Integrand size = 108, antiderivative size = 33 \[ \int \frac {81 x^2-66 x^3+18 x^4-2 x^5+63 x^6-30 x^7+5 x^8+e^x \left (-9 x^2+7 x^3-x^4-7 x^6+x^7\right )}{162+2 e^{2 x}-180 x+86 x^2-20 x^3+2 x^4+e^x \left (-36+20 x-4 x^2\right )} \, dx=-1+\frac {x^3 \left (3-x+x^4\right )}{2 \left (-e^x+(3-x)^2+x\right )} \] Output:
1/2*x^3/(x-exp(x)+(3-x)^2)*(x^4-x+3)-1
Time = 3.11 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.91 \[ \int \frac {81 x^2-66 x^3+18 x^4-2 x^5+63 x^6-30 x^7+5 x^8+e^x \left (-9 x^2+7 x^3-x^4-7 x^6+x^7\right )}{162+2 e^{2 x}-180 x+86 x^2-20 x^3+2 x^4+e^x \left (-36+20 x-4 x^2\right )} \, dx=\frac {x^3 \left (3-x+x^4\right )}{2 \left (9-e^x-5 x+x^2\right )} \] Input:
Integrate[(81*x^2 - 66*x^3 + 18*x^4 - 2*x^5 + 63*x^6 - 30*x^7 + 5*x^8 + E^ x*(-9*x^2 + 7*x^3 - x^4 - 7*x^6 + x^7))/(162 + 2*E^(2*x) - 180*x + 86*x^2 - 20*x^3 + 2*x^4 + E^x*(-36 + 20*x - 4*x^2)),x]
Output:
(x^3*(3 - x + x^4))/(2*(9 - E^x - 5*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.
\(\displaystyle \int \frac {5 x^8-30 x^7+63 x^6-2 x^5+18 x^4-66 x^3+81 x^2+e^x \left (x^7-7 x^6-x^4+7 x^3-9 x^2\right )}{2 x^4-20 x^3+86 x^2+e^x \left (-4 x^2+20 x-36\right )-180 x+2 e^{2 x}+162} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {5 x^8-30 x^7+63 x^6-2 x^5+18 x^4-66 x^3+81 x^2+e^x \left (x^7-7 x^6-x^4+7 x^3-9 x^2\right )}{2 \left (x^2-5 x-e^x+9\right )^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{2} \int \frac {5 x^8-30 x^7+63 x^6-2 x^5+18 x^4-66 x^3+81 x^2-e^x \left (-x^7+7 x^6+x^4-7 x^3+9 x^2\right )}{\left (x^2-5 x-e^x+9\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \frac {1}{2} \int \left (\frac {x^3 \left (x^6-7 x^5+14 x^4-x^3+10 x^2-35 x+42\right )}{\left (x^2-5 x-e^x+9\right )^2}-\frac {x^2 \left (x^5-7 x^4-x^2+7 x-9\right )}{x^2-5 x-e^x+9}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (9 \int \frac {x^2}{x^2-5 x-e^x+9}dx+\int \frac {x^9}{\left (x^2-5 x-e^x+9\right )^2}dx-7 \int \frac {x^8}{\left (x^2-5 x-e^x+9\right )^2}dx+14 \int \frac {x^7}{\left (x^2-5 x-e^x+9\right )^2}dx-\int \frac {x^7}{x^2-5 x-e^x+9}dx-\int \frac {x^6}{\left (x^2-5 x-e^x+9\right )^2}dx+7 \int \frac {x^6}{x^2-5 x-e^x+9}dx+10 \int \frac {x^5}{\left (x^2-5 x-e^x+9\right )^2}dx-35 \int \frac {x^4}{\left (x^2-5 x-e^x+9\right )^2}dx+\int \frac {x^4}{x^2-5 x-e^x+9}dx+42 \int \frac {x^3}{\left (x^2-5 x-e^x+9\right )^2}dx-7 \int \frac {x^3}{x^2-5 x-e^x+9}dx\right )\) |
Input:
Int[(81*x^2 - 66*x^3 + 18*x^4 - 2*x^5 + 63*x^6 - 30*x^7 + 5*x^8 + E^x*(-9* x^2 + 7*x^3 - x^4 - 7*x^6 + x^7))/(162 + 2*E^(2*x) - 180*x + 86*x^2 - 20*x ^3 + 2*x^4 + E^x*(-36 + 20*x - 4*x^2)),x]
Output:
$Aborted
Time = 0.36 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.85
method | result | size |
risch | \(\frac {\left (x^{4}-x +3\right ) x^{3}}{2 x^{2}-10 x -2 \,{\mathrm e}^{x}+18}\) | \(28\) |
parallelrisch | \(\frac {x^{7}-x^{4}+3 x^{3}}{2 x^{2}-10 x -2 \,{\mathrm e}^{x}+18}\) | \(31\) |
norman | \(\frac {\frac {3}{2} x^{3}-\frac {1}{2} x^{4}+\frac {1}{2} x^{7}}{x^{2}-5 x -{\mathrm e}^{x}+9}\) | \(32\) |
Input:
int(((x^7-7*x^6-x^4+7*x^3-9*x^2)*exp(x)+5*x^8-30*x^7+63*x^6-2*x^5+18*x^4-6 6*x^3+81*x^2)/(2*exp(x)^2+(-4*x^2+20*x-36)*exp(x)+2*x^4-20*x^3+86*x^2-180* x+162),x,method=_RETURNVERBOSE)
Output:
1/2*(x^4-x+3)*x^3/(x^2-5*x-exp(x)+9)
Time = 0.08 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.91 \[ \int \frac {81 x^2-66 x^3+18 x^4-2 x^5+63 x^6-30 x^7+5 x^8+e^x \left (-9 x^2+7 x^3-x^4-7 x^6+x^7\right )}{162+2 e^{2 x}-180 x+86 x^2-20 x^3+2 x^4+e^x \left (-36+20 x-4 x^2\right )} \, dx=\frac {x^{7} - x^{4} + 3 \, x^{3}}{2 \, {\left (x^{2} - 5 \, x - e^{x} + 9\right )}} \] Input:
integrate(((x^7-7*x^6-x^4+7*x^3-9*x^2)*exp(x)+5*x^8-30*x^7+63*x^6-2*x^5+18 *x^4-66*x^3+81*x^2)/(2*exp(x)^2+(-4*x^2+20*x-36)*exp(x)+2*x^4-20*x^3+86*x^ 2-180*x+162),x, algorithm="fricas")
Output:
1/2*(x^7 - x^4 + 3*x^3)/(x^2 - 5*x - e^x + 9)
Time = 0.08 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.79 \[ \int \frac {81 x^2-66 x^3+18 x^4-2 x^5+63 x^6-30 x^7+5 x^8+e^x \left (-9 x^2+7 x^3-x^4-7 x^6+x^7\right )}{162+2 e^{2 x}-180 x+86 x^2-20 x^3+2 x^4+e^x \left (-36+20 x-4 x^2\right )} \, dx=\frac {- x^{7} + x^{4} - 3 x^{3}}{- 2 x^{2} + 10 x + 2 e^{x} - 18} \] Input:
integrate(((x**7-7*x**6-x**4+7*x**3-9*x**2)*exp(x)+5*x**8-30*x**7+63*x**6- 2*x**5+18*x**4-66*x**3+81*x**2)/(2*exp(x)**2+(-4*x**2+20*x-36)*exp(x)+2*x* *4-20*x**3+86*x**2-180*x+162),x)
Output:
(-x**7 + x**4 - 3*x**3)/(-2*x**2 + 10*x + 2*exp(x) - 18)
Time = 0.09 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.91 \[ \int \frac {81 x^2-66 x^3+18 x^4-2 x^5+63 x^6-30 x^7+5 x^8+e^x \left (-9 x^2+7 x^3-x^4-7 x^6+x^7\right )}{162+2 e^{2 x}-180 x+86 x^2-20 x^3+2 x^4+e^x \left (-36+20 x-4 x^2\right )} \, dx=\frac {x^{7} - x^{4} + 3 \, x^{3}}{2 \, {\left (x^{2} - 5 \, x - e^{x} + 9\right )}} \] Input:
integrate(((x^7-7*x^6-x^4+7*x^3-9*x^2)*exp(x)+5*x^8-30*x^7+63*x^6-2*x^5+18 *x^4-66*x^3+81*x^2)/(2*exp(x)^2+(-4*x^2+20*x-36)*exp(x)+2*x^4-20*x^3+86*x^ 2-180*x+162),x, algorithm="maxima")
Output:
1/2*(x^7 - x^4 + 3*x^3)/(x^2 - 5*x - e^x + 9)
Time = 0.11 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.91 \[ \int \frac {81 x^2-66 x^3+18 x^4-2 x^5+63 x^6-30 x^7+5 x^8+e^x \left (-9 x^2+7 x^3-x^4-7 x^6+x^7\right )}{162+2 e^{2 x}-180 x+86 x^2-20 x^3+2 x^4+e^x \left (-36+20 x-4 x^2\right )} \, dx=\frac {x^{7} - x^{4} + 3 \, x^{3}}{2 \, {\left (x^{2} - 5 \, x - e^{x} + 9\right )}} \] Input:
integrate(((x^7-7*x^6-x^4+7*x^3-9*x^2)*exp(x)+5*x^8-30*x^7+63*x^6-2*x^5+18 *x^4-66*x^3+81*x^2)/(2*exp(x)^2+(-4*x^2+20*x-36)*exp(x)+2*x^4-20*x^3+86*x^ 2-180*x+162),x, algorithm="giac")
Output:
1/2*(x^7 - x^4 + 3*x^3)/(x^2 - 5*x - e^x + 9)
Time = 7.23 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.97 \[ \int \frac {81 x^2-66 x^3+18 x^4-2 x^5+63 x^6-30 x^7+5 x^8+e^x \left (-9 x^2+7 x^3-x^4-7 x^6+x^7\right )}{162+2 e^{2 x}-180 x+86 x^2-20 x^3+2 x^4+e^x \left (-36+20 x-4 x^2\right )} \, dx=-\frac {x^7-x^4+3\,x^3}{10\,x+2\,{\mathrm {e}}^x-2\,x^2-18} \] Input:
int(-(exp(x)*(9*x^2 - 7*x^3 + x^4 + 7*x^6 - x^7) - 81*x^2 + 66*x^3 - 18*x^ 4 + 2*x^5 - 63*x^6 + 30*x^7 - 5*x^8)/(2*exp(2*x) - 180*x - exp(x)*(4*x^2 - 20*x + 36) + 86*x^2 - 20*x^3 + 2*x^4 + 162),x)
Output:
-(3*x^3 - x^4 + x^7)/(10*x + 2*exp(x) - 2*x^2 - 18)
Time = 0.19 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.88 \[ \int \frac {81 x^2-66 x^3+18 x^4-2 x^5+63 x^6-30 x^7+5 x^8+e^x \left (-9 x^2+7 x^3-x^4-7 x^6+x^7\right )}{162+2 e^{2 x}-180 x+86 x^2-20 x^3+2 x^4+e^x \left (-36+20 x-4 x^2\right )} \, dx=\frac {x^{3} \left (-x^{4}+x -3\right )}{2 e^{x}-2 x^{2}+10 x -18} \] Input:
int(((x^7-7*x^6-x^4+7*x^3-9*x^2)*exp(x)+5*x^8-30*x^7+63*x^6-2*x^5+18*x^4-6 6*x^3+81*x^2)/(2*exp(x)^2+(-4*x^2+20*x-36)*exp(x)+2*x^4-20*x^3+86*x^2-180* x+162),x)
Output:
(x**3*( - x**4 + x - 3))/(2*(e**x - x**2 + 5*x - 9))