+3\right )+3 \left (1+\frac {e^2}{3}\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 3 \int \left (\frac {e^{16-x} \left (-x^3-x^2-3 x+3\right )}{\left (x^2+3\right )^2}+\frac {e^{18-x} x \left (e^{x/2} x^6+x^6+e^{\frac {x}{2}+1} x^5+2 e x^5+2 \left (1+\frac {3}{2 e}\right ) e^{\frac {x}{2}+1} x^4+3 \left (1+\frac {1}{3} e (2+e)\right ) x^4-6 e^{\frac {x}{2}+1} x^3+2 e^2 x^3-36 \left (1+\frac {1}{4 e}\right ) e^{\frac {x}{2}+1} x^2-9 (1+4 e) x^2-27 e^{\frac {x}{2}+1} x-18 e (1+e) x+18 \left (1-\frac {3}{2 e}\right ) e^{\frac {x}{2}+1}-27 \left (1+\frac {1}{3} (-2+e) e\right )\right )}{\left (x^2+3\right )^3 \left (2 e^{x/2} x^2+e^x x^2+x^2+2 e^{\frac {x}{2}+1} x+2 e x+6 e^{x/2}+3 e^x+3 \left (1+\frac {e^2}{3}\right )\right )^2}+\frac {e^{17-x} \left (3 e^{x/2} x^6+2 x^6+4 e^{x/2} x^5+4 x^5+18 e^{x/2} x^4+12 \left (1-\frac {e}{12}\right ) x^4+6 e x^3+27 e^{x/2} x^2+18 (1+e) x^2-36 e^{x/2} x-36 \left (1-\frac {e}{2}\right ) x-9 e\right )}{\left (x^2+3\right )^3 \left (2 e^{x/2} x^2+e^x x^2+x^2+2 e^{\frac {x}{2}+1} x+2 e x+6 e^{x/2}+3 e^x+3 \left (1+\frac {e^2}{3}\right )\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 3 \int \frac {e^{16-x} \left (1+e^{x/2}\right ) \left (-x^3-3 e^{\frac {x}{2}+1} x^2-e^{x+1} x^2-(1+2 e) x^2-3 x+e^{\frac {x}{2}+2}-e^2 (x-1)-3 e^{x/2} \left (x^3+x^2+3 x-3\right )-3 e^x \left (x^3+x^2+3 x-3\right )-e^{3 x/2} \left (x^3+x^2+3 x-3\right )+3\right )}{\left (x^2+2 e^{\frac {x}{2}+1} x+2 e x+2 e^{x/2} \left (x^2+3\right )+e^x \left (x^2+3\right )+3 \left (1+\frac {e^2}{3}\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 3 \int \left (\frac {e^{16-x} \left (-x^3-x^2-3 x+3\right )}{\left (x^2+3\right )^2}+\frac {e^{18-x} x \left (e^{x/2} x^6+x^6+e^{\frac {x}{2}+1} x^5+2 e x^5+2 \left (1+\frac {3}{2 e}\right ) e^{\frac {x}{2}+1} x^4+3 \left (1+\frac {1}{3} e (2+e)\right ) x^4-6 e^{\frac {x}{2}+1} x^3+2 e^2 x^3-36 \left (1+\frac {1}{4 e}\right ) e^{\frac {x}{2}+1} x^2-9 (1+4 e) x^2-27 e^{\frac {x}{2}+1} x-18 e (1+e) x+18 \left (1-\frac {3}{2 e}\right ) e^{\frac {x}{2}+1}-27 \left (1+\frac {1}{3} (-2+e) e\right )\right )}{\left (x^2+3\right )^3 \left (2 e^{x/2} x^2+e^x x^2+x^2+2 e^{\frac {x}{2}+1} x+2 e x+6 e^{x/2}+3 e^x+3 \left (1+\frac {e^2}{3}\right )\right )^2}+\frac {e^{17-x} \left (3 e^{x/2} x^6+2 x^6+4 e^{x/2} x^5+4 x^5+18 e^{x/2} x^4+12 \left (1-\frac {e}{12}\right ) x^4+6 e x^3+27 e^{x/2} x^2+18 (1+e) x^2-36 e^{x/2} x-36 \left (1-\frac {e}{2}\right ) x-9 e\right )}{\left (x^2+3\right )^3 \left (2 e^{x/2} x^2+e^x x^2+x^2+2 e^{\frac {x}{2}+1} x+2 e x+6 e^{x/2}+3 e^x+3 \left (1+\frac {e^2}{3}\right )\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 3 \int \frac {e^{16-x} \left (1+e^{x/2}\right ) \left (-x^3-3 e^{\frac {x}{2}+1} x^2-e^{x+1} x^2-(1+2 e) x^2-3 x+e^{\frac {x}{2}+2}-e^2 (x-1)-3 e^{x/2} \left (x^3+x^2+3 x-3\right )-3 e^x \left (x^3+x^2+3 x-3\right )-e^{3 x/2} \left (x^3+x^2+3 x-3\right )+3\right )}{\left (x^2+2 e^{\frac {x}{2}+1} x+2 e x+2 e^{x/2} \left (x^2+3\right )+e^x \left (x^2+3\right )+3 \left (1+\frac {e^2}{3}\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 3 \int \left (\frac {e^{16-x} \left (-x^3-x^2-3 x+3\right )}{\left (x^2+3\right )^2}+\frac {e^{18-x} x \left (e^{x/2} x^6+x^6+e^{\frac {x}{2}+1} x^5+2 e x^5+2 \left (1+\frac {3}{2 e}\right ) e^{\frac {x}{2}+1} x^4+3 \left (1+\frac {1}{3} e (2+e)\right ) x^4-6 e^{\frac {x}{2}+1} x^3+2 e^2 x^3-36 \left (1+\frac {1}{4 e}\right ) e^{\frac {x}{2}+1} x^2-9 (1+4 e) x^2-27 e^{\frac {x}{2}+1} x-18 e (1+e) x+18 \left (1-\frac {3}{2 e}\right ) e^{\frac {x}{2}+1}-27 \left (1+\frac {1}{3} (-2+e) e\right )\right )}{\left (x^2+3\right )^3 \left (2 e^{x/2} x^2+e^x x^2+x^2+2 e^{\frac {x}{2}+1} x+2 e x+6 e^{x/2}+3 e^x+3 \left (1+\frac {e^2}{3}\right )\right )^2}+\frac {e^{17-x} \left (3 e^{x/2} x^6+2 x^6+4 e^{x/2} x^5+4 x^5+18 e^{x/2} x^4+12 \left (1-\frac {e}{12}\right ) x^4+6 e x^3+27 e^{x/2} x^2+18 (1+e) x^2-36 e^{x/2} x-36 \left (1-\frac {e}{2}\right ) x-9 e\right )}{\left (x^2+3\right )^3 \left (2 e^{x/2} x^2+e^x x^2+x^2+2 e^{\frac {x}{2}+1} x+2 e x+6 e^{x/2}+3 e^x+3 \left (1+\frac {e^2}{3}\right )\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 3 \int \frac {e^{16-x} \left (1+e^{x/2}\right ) \left (-x^3-3 e^{\frac {x}{2}+1} x^2-e^{x+1} x^2-(1+2 e) x^2-3 x+e^{\frac {x}{2}+2}-e^2 (x-1)-3 e^{x/2} \left (x^3+x^2+3 x-3\right )-3 e^x \left (x^3+x^2+3 x-3\right )-e^{3 x/2} \left (x^3+x^2+3 x-3\right )+3\right )}{\left (x^2+2 e^{\frac {x}{2}+1} x+2 e x+2 e^{x/2} \left (x^2+3\right )+e^x \left (x^2+3\right )+3 \left (1+\frac {e^2}{3}\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 3 \int \left (\frac {e^{16-x} \left (-x^3-x^2-3 x+3\right )}{\left (x^2+3\right )^2}+\frac {e^{18-x} x \left (e^{x/2} x^6+x^6+e^{\frac {x}{2}+1} x^5+2 e x^5+2 \left (1+\frac {3}{2 e}\right ) e^{\frac {x}{2}+1} x^4+3 \left (1+\frac {1}{3} e (2+e)\right ) x^4-6 e^{\frac {x}{2}+1} x^3+2 e^2 x^3-36 \left (1+\frac {1}{4 e}\right ) e^{\frac {x}{2}+1} x^2-9 (1+4 e) x^2-27 e^{\frac {x}{2}+1} x-18 e (1+e) x+18 \left (1-\frac {3}{2 e}\right ) e^{\frac {x}{2}+1}-27 \left (1+\frac {1}{3} (-2+e) e\right )\right )}{\left (x^2+3\right )^3 \left (2 e^{x/2} x^2+e^x x^2+x^2+2 e^{\frac {x}{2}+1} x+2 e x+6 e^{x/2}+3 e^x+3 \left (1+\frac {e^2}{3}\right )\right )^2}+\frac {e^{17-x} \left (3 e^{x/2} x^6+2 x^6+4 e^{x/2} x^5+4 x^5+18 e^{x/2} x^4+12 \left (1-\frac {e}{12}\right ) x^4+6 e x^3+27 e^{x/2} x^2+18 (1+e) x^2-36 e^{x/2} x-36 \left (1-\frac {e}{2}\right ) x-9 e\right )}{\left (x^2+3\right )^3 \left (2 e^{x/2} x^2+e^x x^2+x^2+2 e^{\frac {x}{2}+1} x+2 e x+6 e^{x/2}+3 e^x+3 \left (1+\frac {e^2}{3}\right )\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 3 \int \frac {e^{16-x} \left (1+e^{x/2}\right ) \left (-x^3-3 e^{\frac {x}{2}+1} x^2-e^{x+1} x^2-(1+2 e) x^2-3 x+e^{\frac {x}{2}+2}-e^2 (x-1)-3 e^{x/2} \left (x^3+x^2+3 x-3\right )-3 e^x \left (x^3+x^2+3 x-3\right )-e^{3 x/2} \left (x^3+x^2+3 x-3\right )+3\right )}{\left (x^2+2 e^{\frac {x}{2}+1} x+2 e x+2 e^{x/2} \left (x^2+3\right )+e^x \left (x^2+3\right )+3 \left (1+\frac {e^2}{3}\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 3 \int \left (\frac {e^{16-x} \left (-x^3-x^2-3 x+3\right )}{\left (x^2+3\right )^2}+\frac {e^{18-x} x \left (e^{x/2} x^6+x^6+e^{\frac {x}{2}+1} x^5+2 e x^5+2 \left (1+\frac {3}{2 e}\right ) e^{\frac {x}{2}+1} x^4+3 \left (1+\frac {1}{3} e (2+e)\right ) x^4-6 e^{\frac {x}{2}+1} x^3+2 e^2 x^3-36 \left (1+\frac {1}{4 e}\right ) e^{\frac {x}{2}+1} x^2-9 (1+4 e) x^2-27 e^{\frac {x}{2}+1} x-18 e (1+e) x+18 \left (1-\frac {3}{2 e}\right ) e^{\frac {x}{2}+1}-27 \left (1+\frac {1}{3} (-2+e) e\right )\right )}{\left (x^2+3\right )^3 \left (2 e^{x/2} x^2+e^x x^2+x^2+2 e^{\frac {x}{2}+1} x+2 e x+6 e^{x/2}+3 e^x+3 \left (1+\frac {e^2}{3}\right )\right )^2}+\frac {e^{17-x} \left (3 e^{x/2} x^6+2 x^6+4 e^{x/2} x^5+4 x^5+18 e^{x/2} x^4+12 \left (1-\frac {e}{12}\right ) x^4+6 e x^3+27 e^{x/2} x^2+18 (1+e) x^2-36 e^{x/2} x-36 \left (1-\frac {e}{2}\right ) x-9 e\right )}{\left (x^2+3\right )^3 \left (2 e^{x/2} x^2+e^x x^2+x^2+2 e^{\frac {x}{2}+1} x+2 e x+6 e^{x/2}+3 e^x+3 \left (1+\frac {e^2}{3}\right )\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 3 \int \frac {e^{16-x} \left (1+e^{x/2}\right ) \left (-x^3-3 e^{\frac {x}{2}+1} x^2-e^{x+1} x^2-(1+2 e) x^2-3 x+e^{\frac {x}{2}+2}-e^2 (x-1)-3 e^{x/2} \left (x^3+x^2+3 x-3\right )-3 e^x \left (x^3+x^2+3 x-3\right )-e^{3 x/2} \left (x^3+x^2+3 x-3\right )+3\right )}{\left (x^2+2 e^{\frac {x}{2}+1} x+2 e x+2 e^{x/2} \left (x^2+3\right )+e^x \left (x^2+3\right )+3 \left (1+\frac {e^2}{3}\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 3 \int \left (\frac {e^{16-x} \left (-x^3-x^2-3 x+3\right )}{\left (x^2+3\right )^2}+\frac {e^{18-x} x \left (e^{x/2} x^6+x^6+e^{\frac {x}{2}+1} x^5+2 e x^5+2 \left (1+\frac {3}{2 e}\right ) e^{\frac {x}{2}+1} x^4+3 \left (1+\frac {1}{3} e (2+e)\right ) x^4-6 e^{\frac {x}{2}+1} x^3+2 e^2 x^3-36 \left (1+\frac {1}{4 e}\right ) e^{\frac {x}{2}+1} x^2-9 (1+4 e) x^2-27 e^{\frac {x}{2}+1} x-18 e (1+e) x+18 \left (1-\frac {3}{2 e}\right ) e^{\frac {x}{2}+1}-27 \left (1+\frac {1}{3} (-2+e) e\right )\right )}{\left (x^2+3\right )^3 \left (2 e^{x/2} x^2+e^x x^2+x^2+2 e^{\frac {x}{2}+1} x+2 e x+6 e^{x/2}+3 e^x+3 \left (1+\frac {e^2}{3}\right )\right )^2}+\frac {e^{17-x} \left (3 e^{x/2} x^6+2 x^6+4 e^{x/2} x^5+4 x^5+18 e^{x/2} x^4+12 \left (1-\frac {e}{12}\right ) x^4+6 e x^3+27 e^{x/2} x^2+18 (1+e) x^2-36 e^{x/2} x-36 \left (1-\frac {e}{2}\right ) x-9 e\right )}{\left (x^2+3\right )^3 \left (2 e^{x/2} x^2+e^x x^2+x^2+2 e^{\frac {x}{2}+1} x+2 e x+6 e^{x/2}+3 e^x+3 \left (1+\frac {e^2}{3}\right )\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 3 \int \frac {e^{16-x} \left (1+e^{x/2}\right ) \left (-x^3-3 e^{\frac {x}{2}+1} x^2-e^{x+1} x^2-(1+2 e) x^2-3 x+e^{\frac {x}{2}+2}-e^2 (x-1)-3 e^{x/2} \left (x^3+x^2+3 x-3\right )-3 e^x \left (x^3+x^2+3 x-3\right )-e^{3 x/2} \left (x^3+x^2+3 x-3\right )+3\right )}{\left (x^2+2 e^{\frac {x}{2}+1} x+2 e x+2 e^{x/2} \left (x^2+3\right )+e^x \left (x^2+3\right )+3 \left (1+\frac {e^2}{3}\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 3 \int \left (\frac {e^{16-x} \left (-x^3-x^2-3 x+3\right )}{\left (x^2+3\right )^2}+\frac {e^{18-x} x \left (e^{x/2} x^6+x^6+e^{\frac {x}{2}+1} x^5+2 e x^5+2 \left (1+\frac {3}{2 e}\right ) e^{\frac {x}{2}+1} x^4+3 \left (1+\frac {1}{3} e (2+e)\right ) x^4-6 e^{\frac {x}{2}+1} x^3+2 e^2 x^3-36 \left (1+\frac {1}{4 e}\right ) e^{\frac {x}{2}+1} x^2-9 (1+4 e) x^2-27 e^{\frac {x}{2}+1} x-18 e (1+e) x+18 \left (1-\frac {3}{2 e}\right ) e^{\frac {x}{2}+1}-27 \left (1+\frac {1}{3} (-2+e) e\right )\right )}{\left (x^2+3\right )^3 \left (2 e^{x/2} x^2+e^x x^2+x^2+2 e^{\frac {x}{2}+1} x+2 e x+6 e^{x/2}+3 e^x+3 \left (1+\frac {e^2}{3}\right )\right )^2}+\frac {e^{17-x} \left (3 e^{x/2} x^6+2 x^6+4 e^{x/2} x^5+4 x^5+18 e^{x/2} x^4+12 \left (1-\frac {e}{12}\right ) x^4+6 e x^3+27 e^{x/2} x^2+18 (1+e) x^2-36 e^{x/2} x-36 \left (1-\frac {e}{2}\right ) x-9 e\right )}{\left (x^2+3\right )^3 \left (2 e^{x/2} x^2+e^x x^2+x^2+2 e^{\frac {x}{2}+1} x+2 e x+6 e^{x/2}+3 e^x+3 \left (1+\frac {e^2}{3}\right )\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 3 \int \frac {e^{16-x} \left (1+e^{x/2}\right ) \left (-x^3-3 e^{\frac {x}{2}+1} x^2-e^{x+1} x^2-(1+2 e) x^2-3 x+e^{\frac {x}{2}+2}-e^2 (x-1)-3 e^{x/2} \left (x^3+x^2+3 x-3\right )-3 e^x \left (x^3+x^2+3 x-3\right )-e^{3 x/2} \left (x^3+x^2+3 x-3\right )+3\right )}{\left (x^2+2 e^{\frac {x}{2}+1} x+2 e x+2 e^{x/2} \left (x^2+3\right )+e^x \left (x^2+3\right )+3 \left (1+\frac {e^2}{3}\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 3 \int \left (\frac {e^{16-x} \left (-x^3-x^2-3 x+3\right )}{\left (x^2+3\right )^2}+\frac {e^{18-x} x \left (e^{x/2} x^6+x^6+e^{\frac {x}{2}+1} x^5+2 e x^5+2 \left (1+\frac {3}{2 e}\right ) e^{\frac {x}{2}+1} x^4+3 \left (1+\frac {1}{3} e (2+e)\right ) x^4-6 e^{\frac {x}{2}+1} x^3+2 e^2 x^3-36 \left (1+\frac {1}{4 e}\right ) e^{\frac {x}{2}+1} x^2-9 (1+4 e) x^2-27 e^{\frac {x}{2}+1} x-18 e (1+e) x+18 \left (1-\frac {3}{2 e}\right ) e^{\frac {x}{2}+1}-27 \left (1+\frac {1}{3} (-2+e) e\right )\right )}{\left (x^2+3\right )^3 \left (2 e^{x/2} x^2+e^x x^2+x^2+2 e^{\frac {x}{2}+1} x+2 e x+6 e^{x/2}+3 e^x+3 \left (1+\frac {e^2}{3}\right )\right )^2}+\frac {e^{17-x} \left (3 e^{x/2} x^6+2 x^6+4 e^{x/2} x^5+4 x^5+18 e^{x/2} x^4+12 \left (1-\frac {e}{12}\right ) x^4+6 e x^3+27 e^{x/2} x^2+18 (1+e) x^2-36 e^{x/2} x-36 \left (1-\frac {e}{2}\right ) x-9 e\right )}{\left (x^2+3\right )^3 \left (2 e^{x/2} x^2+e^x x^2+x^2+2 e^{\frac {x}{2}+1} x+2 e x+6 e^{x/2}+3 e^x+3 \left (1+\frac {e^2}{3}\right )\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 3 \int \frac {e^{16-x} \left (1+e^{x/2}\right ) \left (-x^3-3 e^{\frac {x}{2}+1} x^2-e^{x+1} x^2-(1+2 e) x^2-3 x+e^{\frac {x}{2}+2}-e^2 (x-1)-3 e^{x/2} \left (x^3+x^2+3 x-3\right )-3 e^x \left (x^3+x^2+3 x-3\right )-e^{3 x/2} \left (x^3+x^2+3 x-3\right )+3\right )}{\left (x^2+2 e^{\frac {x}{2}+1} x+2 e x+2 e^{x/2} \left (x^2+3\right )+e^x \left (x^2+3\right )+3 \left (1+\frac {e^2}{3}\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 3 \int \left (\frac {e^{16-x} \left (-x^3-x^2-3 x+3\right )}{\left (x^2+3\right )^2}+\frac {e^{18-x} x \left (e^{x/2} x^6+x^6+e^{\frac {x}{2}+1} x^5+2 e x^5+2 \left (1+\frac {3}{2 e}\right ) e^{\frac {x}{2}+1} x^4+3 \left (1+\frac {1}{3} e (2+e)\right ) x^4-6 e^{\frac {x}{2}+1} x^3+2 e^2 x^3-36 \left (1+\frac {1}{4 e}\right ) e^{\frac {x}{2}+1} x^2-9 (1+4 e) x^2-27 e^{\frac {x}{2}+1} x-18 e (1+e) x+18 \left (1-\frac {3}{2 e}\right ) e^{\frac {x}{2}+1}-27 \left (1+\frac {1}{3} (-2+e) e\right )\right )}{\left (x^2+3\right )^3 \left (2 e^{x/2} x^2+e^x x^2+x^2+2 e^{\frac {x}{2}+1} x+2 e x+6 e^{x/2}+3 e^x+3 \left (1+\frac {e^2}{3}\right )\right )^2}+\frac {e^{17-x} \left (3 e^{x/2} x^6+2 x^6+4 e^{x/2} x^5+4 x^5+18 e^{x/2} x^4+12 \left (1-\frac {e}{12}\right ) x^4+6 e x^3+27 e^{x/2} x^2+18 (1+e) x^2-36 e^{x/2} x-36 \left (1-\frac {e}{2}\right ) x-9 e\right )}{\left (x^2+3\right )^3 \left (2 e^{x/2} x^2+e^x x^2+x^2+2 e^{\frac {x}{2}+1} x+2 e x+6 e^{x/2}+3 e^x+3 \left (1+\frac {e^2}{3}\right )\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 3 \int \frac {e^{16-x} \left (1+e^{x/2}\right ) \left (-x^3-3 e^{\frac {x}{2}+1} x^2-e^{x+1} x^2-(1+2 e) x^2-3 x+e^{\frac {x}{2}+2}-e^2 (x-1)-3 e^{x/2} \left (x^3+x^2+3 x-3\right )-3 e^x \left (x^3+x^2+3 x-3\right )-e^{3 x/2} \left (x^3+x^2+3 x-3\right )+3\right )}{\left (x^2+2 e^{\frac {x}{2}+1} x+2 e x+2 e^{x/2} \left (x^2+3\right )+e^x \left (x^2+3\right )+3 \left (1+\frac {e^2}{3}\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 3 \int \left (\frac {e^{16-x} \left (-x^3-x^2-3 x+3\right )}{\left (x^2+3\right )^2}+\frac {e^{18-x} x \left (e^{x/2} x^6+x^6+e^{\frac {x}{2}+1} x^5+2 e x^5+2 \left (1+\frac {3}{2 e}\right ) e^{\frac {x}{2}+1} x^4+3 \left (1+\frac {1}{3} e (2+e)\right ) x^4-6 e^{\frac {x}{2}+1} x^3+2 e^2 x^3-36 \left (1+\frac {1}{4 e}\right ) e^{\frac {x}{2}+1} x^2-9 (1+4 e) x^2-27 e^{\frac {x}{2}+1} x-18 e (1+e) x+18 \left (1-\frac {3}{2 e}\right ) e^{\frac {x}{2}+1}-27 \left (1+\frac {1}{3} (-2+e) e\right )\right )}{\left (x^2+3\right )^3 \left (2 e^{x/2} x^2+e^x x^2+x^2+2 e^{\frac {x}{2}+1} x+2 e x+6 e^{x/2}+3 e^x+3 \left (1+\frac {e^2}{3}\right )\right )^2}+\frac {e^{17-x} \left (3 e^{x/2} x^6+2 x^6+4 e^{x/2} x^5+4 x^5+18 e^{x/2} x^4+12 \left (1-\frac {e}{12}\right ) x^4+6 e x^3+27 e^{x/2} x^2+18 (1+e) x^2-36 e^{x/2} x-36 \left (1-\frac {e}{2}\right ) x-9 e\right )}{\left (x^2+3\right )^3 \left (2 e^{x/2} x^2+e^x x^2+x^2+2 e^{\frac {x}{2}+1} x+2 e x+6 e^{x/2}+3 e^x+3 \left (1+\frac {e^2}{3}\right )\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 3 \int \frac {e^{16-x} \left (1+e^{x/2}\right ) \left (-x^3-3 e^{\frac {x}{2}+1} x^2-e^{x+1} x^2-(1+2 e) x^2-3 x+e^{\frac {x}{2}+2}-e^2 (x-1)-3 e^{x/2} \left (x^3+x^2+3 x-3\right )-3 e^x \left (x^3+x^2+3 x-3\right )-e^{3 x/2} \left (x^3+x^2+3 x-3\right )+3\right )}{\left (x^2+2 e^{\frac {x}{2}+1} x+2 e x+2 e^{x/2} \left (x^2+3\right )+e^x \left (x^2+3\right )+3 \left (1+\frac {e^2}{3}\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 3 \int \left (\frac {e^{16-x} \left (-x^3-x^2-3 x+3\right )}{\left (x^2+3\right )^2}+\frac {e^{18-x} x \left (e^{x/2} x^6+x^6+e^{\frac {x}{2}+1} x^5+2 e x^5+2 \left (1+\frac {3}{2 e}\right ) e^{\frac {x}{2}+1} x^4+3 \left (1+\frac {1}{3} e (2+e)\right ) x^4-6 e^{\frac {x}{2}+1} x^3+2 e^2 x^3-36 \left (1+\frac {1}{4 e}\right ) e^{\frac {x}{2}+1} x^2-9 (1+4 e) x^2-27 e^{\frac {x}{2}+1} x-18 e (1+e) x+18 \left (1-\frac {3}{2 e}\right ) e^{\frac {x}{2}+1}-27 \left (1+\frac {1}{3} (-2+e) e\right )\right )}{\left (x^2+3\right )^3 \left (2 e^{x/2} x^2+e^x x^2+x^2+2 e^{\frac {x}{2}+1} x+2 e x+6 e^{x/2}+3 e^x+3 \left (1+\frac {e^2}{3}\right )\right )^2}+\frac {e^{17-x} \left (3 e^{x/2} x^6+2 x^6+4 e^{x/2} x^5+4 x^5+18 e^{x/2} x^4+12 \left (1-\frac {e}{12}\right ) x^4+6 e x^3+27 e^{x/2} x^2+18 (1+e) x^2-36 e^{x/2} x-36 \left (1-\frac {e}{2}\right ) x-9 e\right )}{\left (x^2+3\right )^3 \left (2 e^{x/2} x^2+e^x x^2+x^2+2 e^{\frac {x}{2}+1} x+2 e x+6 e^{x/2}+3 e^x+3 \left (1+\frac {e^2}{3}\right )\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 3 \int \frac {e^{16-x} \left (1+e^{x/2}\right ) \left (-x^3-3 e^{\frac {x}{2}+1} x^2-e^{x+1} x^2-(1+2 e) x^2-3 x+e^{\frac {x}{2}+2}-e^2 (x-1)-3 e^{x/2} \left (x^3+x^2+3 x-3\right )-3 e^x \left (x^3+x^2+3 x-3\right )-e^{3 x/2} \left (x^3+x^2+3 x-3\right )+3\right )}{\left (x^2+2 e^{\frac {x}{2}+1} x+2 e x+2 e^{x/2} \left (x^2+3\right )+e^x \left (x^2+3\right )+3 \left (1+\frac {e^2}{3}\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 3 \int \left (\frac {e^{16-x} \left (-x^3-x^2-3 x+3\right )}{\left (x^2+3\right )^2}+\frac {e^{18-x} x \left (e^{x/2} x^6+x^6+e^{\frac {x}{2}+1} x^5+2 e x^5+2 \left (1+\frac {3}{2 e}\right ) e^{\frac {x}{2}+1} x^4+3 \left (1+\frac {1}{3} e (2+e)\right ) x^4-6 e^{\frac {x}{2}+1} x^3+2 e^2 x^3-36 \left (1+\frac {1}{4 e}\right ) e^{\frac {x}{2}+1} x^2-9 (1+4 e) x^2-27 e^{\frac {x}{2}+1} x-18 e (1+e) x+18 \left (1-\frac {3}{2 e}\right ) e^{\frac {x}{2}+1}-27 \left (1+\frac {1}{3} (-2+e) e\right )\right )}{\left (x^2+3\right )^3 \left (2 e^{x/2} x^2+e^x x^2+x^2+2 e^{\frac {x}{2}+1} x+2 e x+6 e^{x/2}+3 e^x+3 \left (1+\frac {e^2}{3}\right )\right )^2}+\frac {e^{17-x} \left (3 e^{x/2} x^6+2 x^6+4 e^{x/2} x^5+4 x^5+18 e^{x/2} x^4+12 \left (1-\frac {e}{12}\right ) x^4+6 e x^3+27 e^{x/2} x^2+18 (1+e) x^2-36 e^{x/2} x-36 \left (1-\frac {e}{2}\right ) x-9 e\right )}{\left (x^2+3\right )^3 \left (2 e^{x/2} x^2+e^x x^2+x^2+2 e^{\frac {x}{2}+1} x+2 e x+6 e^{x/2}+3 e^x+3 \left (1+\frac {e^2}{3}\right )\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 3 \int \frac {e^{16-x} \left (1+e^{x/2}\right ) \left (-x^3-3 e^{\frac {x}{2}+1} x^2-e^{x+1} x^2-(1+2 e) x^2-3 x+e^{\frac {x}{2}+2}-e^2 (x-1)-3 e^{x/2} \left (x^3+x^2+3 x-3\right )-3 e^x \left (x^3+x^2+3 x-3\right )-e^{3 x/2} \left (x^3+x^2+3 x-3\right )+3\right )}{\left (x^2+2 e^{\frac {x}{2}+1} x+2 e x+2 e^{x/2} \left (x^2+3\right )+e^x \left (x^2+3\right )+3 \left (1+\frac {e^2}{3}\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 3 \int \left (\frac {e^{16-x} \left (-x^3-x^2-3 x+3\right )}{\left (x^2+3\right )^2}+\frac {e^{18-x} x \left (e^{x/2} x^6+x^6+e^{\frac {x}{2}+1} x^5+2 e x^5+2 \left (1+\frac {3}{2 e}\right ) e^{\frac {x}{2}+1} x^4+3 \left (1+\frac {1}{3} e (2+e)\right ) x^4-6 e^{\frac {x}{2}+1} x^3+2 e^2 x^3-36 \left (1+\frac {1}{4 e}\right ) e^{\frac {x}{2}+1} x^2-9 (1+4 e) x^2-27 e^{\frac {x}{2}+1} x-18 e (1+e) x+18 \left (1-\frac {3}{2 e}\right ) e^{\frac {x}{2}+1}-27 \left (1+\frac {1}{3} (-2+e) e\right )\right )}{\left (x^2+3\right )^3 \left (2 e^{x/2} x^2+e^x x^2+x^2+2 e^{\frac {x}{2}+1} x+2 e x+6 e^{x/2}+3 e^x+3 \left (1+\frac {e^2}{3}\right )\right )^2}+\frac {e^{17-x} \left (3 e^{x/2} x^6+2 x^6+4 e^{x/2} x^5+4 x^5+18 e^{x/2} x^4+12 \left (1-\frac {e}{12}\right ) x^4+6 e x^3+27 e^{x/2} x^2+18 (1+e) x^2-36 e^{x/2} x-36 \left (1-\frac {e}{2}\right ) x-9 e\right )}{\left (x^2+3\right )^3 \left (2 e^{x/2} x^2+e^x x^2+x^2+2 e^{\frac {x}{2}+1} x+2 e x+6 e^{x/2}+3 e^x+3 \left (1+\frac {e^2}{3}\right )\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 3 \int \frac {e^{16-x} \left (1+e^{x/2}\right ) \left (-x^3-3 e^{\frac {x}{2}+1} x^2-e^{x+1} x^2-(1+2 e) x^2-3 x+e^{\frac {x}{2}+2}-e^2 (x-1)-3 e^{x/2} \left (x^3+x^2+3 x-3\right )-3 e^x \left (x^3+x^2+3 x-3\right )-e^{3 x/2} \left (x^3+x^2+3 x-3\right )+3\right )}{\left (x^2+2 e^{\frac {x}{2}+1} x+2 e x+2 e^{x/2} \left (x^2+3\right )+e^x \left (x^2+3\right )+3 \left (1+\frac {e^2}{3}\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 3 \int \left (\frac {e^{16-x} \left (-x^3-x^2-3 x+3\right )}{\left (x^2+3\right )^2}+\frac {e^{18-x} x \left (e^{x/2} x^6+x^6+e^{\frac {x}{2}+1} x^5+2 e x^5+2 \left (1+\frac {3}{2 e}\right ) e^{\frac {x}{2}+1} x^4+3 \left (1+\frac {1}{3} e (2+e)\right ) x^4-6 e^{\frac {x}{2}+1} x^3+2 e^2 x^3-36 \left (1+\frac {1}{4 e}\right ) e^{\frac {x}{2}+1} x^2-9 (1+4 e) x^2-27 e^{\frac {x}{2}+1} x-18 e (1+e) x+18 \left (1-\frac {3}{2 e}\right ) e^{\frac {x}{2}+1}-27 \left (1+\frac {1}{3} (-2+e) e\right )\right )}{\left (x^2+3\right )^3 \left (2 e^{x/2} x^2+e^x x^2+x^2+2 e^{\frac {x}{2}+1} x+2 e x+6 e^{x/2}+3 e^x+3 \left (1+\frac {e^2}{3}\right )\right )^2}+\frac {e^{17-x} \left (3 e^{x/2} x^6+2 x^6+4 e^{x/2} x^5+4 x^5+18 e^{x/2} x^4+12 \left (1-\frac {e}{12}\right ) x^4+6 e x^3+27 e^{x/2} x^2+18 (1+e) x^2-36 e^{x/2} x-36 \left (1-\frac {e}{2}\right ) x-9 e\right )}{\left (x^2+3\right )^3 \left (2 e^{x/2} x^2+e^x x^2+x^2+2 e^{\frac {x}{2}+1} x+2 e x+6 e^{x/2}+3 e^x+3 \left (1+\frac {e^2}{3}\right )\right )}\right )dx\) |
Input:
Int[(E^16*(54 + 3*E^2 - 54*x - 18*x^2 - 12*E*x^2 - 18*x^3) + E^(16 - x/2)* (36 + E^2*(6 - 3*x) - 36*x - 12*x^2 - 15*E*x^2 - 12*x^3) + E^(16 + x/2)*(3 6 - 36*x - 12*x^2 - 3*E*x^2 - 12*x^3) + E^(16 + x)*(9 - 9*x - 3*x^2 - 3*x^ 3) + E^(16 - x)*(9 + E^2*(3 - 3*x) - 9*x - 3*x^2 - 6*E*x^2 - 3*x^3))/(9 + E^4 + 4*E^3*x + 6*x^2 + x^4 + E^2*(6 + 6*x^2) + E*(12*x + 4*x^3) + E^(2*x) *(9 + 6*x^2 + x^4) + E^((3*x)/2)*(36 + 24*x^2 + 4*x^4 + E*(12*x + 4*x^3)) + E^x*(54 + 36*x^2 + 6*x^4 + E^2*(6 + 6*x^2) + E*(36*x + 12*x^3)) + E^(x/2 )*(36 + 4*E^3*x + 24*x^2 + 4*x^4 + E^2*(12 + 12*x^2) + E*(36*x + 12*x^3))) ,x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(97\) vs. \(2(28)=56\).
Time = 7.78 (sec) , antiderivative size = 98, normalized size of antiderivative = 3.16
| method | result | size |
| parallelrisch | \(\frac {3 \,{\mathrm e}^{x} x \,{\mathrm e}^{16-x}+6 x \,{\mathrm e}^{\frac {x}{2}} {\mathrm e}^{16-x}+3 x \,{\mathrm e}^{16-x}}{{\mathrm e}^{x} x^{2}+2 \,{\mathrm e} \,{\mathrm e}^{\frac {x}{2}} x +2 x^{2} {\mathrm e}^{\frac {x}{2}}+{\mathrm e}^{2}+2 x \,{\mathrm e}+3 \,{\mathrm e}^{x}+x^{2}+6 \,{\mathrm e}^{\frac {x}{2}}+3}\) | \(98\) |
| risch | \(\frac {6 x \left (x +{\mathrm e}\right ) {\mathrm e}^{17-\frac {x}{2}}}{\left ({\mathrm e}^{2}+2 x \,{\mathrm e}+x^{2}+3\right )^{2}}+\frac {3 x \,{\mathrm e}^{16-x}}{{\mathrm e}^{2}+2 x \,{\mathrm e}+x^{2}+3}+\frac {3 \,{\mathrm e}^{17} x \left (-2 x^{2} {\mathrm e}^{1+\frac {x}{2}}-2 \,{\mathrm e}^{\frac {x}{2}} x^{3}+{\mathrm e}^{3}-3 x^{2} {\mathrm e}-2 x^{3}-6 \,{\mathrm e}^{1+\frac {x}{2}}-6 x \,{\mathrm e}^{\frac {x}{2}}-9 \,{\mathrm e}-6 x \right )}{\left ({\mathrm e}^{2}+2 x \,{\mathrm e}+x^{2}+3\right )^{2} \left ({\mathrm e}^{x} x^{2}+2 x \,{\mathrm e}^{1+\frac {x}{2}}+2 x^{2} {\mathrm e}^{\frac {x}{2}}+{\mathrm e}^{2}+2 x \,{\mathrm e}+3 \,{\mathrm e}^{x}+x^{2}+6 \,{\mathrm e}^{\frac {x}{2}}+3\right )}\) | \(176\) |
Input:
int(((-3*x^3-3*x^2-9*x+9)*exp(16-x)*exp(1/2*x)^4+(-3*x^2*exp(1)-12*x^3-12* x^2-36*x+36)*exp(16-x)*exp(1/2*x)^3+(3*exp(1)^2-12*x^2*exp(1)-18*x^3-18*x^ 2-54*x+54)*exp(16-x)*exp(1/2*x)^2+((-3*x+6)*exp(1)^2-15*x^2*exp(1)-12*x^3- 12*x^2-36*x+36)*exp(16-x)*exp(1/2*x)+((-3*x+3)*exp(1)^2-6*x^2*exp(1)-3*x^3 -3*x^2-9*x+9)*exp(16-x))/((x^4+6*x^2+9)*exp(1/2*x)^4+((4*x^3+12*x)*exp(1)+ 4*x^4+24*x^2+36)*exp(1/2*x)^3+((6*x^2+6)*exp(1)^2+(12*x^3+36*x)*exp(1)+6*x ^4+36*x^2+54)*exp(1/2*x)^2+(4*x*exp(1)^3+(12*x^2+12)*exp(1)^2+(12*x^3+36*x )*exp(1)+4*x^4+24*x^2+36)*exp(1/2*x)+exp(1)^4+4*x*exp(1)^3+(6*x^2+6)*exp(1 )^2+(4*x^3+12*x)*exp(1)+x^4+6*x^2+9),x,method=_RETURNVERBOSE)
Output:
(3*exp(1/2*x)^2*x*exp(16-x)+6*x*exp(1/2*x)*exp(16-x)+3*x*exp(16-x))/(x^2*e xp(1/2*x)^2+2*exp(1)*exp(1/2*x)*x+2*x^2*exp(1/2*x)+exp(1)^2+2*x*exp(1)+3*e xp(1/2*x)^2+x^2+6*exp(1/2*x)+3)
Leaf count of result is larger than twice the leaf count of optimal. 79 vs. \(2 (28) = 56\).
Time = 0.22 (sec) , antiderivative size = 79, normalized size of antiderivative = 2.55 \[ \int \frac {e^{16} \left (54+3 e^2-54 x-18 x^2-12 e x^2-18 x^3\right )+e^{16-\frac {x}{2}} \left (36+e^2 (6-3 x)-36 x-12 x^2-15 e x^2-12 x^3\right )+e^{16+\frac {x}{2}} \left (36-36 x-12 x^2-3 e x^2-12 x^3\right )+e^{16+x} \left (9-9 x-3 x^2-3 x^3\right )+e^{16-x} \left (9+e^2 (3-3 x)-9 x-3 x^2-6 e x^2-3 x^3\right )}{9+e^4+4 e^3 x+6 x^2+x^4+e^2 \left (6+6 x^2\right )+e \left (12 x+4 x^3\right )+e^{2 x} \left (9+6 x^2+x^4\right )+e^{3 x/2} \left (36+24 x^2+4 x^4+e \left (12 x+4 x^3\right )\right )+e^x \left (54+36 x^2+6 x^4+e^2 \left (6+6 x^2\right )+e \left (36 x+12 x^3\right )\right )+e^{x/2} \left (36+4 e^3 x+24 x^2+4 x^4+e^2 \left (12+12 x^2\right )+e \left (36 x+12 x^3\right )\right )} \, dx=\frac {3 \, {\left (x e^{80} + x e^{\left (x + 80\right )} + 2 \, x e^{\left (\frac {1}{2} \, x + 80\right )}\right )}}{{\left (x^{2} + 3\right )} e^{\left (2 \, x + 64\right )} + 2 \, {\left (x e^{17} + {\left (x^{2} + 3\right )} e^{16}\right )} e^{\left (\frac {3}{2} \, x + 48\right )} + {\left (2 \, x e^{33} + {\left (x^{2} + 3\right )} e^{32} + e^{34}\right )} e^{\left (x + 32\right )}} \] Input:
integrate(((-3*x^3-3*x^2-9*x+9)*exp(16-x)*exp(1/2*x)^4+(-3*x^2*exp(1)-12*x ^3-12*x^2-36*x+36)*exp(16-x)*exp(1/2*x)^3+(3*exp(1)^2-12*x^2*exp(1)-18*x^3 -18*x^2-54*x+54)*exp(16-x)*exp(1/2*x)^2+((-3*x+6)*exp(1)^2-15*x^2*exp(1)-1 2*x^3-12*x^2-36*x+36)*exp(16-x)*exp(1/2*x)+((-3*x+3)*exp(1)^2-6*x^2*exp(1) -3*x^3-3*x^2-9*x+9)*exp(16-x))/((x^4+6*x^2+9)*exp(1/2*x)^4+((4*x^3+12*x)*e xp(1)+4*x^4+24*x^2+36)*exp(1/2*x)^3+((6*x^2+6)*exp(1)^2+(12*x^3+36*x)*exp( 1)+6*x^4+36*x^2+54)*exp(1/2*x)^2+(4*x*exp(1)^3+(12*x^2+12)*exp(1)^2+(12*x^ 3+36*x)*exp(1)+4*x^4+24*x^2+36)*exp(1/2*x)+exp(1)^4+4*x*exp(1)^3+(6*x^2+6) *exp(1)^2+(4*x^3+12*x)*exp(1)+x^4+6*x^2+9),x, algorithm="fricas")
Output:
3*(x*e^80 + x*e^(x + 80) + 2*x*e^(1/2*x + 80))/((x^2 + 3)*e^(2*x + 64) + 2 *(x*e^17 + (x^2 + 3)*e^16)*e^(3/2*x + 48) + (2*x*e^33 + (x^2 + 3)*e^32 + e ^34)*e^(x + 32))
Leaf count of result is larger than twice the leaf count of optimal. 600 vs. \(2 (22) = 44\).
Time = 1.41 (sec) , antiderivative size = 600, normalized size of antiderivative = 19.35 \[ \int \frac {e^{16} \left (54+3 e^2-54 x-18 x^2-12 e x^2-18 x^3\right )+e^{16-\frac {x}{2}} \left (36+e^2 (6-3 x)-36 x-12 x^2-15 e x^2-12 x^3\right )+e^{16+\frac {x}{2}} \left (36-36 x-12 x^2-3 e x^2-12 x^3\right )+e^{16+x} \left (9-9 x-3 x^2-3 x^3\right )+e^{16-x} \left (9+e^2 (3-3 x)-9 x-3 x^2-6 e x^2-3 x^3\right )}{9+e^4+4 e^3 x+6 x^2+x^4+e^2 \left (6+6 x^2\right )+e \left (12 x+4 x^3\right )+e^{2 x} \left (9+6 x^2+x^4\right )+e^{3 x/2} \left (36+24 x^2+4 x^4+e \left (12 x+4 x^3\right )\right )+e^x \left (54+36 x^2+6 x^4+e^2 \left (6+6 x^2\right )+e \left (36 x+12 x^3\right )\right )+e^{x/2} \left (36+4 e^3 x+24 x^2+4 x^4+e^2 \left (12+12 x^2\right )+e \left (36 x+12 x^3\right )\right )} \, dx=\frac {\left (6 x^{4} e^{17} + 18 x^{3} e^{18} + 18 x^{2} e^{17} + 18 x^{2} e^{19} + 18 x e^{18} + 6 x e^{20}\right ) e^{- \frac {x}{2}} + \left (3 x^{5} e^{16} + 12 x^{4} e^{17} + 18 x^{3} e^{16} + 18 x^{3} e^{18} + 36 x^{2} e^{17} + 12 x^{2} e^{19} + 27 x e^{16} + 18 x e^{18} + 3 x e^{20}\right ) e^{- x}}{x^{6} + 6 e x^{5} + 9 x^{4} + 15 x^{4} e^{2} + 36 e x^{3} + 20 x^{3} e^{3} + 27 x^{2} + 54 x^{2} e^{2} + 15 x^{2} e^{4} + 54 e x + 36 x e^{3} + 6 x e^{5} + 27 + 27 e^{2} + e^{6} + 9 e^{4}} + \frac {- 6 x^{4} e^{17} - 9 x^{3} e^{18} - 18 x^{2} e^{17} - 27 x e^{18} + 3 x e^{20} + \left (- 6 x^{4} e^{17} - 6 x^{3} e^{18} - 18 x^{2} e^{17} - 18 x e^{18}\right ) e^{\frac {x}{2}}}{x^{6} + 6 e x^{5} + 9 x^{4} + 15 x^{4} e^{2} + 36 e x^{3} + 20 x^{3} e^{3} + 27 x^{2} + 54 x^{2} e^{2} + 15 x^{2} e^{4} + 54 e x + 36 x e^{3} + 6 x e^{5} + \left (x^{6} + 4 e x^{5} + 9 x^{4} + 6 x^{4} e^{2} + 24 e x^{3} + 4 x^{3} e^{3} + 27 x^{2} + x^{2} e^{4} + 24 x^{2} e^{2} + 36 e x + 12 x e^{3} + 27 + 18 e^{2} + 3 e^{4}\right ) e^{x} + \left (2 x^{6} + 10 e x^{5} + 18 x^{4} + 20 x^{4} e^{2} + 60 e x^{3} + 20 x^{3} e^{3} + 54 x^{2} + 72 x^{2} e^{2} + 10 x^{2} e^{4} + 90 e x + 2 x e^{5} + 36 x e^{3} + 54 + 36 e^{2} + 6 e^{4}\right ) e^{\frac {x}{2}} + 27 + 27 e^{2} + e^{6} + 9 e^{4}} \] Input:
integrate(((-3*x**3-3*x**2-9*x+9)*exp(16-x)*exp(1/2*x)**4+(-3*x**2*exp(1)- 12*x**3-12*x**2-36*x+36)*exp(16-x)*exp(1/2*x)**3+(3*exp(1)**2-12*x**2*exp( 1)-18*x**3-18*x**2-54*x+54)*exp(16-x)*exp(1/2*x)**2+((-3*x+6)*exp(1)**2-15 *x**2*exp(1)-12*x**3-12*x**2-36*x+36)*exp(16-x)*exp(1/2*x)+((-3*x+3)*exp(1 )**2-6*x**2*exp(1)-3*x**3-3*x**2-9*x+9)*exp(16-x))/((x**4+6*x**2+9)*exp(1/ 2*x)**4+((4*x**3+12*x)*exp(1)+4*x**4+24*x**2+36)*exp(1/2*x)**3+((6*x**2+6) *exp(1)**2+(12*x**3+36*x)*exp(1)+6*x**4+36*x**2+54)*exp(1/2*x)**2+(4*x*exp (1)**3+(12*x**2+12)*exp(1)**2+(12*x**3+36*x)*exp(1)+4*x**4+24*x**2+36)*exp (1/2*x)+exp(1)**4+4*x*exp(1)**3+(6*x**2+6)*exp(1)**2+(4*x**3+12*x)*exp(1)+ x**4+6*x**2+9),x)
Output:
((6*x**4*exp(17) + 18*x**3*exp(18) + 18*x**2*exp(17) + 18*x**2*exp(19) + 1 8*x*exp(18) + 6*x*exp(20))*exp(-x/2) + (3*x**5*exp(16) + 12*x**4*exp(17) + 18*x**3*exp(16) + 18*x**3*exp(18) + 36*x**2*exp(17) + 12*x**2*exp(19) + 2 7*x*exp(16) + 18*x*exp(18) + 3*x*exp(20))*exp(-x))/(x**6 + 6*E*x**5 + 9*x* *4 + 15*x**4*exp(2) + 36*E*x**3 + 20*x**3*exp(3) + 27*x**2 + 54*x**2*exp(2 ) + 15*x**2*exp(4) + 54*E*x + 36*x*exp(3) + 6*x*exp(5) + 27 + 27*exp(2) + exp(6) + 9*exp(4)) + (-6*x**4*exp(17) - 9*x**3*exp(18) - 18*x**2*exp(17) - 27*x*exp(18) + 3*x*exp(20) + (-6*x**4*exp(17) - 6*x**3*exp(18) - 18*x**2* exp(17) - 18*x*exp(18))*exp(x/2))/(x**6 + 6*E*x**5 + 9*x**4 + 15*x**4*exp( 2) + 36*E*x**3 + 20*x**3*exp(3) + 27*x**2 + 54*x**2*exp(2) + 15*x**2*exp(4 ) + 54*E*x + 36*x*exp(3) + 6*x*exp(5) + (x**6 + 4*E*x**5 + 9*x**4 + 6*x**4 *exp(2) + 24*E*x**3 + 4*x**3*exp(3) + 27*x**2 + x**2*exp(4) + 24*x**2*exp( 2) + 36*E*x + 12*x*exp(3) + 27 + 18*exp(2) + 3*exp(4))*exp(x) + (2*x**6 + 10*E*x**5 + 18*x**4 + 20*x**4*exp(2) + 60*E*x**3 + 20*x**3*exp(3) + 54*x** 2 + 72*x**2*exp(2) + 10*x**2*exp(4) + 90*E*x + 2*x*exp(5) + 36*x*exp(3) + 54 + 36*exp(2) + 6*exp(4))*exp(x/2) + 27 + 27*exp(2) + exp(6) + 9*exp(4))
Exception generated. \[ \int \frac {e^{16} \left (54+3 e^2-54 x-18 x^2-12 e x^2-18 x^3\right )+e^{16-\frac {x}{2}} \left (36+e^2 (6-3 x)-36 x-12 x^2-15 e x^2-12 x^3\right )+e^{16+\frac {x}{2}} \left (36-36 x-12 x^2-3 e x^2-12 x^3\right )+e^{16+x} \left (9-9 x-3 x^2-3 x^3\right )+e^{16-x} \left (9+e^2 (3-3 x)-9 x-3 x^2-6 e x^2-3 x^3\right )}{9+e^4+4 e^3 x+6 x^2+x^4+e^2 \left (6+6 x^2\right )+e \left (12 x+4 x^3\right )+e^{2 x} \left (9+6 x^2+x^4\right )+e^{3 x/2} \left (36+24 x^2+4 x^4+e \left (12 x+4 x^3\right )\right )+e^x \left (54+36 x^2+6 x^4+e^2 \left (6+6 x^2\right )+e \left (36 x+12 x^3\right )\right )+e^{x/2} \left (36+4 e^3 x+24 x^2+4 x^4+e^2 \left (12+12 x^2\right )+e \left (36 x+12 x^3\right )\right )} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(((-3*x^3-3*x^2-9*x+9)*exp(16-x)*exp(1/2*x)^4+(-3*x^2*exp(1)-12*x ^3-12*x^2-36*x+36)*exp(16-x)*exp(1/2*x)^3+(3*exp(1)^2-12*x^2*exp(1)-18*x^3 -18*x^2-54*x+54)*exp(16-x)*exp(1/2*x)^2+((-3*x+6)*exp(1)^2-15*x^2*exp(1)-1 2*x^3-12*x^2-36*x+36)*exp(16-x)*exp(1/2*x)+((-3*x+3)*exp(1)^2-6*x^2*exp(1) -3*x^3-3*x^2-9*x+9)*exp(16-x))/((x^4+6*x^2+9)*exp(1/2*x)^4+((4*x^3+12*x)*e xp(1)+4*x^4+24*x^2+36)*exp(1/2*x)^3+((6*x^2+6)*exp(1)^2+(12*x^3+36*x)*exp( 1)+6*x^4+36*x^2+54)*exp(1/2*x)^2+(4*x*exp(1)^3+(12*x^2+12)*exp(1)^2+(12*x^ 3+36*x)*exp(1)+4*x^4+24*x^2+36)*exp(1/2*x)+exp(1)^4+4*x*exp(1)^3+(6*x^2+6) *exp(1)^2+(4*x^3+12*x)*exp(1)+x^4+6*x^2+9),x, algorithm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is un defined.
Timed out. \[ \int \frac {e^{16} \left (54+3 e^2-54 x-18 x^2-12 e x^2-18 x^3\right )+e^{16-\frac {x}{2}} \left (36+e^2 (6-3 x)-36 x-12 x^2-15 e x^2-12 x^3\right )+e^{16+\frac {x}{2}} \left (36-36 x-12 x^2-3 e x^2-12 x^3\right )+e^{16+x} \left (9-9 x-3 x^2-3 x^3\right )+e^{16-x} \left (9+e^2 (3-3 x)-9 x-3 x^2-6 e x^2-3 x^3\right )}{9+e^4+4 e^3 x+6 x^2+x^4+e^2 \left (6+6 x^2\right )+e \left (12 x+4 x^3\right )+e^{2 x} \left (9+6 x^2+x^4\right )+e^{3 x/2} \left (36+24 x^2+4 x^4+e \left (12 x+4 x^3\right )\right )+e^x \left (54+36 x^2+6 x^4+e^2 \left (6+6 x^2\right )+e \left (36 x+12 x^3\right )\right )+e^{x/2} \left (36+4 e^3 x+24 x^2+4 x^4+e^2 \left (12+12 x^2\right )+e \left (36 x+12 x^3\right )\right )} \, dx=\text {Timed out} \] Input:
integrate(((-3*x^3-3*x^2-9*x+9)*exp(16-x)*exp(1/2*x)^4+(-3*x^2*exp(1)-12*x ^3-12*x^2-36*x+36)*exp(16-x)*exp(1/2*x)^3+(3*exp(1)^2-12*x^2*exp(1)-18*x^3 -18*x^2-54*x+54)*exp(16-x)*exp(1/2*x)^2+((-3*x+6)*exp(1)^2-15*x^2*exp(1)-1 2*x^3-12*x^2-36*x+36)*exp(16-x)*exp(1/2*x)+((-3*x+3)*exp(1)^2-6*x^2*exp(1) -3*x^3-3*x^2-9*x+9)*exp(16-x))/((x^4+6*x^2+9)*exp(1/2*x)^4+((4*x^3+12*x)*e xp(1)+4*x^4+24*x^2+36)*exp(1/2*x)^3+((6*x^2+6)*exp(1)^2+(12*x^3+36*x)*exp( 1)+6*x^4+36*x^2+54)*exp(1/2*x)^2+(4*x*exp(1)^3+(12*x^2+12)*exp(1)^2+(12*x^ 3+36*x)*exp(1)+4*x^4+24*x^2+36)*exp(1/2*x)+exp(1)^4+4*x*exp(1)^3+(6*x^2+6) *exp(1)^2+(4*x^3+12*x)*exp(1)+x^4+6*x^2+9),x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {e^{16} \left (54+3 e^2-54 x-18 x^2-12 e x^2-18 x^3\right )+e^{16-\frac {x}{2}} \left (36+e^2 (6-3 x)-36 x-12 x^2-15 e x^2-12 x^3\right )+e^{16+\frac {x}{2}} \left (36-36 x-12 x^2-3 e x^2-12 x^3\right )+e^{16+x} \left (9-9 x-3 x^2-3 x^3\right )+e^{16-x} \left (9+e^2 (3-3 x)-9 x-3 x^2-6 e x^2-3 x^3\right )}{9+e^4+4 e^3 x+6 x^2+x^4+e^2 \left (6+6 x^2\right )+e \left (12 x+4 x^3\right )+e^{2 x} \left (9+6 x^2+x^4\right )+e^{3 x/2} \left (36+24 x^2+4 x^4+e \left (12 x+4 x^3\right )\right )+e^x \left (54+36 x^2+6 x^4+e^2 \left (6+6 x^2\right )+e \left (36 x+12 x^3\right )\right )+e^{x/2} \left (36+4 e^3 x+24 x^2+4 x^4+e^2 \left (12+12 x^2\right )+e \left (36 x+12 x^3\right )\right )} \, dx=\int -\frac {{\mathrm {e}}^{16-x}\,\left (9\,x+6\,x^2\,\mathrm {e}+3\,x^2+3\,x^3+{\mathrm {e}}^2\,\left (3\,x-3\right )-9\right )+{\mathrm {e}}^{x/2}\,{\mathrm {e}}^{16-x}\,\left (36\,x+15\,x^2\,\mathrm {e}+12\,x^2+12\,x^3+{\mathrm {e}}^2\,\left (3\,x-6\right )-36\right )+{\mathrm {e}}^{\frac {3\,x}{2}}\,{\mathrm {e}}^{16-x}\,\left (36\,x+3\,x^2\,\mathrm {e}+12\,x^2+12\,x^3-36\right )+{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{16-x}\,\left (3\,x^3+3\,x^2+9\,x-9\right )+{\mathrm {e}}^{16-x}\,{\mathrm {e}}^x\,\left (54\,x-3\,{\mathrm {e}}^2+12\,x^2\,\mathrm {e}+18\,x^2+18\,x^3-54\right )}{{\mathrm {e}}^4+{\mathrm {e}}^{2\,x}\,\left (x^4+6\,x^2+9\right )+\mathrm {e}\,\left (4\,x^3+12\,x\right )+{\mathrm {e}}^{x/2}\,\left (\mathrm {e}\,\left (12\,x^3+36\,x\right )+4\,x\,{\mathrm {e}}^3+{\mathrm {e}}^2\,\left (12\,x^2+12\right )+24\,x^2+4\,x^4+36\right )+4\,x\,{\mathrm {e}}^3+{\mathrm {e}}^2\,\left (6\,x^2+6\right )+{\mathrm {e}}^x\,\left (\mathrm {e}\,\left (12\,x^3+36\,x\right )+{\mathrm {e}}^2\,\left (6\,x^2+6\right )+36\,x^2+6\,x^4+54\right )+{\mathrm {e}}^{\frac {3\,x}{2}}\,\left (\mathrm {e}\,\left (4\,x^3+12\,x\right )+24\,x^2+4\,x^4+36\right )+6\,x^2+x^4+9} \,d x \] Input:
int(-(exp(16 - x)*(9*x + 6*x^2*exp(1) + 3*x^2 + 3*x^3 + exp(2)*(3*x - 3) - 9) + exp(x/2)*exp(16 - x)*(36*x + 15*x^2*exp(1) + 12*x^2 + 12*x^3 + exp(2 )*(3*x - 6) - 36) + exp((3*x)/2)*exp(16 - x)*(36*x + 3*x^2*exp(1) + 12*x^2 + 12*x^3 - 36) + exp(2*x)*exp(16 - x)*(9*x + 3*x^2 + 3*x^3 - 9) + exp(16 - x)*exp(x)*(54*x - 3*exp(2) + 12*x^2*exp(1) + 18*x^2 + 18*x^3 - 54))/(exp (4) + exp(2*x)*(6*x^2 + x^4 + 9) + exp(1)*(12*x + 4*x^3) + exp(x/2)*(exp(1 )*(36*x + 12*x^3) + 4*x*exp(3) + exp(2)*(12*x^2 + 12) + 24*x^2 + 4*x^4 + 3 6) + 4*x*exp(3) + exp(2)*(6*x^2 + 6) + exp(x)*(exp(1)*(36*x + 12*x^3) + ex p(2)*(6*x^2 + 6) + 36*x^2 + 6*x^4 + 54) + exp((3*x)/2)*(exp(1)*(12*x + 4*x ^3) + 24*x^2 + 4*x^4 + 36) + 6*x^2 + x^4 + 9),x)
Output:
int(-(exp(16 - x)*(9*x + 6*x^2*exp(1) + 3*x^2 + 3*x^3 + exp(2)*(3*x - 3) - 9) + exp(x/2)*exp(16 - x)*(36*x + 15*x^2*exp(1) + 12*x^2 + 12*x^3 + exp(2 )*(3*x - 6) - 36) + exp((3*x)/2)*exp(16 - x)*(36*x + 3*x^2*exp(1) + 12*x^2 + 12*x^3 - 36) + exp(2*x)*exp(16 - x)*(9*x + 3*x^2 + 3*x^3 - 9) + exp(16 - x)*exp(x)*(54*x - 3*exp(2) + 12*x^2*exp(1) + 18*x^2 + 18*x^3 - 54))/(exp (4) + exp(2*x)*(6*x^2 + x^4 + 9) + exp(1)*(12*x + 4*x^3) + exp(x/2)*(exp(1 )*(36*x + 12*x^3) + 4*x*exp(3) + exp(2)*(12*x^2 + 12) + 24*x^2 + 4*x^4 + 3 6) + 4*x*exp(3) + exp(2)*(6*x^2 + 6) + exp(x)*(exp(1)*(36*x + 12*x^3) + ex p(2)*(6*x^2 + 6) + 36*x^2 + 6*x^4 + 54) + exp((3*x)/2)*(exp(1)*(12*x + 4*x ^3) + 24*x^2 + 4*x^4 + 36) + 6*x^2 + x^4 + 9), x)
\[ \int \frac {e^{16} \left (54+3 e^2-54 x-18 x^2-12 e x^2-18 x^3\right )+e^{16-\frac {x}{2}} \left (36+e^2 (6-3 x)-36 x-12 x^2-15 e x^2-12 x^3\right )+e^{16+\frac {x}{2}} \left (36-36 x-12 x^2-3 e x^2-12 x^3\right )+e^{16+x} \left (9-9 x-3 x^2-3 x^3\right )+e^{16-x} \left (9+e^2 (3-3 x)-9 x-3 x^2-6 e x^2-3 x^3\right )}{9+e^4+4 e^3 x+6 x^2+x^4+e^2 \left (6+6 x^2\right )+e \left (12 x+4 x^3\right )+e^{2 x} \left (9+6 x^2+x^4\right )+e^{3 x/2} \left (36+24 x^2+4 x^4+e \left (12 x+4 x^3\right )\right )+e^x \left (54+36 x^2+6 x^4+e^2 \left (6+6 x^2\right )+e \left (36 x+12 x^3\right )\right )+e^{x/2} \left (36+4 e^3 x+24 x^2+4 x^4+e^2 \left (12+12 x^2\right )+e \left (36 x+12 x^3\right )\right )} \, dx=\int \frac {\left (-3 x^{3}-3 x^{2}-9 x +9\right ) {\mathrm e}^{16-x} \left ({\mathrm e}^{\frac {x}{2}}\right )^{4}+\left (-3 x^{2} {\mathrm e}-12 x^{3}-12 x^{2}-36 x +36\right ) {\mathrm e}^{16-x} \left ({\mathrm e}^{\frac {x}{2}}\right )^{3}+\left (3 \left ({\mathrm e}\right )^{2}-12 x^{2} {\mathrm e}-18 x^{3}-18 x^{2}-54 x +54\right ) {\mathrm e}^{16-x} \left ({\mathrm e}^{\frac {x}{2}}\right )^{2}+\left (\left (-3 x +6\right ) \left ({\mathrm e}\right )^{2}-15 x^{2} {\mathrm e}-12 x^{3}-12 x^{2}-36 x +36\right ) {\mathrm e}^{16-x} {\mathrm e}^{\frac {x}{2}}+\left (\left (-3 x +3\right ) \left ({\mathrm e}\right )^{2}-6 x^{2} {\mathrm e}-3 x^{3}-3 x^{2}-9 x +9\right ) {\mathrm e}^{16-x}}{\left (x^{4}+6 x^{2}+9\right ) \left ({\mathrm e}^{\frac {x}{2}}\right )^{4}+\left (\left (4 x^{3}+12 x \right ) {\mathrm e}+4 x^{4}+24 x^{2}+36\right ) \left ({\mathrm e}^{\frac {x}{2}}\right )^{3}+\left (\left (6 x^{2}+6\right ) \left ({\mathrm e}\right )^{2}+\left (12 x^{3}+36 x \right ) {\mathrm e}+6 x^{4}+36 x^{2}+54\right ) \left ({\mathrm e}^{\frac {x}{2}}\right )^{2}+\left (4 x \left ({\mathrm e}\right )^{3}+\left (12 x^{2}+12\right ) \left ({\mathrm e}\right )^{2}+\left (12 x^{3}+36 x \right ) {\mathrm e}+4 x^{4}+24 x^{2}+36\right ) {\mathrm e}^{\frac {x}{2}}+\left ({\mathrm e}\right )^{4}+4 x \left ({\mathrm e}\right )^{3}+\left (6 x^{2}+6\right ) \left ({\mathrm e}\right )^{2}+\left (4 x^{3}+12 x \right ) {\mathrm e}+x^{4}+6 x^{2}+9}d x \] Input:
int(((-3*x^3-3*x^2-9*x+9)*exp(16-x)*exp(1/2*x)^4+(-3*x^2*exp(1)-12*x^3-12* x^2-36*x+36)*exp(16-x)*exp(1/2*x)^3+(3*exp(1)^2-12*x^2*exp(1)-18*x^3-18*x^ 2-54*x+54)*exp(16-x)*exp(1/2*x)^2+((-3*x+6)*exp(1)^2-15*x^2*exp(1)-12*x^3- 12*x^2-36*x+36)*exp(16-x)*exp(1/2*x)+((-3*x+3)*exp(1)^2-6*x^2*exp(1)-3*x^3 -3*x^2-9*x+9)*exp(16-x))/((x^4+6*x^2+9)*exp(1/2*x)^4+((4*x^3+12*x)*exp(1)+ 4*x^4+24*x^2+36)*exp(1/2*x)^3+((6*x^2+6)*exp(1)^2+(12*x^3+36*x)*exp(1)+6*x ^4+36*x^2+54)*exp(1/2*x)^2+(4*x*exp(1)^3+(12*x^2+12)*exp(1)^2+(12*x^3+36*x )*exp(1)+4*x^4+24*x^2+36)*exp(1/2*x)+exp(1)^4+4*x*exp(1)^3+(6*x^2+6)*exp(1 )^2+(4*x^3+12*x)*exp(1)+x^4+6*x^2+9),x)
Output:
int(((-3*x^3-3*x^2-9*x+9)*exp(16-x)*exp(1/2*x)^4+(-3*x^2*exp(1)-12*x^3-12* x^2-36*x+36)*exp(16-x)*exp(1/2*x)^3+(3*exp(1)^2-12*x^2*exp(1)-18*x^3-18*x^ 2-54*x+54)*exp(16-x)*exp(1/2*x)^2+((-3*x+6)*exp(1)^2-15*x^2*exp(1)-12*x^3- 12*x^2-36*x+36)*exp(16-x)*exp(1/2*x)+((-3*x+3)*exp(1)^2-6*x^2*exp(1)-3*x^3 -3*x^2-9*x+9)*exp(16-x))/((x^4+6*x^2+9)*exp(1/2*x)^4+((4*x^3+12*x)*exp(1)+ 4*x^4+24*x^2+36)*exp(1/2*x)^3+((6*x^2+6)*exp(1)^2+(12*x^3+36*x)*exp(1)+6*x ^4+36*x^2+54)*exp(1/2*x)^2+(4*x*exp(1)^3+(12*x^2+12)*exp(1)^2+(12*x^3+36*x )*exp(1)+4*x^4+24*x^2+36)*exp(1/2*x)+exp(1)^4+4*x*exp(1)^3+(6*x^2+6)*exp(1 )^2+(4*x^3+12*x)*exp(1)+x^4+6*x^2+9),x)