Integrand size = 106, antiderivative size = 34 \[ \int \frac {-9+e^{2 x^2} (-4+16 e)-8 x-4 x^2+e^{x^2} \left (13+e (-72-32 x)+8 x-10 x^2\right )+e \left (81+72 x+16 x^2\right )}{81 x^2+16 e^{2 x^2} x^2+72 x^3+16 x^4+e^{x^2} \left (-72 x^2-32 x^3\right )} \, dx=\frac {-e+x-\frac {x}{x-5 \left (x+\frac {x}{1-e^{x^2}+x}\right )}}{x} \] Output:
(x-exp(1)-x/(-4*x-5*x/(1-exp(x^2)+x)))/x
Time = 4.87 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.82 \[ \int \frac {-9+e^{2 x^2} (-4+16 e)-8 x-4 x^2+e^{x^2} \left (13+e (-72-32 x)+8 x-10 x^2\right )+e \left (81+72 x+16 x^2\right )}{81 x^2+16 e^{2 x^2} x^2+72 x^3+16 x^4+e^{x^2} \left (-72 x^2-32 x^3\right )} \, dx=\frac {1-4 e+\frac {5}{-9+4 e^{x^2}-4 x}}{4 x} \] Input:
Integrate[(-9 + E^(2*x^2)*(-4 + 16*E) - 8*x - 4*x^2 + E^x^2*(13 + E*(-72 - 32*x) + 8*x - 10*x^2) + E*(81 + 72*x + 16*x^2))/(81*x^2 + 16*E^(2*x^2)*x^ 2 + 72*x^3 + 16*x^4 + E^x^2*(-72*x^2 - 32*x^3)),x]
Output:
(1 - 4*E + 5/(-9 + 4*E^x^2 - 4*x))/(4*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.
\(\displaystyle \int \frac {-4 x^2+e^{x^2} \left (-10 x^2+8 x+e (-32 x-72)+13\right )+e \left (16 x^2+72 x+81\right )+(16 e-4) e^{2 x^2}-8 x-9}{16 x^4+72 x^3+16 e^{2 x^2} x^2+81 x^2+e^{x^2} \left (-32 x^3-72 x^2\right )} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {-4 x^2+e^{x^2} \left (-10 x^2+8 x+e (-32 x-72)+13\right )+e \left (16 x^2+72 x+81\right )+(16 e-4) e^{2 x^2}-8 x-9}{x^2 \left (-4 e^{x^2}+4 x+9\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (-\frac {5 \left (2 x^2+1\right )}{4 \left (4 e^{x^2}-4 x-9\right ) x^2}-\frac {5 \left (4 x^2+9 x-2\right )}{2 x \left (-4 e^{x^2}+4 x+9\right )^2}+\frac {4 e-1}{4 x^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {45}{2} \int \frac {1}{\left (-4 x+4 e^{x^2}-9\right )^2}dx-\frac {5}{2} \int \frac {1}{-4 x+4 e^{x^2}-9}dx-\frac {5}{4} \int \frac {1}{\left (-4 x+4 e^{x^2}-9\right ) x^2}dx-10 \int \frac {x}{\left (-4 x+4 e^{x^2}-9\right )^2}dx+5 \int \frac {1}{x \left (4 x-4 e^{x^2}+9\right )^2}dx+\frac {1-4 e}{4 x}\) |
Input:
Int[(-9 + E^(2*x^2)*(-4 + 16*E) - 8*x - 4*x^2 + E^x^2*(13 + E*(-72 - 32*x) + 8*x - 10*x^2) + E*(81 + 72*x + 16*x^2))/(81*x^2 + 16*E^(2*x^2)*x^2 + 72 *x^3 + 16*x^4 + E^x^2*(-72*x^2 - 32*x^3)),x]
Output:
$Aborted
Time = 0.40 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.94
method | result | size |
risch | \(\frac {1}{4 x}-\frac {{\mathrm e}}{x}-\frac {5}{4 x \left (4 x -4 \,{\mathrm e}^{x^{2}}+9\right )}\) | \(32\) |
norman | \(\frac {\left (-1+4 \,{\mathrm e}\right ) {\mathrm e}^{x^{2}}+\left (1-4 \,{\mathrm e}\right ) x +1-9 \,{\mathrm e}}{x \left (4 x -4 \,{\mathrm e}^{x^{2}}+9\right )}\) | \(43\) |
parallelrisch | \(-\frac {-4+16 x \,{\mathrm e}-16 \,{\mathrm e}^{x^{2}} {\mathrm e}+36 \,{\mathrm e}-4 x +4 \,{\mathrm e}^{x^{2}}}{4 x \left (4 x -4 \,{\mathrm e}^{x^{2}}+9\right )}\) | \(47\) |
Input:
int(((16*exp(1)-4)*exp(x^2)^2+((-32*x-72)*exp(1)-10*x^2+8*x+13)*exp(x^2)+( 16*x^2+72*x+81)*exp(1)-4*x^2-8*x-9)/(16*x^2*exp(x^2)^2+(-32*x^3-72*x^2)*ex p(x^2)+16*x^4+72*x^3+81*x^2),x,method=_RETURNVERBOSE)
Output:
1/4/x-exp(1)/x-5/4/x/(4*x-4*exp(x^2)+9)
Time = 0.10 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.32 \[ \int \frac {-9+e^{2 x^2} (-4+16 e)-8 x-4 x^2+e^{x^2} \left (13+e (-72-32 x)+8 x-10 x^2\right )+e \left (81+72 x+16 x^2\right )}{81 x^2+16 e^{2 x^2} x^2+72 x^3+16 x^4+e^{x^2} \left (-72 x^2-32 x^3\right )} \, dx=-\frac {{\left (4 \, x + 9\right )} e - {\left (4 \, e - 1\right )} e^{\left (x^{2}\right )} - x - 1}{4 \, x^{2} - 4 \, x e^{\left (x^{2}\right )} + 9 \, x} \] Input:
integrate(((16*exp(1)-4)*exp(x^2)^2+((-32*x-72)*exp(1)-10*x^2+8*x+13)*exp( x^2)+(16*x^2+72*x+81)*exp(1)-4*x^2-8*x-9)/(16*x^2*exp(x^2)^2+(-32*x^3-72*x ^2)*exp(x^2)+16*x^4+72*x^3+81*x^2),x, algorithm="fricas")
Output:
-((4*x + 9)*e - (4*e - 1)*e^(x^2) - x - 1)/(4*x^2 - 4*x*e^(x^2) + 9*x)
Time = 0.08 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.76 \[ \int \frac {-9+e^{2 x^2} (-4+16 e)-8 x-4 x^2+e^{x^2} \left (13+e (-72-32 x)+8 x-10 x^2\right )+e \left (81+72 x+16 x^2\right )}{81 x^2+16 e^{2 x^2} x^2+72 x^3+16 x^4+e^{x^2} \left (-72 x^2-32 x^3\right )} \, dx=\frac {5}{- 16 x^{2} + 16 x e^{x^{2}} - 36 x} - \frac {- \frac {1}{4} + e}{x} \] Input:
integrate(((16*exp(1)-4)*exp(x**2)**2+((-32*x-72)*exp(1)-10*x**2+8*x+13)*e xp(x**2)+(16*x**2+72*x+81)*exp(1)-4*x**2-8*x-9)/(16*x**2*exp(x**2)**2+(-32 *x**3-72*x**2)*exp(x**2)+16*x**4+72*x**3+81*x**2),x)
Output:
5/(-16*x**2 + 16*x*exp(x**2) - 36*x) - (-1/4 + E)/x
Time = 0.08 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.35 \[ \int \frac {-9+e^{2 x^2} (-4+16 e)-8 x-4 x^2+e^{x^2} \left (13+e (-72-32 x)+8 x-10 x^2\right )+e \left (81+72 x+16 x^2\right )}{81 x^2+16 e^{2 x^2} x^2+72 x^3+16 x^4+e^{x^2} \left (-72 x^2-32 x^3\right )} \, dx=-\frac {x {\left (4 \, e - 1\right )} - {\left (4 \, e - 1\right )} e^{\left (x^{2}\right )} + 9 \, e - 1}{4 \, x^{2} - 4 \, x e^{\left (x^{2}\right )} + 9 \, x} \] Input:
integrate(((16*exp(1)-4)*exp(x^2)^2+((-32*x-72)*exp(1)-10*x^2+8*x+13)*exp( x^2)+(16*x^2+72*x+81)*exp(1)-4*x^2-8*x-9)/(16*x^2*exp(x^2)^2+(-32*x^3-72*x ^2)*exp(x^2)+16*x^4+72*x^3+81*x^2),x, algorithm="maxima")
Output:
-(x*(4*e - 1) - (4*e - 1)*e^(x^2) + 9*e - 1)/(4*x^2 - 4*x*e^(x^2) + 9*x)
Time = 0.12 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.35 \[ \int \frac {-9+e^{2 x^2} (-4+16 e)-8 x-4 x^2+e^{x^2} \left (13+e (-72-32 x)+8 x-10 x^2\right )+e \left (81+72 x+16 x^2\right )}{81 x^2+16 e^{2 x^2} x^2+72 x^3+16 x^4+e^{x^2} \left (-72 x^2-32 x^3\right )} \, dx=-\frac {4 \, x e - x + 9 \, e - 4 \, e^{\left (x^{2} + 1\right )} + e^{\left (x^{2}\right )} - 1}{4 \, x^{2} - 4 \, x e^{\left (x^{2}\right )} + 9 \, x} \] Input:
integrate(((16*exp(1)-4)*exp(x^2)^2+((-32*x-72)*exp(1)-10*x^2+8*x+13)*exp( x^2)+(16*x^2+72*x+81)*exp(1)-4*x^2-8*x-9)/(16*x^2*exp(x^2)^2+(-32*x^3-72*x ^2)*exp(x^2)+16*x^4+72*x^3+81*x^2),x, algorithm="giac")
Output:
-(4*x*e - x + 9*e - 4*e^(x^2 + 1) + e^(x^2) - 1)/(4*x^2 - 4*x*e^(x^2) + 9* x)
Time = 7.30 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.74 \[ \int \frac {-9+e^{2 x^2} (-4+16 e)-8 x-4 x^2+e^{x^2} \left (13+e (-72-32 x)+8 x-10 x^2\right )+e \left (81+72 x+16 x^2\right )}{81 x^2+16 e^{2 x^2} x^2+72 x^3+16 x^4+e^{x^2} \left (-72 x^2-32 x^3\right )} \, dx=\frac {x^2\,\left (\frac {16\,\mathrm {e}}{9}-\frac {4}{9}\right )-9\,\mathrm {e}+{\mathrm {e}}^{x^2}\,\left (4\,\mathrm {e}-1\right )-x\,{\mathrm {e}}^{x^2}\,\left (\frac {16\,\mathrm {e}}{9}-\frac {4}{9}\right )+1}{9\,x-4\,x\,{\mathrm {e}}^{x^2}+4\,x^2} \] Input:
int(-(8*x - exp(x^2)*(8*x - 10*x^2 - exp(1)*(32*x + 72) + 13) - exp(2*x^2) *(16*exp(1) - 4) - exp(1)*(72*x + 16*x^2 + 81) + 4*x^2 + 9)/(16*x^2*exp(2* x^2) - exp(x^2)*(72*x^2 + 32*x^3) + 81*x^2 + 72*x^3 + 16*x^4),x)
Output:
(x^2*((16*exp(1))/9 - 4/9) - 9*exp(1) + exp(x^2)*(4*exp(1) - 1) - x*exp(x^ 2)*((16*exp(1))/9 - 4/9) + 1)/(9*x - 4*x*exp(x^2) + 4*x^2)
Time = 0.16 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.74 \[ \int \frac {-9+e^{2 x^2} (-4+16 e)-8 x-4 x^2+e^{x^2} \left (13+e (-72-32 x)+8 x-10 x^2\right )+e \left (81+72 x+16 x^2\right )}{81 x^2+16 e^{2 x^2} x^2+72 x^3+16 x^4+e^{x^2} \left (-72 x^2-32 x^3\right )} \, dx=\frac {-324 e^{x^{2}} e -20 e^{x^{2}} x +81 e^{x^{2}}+324 e x +729 e +20 x^{2}-36 x -81}{81 x \left (4 e^{x^{2}}-4 x -9\right )} \] Input:
int(((16*exp(1)-4)*exp(x^2)^2+((-32*x-72)*exp(1)-10*x^2+8*x+13)*exp(x^2)+( 16*x^2+72*x+81)*exp(1)-4*x^2-8*x-9)/(16*x^2*exp(x^2)^2+(-32*x^3-72*x^2)*ex p(x^2)+16*x^4+72*x^3+81*x^2),x)
Output:
( - 324*e**(x**2)*e - 20*e**(x**2)*x + 81*e**(x**2) + 324*e*x + 729*e + 20 *x**2 - 36*x - 81)/(81*x*(4*e**(x**2) - 4*x - 9))