Integrand size = 289, antiderivative size = 35 \[ \int \frac {2 e^{4+4 x}-2 x-4 e^{4+3 x} x+e^{4+2 x} \left (-10+2 x^2\right )+e^{4+x} \left (20 x+4 x^2\right )+e^4 \left (-x-10 x^2-2 x^3\right )+\left (8 x^3+e^{4+3 x} \left (-4 x+4 x^2\right )+e^4 \left (-20 x^2-2 x^3\right )+e^{2 x} \left (8 x+e^4 \left (-18 x-4 x^3\right )\right )+e^x \left (-16 x^2+e^4 \left (20 x+20 x^2+4 x^3\right )\right )\right ) \log (x)+\left (-8 e^{4 x} x+32 e^{3 x} x^2-48 e^{2 x} x^3+32 e^x x^4-8 x^5\right ) \log ^2(x)}{2 x+\left (-8 e^{2 x} x+16 e^x x^2-8 x^3\right ) \log (x)+\left (8 e^{4 x} x-32 e^{3 x} x^2+48 e^{2 x} x^3-32 e^x x^4+8 x^5\right ) \log ^2(x)} \, dx=-x+\frac {e^4 \left (-5+e^{2 x}-x\right )}{2-4 \left (-e^x+x\right )^2 \log (x)} \]
Time = 0.23 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.14 \[ \int \frac {2 e^{4+4 x}-2 x-4 e^{4+3 x} x+e^{4+2 x} \left (-10+2 x^2\right )+e^{4+x} \left (20 x+4 x^2\right )+e^4 \left (-x-10 x^2-2 x^3\right )+\left (8 x^3+e^{4+3 x} \left (-4 x+4 x^2\right )+e^4 \left (-20 x^2-2 x^3\right )+e^{2 x} \left (8 x+e^4 \left (-18 x-4 x^3\right )\right )+e^x \left (-16 x^2+e^4 \left (20 x+20 x^2+4 x^3\right )\right )\right ) \log (x)+\left (-8 e^{4 x} x+32 e^{3 x} x^2-48 e^{2 x} x^3+32 e^x x^4-8 x^5\right ) \log ^2(x)}{2 x+\left (-8 e^{2 x} x+16 e^x x^2-8 x^3\right ) \log (x)+\left (8 e^{4 x} x-32 e^{3 x} x^2+48 e^{2 x} x^3-32 e^x x^4+8 x^5\right ) \log ^2(x)} \, dx=\frac {1}{2} \left (-2 x-\frac {e^4 \left (-5+e^{2 x}-x\right )}{-1+2 \left (e^x-x\right )^2 \log (x)}\right ) \]
Integrate[(2*E^(4 + 4*x) - 2*x - 4*E^(4 + 3*x)*x + E^(4 + 2*x)*(-10 + 2*x^ 2) + E^(4 + x)*(20*x + 4*x^2) + E^4*(-x - 10*x^2 - 2*x^3) + (8*x^3 + E^(4 + 3*x)*(-4*x + 4*x^2) + E^4*(-20*x^2 - 2*x^3) + E^(2*x)*(8*x + E^4*(-18*x - 4*x^3)) + E^x*(-16*x^2 + E^4*(20*x + 20*x^2 + 4*x^3)))*Log[x] + (-8*E^(4 *x)*x + 32*E^(3*x)*x^2 - 48*E^(2*x)*x^3 + 32*E^x*x^4 - 8*x^5)*Log[x]^2)/(2 *x + (-8*E^(2*x)*x + 16*E^x*x^2 - 8*x^3)*Log[x] + (8*E^(4*x)*x - 32*E^(3*x )*x^2 + 48*E^(2*x)*x^3 - 32*E^x*x^4 + 8*x^5)*Log[x]^2),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.
| \(\displaystyle \int \frac {e^{2 x+4} \left (2 x^2-10\right )+e^{x+4} \left (4 x^2+20 x\right )+e^4 \left (-2 x^3-10 x^2-x\right )+\left (8 x^3+e^{2 x} \left (e^4 \left (-4 x^3-18 x\right )+8 x\right )+e^{3 x+4} \left (4 x^2-4 x\right )+e^4 \left (-2 x^3-20 x^2\right )+e^x \left (e^4 \left (4 x^3+20 x^2+20 x\right )-16 x^2\right )\right ) \log (x)+\left (-8 x^5+32 e^x x^4-48 e^{2 x} x^3+32 e^{3 x} x^2-8 e^{4 x} x\right ) \log ^2(x)+2 e^{4 x+4}-4 e^{3 x+4} x-2 x}{\left (-8 x^3+16 e^x x^2-8 e^{2 x} x\right ) \log (x)+\left (8 x^5-32 e^x x^4+48 e^{2 x} x^3-32 e^{3 x} x^2+8 e^{4 x} x\right ) \log ^2(x)+2 x} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {2 e^{2 x+4} \left (x^2-5\right )-e^4 x \left (2 x^2+10 x+1\right )+2 e^{4 x+4}-4 e^{3 x+4} x-2 x+4 e^{x+4} x (x+5)-8 x \left (e^x-x\right )^4 \log ^2(x)+2 x \left (2 e^{2 x+4} (x-1)+4 e^x-4 x+e^4 (x+10)-e^{x+4} (2 x+9)\right ) \left (e^x-x\right ) \log (x)}{2 x \left (1-2 \left (e^x-x\right )^2 \log (x)\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \int \frac {-8 x \log ^2(x) \left (e^x-x\right )^4+2 x \left (-2 e^{2 x+4} (1-x)+4 e^x-4 x+e^4 (x+10)-e^{x+4} (2 x+9)\right ) \log (x) \left (e^x-x\right )+2 e^{4 x+4}-4 e^{3 x+4} x-2 x+4 e^{x+4} x (x+5)-2 e^{2 x+4} \left (5-x^2\right )-e^4 x \left (2 x^2+10 x+1\right )}{x \left (1-2 \left (e^x-x\right )^2 \log (x)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{2} \int \left (\frac {e^4-4 x \log ^2(x)}{2 x \log ^2(x)}+\frac {e^4 \left (2 \log ^2(x) x^3+2 e^x \log ^2(x) x^2-4 \log ^2(x) x^2-\log (x) x^2-2 e^x \log ^2(x) x-9 \log ^2(x) x+2 e^x \log (x) x-5 \log (x)+1\right )}{x \log ^2(x) \left (2 \log (x) x^2-4 e^x \log (x) x+2 e^{2 x} \log (x)-1\right )}-\frac {e^4 \left (8 \log ^3(x) x^5-8 e^x \log ^3(x) x^4-16 \log ^3(x) x^4+16 e^x \log ^3(x) x^3-32 \log ^3(x) x^3+32 e^x \log ^3(x) x^2+40 \log ^3(x) x^2-12 e^x \log ^2(x) x^2+8 \log ^2(x) x^2+2 \log (x) x^2-40 e^x \log ^3(x) x+4 e^x \log ^2(x) x+20 \log ^2(x) x-4 e^x \log (x) x+10 \log (x)-1\right )}{2 x \log ^2(x) \left (2 \log (x) x^2-4 e^x \log (x) x+2 e^{2 x} \log (x)-1\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} \left (-2 x-10 e^4 \int \frac {1}{\left (2 \log (x) x^2-4 e^x \log (x) x+2 e^{2 x} \log (x)-1\right )^2}dx-2 e^4 \int \frac {e^x}{\left (2 \log (x) x^2-4 e^x \log (x) x+2 e^{2 x} \log (x)-1\right )^2}dx-4 e^4 \int \frac {x}{\left (2 \log (x) x^2-4 e^x \log (x) x+2 e^{2 x} \log (x)-1\right )^2}dx+6 e^4 \int \frac {e^x x}{\left (2 \log (x) x^2-4 e^x \log (x) x+2 e^{2 x} \log (x)-1\right )^2}dx+\frac {1}{2} e^4 \int \frac {1}{x \log ^2(x) \left (2 \log (x) x^2-4 e^x \log (x) x+2 e^{2 x} \log (x)-1\right )^2}dx+2 e^4 \int \frac {e^x}{\log (x) \left (2 \log (x) x^2-4 e^x \log (x) x+2 e^{2 x} \log (x)-1\right )^2}dx-5 e^4 \int \frac {1}{x \log (x) \left (2 \log (x) x^2-4 e^x \log (x) x+2 e^{2 x} \log (x)-1\right )^2}dx-e^4 \int \frac {x}{\log (x) \left (2 \log (x) x^2-4 e^x \log (x) x+2 e^{2 x} \log (x)-1\right )^2}dx+20 e^4 \int \frac {e^x \log (x)}{\left (2 \log (x) x^2-4 e^x \log (x) x+2 e^{2 x} \log (x)-1\right )^2}dx-20 e^4 \int \frac {x \log (x)}{\left (2 \log (x) x^2-4 e^x \log (x) x+2 e^{2 x} \log (x)-1\right )^2}dx-16 e^4 \int \frac {e^x x \log (x)}{\left (2 \log (x) x^2-4 e^x \log (x) x+2 e^{2 x} \log (x)-1\right )^2}dx+16 e^4 \int \frac {x^2 \log (x)}{\left (2 \log (x) x^2-4 e^x \log (x) x+2 e^{2 x} \log (x)-1\right )^2}dx-8 e^4 \int \frac {e^x x^2 \log (x)}{\left (2 \log (x) x^2-4 e^x \log (x) x+2 e^{2 x} \log (x)-1\right )^2}dx+8 e^4 \int \frac {x^3 \log (x)}{\left (2 \log (x) x^2-4 e^x \log (x) x+2 e^{2 x} \log (x)-1\right )^2}dx+4 e^4 \int \frac {e^x x^3 \log (x)}{\left (2 \log (x) x^2-4 e^x \log (x) x+2 e^{2 x} \log (x)-1\right )^2}dx-4 e^4 \int \frac {x^4 \log (x)}{\left (2 \log (x) x^2-4 e^x \log (x) x+2 e^{2 x} \log (x)-1\right )^2}dx-9 e^4 \int \frac {1}{2 \log (x) x^2-4 e^x \log (x) x+2 e^{2 x} \log (x)-1}dx-2 e^4 \int \frac {e^x}{2 \log (x) x^2-4 e^x \log (x) x+2 e^{2 x} \log (x)-1}dx-4 e^4 \int \frac {x}{2 \log (x) x^2-4 e^x \log (x) x+2 e^{2 x} \log (x)-1}dx+2 e^4 \int \frac {e^x x}{2 \log (x) x^2-4 e^x \log (x) x+2 e^{2 x} \log (x)-1}dx+2 e^4 \int \frac {x^2}{2 \log (x) x^2-4 e^x \log (x) x+2 e^{2 x} \log (x)-1}dx+e^4 \int \frac {1}{x \log ^2(x) \left (2 \log (x) x^2-4 e^x \log (x) x+2 e^{2 x} \log (x)-1\right )}dx+2 e^4 \int \frac {e^x}{\log (x) \left (2 \log (x) x^2-4 e^x \log (x) x+2 e^{2 x} \log (x)-1\right )}dx-5 e^4 \int \frac {1}{x \log (x) \left (2 \log (x) x^2-4 e^x \log (x) x+2 e^{2 x} \log (x)-1\right )}dx-e^4 \int \frac {x}{\log (x) \left (2 \log (x) x^2-4 e^x \log (x) x+2 e^{2 x} \log (x)-1\right )}dx-\frac {e^4}{2 \log (x)}\right )\) |
Int[(2*E^(4 + 4*x) - 2*x - 4*E^(4 + 3*x)*x + E^(4 + 2*x)*(-10 + 2*x^2) + E ^(4 + x)*(20*x + 4*x^2) + E^4*(-x - 10*x^2 - 2*x^3) + (8*x^3 + E^(4 + 3*x) *(-4*x + 4*x^2) + E^4*(-20*x^2 - 2*x^3) + E^(2*x)*(8*x + E^4*(-18*x - 4*x^ 3)) + E^x*(-16*x^2 + E^4*(20*x + 20*x^2 + 4*x^3)))*Log[x] + (-8*E^(4*x)*x + 32*E^(3*x)*x^2 - 48*E^(2*x)*x^3 + 32*E^x*x^4 - 8*x^5)*Log[x]^2)/(2*x + ( -8*E^(2*x)*x + 16*E^x*x^2 - 8*x^3)*Log[x] + (8*E^(4*x)*x - 32*E^(3*x)*x^2 + 48*E^(2*x)*x^3 - 32*E^x*x^4 + 8*x^5)*Log[x]^2),x]
3.7.11.3.1 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]]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 1.43 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.26
| method | result | size |
| risch | \(-x +\frac {\left (5+x -{\mathrm e}^{2 x}\right ) {\mathrm e}^{4}}{4 \,{\mathrm e}^{2 x} \ln \left (x \right )-8 x \,{\mathrm e}^{x} \ln \left (x \right )+4 x^{2} \ln \left (x \right )-2}\) | \(44\) |
| parallelrisch | \(\frac {-8 x^{3} \ln \left (x \right )+16 x^{2} {\mathrm e}^{x} \ln \left (x \right )-8 \,{\mathrm e}^{2 x} \ln \left (x \right ) x -2 \,{\mathrm e}^{2 x} {\mathrm e}^{4}+2 x \,{\mathrm e}^{4}+10 \,{\mathrm e}^{4}+4 x}{8 \,{\mathrm e}^{2 x} \ln \left (x \right )-16 x \,{\mathrm e}^{x} \ln \left (x \right )+8 x^{2} \ln \left (x \right )-4}\) | \(75\) |
int(((-8*x*exp(x)^4+32*x^2*exp(x)^3-48*exp(x)^2*x^3+32*exp(x)*x^4-8*x^5)*l n(x)^2+((4*x^2-4*x)*exp(4)*exp(x)^3+((-4*x^3-18*x)*exp(4)+8*x)*exp(x)^2+(( 4*x^3+20*x^2+20*x)*exp(4)-16*x^2)*exp(x)+(-2*x^3-20*x^2)*exp(4)+8*x^3)*ln( x)+2*exp(4)*exp(x)^4-4*x*exp(4)*exp(x)^3+(2*x^2-10)*exp(4)*exp(x)^2+(4*x^2 +20*x)*exp(4)*exp(x)+(-2*x^3-10*x^2-x)*exp(4)-2*x)/((8*x*exp(x)^4-32*x^2*e xp(x)^3+48*exp(x)^2*x^3-32*exp(x)*x^4+8*x^5)*ln(x)^2+(-8*x*exp(x)^2+16*exp (x)*x^2-8*x^3)*ln(x)+2*x),x,method=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 81 vs. \(2 (33) = 66\).
Time = 0.38 (sec) , antiderivative size = 81, normalized size of antiderivative = 2.31 \[ \int \frac {2 e^{4+4 x}-2 x-4 e^{4+3 x} x+e^{4+2 x} \left (-10+2 x^2\right )+e^{4+x} \left (20 x+4 x^2\right )+e^4 \left (-x-10 x^2-2 x^3\right )+\left (8 x^3+e^{4+3 x} \left (-4 x+4 x^2\right )+e^4 \left (-20 x^2-2 x^3\right )+e^{2 x} \left (8 x+e^4 \left (-18 x-4 x^3\right )\right )+e^x \left (-16 x^2+e^4 \left (20 x+20 x^2+4 x^3\right )\right )\right ) \log (x)+\left (-8 e^{4 x} x+32 e^{3 x} x^2-48 e^{2 x} x^3+32 e^x x^4-8 x^5\right ) \log ^2(x)}{2 x+\left (-8 e^{2 x} x+16 e^x x^2-8 x^3\right ) \log (x)+\left (8 e^{4 x} x-32 e^{3 x} x^2+48 e^{2 x} x^3-32 e^x x^4+8 x^5\right ) \log ^2(x)} \, dx=\frac {{\left (x + 5\right )} e^{12} + 2 \, x e^{8} - 4 \, {\left (x^{3} e^{8} - 2 \, x^{2} e^{\left (x + 8\right )} + x e^{\left (2 \, x + 8\right )}\right )} \log \left (x\right ) - e^{\left (2 \, x + 12\right )}}{2 \, {\left (2 \, {\left (x^{2} e^{8} - 2 \, x e^{\left (x + 8\right )} + e^{\left (2 \, x + 8\right )}\right )} \log \left (x\right ) - e^{8}\right )}} \]
integrate(((-8*x*exp(x)^4+32*x^2*exp(x)^3-48*exp(x)^2*x^3+32*exp(x)*x^4-8* x^5)*log(x)^2+((4*x^2-4*x)*exp(4)*exp(x)^3+((-4*x^3-18*x)*exp(4)+8*x)*exp( x)^2+((4*x^3+20*x^2+20*x)*exp(4)-16*x^2)*exp(x)+(-2*x^3-20*x^2)*exp(4)+8*x ^3)*log(x)+2*exp(4)*exp(x)^4-4*x*exp(4)*exp(x)^3+(2*x^2-10)*exp(4)*exp(x)^ 2+(4*x^2+20*x)*exp(4)*exp(x)+(-2*x^3-10*x^2-x)*exp(4)-2*x)/((8*x*exp(x)^4- 32*x^2*exp(x)^3+48*exp(x)^2*x^3-32*exp(x)*x^4+8*x^5)*log(x)^2+(-8*x*exp(x) ^2+16*exp(x)*x^2-8*x^3)*log(x)+2*x),x, algorithm=\
1/2*((x + 5)*e^12 + 2*x*e^8 - 4*(x^3*e^8 - 2*x^2*e^(x + 8) + x*e^(2*x + 8) )*log(x) - e^(2*x + 12))/(2*(x^2*e^8 - 2*x*e^(x + 8) + e^(2*x + 8))*log(x) - e^8)
Leaf count of result is larger than twice the leaf count of optimal. 95 vs. \(2 (26) = 52\).
Time = 0.23 (sec) , antiderivative size = 95, normalized size of antiderivative = 2.71 \[ \int \frac {2 e^{4+4 x}-2 x-4 e^{4+3 x} x+e^{4+2 x} \left (-10+2 x^2\right )+e^{4+x} \left (20 x+4 x^2\right )+e^4 \left (-x-10 x^2-2 x^3\right )+\left (8 x^3+e^{4+3 x} \left (-4 x+4 x^2\right )+e^4 \left (-20 x^2-2 x^3\right )+e^{2 x} \left (8 x+e^4 \left (-18 x-4 x^3\right )\right )+e^x \left (-16 x^2+e^4 \left (20 x+20 x^2+4 x^3\right )\right )\right ) \log (x)+\left (-8 e^{4 x} x+32 e^{3 x} x^2-48 e^{2 x} x^3+32 e^x x^4-8 x^5\right ) \log ^2(x)}{2 x+\left (-8 e^{2 x} x+16 e^x x^2-8 x^3\right ) \log (x)+\left (8 e^{4 x} x-32 e^{3 x} x^2+48 e^{2 x} x^3-32 e^x x^4+8 x^5\right ) \log ^2(x)} \, dx=- x - \frac {e^{4}}{4 \log {\left (x \right )}} + \frac {2 x^{2} e^{4} \log {\left (x \right )} - 4 x e^{4} e^{x} \log {\left (x \right )} + 2 x e^{4} \log {\left (x \right )} + 10 e^{4} \log {\left (x \right )} - e^{4}}{8 x^{2} \log {\left (x \right )}^{2} - 16 x e^{x} \log {\left (x \right )}^{2} + 8 e^{2 x} \log {\left (x \right )}^{2} - 4 \log {\left (x \right )}} \]
integrate(((-8*x*exp(x)**4+32*x**2*exp(x)**3-48*exp(x)**2*x**3+32*exp(x)*x **4-8*x**5)*ln(x)**2+((4*x**2-4*x)*exp(4)*exp(x)**3+((-4*x**3-18*x)*exp(4) +8*x)*exp(x)**2+((4*x**3+20*x**2+20*x)*exp(4)-16*x**2)*exp(x)+(-2*x**3-20* x**2)*exp(4)+8*x**3)*ln(x)+2*exp(4)*exp(x)**4-4*x*exp(4)*exp(x)**3+(2*x**2 -10)*exp(4)*exp(x)**2+(4*x**2+20*x)*exp(4)*exp(x)+(-2*x**3-10*x**2-x)*exp( 4)-2*x)/((8*x*exp(x)**4-32*x**2*exp(x)**3+48*exp(x)**2*x**3-32*exp(x)*x**4 +8*x**5)*ln(x)**2+(-8*x*exp(x)**2+16*exp(x)*x**2-8*x**3)*ln(x)+2*x),x)
-x - exp(4)/(4*log(x)) + (2*x**2*exp(4)*log(x) - 4*x*exp(4)*exp(x)*log(x) + 2*x*exp(4)*log(x) + 10*exp(4)*log(x) - exp(4))/(8*x**2*log(x)**2 - 16*x* exp(x)*log(x)**2 + 8*exp(2*x)*log(x)**2 - 4*log(x))
Leaf count of result is larger than twice the leaf count of optimal. 69 vs. \(2 (33) = 66\).
Time = 0.31 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.97 \[ \int \frac {2 e^{4+4 x}-2 x-4 e^{4+3 x} x+e^{4+2 x} \left (-10+2 x^2\right )+e^{4+x} \left (20 x+4 x^2\right )+e^4 \left (-x-10 x^2-2 x^3\right )+\left (8 x^3+e^{4+3 x} \left (-4 x+4 x^2\right )+e^4 \left (-20 x^2-2 x^3\right )+e^{2 x} \left (8 x+e^4 \left (-18 x-4 x^3\right )\right )+e^x \left (-16 x^2+e^4 \left (20 x+20 x^2+4 x^3\right )\right )\right ) \log (x)+\left (-8 e^{4 x} x+32 e^{3 x} x^2-48 e^{2 x} x^3+32 e^x x^4-8 x^5\right ) \log ^2(x)}{2 x+\left (-8 e^{2 x} x+16 e^x x^2-8 x^3\right ) \log (x)+\left (8 e^{4 x} x-32 e^{3 x} x^2+48 e^{2 x} x^3-32 e^x x^4+8 x^5\right ) \log ^2(x)} \, dx=-\frac {4 \, x^{3} \log \left (x\right ) - 8 \, x^{2} e^{x} \log \left (x\right ) - x {\left (e^{4} + 2\right )} + {\left (4 \, x \log \left (x\right ) + e^{4}\right )} e^{\left (2 \, x\right )} - 5 \, e^{4}}{2 \, {\left (2 \, x^{2} \log \left (x\right ) - 4 \, x e^{x} \log \left (x\right ) + 2 \, e^{\left (2 \, x\right )} \log \left (x\right ) - 1\right )}} \]
integrate(((-8*x*exp(x)^4+32*x^2*exp(x)^3-48*exp(x)^2*x^3+32*exp(x)*x^4-8* x^5)*log(x)^2+((4*x^2-4*x)*exp(4)*exp(x)^3+((-4*x^3-18*x)*exp(4)+8*x)*exp( x)^2+((4*x^3+20*x^2+20*x)*exp(4)-16*x^2)*exp(x)+(-2*x^3-20*x^2)*exp(4)+8*x ^3)*log(x)+2*exp(4)*exp(x)^4-4*x*exp(4)*exp(x)^3+(2*x^2-10)*exp(4)*exp(x)^ 2+(4*x^2+20*x)*exp(4)*exp(x)+(-2*x^3-10*x^2-x)*exp(4)-2*x)/((8*x*exp(x)^4- 32*x^2*exp(x)^3+48*exp(x)^2*x^3-32*exp(x)*x^4+8*x^5)*log(x)^2+(-8*x*exp(x) ^2+16*exp(x)*x^2-8*x^3)*log(x)+2*x),x, algorithm=\
-1/2*(4*x^3*log(x) - 8*x^2*e^x*log(x) - x*(e^4 + 2) + (4*x*log(x) + e^4)*e ^(2*x) - 5*e^4)/(2*x^2*log(x) - 4*x*e^x*log(x) + 2*e^(2*x)*log(x) - 1)
Leaf count of result is larger than twice the leaf count of optimal. 842 vs. \(2 (33) = 66\).
Time = 0.85 (sec) , antiderivative size = 842, normalized size of antiderivative = 24.06 \[ \int \frac {2 e^{4+4 x}-2 x-4 e^{4+3 x} x+e^{4+2 x} \left (-10+2 x^2\right )+e^{4+x} \left (20 x+4 x^2\right )+e^4 \left (-x-10 x^2-2 x^3\right )+\left (8 x^3+e^{4+3 x} \left (-4 x+4 x^2\right )+e^4 \left (-20 x^2-2 x^3\right )+e^{2 x} \left (8 x+e^4 \left (-18 x-4 x^3\right )\right )+e^x \left (-16 x^2+e^4 \left (20 x+20 x^2+4 x^3\right )\right )\right ) \log (x)+\left (-8 e^{4 x} x+32 e^{3 x} x^2-48 e^{2 x} x^3+32 e^x x^4-8 x^5\right ) \log ^2(x)}{2 x+\left (-8 e^{2 x} x+16 e^x x^2-8 x^3\right ) \log (x)+\left (8 e^{4 x} x-32 e^{3 x} x^2+48 e^{2 x} x^3-32 e^x x^4+8 x^5\right ) \log ^2(x)} \, dx=\text {Too large to display} \]
integrate(((-8*x*exp(x)^4+32*x^2*exp(x)^3-48*exp(x)^2*x^3+32*exp(x)*x^4-8* x^5)*log(x)^2+((4*x^2-4*x)*exp(4)*exp(x)^3+((-4*x^3-18*x)*exp(4)+8*x)*exp( x)^2+((4*x^3+20*x^2+20*x)*exp(4)-16*x^2)*exp(x)+(-2*x^3-20*x^2)*exp(4)+8*x ^3)*log(x)+2*exp(4)*exp(x)^4-4*x*exp(4)*exp(x)^3+(2*x^2-10)*exp(4)*exp(x)^ 2+(4*x^2+20*x)*exp(4)*exp(x)+(-2*x^3-10*x^2-x)*exp(4)-2*x)/((8*x*exp(x)^4- 32*x^2*exp(x)^3+48*exp(x)^2*x^3-32*exp(x)*x^4+8*x^5)*log(x)^2+(-8*x*exp(x) ^2+16*exp(x)*x^2-8*x^3)*log(x)+2*x),x, algorithm=\
-1/2*(32*x^8*log(x)^5 - 32*x^7*e^x*log(x)^5 - 32*x^7*log(x)^5 + 32*x^6*e^x *log(x)^5 + 32*x^7*log(x)^4 + 8*x^6*e^4*log(x)^4 - 64*x^6*e^x*log(x)^4 + 8 *x^6*e^4*log(x)^3 - 80*x^6*log(x)^4 + 72*x^5*e^4*log(x)^4 + 32*x^5*e^(2*x) *log(x)^4 - 8*x^5*e^(x + 4)*log(x)^4 + 160*x^5*e^x*log(x)^4 + 24*x^5*e^4*l og(x)^3 - 8*x^5*e^(x + 4)*log(x)^3 + 64*x^5*log(x)^4 - 80*x^4*e^4*log(x)^4 - 64*x^4*e^(2*x)*log(x)^4 - 72*x^4*e^(x + 4)*log(x)^4 - 96*x^4*e^x*log(x) ^4 - 24*x^5*log(x)^3 - 56*x^4*e^4*log(x)^3 + 8*x^4*e^(2*x + 4)*log(x)^3 - 32*x^4*e^(x + 4)*log(x)^3 + 32*x^4*e^x*log(x)^3 + 32*x^3*e^(2*x)*log(x)^4 + 80*x^3*e^(x + 4)*log(x)^4 + 4*x^4*e^4*log(x)^2 + 24*x^4*log(x)^3 + 108*x ^3*e^4*log(x)^3 - 16*x^3*e^(2*x)*log(x)^3 - 16*x^3*e^(2*x + 4)*log(x)^3 + 36*x^3*e^(x + 4)*log(x)^3 - 8*x^3*e^x*log(x)^3 - 16*x^4*log(x)^2 + 26*x^3* e^4*log(x)^2 + 32*x^3*e^x*log(x)^2 - 24*x^3*log(x)^3 - 40*x^2*e^4*log(x)^3 + 8*x^2*e^(2*x + 4)*log(x)^3 + 4*x^2*e^(x + 4)*log(x)^3 + 8*x^2*e^x*log(x )^3 + 2*x^3*e^4*log(x) + 42*x^2*e^4*log(x)^2 - 16*x^2*e^(2*x)*log(x)^2 - 4 *x^2*e^(2*x + 4)*log(x)^2 - 4*x^3*log(x) + 14*x^2*e^4*log(x) + 8*x^2*e^x*l og(x) + 4*x^2*log(x)^2 + 8*x^2*log(x) + 21*x*e^4*log(x) - 4*x*e^(2*x)*log( x) - 4*x*e^(2*x + 4)*log(x) + x*e^4 + 2*x*log(x) + 2*x + 5*e^4 - e^(2*x + 4))/(16*x^6*log(x)^4 - 32*x^5*e^x*log(x)^4 - 32*x^5*log(x)^4 + 16*x^4*e^(2 *x)*log(x)^4 + 64*x^4*e^x*log(x)^4 + 16*x^4*log(x)^4 - 32*x^3*e^(2*x)*log( x)^4 - 32*x^3*e^x*log(x)^4 - 16*x^4*log(x)^3 + 16*x^3*e^x*log(x)^3 + 16...
Timed out. \[ \int \frac {2 e^{4+4 x}-2 x-4 e^{4+3 x} x+e^{4+2 x} \left (-10+2 x^2\right )+e^{4+x} \left (20 x+4 x^2\right )+e^4 \left (-x-10 x^2-2 x^3\right )+\left (8 x^3+e^{4+3 x} \left (-4 x+4 x^2\right )+e^4 \left (-20 x^2-2 x^3\right )+e^{2 x} \left (8 x+e^4 \left (-18 x-4 x^3\right )\right )+e^x \left (-16 x^2+e^4 \left (20 x+20 x^2+4 x^3\right )\right )\right ) \log (x)+\left (-8 e^{4 x} x+32 e^{3 x} x^2-48 e^{2 x} x^3+32 e^x x^4-8 x^5\right ) \log ^2(x)}{2 x+\left (-8 e^{2 x} x+16 e^x x^2-8 x^3\right ) \log (x)+\left (8 e^{4 x} x-32 e^{3 x} x^2+48 e^{2 x} x^3-32 e^x x^4+8 x^5\right ) \log ^2(x)} \, dx=\int -\frac {\left (8\,x\,{\mathrm {e}}^{4\,x}-32\,x^4\,{\mathrm {e}}^x-32\,x^2\,{\mathrm {e}}^{3\,x}+48\,x^3\,{\mathrm {e}}^{2\,x}+8\,x^5\right )\,{\ln \left (x\right )}^2+\left ({\mathrm {e}}^4\,\left (2\,x^3+20\,x^2\right )-{\mathrm {e}}^x\,\left ({\mathrm {e}}^4\,\left (4\,x^3+20\,x^2+20\,x\right )-16\,x^2\right )-{\mathrm {e}}^{2\,x}\,\left (8\,x-{\mathrm {e}}^4\,\left (4\,x^3+18\,x\right )\right )-8\,x^3+{\mathrm {e}}^{3\,x}\,{\mathrm {e}}^4\,\left (4\,x-4\,x^2\right )\right )\,\ln \left (x\right )+2\,x-2\,{\mathrm {e}}^{4\,x}\,{\mathrm {e}}^4+{\mathrm {e}}^4\,\left (2\,x^3+10\,x^2+x\right )-{\mathrm {e}}^4\,{\mathrm {e}}^x\,\left (4\,x^2+20\,x\right )+4\,x\,{\mathrm {e}}^{3\,x}\,{\mathrm {e}}^4-{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^4\,\left (2\,x^2-10\right )}{\left (8\,x\,{\mathrm {e}}^{4\,x}-32\,x^4\,{\mathrm {e}}^x-32\,x^2\,{\mathrm {e}}^{3\,x}+48\,x^3\,{\mathrm {e}}^{2\,x}+8\,x^5\right )\,{\ln \left (x\right )}^2+\left (16\,x^2\,{\mathrm {e}}^x-8\,x\,{\mathrm {e}}^{2\,x}-8\,x^3\right )\,\ln \left (x\right )+2\,x} \,d x \]
int(-(2*x - 2*exp(4*x)*exp(4) + exp(4)*(x + 10*x^2 + 2*x^3) - log(x)*(exp( 2*x)*(8*x - exp(4)*(18*x + 4*x^3)) + exp(x)*(exp(4)*(20*x + 20*x^2 + 4*x^3 ) - 16*x^2) - exp(4)*(20*x^2 + 2*x^3) + 8*x^3 - exp(3*x)*exp(4)*(4*x - 4*x ^2)) + log(x)^2*(8*x*exp(4*x) - 32*x^4*exp(x) - 32*x^2*exp(3*x) + 48*x^3*e xp(2*x) + 8*x^5) - exp(4)*exp(x)*(20*x + 4*x^2) + 4*x*exp(3*x)*exp(4) - ex p(2*x)*exp(4)*(2*x^2 - 10))/(2*x - log(x)*(8*x*exp(2*x) - 16*x^2*exp(x) + 8*x^3) + log(x)^2*(8*x*exp(4*x) - 32*x^4*exp(x) - 32*x^2*exp(3*x) + 48*x^3 *exp(2*x) + 8*x^5)),x)
int(-(2*x - 2*exp(4*x)*exp(4) + exp(4)*(x + 10*x^2 + 2*x^3) - log(x)*(exp( 2*x)*(8*x - exp(4)*(18*x + 4*x^3)) + exp(x)*(exp(4)*(20*x + 20*x^2 + 4*x^3 ) - 16*x^2) - exp(4)*(20*x^2 + 2*x^3) + 8*x^3 - exp(3*x)*exp(4)*(4*x - 4*x ^2)) + log(x)^2*(8*x*exp(4*x) - 32*x^4*exp(x) - 32*x^2*exp(3*x) + 48*x^3*e xp(2*x) + 8*x^5) - exp(4)*exp(x)*(20*x + 4*x^2) + 4*x*exp(3*x)*exp(4) - ex p(2*x)*exp(4)*(2*x^2 - 10))/(2*x - log(x)*(8*x*exp(2*x) - 16*x^2*exp(x) + 8*x^3) + log(x)^2*(8*x*exp(4*x) - 32*x^4*exp(x) - 32*x^2*exp(3*x) + 48*x^3 *exp(2*x) + 8*x^5)), x)