Integrand size = 286, antiderivative size = 37 \[ \int \frac {25 e^{2 e^3}+125 x^4-100 x^5+e^{e^3} \left (150 x^2-100 x^3\right )+e^{2 e^{x+x^2}} \left (5 x^4-4 x^5+e^{x+x^2} \left (2 x^5+2 x^6-4 x^7\right )+e^{2 e^3} \left (1+e^{x+x^2} \left (2 x+2 x^2-4 x^3\right )\right )+e^{e^3} \left (6 x^2-4 x^3+e^{x+x^2} \left (4 x^3+4 x^4-8 x^5\right )\right )\right )+e^{e^{x+x^2}} \left (-50 x^4+40 x^5+e^{x+x^2} \left (-10 x^5-10 x^6+20 x^7\right )+e^{2 e^3} \left (-10+e^{x+x^2} \left (-10 x-10 x^2+20 x^3\right )\right )+e^{e^3} \left (-60 x^2+40 x^3+e^{x+x^2} \left (-20 x^3-20 x^4+40 x^5\right )\right )\right )}{1-2 x+x^2} \, dx=\frac {\left (-5+e^{e^{x+x^2}}\right )^2 x^2 \left (e^{e^3}+x^2\right )^2}{x-x^2} \] Output:
(x^2+exp(exp(3)))^2*(exp(exp(x^2+x))-5)^2*x^2/(-x^2+x)
Leaf count is larger than twice the leaf count of optimal. \(144\) vs. \(2(37)=74\).
Time = 0.15 (sec) , antiderivative size = 144, normalized size of antiderivative = 3.89 \[ \int \frac {25 e^{2 e^3}+125 x^4-100 x^5+e^{e^3} \left (150 x^2-100 x^3\right )+e^{2 e^{x+x^2}} \left (5 x^4-4 x^5+e^{x+x^2} \left (2 x^5+2 x^6-4 x^7\right )+e^{2 e^3} \left (1+e^{x+x^2} \left (2 x+2 x^2-4 x^3\right )\right )+e^{e^3} \left (6 x^2-4 x^3+e^{x+x^2} \left (4 x^3+4 x^4-8 x^5\right )\right )\right )+e^{e^{x+x^2}} \left (-50 x^4+40 x^5+e^{x+x^2} \left (-10 x^5-10 x^6+20 x^7\right )+e^{2 e^3} \left (-10+e^{x+x^2} \left (-10 x-10 x^2+20 x^3\right )\right )+e^{e^3} \left (-60 x^2+40 x^3+e^{x+x^2} \left (-20 x^3-20 x^4+40 x^5\right )\right )\right )}{1-2 x+x^2} \, dx=-\frac {25 e^{2 e^3}+e^{2 \left (e^3+e^{x+x^2}\right )} x-10 e^{2 e^3+e^{x+x^2}} x-20 e^{e^3+e^{x+x^2}} x^3+2 e^{e^3+2 e^{x+x^2}} x^3-10 e^{e^{x+x^2}} x^5+e^{2 e^{x+x^2}} x^5+50 e^{e^3} \left (1-x+x^3\right )+25 \left (1-x+x^5\right )}{-1+x} \] Input:
Integrate[(25*E^(2*E^3) + 125*x^4 - 100*x^5 + E^E^3*(150*x^2 - 100*x^3) + E^(2*E^(x + x^2))*(5*x^4 - 4*x^5 + E^(x + x^2)*(2*x^5 + 2*x^6 - 4*x^7) + E ^(2*E^3)*(1 + E^(x + x^2)*(2*x + 2*x^2 - 4*x^3)) + E^E^3*(6*x^2 - 4*x^3 + E^(x + x^2)*(4*x^3 + 4*x^4 - 8*x^5))) + E^E^(x + x^2)*(-50*x^4 + 40*x^5 + E^(x + x^2)*(-10*x^5 - 10*x^6 + 20*x^7) + E^(2*E^3)*(-10 + E^(x + x^2)*(-1 0*x - 10*x^2 + 20*x^3)) + E^E^3*(-60*x^2 + 40*x^3 + E^(x + x^2)*(-20*x^3 - 20*x^4 + 40*x^5))))/(1 - 2*x + x^2),x]
Output:
-((25*E^(2*E^3) + E^(2*(E^3 + E^(x + x^2)))*x - 10*E^(2*E^3 + E^(x + x^2)) *x - 20*E^(E^3 + E^(x + x^2))*x^3 + 2*E^(E^3 + 2*E^(x + x^2))*x^3 - 10*E^E ^(x + x^2)*x^5 + E^(2*E^(x + x^2))*x^5 + 50*E^E^3*(1 - x + x^3) + 25*(1 - x + x^5))/(-1 + 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.
| \(\displaystyle \int \frac {-100 x^5+125 x^4+e^{e^3} \left (150 x^2-100 x^3\right )+e^{2 e^{x^2+x}} \left (-4 x^5+5 x^4+e^{2 e^3} \left (e^{x^2+x} \left (-4 x^3+2 x^2+2 x\right )+1\right )+e^{x^2+x} \left (-4 x^7+2 x^6+2 x^5\right )+e^{e^3} \left (-4 x^3+6 x^2+e^{x^2+x} \left (-8 x^5+4 x^4+4 x^3\right )\right )\right )+e^{e^{x^2+x}} \left (40 x^5-50 x^4+e^{2 e^3} \left (e^{x^2+x} \left (20 x^3-10 x^2-10 x\right )-10\right )+e^{x^2+x} \left (20 x^7-10 x^6-10 x^5\right )+e^{e^3} \left (40 x^3-60 x^2+e^{x^2+x} \left (40 x^5-20 x^4-20 x^3\right )\right )\right )+25 e^{2 e^3}}{x^2-2 x+1} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (5-e^{e^{x^2+x}}\right ) \left (x^2+e^{e^3}\right ) \left (5 (5-4 x) x^2+e^{e^{x^2+x}} (4 x-5) x^2+2 e^{x^2+e^{x^2+x}+x+e^3} \left (2 x^2-x-1\right ) x-e^{e^{x^2+x}+e^3}+2 e^{x^2+e^{x^2+x}+x} \left (2 x^2-x-1\right ) x^3+5 e^{e^3}\right )}{(1-x)^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {e^{e^{x (x+1)}} \left (5-e^{e^{x^2+x}}\right ) (5-4 x) \left (-x^2-e^{e^3}\right ) x^2}{(1-x)^2}+\frac {5 \left (e^{e^{x^2+x}}-5\right ) (4 x-5) \left (x^2+e^{e^3}\right ) x^2}{(x-1)^2}-\frac {2 e^{x^2+e^{x^2+x}+x} \left (e^{e^{x^2+x}}-5\right ) (2 x+1) \left (x^2+e^{e^3}\right )^2 x}{x-1}+\frac {e^{e^{x^2+x}+e^3} \left (e^{e^{x^2+x}}-5\right ) \left (x^2+e^{e^3}\right )}{(x-1)^2}-\frac {5 e^{e^3} \left (e^{e^{x^2+x}}-5\right ) \left (x^2+e^{e^3}\right )}{(x-1)^2}\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \int \left (\frac {e^{e^{x (x+1)}} \left (5-e^{e^{x^2+x}}\right ) (5-4 x) \left (-x^2-e^{e^3}\right ) x^2}{(1-x)^2}+\frac {5 \left (e^{e^{x^2+x}}-5\right ) (4 x-5) \left (x^2+e^{e^3}\right ) x^2}{(x-1)^2}-\frac {2 e^{x^2+e^{x^2+x}+x} \left (e^{e^{x^2+x}}-5\right ) (2 x+1) \left (x^2+e^{e^3}\right )^2 x}{x-1}+\frac {e^{e^{x^2+x}+e^3} \left (e^{e^{x^2+x}}-5\right ) \left (x^2+e^{e^3}\right )}{(x-1)^2}-\frac {5 e^{e^3} \left (e^{e^{x^2+x}}-5\right ) \left (x^2+e^{e^3}\right )}{(x-1)^2}\right )dx\) |
Input:
Int[(25*E^(2*E^3) + 125*x^4 - 100*x^5 + E^E^3*(150*x^2 - 100*x^3) + E^(2*E ^(x + x^2))*(5*x^4 - 4*x^5 + E^(x + x^2)*(2*x^5 + 2*x^6 - 4*x^7) + E^(2*E^ 3)*(1 + E^(x + x^2)*(2*x + 2*x^2 - 4*x^3)) + E^E^3*(6*x^2 - 4*x^3 + E^(x + x^2)*(4*x^3 + 4*x^4 - 8*x^5))) + E^E^(x + x^2)*(-50*x^4 + 40*x^5 + E^(x + x^2)*(-10*x^5 - 10*x^6 + 20*x^7) + E^(2*E^3)*(-10 + E^(x + x^2)*(-10*x - 10*x^2 + 20*x^3)) + E^E^3*(-60*x^2 + 40*x^3 + E^(x + x^2)*(-20*x^3 - 20*x^ 4 + 40*x^5))))/(1 - 2*x + x^2),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(116\) vs. \(2(33)=66\).
Time = 3.26 (sec) , antiderivative size = 117, normalized size of antiderivative = 3.16
| method | result | size |
| parallelrisch | \(-\frac {{\mathrm e}^{2 \,{\mathrm e}^{x^{2}+x}} x^{5}+2 \,{\mathrm e}^{{\mathrm e}^{3}} {\mathrm e}^{2 \,{\mathrm e}^{x^{2}+x}} x^{3}-10 \,{\mathrm e}^{{\mathrm e}^{x^{2}+x}} x^{5}+{\mathrm e}^{2 \,{\mathrm e}^{3}} {\mathrm e}^{2 \,{\mathrm e}^{x^{2}+x}} x -20 \,{\mathrm e}^{{\mathrm e}^{3}} {\mathrm e}^{{\mathrm e}^{x^{2}+x}} x^{3}+25 x^{5}-10 \,{\mathrm e}^{2 \,{\mathrm e}^{3}} {\mathrm e}^{{\mathrm e}^{x^{2}+x}} x +50 \,{\mathrm e}^{{\mathrm e}^{3}} x^{3}+25 \,{\mathrm e}^{2 \,{\mathrm e}^{3}}}{-1+x}\) | \(117\) |
| risch | \(-25 x^{4}-50 \,{\mathrm e}^{{\mathrm e}^{3}} x^{2}-25 x^{3}-50 x \,{\mathrm e}^{{\mathrm e}^{3}}-25 x^{2}-25 x -\frac {25 \,{\mathrm e}^{2 \,{\mathrm e}^{3}}}{-1+x}-\frac {50 \,{\mathrm e}^{{\mathrm e}^{3}}}{-1+x}-\frac {25}{-1+x}-\frac {x \left (x^{4}+2 \,{\mathrm e}^{{\mathrm e}^{3}} x^{2}+{\mathrm e}^{2 \,{\mathrm e}^{3}}\right ) {\mathrm e}^{2 \,{\mathrm e}^{\left (1+x \right ) x}}}{-1+x}+\frac {10 x \left (x^{4}+2 \,{\mathrm e}^{{\mathrm e}^{3}} x^{2}+{\mathrm e}^{2 \,{\mathrm e}^{3}}\right ) {\mathrm e}^{{\mathrm e}^{\left (1+x \right ) x}}}{-1+x}\) | \(129\) |
Input:
int(((((-4*x^3+2*x^2+2*x)*exp(x^2+x)+1)*exp(exp(3))^2+((-8*x^5+4*x^4+4*x^3 )*exp(x^2+x)-4*x^3+6*x^2)*exp(exp(3))+(-4*x^7+2*x^6+2*x^5)*exp(x^2+x)-4*x^ 5+5*x^4)*exp(exp(x^2+x))^2+(((20*x^3-10*x^2-10*x)*exp(x^2+x)-10)*exp(exp(3 ))^2+((40*x^5-20*x^4-20*x^3)*exp(x^2+x)+40*x^3-60*x^2)*exp(exp(3))+(20*x^7 -10*x^6-10*x^5)*exp(x^2+x)+40*x^5-50*x^4)*exp(exp(x^2+x))+25*exp(exp(3))^2 +(-100*x^3+150*x^2)*exp(exp(3))-100*x^5+125*x^4)/(x^2-2*x+1),x,method=_RET URNVERBOSE)
Output:
-(exp(exp(x^2+x))^2*x^5+2*exp(exp(3))*exp(exp(x^2+x))^2*x^3-10*exp(exp(x^2 +x))*x^5+exp(exp(3))^2*exp(exp(x^2+x))^2*x-20*exp(exp(3))*exp(exp(x^2+x))* x^3+25*x^5-10*exp(exp(3))^2*exp(exp(x^2+x))*x+50*exp(exp(3))*x^3+25*exp(ex p(3))^2)/(-1+x)
Leaf count of result is larger than twice the leaf count of optimal. 94 vs. \(2 (34) = 68\).
Time = 0.09 (sec) , antiderivative size = 94, normalized size of antiderivative = 2.54 \[ \int \frac {25 e^{2 e^3}+125 x^4-100 x^5+e^{e^3} \left (150 x^2-100 x^3\right )+e^{2 e^{x+x^2}} \left (5 x^4-4 x^5+e^{x+x^2} \left (2 x^5+2 x^6-4 x^7\right )+e^{2 e^3} \left (1+e^{x+x^2} \left (2 x+2 x^2-4 x^3\right )\right )+e^{e^3} \left (6 x^2-4 x^3+e^{x+x^2} \left (4 x^3+4 x^4-8 x^5\right )\right )\right )+e^{e^{x+x^2}} \left (-50 x^4+40 x^5+e^{x+x^2} \left (-10 x^5-10 x^6+20 x^7\right )+e^{2 e^3} \left (-10+e^{x+x^2} \left (-10 x-10 x^2+20 x^3\right )\right )+e^{e^3} \left (-60 x^2+40 x^3+e^{x+x^2} \left (-20 x^3-20 x^4+40 x^5\right )\right )\right )}{1-2 x+x^2} \, dx=-\frac {25 \, x^{5} + {\left (x^{5} + 2 \, x^{3} e^{\left (e^{3}\right )} + x e^{\left (2 \, e^{3}\right )}\right )} e^{\left (2 \, e^{\left (x^{2} + x\right )}\right )} + 50 \, {\left (x^{3} - x + 1\right )} e^{\left (e^{3}\right )} - 10 \, {\left (x^{5} + 2 \, x^{3} e^{\left (e^{3}\right )} + x e^{\left (2 \, e^{3}\right )}\right )} e^{\left (e^{\left (x^{2} + x\right )}\right )} - 25 \, x + 25 \, e^{\left (2 \, e^{3}\right )} + 25}{x - 1} \] Input:
integrate(((((-4*x^3+2*x^2+2*x)*exp(x^2+x)+1)*exp(exp(3))^2+((-8*x^5+4*x^4 +4*x^3)*exp(x^2+x)-4*x^3+6*x^2)*exp(exp(3))+(-4*x^7+2*x^6+2*x^5)*exp(x^2+x )-4*x^5+5*x^4)*exp(exp(x^2+x))^2+(((20*x^3-10*x^2-10*x)*exp(x^2+x)-10)*exp (exp(3))^2+((40*x^5-20*x^4-20*x^3)*exp(x^2+x)+40*x^3-60*x^2)*exp(exp(3))+( 20*x^7-10*x^6-10*x^5)*exp(x^2+x)+40*x^5-50*x^4)*exp(exp(x^2+x))+25*exp(exp (3))^2+(-100*x^3+150*x^2)*exp(exp(3))-100*x^5+125*x^4)/(x^2-2*x+1),x, algo rithm="fricas")
Output:
-(25*x^5 + (x^5 + 2*x^3*e^(e^3) + x*e^(2*e^3))*e^(2*e^(x^2 + x)) + 50*(x^3 - x + 1)*e^(e^3) - 10*(x^5 + 2*x^3*e^(e^3) + x*e^(2*e^3))*e^(e^(x^2 + x)) - 25*x + 25*e^(2*e^3) + 25)/(x - 1)
Leaf count of result is larger than twice the leaf count of optimal. 177 vs. \(2 (29) = 58\).
Time = 0.31 (sec) , antiderivative size = 177, normalized size of antiderivative = 4.78 \[ \int \frac {25 e^{2 e^3}+125 x^4-100 x^5+e^{e^3} \left (150 x^2-100 x^3\right )+e^{2 e^{x+x^2}} \left (5 x^4-4 x^5+e^{x+x^2} \left (2 x^5+2 x^6-4 x^7\right )+e^{2 e^3} \left (1+e^{x+x^2} \left (2 x+2 x^2-4 x^3\right )\right )+e^{e^3} \left (6 x^2-4 x^3+e^{x+x^2} \left (4 x^3+4 x^4-8 x^5\right )\right )\right )+e^{e^{x+x^2}} \left (-50 x^4+40 x^5+e^{x+x^2} \left (-10 x^5-10 x^6+20 x^7\right )+e^{2 e^3} \left (-10+e^{x+x^2} \left (-10 x-10 x^2+20 x^3\right )\right )+e^{e^3} \left (-60 x^2+40 x^3+e^{x+x^2} \left (-20 x^3-20 x^4+40 x^5\right )\right )\right )}{1-2 x+x^2} \, dx=- 25 x^{4} - 25 x^{3} - x^{2} \cdot \left (25 + 50 e^{e^{3}}\right ) - x \left (25 + 50 e^{e^{3}}\right ) + \frac {\left (- x^{6} + x^{5} - 2 x^{4} e^{e^{3}} + 2 x^{3} e^{e^{3}} - x^{2} e^{2 e^{3}} + x e^{2 e^{3}}\right ) e^{2 e^{x^{2} + x}} + \left (10 x^{6} - 10 x^{5} + 20 x^{4} e^{e^{3}} - 20 x^{3} e^{e^{3}} + 10 x^{2} e^{2 e^{3}} - 10 x e^{2 e^{3}}\right ) e^{e^{x^{2} + x}}}{x^{2} - 2 x + 1} - \frac {25 + 50 e^{e^{3}} + 25 e^{2 e^{3}}}{x - 1} \] Input:
integrate(((((-4*x**3+2*x**2+2*x)*exp(x**2+x)+1)*exp(exp(3))**2+((-8*x**5+ 4*x**4+4*x**3)*exp(x**2+x)-4*x**3+6*x**2)*exp(exp(3))+(-4*x**7+2*x**6+2*x* *5)*exp(x**2+x)-4*x**5+5*x**4)*exp(exp(x**2+x))**2+(((20*x**3-10*x**2-10*x )*exp(x**2+x)-10)*exp(exp(3))**2+((40*x**5-20*x**4-20*x**3)*exp(x**2+x)+40 *x**3-60*x**2)*exp(exp(3))+(20*x**7-10*x**6-10*x**5)*exp(x**2+x)+40*x**5-5 0*x**4)*exp(exp(x**2+x))+25*exp(exp(3))**2+(-100*x**3+150*x**2)*exp(exp(3) )-100*x**5+125*x**4)/(x**2-2*x+1),x)
Output:
-25*x**4 - 25*x**3 - x**2*(25 + 50*exp(exp(3))) - x*(25 + 50*exp(exp(3))) + ((-x**6 + x**5 - 2*x**4*exp(exp(3)) + 2*x**3*exp(exp(3)) - x**2*exp(2*ex p(3)) + x*exp(2*exp(3)))*exp(2*exp(x**2 + x)) + (10*x**6 - 10*x**5 + 20*x* *4*exp(exp(3)) - 20*x**3*exp(exp(3)) + 10*x**2*exp(2*exp(3)) - 10*x*exp(2* exp(3)))*exp(exp(x**2 + x)))/(x**2 - 2*x + 1) - (25 + 50*exp(exp(3)) + 25* exp(2*exp(3)))/(x - 1)
Leaf count of result is larger than twice the leaf count of optimal. 148 vs. \(2 (34) = 68\).
Time = 0.09 (sec) , antiderivative size = 148, normalized size of antiderivative = 4.00 \[ \int \frac {25 e^{2 e^3}+125 x^4-100 x^5+e^{e^3} \left (150 x^2-100 x^3\right )+e^{2 e^{x+x^2}} \left (5 x^4-4 x^5+e^{x+x^2} \left (2 x^5+2 x^6-4 x^7\right )+e^{2 e^3} \left (1+e^{x+x^2} \left (2 x+2 x^2-4 x^3\right )\right )+e^{e^3} \left (6 x^2-4 x^3+e^{x+x^2} \left (4 x^3+4 x^4-8 x^5\right )\right )\right )+e^{e^{x+x^2}} \left (-50 x^4+40 x^5+e^{x+x^2} \left (-10 x^5-10 x^6+20 x^7\right )+e^{2 e^3} \left (-10+e^{x+x^2} \left (-10 x-10 x^2+20 x^3\right )\right )+e^{e^3} \left (-60 x^2+40 x^3+e^{x+x^2} \left (-20 x^3-20 x^4+40 x^5\right )\right )\right )}{1-2 x+x^2} \, dx=-25 \, x^{4} - 25 \, x^{3} - 25 \, x^{2} - 50 \, {\left (x^{2} + 4 \, x - \frac {2}{x - 1} + 6 \, \log \left (x - 1\right )\right )} e^{\left (e^{3}\right )} + 150 \, {\left (x - \frac {1}{x - 1} + 2 \, \log \left (x - 1\right )\right )} e^{\left (e^{3}\right )} - 25 \, x - \frac {{\left (x^{5} + 2 \, x^{3} e^{\left (e^{3}\right )} + x e^{\left (2 \, e^{3}\right )}\right )} e^{\left (2 \, e^{\left (x^{2} + x\right )}\right )} - 10 \, {\left (x^{5} + 2 \, x^{3} e^{\left (e^{3}\right )} + x e^{\left (2 \, e^{3}\right )}\right )} e^{\left (e^{\left (x^{2} + x\right )}\right )}}{x - 1} - \frac {25 \, e^{\left (2 \, e^{3}\right )}}{x - 1} - \frac {25}{x - 1} \] Input:
integrate(((((-4*x^3+2*x^2+2*x)*exp(x^2+x)+1)*exp(exp(3))^2+((-8*x^5+4*x^4 +4*x^3)*exp(x^2+x)-4*x^3+6*x^2)*exp(exp(3))+(-4*x^7+2*x^6+2*x^5)*exp(x^2+x )-4*x^5+5*x^4)*exp(exp(x^2+x))^2+(((20*x^3-10*x^2-10*x)*exp(x^2+x)-10)*exp (exp(3))^2+((40*x^5-20*x^4-20*x^3)*exp(x^2+x)+40*x^3-60*x^2)*exp(exp(3))+( 20*x^7-10*x^6-10*x^5)*exp(x^2+x)+40*x^5-50*x^4)*exp(exp(x^2+x))+25*exp(exp (3))^2+(-100*x^3+150*x^2)*exp(exp(3))-100*x^5+125*x^4)/(x^2-2*x+1),x, algo rithm="maxima")
Output:
-25*x^4 - 25*x^3 - 25*x^2 - 50*(x^2 + 4*x - 2/(x - 1) + 6*log(x - 1))*e^(e ^3) + 150*(x - 1/(x - 1) + 2*log(x - 1))*e^(e^3) - 25*x - ((x^5 + 2*x^3*e^ (e^3) + x*e^(2*e^3))*e^(2*e^(x^2 + x)) - 10*(x^5 + 2*x^3*e^(e^3) + x*e^(2* e^3))*e^(e^(x^2 + x)))/(x - 1) - 25*e^(2*e^3)/(x - 1) - 25/(x - 1)
\[ \int \frac {25 e^{2 e^3}+125 x^4-100 x^5+e^{e^3} \left (150 x^2-100 x^3\right )+e^{2 e^{x+x^2}} \left (5 x^4-4 x^5+e^{x+x^2} \left (2 x^5+2 x^6-4 x^7\right )+e^{2 e^3} \left (1+e^{x+x^2} \left (2 x+2 x^2-4 x^3\right )\right )+e^{e^3} \left (6 x^2-4 x^3+e^{x+x^2} \left (4 x^3+4 x^4-8 x^5\right )\right )\right )+e^{e^{x+x^2}} \left (-50 x^4+40 x^5+e^{x+x^2} \left (-10 x^5-10 x^6+20 x^7\right )+e^{2 e^3} \left (-10+e^{x+x^2} \left (-10 x-10 x^2+20 x^3\right )\right )+e^{e^3} \left (-60 x^2+40 x^3+e^{x+x^2} \left (-20 x^3-20 x^4+40 x^5\right )\right )\right )}{1-2 x+x^2} \, dx=\int { -\frac {100 \, x^{5} - 125 \, x^{4} + {\left (4 \, x^{5} - 5 \, x^{4} + 2 \, {\left (2 \, x^{7} - x^{6} - x^{5}\right )} e^{\left (x^{2} + x\right )} + {\left (2 \, {\left (2 \, x^{3} - x^{2} - x\right )} e^{\left (x^{2} + x\right )} - 1\right )} e^{\left (2 \, e^{3}\right )} + 2 \, {\left (2 \, x^{3} - 3 \, x^{2} + 2 \, {\left (2 \, x^{5} - x^{4} - x^{3}\right )} e^{\left (x^{2} + x\right )}\right )} e^{\left (e^{3}\right )}\right )} e^{\left (2 \, e^{\left (x^{2} + x\right )}\right )} + 50 \, {\left (2 \, x^{3} - 3 \, x^{2}\right )} e^{\left (e^{3}\right )} - 10 \, {\left (4 \, x^{5} - 5 \, x^{4} + {\left (2 \, x^{7} - x^{6} - x^{5}\right )} e^{\left (x^{2} + x\right )} + {\left ({\left (2 \, x^{3} - x^{2} - x\right )} e^{\left (x^{2} + x\right )} - 1\right )} e^{\left (2 \, e^{3}\right )} + 2 \, {\left (2 \, x^{3} - 3 \, x^{2} + {\left (2 \, x^{5} - x^{4} - x^{3}\right )} e^{\left (x^{2} + x\right )}\right )} e^{\left (e^{3}\right )}\right )} e^{\left (e^{\left (x^{2} + x\right )}\right )} - 25 \, e^{\left (2 \, e^{3}\right )}}{x^{2} - 2 \, x + 1} \,d x } \] Input:
integrate(((((-4*x^3+2*x^2+2*x)*exp(x^2+x)+1)*exp(exp(3))^2+((-8*x^5+4*x^4 +4*x^3)*exp(x^2+x)-4*x^3+6*x^2)*exp(exp(3))+(-4*x^7+2*x^6+2*x^5)*exp(x^2+x )-4*x^5+5*x^4)*exp(exp(x^2+x))^2+(((20*x^3-10*x^2-10*x)*exp(x^2+x)-10)*exp (exp(3))^2+((40*x^5-20*x^4-20*x^3)*exp(x^2+x)+40*x^3-60*x^2)*exp(exp(3))+( 20*x^7-10*x^6-10*x^5)*exp(x^2+x)+40*x^5-50*x^4)*exp(exp(x^2+x))+25*exp(exp (3))^2+(-100*x^3+150*x^2)*exp(exp(3))-100*x^5+125*x^4)/(x^2-2*x+1),x, algo rithm="giac")
Output:
integrate(-(100*x^5 - 125*x^4 + (4*x^5 - 5*x^4 + 2*(2*x^7 - x^6 - x^5)*e^( x^2 + x) + (2*(2*x^3 - x^2 - x)*e^(x^2 + x) - 1)*e^(2*e^3) + 2*(2*x^3 - 3* x^2 + 2*(2*x^5 - x^4 - x^3)*e^(x^2 + x))*e^(e^3))*e^(2*e^(x^2 + x)) + 50*( 2*x^3 - 3*x^2)*e^(e^3) - 10*(4*x^5 - 5*x^4 + (2*x^7 - x^6 - x^5)*e^(x^2 + x) + ((2*x^3 - x^2 - x)*e^(x^2 + x) - 1)*e^(2*e^3) + 2*(2*x^3 - 3*x^2 + (2 *x^5 - x^4 - x^3)*e^(x^2 + x))*e^(e^3))*e^(e^(x^2 + x)) - 25*e^(2*e^3))/(x ^2 - 2*x + 1), x)
Time = 0.43 (sec) , antiderivative size = 125, normalized size of antiderivative = 3.38 \[ \int \frac {25 e^{2 e^3}+125 x^4-100 x^5+e^{e^3} \left (150 x^2-100 x^3\right )+e^{2 e^{x+x^2}} \left (5 x^4-4 x^5+e^{x+x^2} \left (2 x^5+2 x^6-4 x^7\right )+e^{2 e^3} \left (1+e^{x+x^2} \left (2 x+2 x^2-4 x^3\right )\right )+e^{e^3} \left (6 x^2-4 x^3+e^{x+x^2} \left (4 x^3+4 x^4-8 x^5\right )\right )\right )+e^{e^{x+x^2}} \left (-50 x^4+40 x^5+e^{x+x^2} \left (-10 x^5-10 x^6+20 x^7\right )+e^{2 e^3} \left (-10+e^{x+x^2} \left (-10 x-10 x^2+20 x^3\right )\right )+e^{e^3} \left (-60 x^2+40 x^3+e^{x+x^2} \left (-20 x^3-20 x^4+40 x^5\right )\right )\right )}{1-2 x+x^2} \, dx=\frac {{\mathrm {e}}^{{\mathrm {e}}^{x^2}\,{\mathrm {e}}^x}\,\left (10\,x^5+20\,{\mathrm {e}}^{{\mathrm {e}}^3}\,x^3+10\,{\mathrm {e}}^{2\,{\mathrm {e}}^3}\,x\right )}{x-1}-\frac {25\,{\mathrm {e}}^{2\,{\mathrm {e}}^3}+50\,{\mathrm {e}}^{{\mathrm {e}}^3}+25}{x-1}-x^2\,\left (50\,{\mathrm {e}}^{{\mathrm {e}}^3}+25\right )-25\,x^3-25\,x^4-x\,\left (50\,{\mathrm {e}}^{{\mathrm {e}}^3}+25\right )-\frac {{\mathrm {e}}^{2\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^x}\,\left (x^5+2\,{\mathrm {e}}^{{\mathrm {e}}^3}\,x^3+{\mathrm {e}}^{2\,{\mathrm {e}}^3}\,x\right )}{x-1} \] Input:
int((25*exp(2*exp(3)) - exp(exp(x + x^2))*(exp(2*exp(3))*(exp(x + x^2)*(10 *x + 10*x^2 - 20*x^3) + 10) + exp(exp(3))*(exp(x + x^2)*(20*x^3 + 20*x^4 - 40*x^5) + 60*x^2 - 40*x^3) + exp(x + x^2)*(10*x^5 + 10*x^6 - 20*x^7) + 50 *x^4 - 40*x^5) + 125*x^4 - 100*x^5 + exp(2*exp(x + x^2))*(exp(2*exp(3))*(e xp(x + x^2)*(2*x + 2*x^2 - 4*x^3) + 1) + exp(exp(3))*(exp(x + x^2)*(4*x^3 + 4*x^4 - 8*x^5) + 6*x^2 - 4*x^3) + exp(x + x^2)*(2*x^5 + 2*x^6 - 4*x^7) + 5*x^4 - 4*x^5) + exp(exp(3))*(150*x^2 - 100*x^3))/(x^2 - 2*x + 1),x)
Output:
(exp(exp(x^2)*exp(x))*(10*x*exp(2*exp(3)) + 20*x^3*exp(exp(3)) + 10*x^5))/ (x - 1) - (25*exp(2*exp(3)) + 50*exp(exp(3)) + 25)/(x - 1) - x^2*(50*exp(e xp(3)) + 25) - 25*x^3 - 25*x^4 - x*(50*exp(exp(3)) + 25) - (exp(2*exp(x^2) *exp(x))*(x*exp(2*exp(3)) + 2*x^3*exp(exp(3)) + x^5))/(x - 1)
Time = 0.34 (sec) , antiderivative size = 136, normalized size of antiderivative = 3.68 \[ \int \frac {25 e^{2 e^3}+125 x^4-100 x^5+e^{e^3} \left (150 x^2-100 x^3\right )+e^{2 e^{x+x^2}} \left (5 x^4-4 x^5+e^{x+x^2} \left (2 x^5+2 x^6-4 x^7\right )+e^{2 e^3} \left (1+e^{x+x^2} \left (2 x+2 x^2-4 x^3\right )\right )+e^{e^3} \left (6 x^2-4 x^3+e^{x+x^2} \left (4 x^3+4 x^4-8 x^5\right )\right )\right )+e^{e^{x+x^2}} \left (-50 x^4+40 x^5+e^{x+x^2} \left (-10 x^5-10 x^6+20 x^7\right )+e^{2 e^3} \left (-10+e^{x+x^2} \left (-10 x-10 x^2+20 x^3\right )\right )+e^{e^3} \left (-60 x^2+40 x^3+e^{x+x^2} \left (-20 x^3-20 x^4+40 x^5\right )\right )\right )}{1-2 x+x^2} \, dx=\frac {x \left (-e^{2 e^{x^{2}+x}+2 e^{3}}-2 e^{2 e^{x^{2}+x}+e^{3}} x^{2}-e^{2 e^{x^{2}+x}} x^{4}+10 e^{e^{x^{2}+x}+2 e^{3}}+20 e^{e^{x^{2}+x}+e^{3}} x^{2}+10 e^{e^{x^{2}+x}} x^{4}-25 e^{2 e^{3}}-50 e^{e^{3}} x^{2}-25 x^{4}\right )}{x -1} \] Input:
int(((((-4*x^3+2*x^2+2*x)*exp(x^2+x)+1)*exp(exp(3))^2+((-8*x^5+4*x^4+4*x^3 )*exp(x^2+x)-4*x^3+6*x^2)*exp(exp(3))+(-4*x^7+2*x^6+2*x^5)*exp(x^2+x)-4*x^ 5+5*x^4)*exp(exp(x^2+x))^2+(((20*x^3-10*x^2-10*x)*exp(x^2+x)-10)*exp(exp(3 ))^2+((40*x^5-20*x^4-20*x^3)*exp(x^2+x)+40*x^3-60*x^2)*exp(exp(3))+(20*x^7 -10*x^6-10*x^5)*exp(x^2+x)+40*x^5-50*x^4)*exp(exp(x^2+x))+25*exp(exp(3))^2 +(-100*x^3+150*x^2)*exp(exp(3))-100*x^5+125*x^4)/(x^2-2*x+1),x)
Output:
(x*( - e**(2*e**(x**2 + x) + 2*e**3) - 2*e**(2*e**(x**2 + x) + e**3)*x**2 - e**(2*e**(x**2 + x))*x**4 + 10*e**(e**(x**2 + x) + 2*e**3) + 20*e**(e**( x**2 + x) + e**3)*x**2 + 10*e**(e**(x**2 + x))*x**4 - 25*e**(2*e**3) - 50* e**(e**3)*x**2 - 25*x**4))/(x - 1)