Integrand size = 201, antiderivative size = 33 \[ \int \frac {e^x \left (-2-2 x+2 x^2+2 x^3\right )+e^{\frac {5 e^{e^x} x-2 x^2-2 x^3}{2+2 x}} \left (e^x \left (2+8 x+10 x^2+4 x^3\right )+e^{e^x} \left (-5 e^x+e^{2 x} \left (-5 x-5 x^2\right )\right )\right )}{8 x^2+16 x^3+8 x^4+e^{\frac {2 \left (5 e^{e^x} x-2 x^2-2 x^3\right )}{2+2 x}} \left (8+16 x+8 x^2\right )+e^{\frac {5 e^{e^x} x-2 x^2-2 x^3}{2+2 x}} \left (16 x+32 x^2+16 x^3\right )} \, dx=\frac {e^x}{4 \left (e^{x \left (-x+\frac {5 e^{e^x}}{2 (1+x)}\right )}+x\right )} \]
Time = 0.64 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.15 \[ \int \frac {e^x \left (-2-2 x+2 x^2+2 x^3\right )+e^{\frac {5 e^{e^x} x-2 x^2-2 x^3}{2+2 x}} \left (e^x \left (2+8 x+10 x^2+4 x^3\right )+e^{e^x} \left (-5 e^x+e^{2 x} \left (-5 x-5 x^2\right )\right )\right )}{8 x^2+16 x^3+8 x^4+e^{\frac {2 \left (5 e^{e^x} x-2 x^2-2 x^3\right )}{2+2 x}} \left (8+16 x+8 x^2\right )+e^{\frac {5 e^{e^x} x-2 x^2-2 x^3}{2+2 x}} \left (16 x+32 x^2+16 x^3\right )} \, dx=\frac {e^{x+x^2}}{4 \left (e^{\frac {5 e^{e^x} x}{2 (1+x)}}+e^{x^2} x\right )} \]
Integrate[(E^x*(-2 - 2*x + 2*x^2 + 2*x^3) + E^((5*E^E^x*x - 2*x^2 - 2*x^3) /(2 + 2*x))*(E^x*(2 + 8*x + 10*x^2 + 4*x^3) + E^E^x*(-5*E^x + E^(2*x)*(-5* x - 5*x^2))))/(8*x^2 + 16*x^3 + 8*x^4 + E^((2*(5*E^E^x*x - 2*x^2 - 2*x^3)) /(2 + 2*x))*(8 + 16*x + 8*x^2) + E^((5*E^E^x*x - 2*x^2 - 2*x^3)/(2 + 2*x)) *(16*x + 32*x^2 + 16*x^3)),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.
| \(\displaystyle \int \frac {e^x \left (2 x^3+2 x^2-2 x-2\right )+e^{\frac {-2 x^3-2 x^2+5 e^{e^x} x}{2 x+2}} \left (e^{e^x} \left (e^{2 x} \left (-5 x^2-5 x\right )-5 e^x\right )+e^x \left (4 x^3+10 x^2+8 x+2\right )\right )}{\left (8 x^2+16 x+8\right ) \exp \left (\frac {2 \left (-2 x^3-2 x^2+5 e^{e^x} x\right )}{2 x+2}\right )+8 x^4+16 x^3+8 x^2+e^{\frac {-2 x^3-2 x^2+5 e^{e^x} x}{2 x+2}} \left (16 x^3+32 x^2+16 x\right )} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {e^{2 x^2} \left (e^x \left (2 x^3+2 x^2-2 x-2\right )+e^{\frac {-2 x^3-2 x^2+5 e^{e^x} x}{2 x+2}} \left (e^{e^x} \left (e^{2 x} \left (-5 x^2-5 x\right )-5 e^x\right )+e^x \left (4 x^3+10 x^2+8 x+2\right )\right )\right )}{8 (x+1)^2 \left (e^{x^2} x+e^{\frac {5 e^{e^x} x}{2 x+2}}\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{8} \int -\frac {e^{2 x^2} \left (2 e^x \left (-x^3-x^2+x+1\right )-e^{\frac {-2 x^3-2 x^2+5 e^{e^x} x}{2 (x+1)}} \left (2 e^x \left (2 x^3+5 x^2+4 x+1\right )-5 e^{e^x} \left (e^{2 x} \left (x^2+x\right )+e^x\right )\right )\right )}{(x+1)^2 \left (e^{x^2} x+e^{\frac {5 e^{e^x} x}{2 (x+1)}}\right )^2}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {1}{8} \int \frac {e^{2 x^2} \left (2 e^x \left (-x^3-x^2+x+1\right )-e^{\frac {-2 x^3-2 x^2+5 e^{e^x} x}{2 (x+1)}} \left (2 e^x \left (2 x^3+5 x^2+4 x+1\right )-5 e^{e^x} \left (e^{2 x} \left (x^2+x\right )+e^x\right )\right )\right )}{(x+1)^2 \left (e^{x^2} x+e^{\frac {5 e^{e^x} x}{2 (x+1)}}\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {1}{8} \int \left (-\frac {2 e^{2 x^2+x} (x-1)}{\left (e^{x^2} x+e^{\frac {5 e^{e^x} x}{2 x+2}}\right )^2}-\frac {\exp \left (2 x^2+\frac {\left (-2 x^2+5 e^{e^x}+2\right ) x}{2 (x+1)}\right ) \left (4 x^3-5 e^{x+e^x} x^2+10 x^2-5 e^{x+e^x} x+8 x-5 e^{e^x}+2\right )}{(x+1)^2 \left (e^{x^2} x+e^{\frac {5 e^{e^x} x}{2 x+2}}\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle -\frac {1}{8} \int \left (-\frac {2 e^{2 x^2+x} (x-1)}{\left (e^{x^2} x+e^{\frac {5 e^{e^x} x}{2 x+2}}\right )^2}-\frac {e^{\frac {x \left (2 x^2+4 x+5 e^{e^x}+2\right )}{2 (x+1)}} \left (4 x^3-5 e^{x+e^x} x^2+10 x^2-5 e^{x+e^x} x+8 x-5 e^{e^x}+2\right )}{(x+1)^2 \left (e^{x^2} x+e^{\frac {5 e^{e^x} x}{2 x+2}}\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {1}{8} \int \left (-\frac {2 e^{2 x^2+x} (x-1)}{\left (e^{x^2} x+e^{\frac {5 e^{e^x} x}{2 x+2}}\right )^2}-\frac {e^{\frac {x \left (2 (x+1)^2+5 e^{e^x}\right )}{2 (x+1)}} \left (4 x^3-5 e^{x+e^x} x^2+10 x^2-5 e^{x+e^x} x+8 x-5 e^{e^x}+2\right )}{(x+1)^2 \left (e^{x^2} x+e^{\frac {5 e^{e^x} x}{2 x+2}}\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle -\frac {1}{8} \int \left (-\frac {2 e^{2 x^2+x} (x-1)}{\left (e^{x^2} x+e^{\frac {5 e^{e^x} x}{2 x+2}}\right )^2}-\frac {e^{\frac {x \left (2 (x+1)^2+5 e^{e^x}\right )}{2 (x+1)}} \left (4 x^3-5 e^{x+e^x} x^2+10 x^2-5 e^{x+e^x} x+8 x-5 e^{e^x}+2\right )}{(x+1)^2 \left (e^{x^2} x+e^{\frac {5 e^{e^x} x}{2 x+2}}\right )^2}\right )dx\) |
Int[(E^x*(-2 - 2*x + 2*x^2 + 2*x^3) + E^((5*E^E^x*x - 2*x^2 - 2*x^3)/(2 + 2*x))*(E^x*(2 + 8*x + 10*x^2 + 4*x^3) + E^E^x*(-5*E^x + E^(2*x)*(-5*x - 5* x^2))))/(8*x^2 + 16*x^3 + 8*x^4 + E^((2*(5*E^E^x*x - 2*x^2 - 2*x^3))/(2 + 2*x))*(8 + 16*x + 8*x^2) + E^((5*E^E^x*x - 2*x^2 - 2*x^3)/(2 + 2*x))*(16*x + 32*x^2 + 16*x^3)),x]
3.19.72.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Time = 5.57 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.97
| method | result | size |
| risch | \(\frac {{\mathrm e}^{x}}{4 x +4 \,{\mathrm e}^{-\frac {x \left (2 x^{2}-5 \,{\mathrm e}^{{\mathrm e}^{x}}+2 x \right )}{2 \left (1+x \right )}}}\) | \(32\) |
| parallelrisch | \(\frac {{\mathrm e}^{x}}{4 \,{\mathrm e}^{\frac {x \left (-2 x^{2}+5 \,{\mathrm e}^{{\mathrm e}^{x}}-2 x \right )}{2+2 x}}+4 x}\) | \(32\) |
int(((((-5*x^2-5*x)*exp(x)^2-5*exp(x))*exp(exp(x))+(4*x^3+10*x^2+8*x+2)*ex p(x))*exp((5*x*exp(exp(x))-2*x^3-2*x^2)/(2+2*x))+(2*x^3+2*x^2-2*x-2)*exp(x ))/((8*x^2+16*x+8)*exp((5*x*exp(exp(x))-2*x^3-2*x^2)/(2+2*x))^2+(16*x^3+32 *x^2+16*x)*exp((5*x*exp(exp(x))-2*x^3-2*x^2)/(2+2*x))+8*x^4+16*x^3+8*x^2), x,method=_RETURNVERBOSE)
Time = 0.24 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00 \[ \int \frac {e^x \left (-2-2 x+2 x^2+2 x^3\right )+e^{\frac {5 e^{e^x} x-2 x^2-2 x^3}{2+2 x}} \left (e^x \left (2+8 x+10 x^2+4 x^3\right )+e^{e^x} \left (-5 e^x+e^{2 x} \left (-5 x-5 x^2\right )\right )\right )}{8 x^2+16 x^3+8 x^4+e^{\frac {2 \left (5 e^{e^x} x-2 x^2-2 x^3\right )}{2+2 x}} \left (8+16 x+8 x^2\right )+e^{\frac {5 e^{e^x} x-2 x^2-2 x^3}{2+2 x}} \left (16 x+32 x^2+16 x^3\right )} \, dx=\frac {e^{x}}{4 \, {\left (x + e^{\left (-\frac {2 \, x^{3} + 2 \, x^{2} - 5 \, x e^{\left (e^{x}\right )}}{2 \, {\left (x + 1\right )}}\right )}\right )}} \]
integrate(((((-5*x^2-5*x)*exp(x)^2-5*exp(x))*exp(exp(x))+(4*x^3+10*x^2+8*x +2)*exp(x))*exp((5*x*exp(exp(x))-2*x^3-2*x^2)/(2+2*x))+(2*x^3+2*x^2-2*x-2) *exp(x))/((8*x^2+16*x+8)*exp((5*x*exp(exp(x))-2*x^3-2*x^2)/(2+2*x))^2+(16* x^3+32*x^2+16*x)*exp((5*x*exp(exp(x))-2*x^3-2*x^2)/(2+2*x))+8*x^4+16*x^3+8 *x^2),x, algorithm=\
Time = 0.35 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.97 \[ \int \frac {e^x \left (-2-2 x+2 x^2+2 x^3\right )+e^{\frac {5 e^{e^x} x-2 x^2-2 x^3}{2+2 x}} \left (e^x \left (2+8 x+10 x^2+4 x^3\right )+e^{e^x} \left (-5 e^x+e^{2 x} \left (-5 x-5 x^2\right )\right )\right )}{8 x^2+16 x^3+8 x^4+e^{\frac {2 \left (5 e^{e^x} x-2 x^2-2 x^3\right )}{2+2 x}} \left (8+16 x+8 x^2\right )+e^{\frac {5 e^{e^x} x-2 x^2-2 x^3}{2+2 x}} \left (16 x+32 x^2+16 x^3\right )} \, dx=\frac {e^{x}}{4 x + 4 e^{\frac {- 2 x^{3} - 2 x^{2} + 5 x e^{e^{x}}}{2 x + 2}}} \]
integrate(((((-5*x**2-5*x)*exp(x)**2-5*exp(x))*exp(exp(x))+(4*x**3+10*x**2 +8*x+2)*exp(x))*exp((5*x*exp(exp(x))-2*x**3-2*x**2)/(2+2*x))+(2*x**3+2*x** 2-2*x-2)*exp(x))/((8*x**2+16*x+8)*exp((5*x*exp(exp(x))-2*x**3-2*x**2)/(2+2 *x))**2+(16*x**3+32*x**2+16*x)*exp((5*x*exp(exp(x))-2*x**3-2*x**2)/(2+2*x) )+8*x**4+16*x**3+8*x**2),x)
Leaf count of result is larger than twice the leaf count of optimal. 161 vs. \(2 (26) = 52\).
Time = 0.38 (sec) , antiderivative size = 161, normalized size of antiderivative = 4.88 \[ \int \frac {e^x \left (-2-2 x+2 x^2+2 x^3\right )+e^{\frac {5 e^{e^x} x-2 x^2-2 x^3}{2+2 x}} \left (e^x \left (2+8 x+10 x^2+4 x^3\right )+e^{e^x} \left (-5 e^x+e^{2 x} \left (-5 x-5 x^2\right )\right )\right )}{8 x^2+16 x^3+8 x^4+e^{\frac {2 \left (5 e^{e^x} x-2 x^2-2 x^3\right )}{2+2 x}} \left (8+16 x+8 x^2\right )+e^{\frac {5 e^{e^x} x-2 x^2-2 x^3}{2+2 x}} \left (16 x+32 x^2+16 x^3\right )} \, dx=\frac {{\left (4 \, x^{4} + 8 \, x^{3} + 6 \, x^{2} - 5 \, {\left ({\left (x^{3} + x^{2}\right )} e^{x} + x\right )} e^{\left (e^{x}\right )} + 4 \, x + 2\right )} e^{\left (x^{2} + x + \frac {5 \, e^{\left (e^{x}\right )}}{2 \, {\left (x + 1\right )}}\right )}}{4 \, {\left ({\left (4 \, x^{5} + 8 \, x^{4} + 6 \, x^{3} + 4 \, x^{2} - 5 \, {\left (x^{2} + {\left (x^{4} + x^{3}\right )} e^{x}\right )} e^{\left (e^{x}\right )} + 2 \, x\right )} e^{\left (x^{2} + \frac {5 \, e^{\left (e^{x}\right )}}{2 \, {\left (x + 1\right )}}\right )} + {\left (4 \, x^{4} + 8 \, x^{3} + 6 \, x^{2} - 5 \, {\left ({\left (x^{3} + x^{2}\right )} e^{x} + x\right )} e^{\left (e^{x}\right )} + 4 \, x + 2\right )} e^{\left (\frac {5}{2} \, e^{\left (e^{x}\right )}\right )}\right )}} \]
integrate(((((-5*x^2-5*x)*exp(x)^2-5*exp(x))*exp(exp(x))+(4*x^3+10*x^2+8*x +2)*exp(x))*exp((5*x*exp(exp(x))-2*x^3-2*x^2)/(2+2*x))+(2*x^3+2*x^2-2*x-2) *exp(x))/((8*x^2+16*x+8)*exp((5*x*exp(exp(x))-2*x^3-2*x^2)/(2+2*x))^2+(16* x^3+32*x^2+16*x)*exp((5*x*exp(exp(x))-2*x^3-2*x^2)/(2+2*x))+8*x^4+16*x^3+8 *x^2),x, algorithm=\
1/4*(4*x^4 + 8*x^3 + 6*x^2 - 5*((x^3 + x^2)*e^x + x)*e^(e^x) + 4*x + 2)*e^ (x^2 + x + 5/2*e^(e^x)/(x + 1))/((4*x^5 + 8*x^4 + 6*x^3 + 4*x^2 - 5*(x^2 + (x^4 + x^3)*e^x)*e^(e^x) + 2*x)*e^(x^2 + 5/2*e^(e^x)/(x + 1)) + (4*x^4 + 8*x^3 + 6*x^2 - 5*((x^3 + x^2)*e^x + x)*e^(e^x) + 4*x + 2)*e^(5/2*e^(e^x)) )
\[ \int \frac {e^x \left (-2-2 x+2 x^2+2 x^3\right )+e^{\frac {5 e^{e^x} x-2 x^2-2 x^3}{2+2 x}} \left (e^x \left (2+8 x+10 x^2+4 x^3\right )+e^{e^x} \left (-5 e^x+e^{2 x} \left (-5 x-5 x^2\right )\right )\right )}{8 x^2+16 x^3+8 x^4+e^{\frac {2 \left (5 e^{e^x} x-2 x^2-2 x^3\right )}{2+2 x}} \left (8+16 x+8 x^2\right )+e^{\frac {5 e^{e^x} x-2 x^2-2 x^3}{2+2 x}} \left (16 x+32 x^2+16 x^3\right )} \, dx=\int { \frac {2 \, {\left (x^{3} + x^{2} - x - 1\right )} e^{x} + {\left (2 \, {\left (2 \, x^{3} + 5 \, x^{2} + 4 \, x + 1\right )} e^{x} - 5 \, {\left ({\left (x^{2} + x\right )} e^{\left (2 \, x\right )} + e^{x}\right )} e^{\left (e^{x}\right )}\right )} e^{\left (-\frac {2 \, x^{3} + 2 \, x^{2} - 5 \, x e^{\left (e^{x}\right )}}{2 \, {\left (x + 1\right )}}\right )}}{8 \, {\left (x^{4} + 2 \, x^{3} + x^{2} + 2 \, {\left (x^{3} + 2 \, x^{2} + x\right )} e^{\left (-\frac {2 \, x^{3} + 2 \, x^{2} - 5 \, x e^{\left (e^{x}\right )}}{2 \, {\left (x + 1\right )}}\right )} + {\left (x^{2} + 2 \, x + 1\right )} e^{\left (-\frac {2 \, x^{3} + 2 \, x^{2} - 5 \, x e^{\left (e^{x}\right )}}{x + 1}\right )}\right )}} \,d x } \]
integrate(((((-5*x^2-5*x)*exp(x)^2-5*exp(x))*exp(exp(x))+(4*x^3+10*x^2+8*x +2)*exp(x))*exp((5*x*exp(exp(x))-2*x^3-2*x^2)/(2+2*x))+(2*x^3+2*x^2-2*x-2) *exp(x))/((8*x^2+16*x+8)*exp((5*x*exp(exp(x))-2*x^3-2*x^2)/(2+2*x))^2+(16* x^3+32*x^2+16*x)*exp((5*x*exp(exp(x))-2*x^3-2*x^2)/(2+2*x))+8*x^4+16*x^3+8 *x^2),x, algorithm=\
integrate(1/8*(2*(x^3 + x^2 - x - 1)*e^x + (2*(2*x^3 + 5*x^2 + 4*x + 1)*e^ x - 5*((x^2 + x)*e^(2*x) + e^x)*e^(e^x))*e^(-1/2*(2*x^3 + 2*x^2 - 5*x*e^(e ^x))/(x + 1)))/(x^4 + 2*x^3 + x^2 + 2*(x^3 + 2*x^2 + x)*e^(-1/2*(2*x^3 + 2 *x^2 - 5*x*e^(e^x))/(x + 1)) + (x^2 + 2*x + 1)*e^(-(2*x^3 + 2*x^2 - 5*x*e^ (e^x))/(x + 1))), x)
Time = 11.49 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.45 \[ \int \frac {e^x \left (-2-2 x+2 x^2+2 x^3\right )+e^{\frac {5 e^{e^x} x-2 x^2-2 x^3}{2+2 x}} \left (e^x \left (2+8 x+10 x^2+4 x^3\right )+e^{e^x} \left (-5 e^x+e^{2 x} \left (-5 x-5 x^2\right )\right )\right )}{8 x^2+16 x^3+8 x^4+e^{\frac {2 \left (5 e^{e^x} x-2 x^2-2 x^3\right )}{2+2 x}} \left (8+16 x+8 x^2\right )+e^{\frac {5 e^{e^x} x-2 x^2-2 x^3}{2+2 x}} \left (16 x+32 x^2+16 x^3\right )} \, dx=\frac {{\mathrm {e}}^x}{4\,x}-\frac {{\mathrm {e}}^x}{x\,\left (4\,x\,{\mathrm {e}}^{\frac {2\,x^2-5\,x\,{\mathrm {e}}^{{\mathrm {e}}^x}+2\,x^3}{2\,x+2}}+4\right )} \]
int(-(exp(-(2*x^2 - 5*x*exp(exp(x)) + 2*x^3)/(2*x + 2))*(exp(exp(x))*(5*ex p(x) + exp(2*x)*(5*x + 5*x^2)) - exp(x)*(8*x + 10*x^2 + 4*x^3 + 2)) + exp( x)*(2*x - 2*x^2 - 2*x^3 + 2))/(exp(-(2*(2*x^2 - 5*x*exp(exp(x)) + 2*x^3))/ (2*x + 2))*(16*x + 8*x^2 + 8) + exp(-(2*x^2 - 5*x*exp(exp(x)) + 2*x^3)/(2* x + 2))*(16*x + 32*x^2 + 16*x^3) + 8*x^2 + 16*x^3 + 8*x^4),x)