Integrand size = 170, antiderivative size = 34 \[ \int \frac {360-1800 x-744 x^2-408 x^3-528 x^4-104 x^5-32 x^6-32 x^7+e^{2 x} \left (-72 x^3-72 x^4-18 x^5\right )+e^x \left (828 x^2+540 x^3+120 x^4+120 x^5+48 x^6\right )}{900+360 x+156 x^2+264 x^3+52 x^4+16 x^5+16 x^6+e^{2 x} \left (36 x^2+36 x^3+9 x^4\right )+e^x \left (-360 x-252 x^2-60 x^3-60 x^4-24 x^5\right )} \, dx=-x^2+\frac {4}{-3+x^2-\left (e^x-\frac {5}{x}-\frac {x}{3}\right ) (2+x)} \] Output:
4/(x^2-3-(exp(x)-5/x-1/3*x)*(2+x))-x^2
Time = 6.67 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.03 \[ \int \frac {360-1800 x-744 x^2-408 x^3-528 x^4-104 x^5-32 x^6-32 x^7+e^{2 x} \left (-72 x^3-72 x^4-18 x^5\right )+e^x \left (828 x^2+540 x^3+120 x^4+120 x^5+48 x^6\right )}{900+360 x+156 x^2+264 x^3+52 x^4+16 x^5+16 x^6+e^{2 x} \left (36 x^2+36 x^3+9 x^4\right )+e^x \left (-360 x-252 x^2-60 x^3-60 x^4-24 x^5\right )} \, dx=-x \left (x-\frac {12}{30-6 \left (-1+e^x\right ) x+\left (2-3 e^x\right ) x^2+4 x^3}\right ) \] Input:
Integrate[(360 - 1800*x - 744*x^2 - 408*x^3 - 528*x^4 - 104*x^5 - 32*x^6 - 32*x^7 + E^(2*x)*(-72*x^3 - 72*x^4 - 18*x^5) + E^x*(828*x^2 + 540*x^3 + 1 20*x^4 + 120*x^5 + 48*x^6))/(900 + 360*x + 156*x^2 + 264*x^3 + 52*x^4 + 16 *x^5 + 16*x^6 + E^(2*x)*(36*x^2 + 36*x^3 + 9*x^4) + E^x*(-360*x - 252*x^2 - 60*x^3 - 60*x^4 - 24*x^5)),x]
Output:
-(x*(x - 12/(30 - 6*(-1 + E^x)*x + (2 - 3*E^x)*x^2 + 4*x^3)))
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.
| \(\displaystyle \int \frac {-32 x^7-32 x^6-104 x^5-528 x^4-408 x^3-744 x^2+e^{2 x} \left (-18 x^5-72 x^4-72 x^3\right )+e^x \left (48 x^6+120 x^5+120 x^4+540 x^3+828 x^2\right )-1800 x+360}{16 x^6+16 x^5+52 x^4+264 x^3+156 x^2+e^{2 x} \left (9 x^4+36 x^3+36 x^2\right )+e^x \left (-24 x^5-60 x^4-60 x^3-252 x^2-360 x\right )+360 x+900} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {-32 x^7-32 x^6-104 x^5-528 x^4-408 x^3-744 x^2+e^{2 x} \left (-18 x^5-72 x^4-72 x^3\right )+e^x \left (48 x^6+120 x^5+120 x^4+540 x^3+828 x^2\right )-1800 x+360}{\left (4 x^3-3 e^x x^2+2 x^2-6 e^x x+6 x+30\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {12 (x+3) x}{(x+2) \left (4 x^3-3 e^x x^2+2 x^2-6 e^x x+6 x+30\right )}+\frac {24 \left (2 x^5+3 x^4-3 x^3+22 x^2+60 x+30\right )}{(x+2) \left (4 x^3-3 e^x x^2+2 x^2-6 e^x x+6 x+30\right )^2}-2 x\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 288 \int \frac {1}{\left (4 x^3-3 e^x x^2+2 x^2-6 e^x x+6 x+30\right )^2}dx+576 \int \frac {x}{\left (4 x^3-3 e^x x^2+2 x^2-6 e^x x+6 x+30\right )^2}dx-24 \int \frac {x^2}{\left (4 x^3-3 e^x x^2+2 x^2-6 e^x x+6 x+30\right )^2}dx-24 \int \frac {x^3}{\left (4 x^3-3 e^x x^2+2 x^2-6 e^x x+6 x+30\right )^2}dx+144 \int \frac {1}{(x+2) \left (4 x^3-3 e^x x^2+2 x^2-6 e^x x+6 x+30\right )^2}dx-12 \int \frac {1}{4 x^3-3 e^x x^2+2 x^2-6 e^x x+6 x+30}dx-12 \int \frac {x}{4 x^3-3 e^x x^2+2 x^2-6 e^x x+6 x+30}dx+24 \int \frac {1}{(x+2) \left (4 x^3-3 e^x x^2+2 x^2-6 e^x x+6 x+30\right )}dx+48 \int \frac {x^4}{\left (4 x^3-3 e^x x^2+2 x^2-6 e^x x+6 x+30\right )^2}dx-x^2\) |
Input:
Int[(360 - 1800*x - 744*x^2 - 408*x^3 - 528*x^4 - 104*x^5 - 32*x^6 - 32*x^ 7 + E^(2*x)*(-72*x^3 - 72*x^4 - 18*x^5) + E^x*(828*x^2 + 540*x^3 + 120*x^4 + 120*x^5 + 48*x^6))/(900 + 360*x + 156*x^2 + 264*x^3 + 52*x^4 + 16*x^5 + 16*x^6 + E^(2*x)*(36*x^2 + 36*x^3 + 9*x^4) + E^x*(-360*x - 252*x^2 - 60*x ^3 - 60*x^4 - 24*x^5)),x]
Output:
$Aborted
Time = 0.68 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.15
| method | result | size |
| risch | \(-x^{2}+\frac {12 x}{4 x^{3}-3 \,{\mathrm e}^{x} x^{2}+2 x^{2}-6 \,{\mathrm e}^{x} x +6 x +30}\) | \(39\) |
| parallelrisch | \(-\frac {12 x^{5}-9 \,{\mathrm e}^{x} x^{4}+6 x^{4}-18 \,{\mathrm e}^{x} x^{3}+18 x^{3}+90 x^{2}-36 x}{3 \left (4 x^{3}-3 \,{\mathrm e}^{x} x^{2}+2 x^{2}-6 \,{\mathrm e}^{x} x +6 x +30\right )}\) | \(70\) |
| norman | \(\frac {-27 x^{2}+21 x -9 \,{\mathrm e}^{x} x -\frac {9 \,{\mathrm e}^{x} x^{2}}{2}-2 x^{4}-4 x^{5}+6 \,{\mathrm e}^{x} x^{3}+3 \,{\mathrm e}^{x} x^{4}+45}{4 x^{3}-3 \,{\mathrm e}^{x} x^{2}+2 x^{2}-6 \,{\mathrm e}^{x} x +6 x +30}\) | \(77\) |
Input:
int(((-18*x^5-72*x^4-72*x^3)*exp(x)^2+(48*x^6+120*x^5+120*x^4+540*x^3+828* x^2)*exp(x)-32*x^7-32*x^6-104*x^5-528*x^4-408*x^3-744*x^2-1800*x+360)/((9* x^4+36*x^3+36*x^2)*exp(x)^2+(-24*x^5-60*x^4-60*x^3-252*x^2-360*x)*exp(x)+1 6*x^6+16*x^5+52*x^4+264*x^3+156*x^2+360*x+900),x,method=_RETURNVERBOSE)
Output:
-x^2+12*x/(4*x^3-3*exp(x)*x^2+2*x^2-6*exp(x)*x+6*x+30)
Leaf count of result is larger than twice the leaf count of optimal. 67 vs. \(2 (32) = 64\).
Time = 0.10 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.97 \[ \int \frac {360-1800 x-744 x^2-408 x^3-528 x^4-104 x^5-32 x^6-32 x^7+e^{2 x} \left (-72 x^3-72 x^4-18 x^5\right )+e^x \left (828 x^2+540 x^3+120 x^4+120 x^5+48 x^6\right )}{900+360 x+156 x^2+264 x^3+52 x^4+16 x^5+16 x^6+e^{2 x} \left (36 x^2+36 x^3+9 x^4\right )+e^x \left (-360 x-252 x^2-60 x^3-60 x^4-24 x^5\right )} \, dx=-\frac {4 \, x^{5} + 2 \, x^{4} + 6 \, x^{3} + 30 \, x^{2} - 3 \, {\left (x^{4} + 2 \, x^{3}\right )} e^{x} - 12 \, x}{4 \, x^{3} + 2 \, x^{2} - 3 \, {\left (x^{2} + 2 \, x\right )} e^{x} + 6 \, x + 30} \] Input:
integrate(((-18*x^5-72*x^4-72*x^3)*exp(x)^2+(48*x^6+120*x^5+120*x^4+540*x^ 3+828*x^2)*exp(x)-32*x^7-32*x^6-104*x^5-528*x^4-408*x^3-744*x^2-1800*x+360 )/((9*x^4+36*x^3+36*x^2)*exp(x)^2+(-24*x^5-60*x^4-60*x^3-252*x^2-360*x)*ex p(x)+16*x^6+16*x^5+52*x^4+264*x^3+156*x^2+360*x+900),x, algorithm="fricas" )
Output:
-(4*x^5 + 2*x^4 + 6*x^3 + 30*x^2 - 3*(x^4 + 2*x^3)*e^x - 12*x)/(4*x^3 + 2* x^2 - 3*(x^2 + 2*x)*e^x + 6*x + 30)
Time = 0.19 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00 \[ \int \frac {360-1800 x-744 x^2-408 x^3-528 x^4-104 x^5-32 x^6-32 x^7+e^{2 x} \left (-72 x^3-72 x^4-18 x^5\right )+e^x \left (828 x^2+540 x^3+120 x^4+120 x^5+48 x^6\right )}{900+360 x+156 x^2+264 x^3+52 x^4+16 x^5+16 x^6+e^{2 x} \left (36 x^2+36 x^3+9 x^4\right )+e^x \left (-360 x-252 x^2-60 x^3-60 x^4-24 x^5\right )} \, dx=- x^{2} - \frac {12 x}{- 4 x^{3} - 2 x^{2} - 6 x + \left (3 x^{2} + 6 x\right ) e^{x} - 30} \] Input:
integrate(((-18*x**5-72*x**4-72*x**3)*exp(x)**2+(48*x**6+120*x**5+120*x**4 +540*x**3+828*x**2)*exp(x)-32*x**7-32*x**6-104*x**5-528*x**4-408*x**3-744* x**2-1800*x+360)/((9*x**4+36*x**3+36*x**2)*exp(x)**2+(-24*x**5-60*x**4-60* x**3-252*x**2-360*x)*exp(x)+16*x**6+16*x**5+52*x**4+264*x**3+156*x**2+360* x+900),x)
Output:
-x**2 - 12*x/(-4*x**3 - 2*x**2 - 6*x + (3*x**2 + 6*x)*exp(x) - 30)
Leaf count of result is larger than twice the leaf count of optimal. 67 vs. \(2 (32) = 64\).
Time = 0.10 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.97 \[ \int \frac {360-1800 x-744 x^2-408 x^3-528 x^4-104 x^5-32 x^6-32 x^7+e^{2 x} \left (-72 x^3-72 x^4-18 x^5\right )+e^x \left (828 x^2+540 x^3+120 x^4+120 x^5+48 x^6\right )}{900+360 x+156 x^2+264 x^3+52 x^4+16 x^5+16 x^6+e^{2 x} \left (36 x^2+36 x^3+9 x^4\right )+e^x \left (-360 x-252 x^2-60 x^3-60 x^4-24 x^5\right )} \, dx=-\frac {4 \, x^{5} + 2 \, x^{4} + 6 \, x^{3} + 30 \, x^{2} - 3 \, {\left (x^{4} + 2 \, x^{3}\right )} e^{x} - 12 \, x}{4 \, x^{3} + 2 \, x^{2} - 3 \, {\left (x^{2} + 2 \, x\right )} e^{x} + 6 \, x + 30} \] Input:
integrate(((-18*x^5-72*x^4-72*x^3)*exp(x)^2+(48*x^6+120*x^5+120*x^4+540*x^ 3+828*x^2)*exp(x)-32*x^7-32*x^6-104*x^5-528*x^4-408*x^3-744*x^2-1800*x+360 )/((9*x^4+36*x^3+36*x^2)*exp(x)^2+(-24*x^5-60*x^4-60*x^3-252*x^2-360*x)*ex p(x)+16*x^6+16*x^5+52*x^4+264*x^3+156*x^2+360*x+900),x, algorithm="maxima" )
Output:
-(4*x^5 + 2*x^4 + 6*x^3 + 30*x^2 - 3*(x^4 + 2*x^3)*e^x - 12*x)/(4*x^3 + 2* x^2 - 3*(x^2 + 2*x)*e^x + 6*x + 30)
Leaf count of result is larger than twice the leaf count of optimal. 69 vs. \(2 (32) = 64\).
Time = 0.13 (sec) , antiderivative size = 69, normalized size of antiderivative = 2.03 \[ \int \frac {360-1800 x-744 x^2-408 x^3-528 x^4-104 x^5-32 x^6-32 x^7+e^{2 x} \left (-72 x^3-72 x^4-18 x^5\right )+e^x \left (828 x^2+540 x^3+120 x^4+120 x^5+48 x^6\right )}{900+360 x+156 x^2+264 x^3+52 x^4+16 x^5+16 x^6+e^{2 x} \left (36 x^2+36 x^3+9 x^4\right )+e^x \left (-360 x-252 x^2-60 x^3-60 x^4-24 x^5\right )} \, dx=-\frac {4 \, x^{5} - 3 \, x^{4} e^{x} + 2 \, x^{4} - 6 \, x^{3} e^{x} + 6 \, x^{3} + 30 \, x^{2} - 12 \, x}{4 \, x^{3} - 3 \, x^{2} e^{x} + 2 \, x^{2} - 6 \, x e^{x} + 6 \, x + 30} \] Input:
integrate(((-18*x^5-72*x^4-72*x^3)*exp(x)^2+(48*x^6+120*x^5+120*x^4+540*x^ 3+828*x^2)*exp(x)-32*x^7-32*x^6-104*x^5-528*x^4-408*x^3-744*x^2-1800*x+360 )/((9*x^4+36*x^3+36*x^2)*exp(x)^2+(-24*x^5-60*x^4-60*x^3-252*x^2-360*x)*ex p(x)+16*x^6+16*x^5+52*x^4+264*x^3+156*x^2+360*x+900),x, algorithm="giac")
Output:
-(4*x^5 - 3*x^4*e^x + 2*x^4 - 6*x^3*e^x + 6*x^3 + 30*x^2 - 12*x)/(4*x^3 - 3*x^2*e^x + 2*x^2 - 6*x*e^x + 6*x + 30)
Timed out. \[ \int \frac {360-1800 x-744 x^2-408 x^3-528 x^4-104 x^5-32 x^6-32 x^7+e^{2 x} \left (-72 x^3-72 x^4-18 x^5\right )+e^x \left (828 x^2+540 x^3+120 x^4+120 x^5+48 x^6\right )}{900+360 x+156 x^2+264 x^3+52 x^4+16 x^5+16 x^6+e^{2 x} \left (36 x^2+36 x^3+9 x^4\right )+e^x \left (-360 x-252 x^2-60 x^3-60 x^4-24 x^5\right )} \, dx=\int -\frac {1800\,x+{\mathrm {e}}^{2\,x}\,\left (18\,x^5+72\,x^4+72\,x^3\right )-{\mathrm {e}}^x\,\left (48\,x^6+120\,x^5+120\,x^4+540\,x^3+828\,x^2\right )+744\,x^2+408\,x^3+528\,x^4+104\,x^5+32\,x^6+32\,x^7-360}{360\,x-{\mathrm {e}}^x\,\left (24\,x^5+60\,x^4+60\,x^3+252\,x^2+360\,x\right )+{\mathrm {e}}^{2\,x}\,\left (9\,x^4+36\,x^3+36\,x^2\right )+156\,x^2+264\,x^3+52\,x^4+16\,x^5+16\,x^6+900} \,d x \] Input:
int(-(1800*x + exp(2*x)*(72*x^3 + 72*x^4 + 18*x^5) - exp(x)*(828*x^2 + 540 *x^3 + 120*x^4 + 120*x^5 + 48*x^6) + 744*x^2 + 408*x^3 + 528*x^4 + 104*x^5 + 32*x^6 + 32*x^7 - 360)/(360*x - exp(x)*(360*x + 252*x^2 + 60*x^3 + 60*x ^4 + 24*x^5) + exp(2*x)*(36*x^2 + 36*x^3 + 9*x^4) + 156*x^2 + 264*x^3 + 52 *x^4 + 16*x^5 + 16*x^6 + 900),x)
Output:
int(-(1800*x + exp(2*x)*(72*x^3 + 72*x^4 + 18*x^5) - exp(x)*(828*x^2 + 540 *x^3 + 120*x^4 + 120*x^5 + 48*x^6) + 744*x^2 + 408*x^3 + 528*x^4 + 104*x^5 + 32*x^6 + 32*x^7 - 360)/(360*x - exp(x)*(360*x + 252*x^2 + 60*x^3 + 60*x ^4 + 24*x^5) + exp(2*x)*(36*x^2 + 36*x^3 + 9*x^4) + 156*x^2 + 264*x^3 + 52 *x^4 + 16*x^5 + 16*x^6 + 900), x)
Time = 0.21 (sec) , antiderivative size = 84, normalized size of antiderivative = 2.47 \[ \int \frac {360-1800 x-744 x^2-408 x^3-528 x^4-104 x^5-32 x^6-32 x^7+e^{2 x} \left (-72 x^3-72 x^4-18 x^5\right )+e^x \left (828 x^2+540 x^3+120 x^4+120 x^5+48 x^6\right )}{900+360 x+156 x^2+264 x^3+52 x^4+16 x^5+16 x^6+e^{2 x} \left (36 x^2+36 x^3+9 x^4\right )+e^x \left (-360 x-252 x^2-60 x^3-60 x^4-24 x^5\right )} \, dx=\frac {-3 e^{x} x^{4}-6 e^{x} x^{3}-6 e^{x} x^{2}-12 e^{x} x +4 x^{5}+2 x^{4}+14 x^{3}+34 x^{2}+60}{3 e^{x} x^{2}+6 e^{x} x -4 x^{3}-2 x^{2}-6 x -30} \] Input:
int(((-18*x^5-72*x^4-72*x^3)*exp(x)^2+(48*x^6+120*x^5+120*x^4+540*x^3+828* x^2)*exp(x)-32*x^7-32*x^6-104*x^5-528*x^4-408*x^3-744*x^2-1800*x+360)/((9* x^4+36*x^3+36*x^2)*exp(x)^2+(-24*x^5-60*x^4-60*x^3-252*x^2-360*x)*exp(x)+1 6*x^6+16*x^5+52*x^4+264*x^3+156*x^2+360*x+900),x)
Output:
( - 3*e**x*x**4 - 6*e**x*x**3 - 6*e**x*x**2 - 12*e**x*x + 4*x**5 + 2*x**4 + 14*x**3 + 34*x**2 + 60)/(3*e**x*x**2 + 6*e**x*x - 4*x**3 - 2*x**2 - 6*x - 30)