Integrand size = 146, antiderivative size = 34 \[ \int \frac {-1800+2700 x-3148 x^2+2925 x^3-1350 x^4+225 x^5+e^{2 x^2} \left (-1800+2700 x-1350 x^2+225 x^3\right )+e^{x^2} \left (-2+7199 x-10796 x^2+5398 x^3-900 x^4\right )}{-1800 x^2+2700 x^3-1350 x^4+225 x^5+e^{2 x^2} \left (-1800+2700 x-1350 x^2+225 x^3\right )+e^{x^2} \left (3600 x-5400 x^2+2700 x^3-450 x^4\right )} \, dx=x-\frac {x+\frac {4 x^2}{225 (-4+2 x)^2}}{x \left (-e^{x^2}+x\right )} \] Output:
x-(x+1/225*x^2/(-2+x)^2)/(-exp(x^2)+x)/x
Time = 12.54 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.09 \[ \int \frac {-1800+2700 x-3148 x^2+2925 x^3-1350 x^4+225 x^5+e^{2 x^2} \left (-1800+2700 x-1350 x^2+225 x^3\right )+e^{x^2} \left (-2+7199 x-10796 x^2+5398 x^3-900 x^4\right )}{-1800 x^2+2700 x^3-1350 x^4+225 x^5+e^{2 x^2} \left (-1800+2700 x-1350 x^2+225 x^3\right )+e^{x^2} \left (3600 x-5400 x^2+2700 x^3-450 x^4\right )} \, dx=\frac {1}{225} \left (225 (-2+x)+\frac {900-899 x+225 x^2}{\left (e^{x^2}-x\right ) (-2+x)^2}\right ) \] Input:
Integrate[(-1800 + 2700*x - 3148*x^2 + 2925*x^3 - 1350*x^4 + 225*x^5 + E^( 2*x^2)*(-1800 + 2700*x - 1350*x^2 + 225*x^3) + E^x^2*(-2 + 7199*x - 10796* x^2 + 5398*x^3 - 900*x^4))/(-1800*x^2 + 2700*x^3 - 1350*x^4 + 225*x^5 + E^ (2*x^2)*(-1800 + 2700*x - 1350*x^2 + 225*x^3) + E^x^2*(3600*x - 5400*x^2 + 2700*x^3 - 450*x^4)),x]
Output:
(225*(-2 + x) + (900 - 899*x + 225*x^2)/((E^x^2 - x)*(-2 + x)^2))/225
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.
\(\displaystyle \int \frac {225 x^5-1350 x^4+2925 x^3-3148 x^2+e^{2 x^2} \left (225 x^3-1350 x^2+2700 x-1800\right )+e^{x^2} \left (-900 x^4+5398 x^3-10796 x^2+7199 x-2\right )+2700 x-1800}{225 x^5-1350 x^4+2700 x^3-1800 x^2+e^{2 x^2} \left (225 x^3-1350 x^2+2700 x-1800\right )+e^{x^2} \left (-450 x^4+2700 x^3-5400 x^2+3600 x\right )} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {-225 x^5+1350 x^4-2925 x^3+3148 x^2-e^{2 x^2} \left (225 x^3-1350 x^2+2700 x-1800\right )-e^{x^2} \left (-900 x^4+5398 x^3-10796 x^2+7199 x-2\right )-2700 x+1800}{225 (2-x)^3 \left (e^{x^2}-x\right )^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{225} \int \frac {-225 x^5+1350 x^4-2925 x^3+3148 x^2-2700 x+225 e^{2 x^2} \left (-x^3+6 x^2-12 x+8\right )+e^{x^2} \left (900 x^4-5398 x^3+10796 x^2-7199 x+2\right )+1800}{(2-x)^3 \left (e^{x^2}-x\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \frac {1}{225} \int \left (-\frac {450 x^4-2698 x^3+5396 x^2-3599 x+2}{\left (e^{x^2}-x\right ) (x-2)^3}-\frac {450 x^4-1798 x^3+1575 x^2+899 x-900}{\left (e^{x^2}-x\right )^2 (x-2)^2}+225\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{225} \left (217 \int \frac {1}{\left (e^{x^2}-x\right )^2}dx-2 \int \frac {1}{e^{x^2}-x}dx-4 \int \frac {1}{\left (e^{x^2}-x\right ) (x-2)^3}dx-14 \int \frac {1}{\left (e^{x^2}-x\right )^2 (x-2)^2}dx-9 \int \frac {1}{\left (e^{x^2}-x\right ) (x-2)^2}dx-23 \int \frac {1}{\left (e^{x^2}-x\right )^2 (x-2)}dx-8 \int \frac {1}{\left (e^{x^2}-x\right ) (x-2)}dx-2 \int \frac {x}{\left (e^{x^2}-x\right )^2}dx-450 \int \frac {x}{e^{x^2}-x}dx-450 \int \frac {x^2}{\left (e^{x^2}-x\right )^2}dx+225 x\right )\) |
Input:
Int[(-1800 + 2700*x - 3148*x^2 + 2925*x^3 - 1350*x^4 + 225*x^5 + E^(2*x^2) *(-1800 + 2700*x - 1350*x^2 + 225*x^3) + E^x^2*(-2 + 7199*x - 10796*x^2 + 5398*x^3 - 900*x^4))/(-1800*x^2 + 2700*x^3 - 1350*x^4 + 225*x^5 + E^(2*x^2 )*(-1800 + 2700*x - 1350*x^2 + 225*x^3) + E^x^2*(3600*x - 5400*x^2 + 2700* x^3 - 450*x^4)),x]
Output:
$Aborted
Time = 0.45 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.03
method | result | size |
risch | \(x -\frac {225 x^{2}-899 x +900}{225 \left (x^{2}-4 x +4\right ) \left (-{\mathrm e}^{x^{2}}+x \right )}\) | \(35\) |
norman | \(\frac {-4+x^{4}-16 \,{\mathrm e}^{x^{2}}-13 x^{2}+\frac {4499 x}{225}+12 \,{\mathrm e}^{x^{2}} x -x^{3} {\mathrm e}^{x^{2}}}{\left (-{\mathrm e}^{x^{2}}+x \right ) \left (-2+x \right )^{2}}\) | \(52\) |
parallelrisch | \(\frac {225 x^{4}-225 x^{3} {\mathrm e}^{x^{2}}-900-2925 x^{2}+2700 \,{\mathrm e}^{x^{2}} x +4499 x -3600 \,{\mathrm e}^{x^{2}}}{225 x^{3}-225 x^{2} {\mathrm e}^{x^{2}}-900 x^{2}+900 \,{\mathrm e}^{x^{2}} x +900 x -900 \,{\mathrm e}^{x^{2}}}\) | \(76\) |
Input:
int(((225*x^3-1350*x^2+2700*x-1800)*exp(x^2)^2+(-900*x^4+5398*x^3-10796*x^ 2+7199*x-2)*exp(x^2)+225*x^5-1350*x^4+2925*x^3-3148*x^2+2700*x-1800)/((225 *x^3-1350*x^2+2700*x-1800)*exp(x^2)^2+(-450*x^4+2700*x^3-5400*x^2+3600*x)* exp(x^2)+225*x^5-1350*x^4+2700*x^3-1800*x^2),x,method=_RETURNVERBOSE)
Output:
x-1/225*(225*x^2-899*x+900)/(x^2-4*x+4)/(-exp(x^2)+x)
Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (30) = 60\).
Time = 0.09 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.00 \[ \int \frac {-1800+2700 x-3148 x^2+2925 x^3-1350 x^4+225 x^5+e^{2 x^2} \left (-1800+2700 x-1350 x^2+225 x^3\right )+e^{x^2} \left (-2+7199 x-10796 x^2+5398 x^3-900 x^4\right )}{-1800 x^2+2700 x^3-1350 x^4+225 x^5+e^{2 x^2} \left (-1800+2700 x-1350 x^2+225 x^3\right )+e^{x^2} \left (3600 x-5400 x^2+2700 x^3-450 x^4\right )} \, dx=\frac {225 \, x^{4} - 900 \, x^{3} + 675 \, x^{2} - 225 \, {\left (x^{3} - 4 \, x^{2} + 4 \, x\right )} e^{\left (x^{2}\right )} + 899 \, x - 900}{225 \, {\left (x^{3} - 4 \, x^{2} - {\left (x^{2} - 4 \, x + 4\right )} e^{\left (x^{2}\right )} + 4 \, x\right )}} \] Input:
integrate(((225*x^3-1350*x^2+2700*x-1800)*exp(x^2)^2+(-900*x^4+5398*x^3-10 796*x^2+7199*x-2)*exp(x^2)+225*x^5-1350*x^4+2925*x^3-3148*x^2+2700*x-1800) /((225*x^3-1350*x^2+2700*x-1800)*exp(x^2)^2+(-450*x^4+2700*x^3-5400*x^2+36 00*x)*exp(x^2)+225*x^5-1350*x^4+2700*x^3-1800*x^2),x, algorithm="fricas")
Output:
1/225*(225*x^4 - 900*x^3 + 675*x^2 - 225*(x^3 - 4*x^2 + 4*x)*e^(x^2) + 899 *x - 900)/(x^3 - 4*x^2 - (x^2 - 4*x + 4)*e^(x^2) + 4*x)
Time = 0.11 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.15 \[ \int \frac {-1800+2700 x-3148 x^2+2925 x^3-1350 x^4+225 x^5+e^{2 x^2} \left (-1800+2700 x-1350 x^2+225 x^3\right )+e^{x^2} \left (-2+7199 x-10796 x^2+5398 x^3-900 x^4\right )}{-1800 x^2+2700 x^3-1350 x^4+225 x^5+e^{2 x^2} \left (-1800+2700 x-1350 x^2+225 x^3\right )+e^{x^2} \left (3600 x-5400 x^2+2700 x^3-450 x^4\right )} \, dx=x + \frac {225 x^{2} - 899 x + 900}{- 225 x^{3} + 900 x^{2} - 900 x + \left (225 x^{2} - 900 x + 900\right ) e^{x^{2}}} \] Input:
integrate(((225*x**3-1350*x**2+2700*x-1800)*exp(x**2)**2+(-900*x**4+5398*x **3-10796*x**2+7199*x-2)*exp(x**2)+225*x**5-1350*x**4+2925*x**3-3148*x**2+ 2700*x-1800)/((225*x**3-1350*x**2+2700*x-1800)*exp(x**2)**2+(-450*x**4+270 0*x**3-5400*x**2+3600*x)*exp(x**2)+225*x**5-1350*x**4+2700*x**3-1800*x**2) ,x)
Output:
x + (225*x**2 - 899*x + 900)/(-225*x**3 + 900*x**2 - 900*x + (225*x**2 - 9 00*x + 900)*exp(x**2))
Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (30) = 60\).
Time = 0.07 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.00 \[ \int \frac {-1800+2700 x-3148 x^2+2925 x^3-1350 x^4+225 x^5+e^{2 x^2} \left (-1800+2700 x-1350 x^2+225 x^3\right )+e^{x^2} \left (-2+7199 x-10796 x^2+5398 x^3-900 x^4\right )}{-1800 x^2+2700 x^3-1350 x^4+225 x^5+e^{2 x^2} \left (-1800+2700 x-1350 x^2+225 x^3\right )+e^{x^2} \left (3600 x-5400 x^2+2700 x^3-450 x^4\right )} \, dx=\frac {225 \, x^{4} - 900 \, x^{3} + 675 \, x^{2} - 225 \, {\left (x^{3} - 4 \, x^{2} + 4 \, x\right )} e^{\left (x^{2}\right )} + 899 \, x - 900}{225 \, {\left (x^{3} - 4 \, x^{2} - {\left (x^{2} - 4 \, x + 4\right )} e^{\left (x^{2}\right )} + 4 \, x\right )}} \] Input:
integrate(((225*x^3-1350*x^2+2700*x-1800)*exp(x^2)^2+(-900*x^4+5398*x^3-10 796*x^2+7199*x-2)*exp(x^2)+225*x^5-1350*x^4+2925*x^3-3148*x^2+2700*x-1800) /((225*x^3-1350*x^2+2700*x-1800)*exp(x^2)^2+(-450*x^4+2700*x^3-5400*x^2+36 00*x)*exp(x^2)+225*x^5-1350*x^4+2700*x^3-1800*x^2),x, algorithm="maxima")
Output:
1/225*(225*x^4 - 900*x^3 + 675*x^2 - 225*(x^3 - 4*x^2 + 4*x)*e^(x^2) + 899 *x - 900)/(x^3 - 4*x^2 - (x^2 - 4*x + 4)*e^(x^2) + 4*x)
Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (30) = 60\).
Time = 0.13 (sec) , antiderivative size = 83, normalized size of antiderivative = 2.44 \[ \int \frac {-1800+2700 x-3148 x^2+2925 x^3-1350 x^4+225 x^5+e^{2 x^2} \left (-1800+2700 x-1350 x^2+225 x^3\right )+e^{x^2} \left (-2+7199 x-10796 x^2+5398 x^3-900 x^4\right )}{-1800 x^2+2700 x^3-1350 x^4+225 x^5+e^{2 x^2} \left (-1800+2700 x-1350 x^2+225 x^3\right )+e^{x^2} \left (3600 x-5400 x^2+2700 x^3-450 x^4\right )} \, dx=\frac {225 \, x^{4} - 225 \, x^{3} e^{\left (x^{2}\right )} - 900 \, x^{3} + 900 \, x^{2} e^{\left (x^{2}\right )} + 675 \, x^{2} - 900 \, x e^{\left (x^{2}\right )} + 899 \, x - 900}{225 \, {\left (x^{3} - x^{2} e^{\left (x^{2}\right )} - 4 \, x^{2} + 4 \, x e^{\left (x^{2}\right )} + 4 \, x - 4 \, e^{\left (x^{2}\right )}\right )}} \] Input:
integrate(((225*x^3-1350*x^2+2700*x-1800)*exp(x^2)^2+(-900*x^4+5398*x^3-10 796*x^2+7199*x-2)*exp(x^2)+225*x^5-1350*x^4+2925*x^3-3148*x^2+2700*x-1800) /((225*x^3-1350*x^2+2700*x-1800)*exp(x^2)^2+(-450*x^4+2700*x^3-5400*x^2+36 00*x)*exp(x^2)+225*x^5-1350*x^4+2700*x^3-1800*x^2),x, algorithm="giac")
Output:
1/225*(225*x^4 - 225*x^3*e^(x^2) - 900*x^3 + 900*x^2*e^(x^2) + 675*x^2 - 9 00*x*e^(x^2) + 899*x - 900)/(x^3 - x^2*e^(x^2) - 4*x^2 + 4*x*e^(x^2) + 4*x - 4*e^(x^2))
Time = 3.00 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.79 \[ \int \frac {-1800+2700 x-3148 x^2+2925 x^3-1350 x^4+225 x^5+e^{2 x^2} \left (-1800+2700 x-1350 x^2+225 x^3\right )+e^{x^2} \left (-2+7199 x-10796 x^2+5398 x^3-900 x^4\right )}{-1800 x^2+2700 x^3-1350 x^4+225 x^5+e^{2 x^2} \left (-1800+2700 x-1350 x^2+225 x^3\right )+e^{x^2} \left (3600 x-5400 x^2+2700 x^3-450 x^4\right )} \, dx=x-\frac {x^2-\frac {899\,x}{225}+4}{\left (x-{\mathrm {e}}^{x^2}\right )\,{\left (x-2\right )}^2} \] Input:
int((2700*x - exp(x^2)*(10796*x^2 - 7199*x - 5398*x^3 + 900*x^4 + 2) + exp (2*x^2)*(2700*x - 1350*x^2 + 225*x^3 - 1800) - 3148*x^2 + 2925*x^3 - 1350* x^4 + 225*x^5 - 1800)/(exp(x^2)*(3600*x - 5400*x^2 + 2700*x^3 - 450*x^4) + exp(2*x^2)*(2700*x - 1350*x^2 + 225*x^3 - 1800) - 1800*x^2 + 2700*x^3 - 1 350*x^4 + 225*x^5),x)
Output:
x - (x^2 - (899*x)/225 + 4)/((x - exp(x^2))*(x - 2)^2)
Time = 0.17 (sec) , antiderivative size = 90, normalized size of antiderivative = 2.65 \[ \int \frac {-1800+2700 x-3148 x^2+2925 x^3-1350 x^4+225 x^5+e^{2 x^2} \left (-1800+2700 x-1350 x^2+225 x^3\right )+e^{x^2} \left (-2+7199 x-10796 x^2+5398 x^3-900 x^4\right )}{-1800 x^2+2700 x^3-1350 x^4+225 x^5+e^{2 x^2} \left (-1800+2700 x-1350 x^2+225 x^3\right )+e^{x^2} \left (3600 x-5400 x^2+2700 x^3-450 x^4\right )} \, dx=\frac {225 e^{x^{2}} x^{3}-900 e^{x^{2}} x^{2}+900 e^{x^{2}} x -225 x^{4}+900 x^{3}-675 x^{2}-899 x +900}{225 e^{x^{2}} x^{2}-900 e^{x^{2}} x +900 e^{x^{2}}-225 x^{3}+900 x^{2}-900 x} \] Input:
int(((225*x^3-1350*x^2+2700*x-1800)*exp(x^2)^2+(-900*x^4+5398*x^3-10796*x^ 2+7199*x-2)*exp(x^2)+225*x^5-1350*x^4+2925*x^3-3148*x^2+2700*x-1800)/((225 *x^3-1350*x^2+2700*x-1800)*exp(x^2)^2+(-450*x^4+2700*x^3-5400*x^2+3600*x)* exp(x^2)+225*x^5-1350*x^4+2700*x^3-1800*x^2),x)
Output:
(225*e**(x**2)*x**3 - 900*e**(x**2)*x**2 + 900*e**(x**2)*x - 225*x**4 + 90 0*x**3 - 675*x**2 - 899*x + 900)/(225*(e**(x**2)*x**2 - 4*e**(x**2)*x + 4* e**(x**2) - x**3 + 4*x**2 - 4*x))