Integrand size = 94, antiderivative size = 28 \[ \int \frac {1}{2} e^{8-2 x} \left (64 x-112 x^2+40 x^3-4 x^4+e^{-4+x} \left (16-24 x+4 x^2\right )+3^{2 x} (-1+\log (3))+3^x \left (8-20 x+4 x^2+\left (8 x-2 x^2\right ) \log (3)+e^{-4+x} (-2+2 \log (3))\right )\right ) \, dx=\left (1+e^{4-x} \left (\frac {3^x}{2}+4 x-x^2\right )\right )^2 \]
Time = 1.04 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.61 \[ \int \frac {1}{2} e^{8-2 x} \left (64 x-112 x^2+40 x^3-4 x^4+e^{-4+x} \left (16-24 x+4 x^2\right )+3^{2 x} (-1+\log (3))+3^x \left (8-20 x+4 x^2+\left (8 x-2 x^2\right ) \log (3)+e^{-4+x} (-2+2 \log (3))\right )\right ) \, dx=\frac {1}{4} e^{4-2 x} \left (3^x+8 x-2 x^2\right ) \left (4 e^x+e^4 \left (3^x+8 x-2 x^2\right )\right ) \]
Integrate[(E^(8 - 2*x)*(64*x - 112*x^2 + 40*x^3 - 4*x^4 + E^(-4 + x)*(16 - 24*x + 4*x^2) + 3^(2*x)*(-1 + Log[3]) + 3^x*(8 - 20*x + 4*x^2 + (8*x - 2* x^2)*Log[3] + E^(-4 + x)*(-2 + 2*Log[3]))))/2,x]
Leaf count is larger than twice the leaf count of optimal. \(253\) vs. \(2(28)=56\).
Time = 2.64 (sec) , antiderivative size = 253, normalized size of antiderivative = 9.04, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {27, 7239, 7293, 2009}
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.
| \(\displaystyle \int \frac {1}{2} e^{8-2 x} \left (-4 x^4+40 x^3-112 x^2+e^{x-4} \left (4 x^2-24 x+16\right )+3^x \left (4 x^2+\left (8 x-2 x^2\right ) \log (3)-20 x+e^{x-4} (2 \log (3)-2)+8\right )+64 x+3^{2 x} (\log (3)-1)\right ) \, dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \int e^{8-2 x} \left (-4 x^4+40 x^3-112 x^2+64 x+4 e^{x-4} \left (x^2-6 x+4\right )+2\ 3^x \left (2 x^2-10 x+\left (4 x-x^2\right ) \log (3)-e^{x-4} (1-\log (3))+4\right )-3^{2 x} (1-\log (3))\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {1}{2} \int e^{4-2 x} \left (e^4 \left (-2 x^2+8 x+3^x\right )+2 e^x\right ) \left (2 x^2-12 x-3^x (1-\log (3))+8\right )dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{2} \int \left (2 e^{4-x} \left (2 x^2-12 x-3^x (1-\log (3))+8\right )+e^{8-2 x} \left (-2 x^2+8 x+3^x\right ) \left (2 x^2-12 x-3^x (1-\log (3))+8\right )\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} \left (2 e^{8-2 x} x^4-16 e^{8-2 x} x^3+32 e^{8-2 x} x^2-4 e^{4-x} x^2-2 x^2 e^{8-x (2-\log (3))}+16 e^{4-x} x+\frac {4 x (5-\log (9)) e^{8-x (2-\log (3))}}{2-\log (3)}-\frac {4 x e^{8-x (2-\log (3))}}{2-\log (3)}+2 e^{4-x (1-\log (3))}+\frac {4 (5-\log (9)) e^{8-x (2-\log (3))}}{(2-\log (3))^2}+\frac {(1-\log (3)) e^{8-x (2-\log (9))}}{2-\log (9)}-\frac {8 e^{8-x (2-\log (3))}}{2-\log (3)}-\frac {4 e^{8-x (2-\log (3))}}{(2-\log (3))^2}\right )\) |
Int[(E^(8 - 2*x)*(64*x - 112*x^2 + 40*x^3 - 4*x^4 + E^(-4 + x)*(16 - 24*x + 4*x^2) + 3^(2*x)*(-1 + Log[3]) + 3^x*(8 - 20*x + 4*x^2 + (8*x - 2*x^2)*L og[3] + E^(-4 + x)*(-2 + 2*Log[3]))))/2,x]
(2*E^(4 - x*(1 - Log[3])) + 16*E^(4 - x)*x + 32*E^(8 - 2*x)*x^2 - 4*E^(4 - x)*x^2 - 2*E^(8 - x*(2 - Log[3]))*x^2 - 16*E^(8 - 2*x)*x^3 + 2*E^(8 - 2*x )*x^4 - (4*E^(8 - x*(2 - Log[3])))/(2 - Log[3])^2 - (8*E^(8 - x*(2 - Log[3 ])))/(2 - Log[3]) - (4*E^(8 - x*(2 - Log[3]))*x)/(2 - Log[3]) + (E^(8 - x* (2 - Log[9]))*(1 - Log[3]))/(2 - Log[9]) + (4*E^(8 - x*(2 - Log[3]))*(5 - Log[9]))/(2 - Log[3])^2 + (4*E^(8 - x*(2 - Log[3]))*x*(5 - Log[9]))/(2 - L og[3]))/2
3.5.34.3.1 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]]
Leaf count of result is larger than twice the leaf count of optimal. \(76\) vs. \(2(27)=54\).
Time = 3.49 (sec) , antiderivative size = 77, normalized size of antiderivative = 2.75
| method | result | size |
| parallelrisch | \(\frac {\left (4 x^{4}-32 x^{3}-8 x^{2} {\mathrm e}^{x -4}-4 \,{\mathrm e}^{x \ln \left (3\right )} x^{2}+64 x^{2}+32 x \,{\mathrm e}^{x -4}+16 \,{\mathrm e}^{x \ln \left (3\right )} x +4 \,{\mathrm e}^{x \ln \left (3\right )} {\mathrm e}^{x -4}+{\mathrm e}^{2 x \ln \left (3\right )}\right ) {\mathrm e}^{-2 x +8}}{4}\) | \(77\) |
| risch | \(\frac {\left (-4 x^{2}+16 x \right ) {\mathrm e}^{-x +4}}{2}+\frac {\left (2 x^{4}-16 x^{3}+32 x^{2}\right ) {\mathrm e}^{-2 x +8}}{2}+\frac {{\mathrm e}^{-2 x +8} 3^{2 x}}{4}-\left (x^{2}-4 x -{\mathrm e}^{x -4}\right ) {\mathrm e}^{-2 x +8} 3^{x}\) | \(80\) |
| parts | \(\frac {{\mathrm e}^{4} \ln \left (3\right ) {\mathrm e}^{x \ln \left (3\right )} {\mathrm e}^{-x}}{\ln \left (3\right )-1}+\frac {4 \,{\mathrm e}^{8} {\mathrm e}^{x \ln \left (3\right )} {\mathrm e}^{-2 x}}{\ln \left (3\right )-2}-\frac {{\mathrm e}^{4} {\mathrm e}^{x \ln \left (3\right )} {\mathrm e}^{-x}}{\ln \left (3\right )-1}+\left (\frac {10 \,{\mathrm e}^{8} {\mathrm e}^{x \ln \left (3\right )}}{\ln \left (3\right )^{2}-4 \ln \left (3\right )+4}-\frac {10 \,{\mathrm e}^{8} x \,{\mathrm e}^{x \ln \left (3\right )}}{\ln \left (3\right )-2}\right ) {\mathrm e}^{-2 x}+\left (\frac {2 \,{\mathrm e}^{8} x^{2} {\mathrm e}^{x \ln \left (3\right )}}{\ln \left (3\right )-2}+\frac {4 \,{\mathrm e}^{8} {\mathrm e}^{x \ln \left (3\right )}}{\left (\ln \left (3\right )^{2}-4 \ln \left (3\right )+4\right ) \left (\ln \left (3\right )-2\right )}-\frac {4 \,{\mathrm e}^{8} x \,{\mathrm e}^{x \ln \left (3\right )}}{\ln \left (3\right )^{2}-4 \ln \left (3\right )+4}\right ) {\mathrm e}^{-2 x}+\left (-\frac {4 \,{\mathrm e}^{8} \ln \left (3\right ) {\mathrm e}^{x \ln \left (3\right )}}{\ln \left (3\right )^{2}-4 \ln \left (3\right )+4}+\frac {4 \,{\mathrm e}^{8} \ln \left (3\right ) x \,{\mathrm e}^{x \ln \left (3\right )}}{\ln \left (3\right )-2}\right ) {\mathrm e}^{-2 x}+\left (-\frac {2 \,{\mathrm e}^{8} \ln \left (3\right ) {\mathrm e}^{x \ln \left (3\right )}}{\left (\ln \left (3\right )^{2}-4 \ln \left (3\right )+4\right ) \left (\ln \left (3\right )-2\right )}+\frac {2 \,{\mathrm e}^{8} \ln \left (3\right ) x \,{\mathrm e}^{x \ln \left (3\right )}}{\ln \left (3\right )^{2}-4 \ln \left (3\right )+4}-\frac {{\mathrm e}^{8} \ln \left (3\right ) x^{2} {\mathrm e}^{x \ln \left (3\right )}}{\ln \left (3\right )-2}\right ) {\mathrm e}^{-2 x}-2 \left (x -4\right )^{2} {\mathrm e}^{-x +4}-8 \left (x -4\right ) {\mathrm e}^{-x +4}+16 \left (x -4\right )^{2} {\mathrm e}^{-2 x +8}+8 \,{\mathrm e}^{-2 x +8} \left (x -4\right )^{3}+{\mathrm e}^{-2 x +8} \left (x -4\right )^{4}+\frac {\left (-\frac {1}{2}+\frac {\ln \left (3\right )}{2}\right ) {\mathrm e}^{-2 x +8+2 x \ln \left (3\right )}}{2 \ln \left (3\right )-2}\) | \(412\) |
| default | \(\frac {{\mathrm e}^{8} \ln \left (3\right ) {\mathrm e}^{2 x \ln \left (3\right )} {\mathrm e}^{-2 x}}{4 \ln \left (3\right )-4}-8 \,{\mathrm e}^{-x} {\mathrm e}^{4}+32 \,{\mathrm e}^{8} \left (-\frac {x \,{\mathrm e}^{-2 x}}{2}-\frac {{\mathrm e}^{-2 x}}{4}\right )-56 \,{\mathrm e}^{8} \left (-\frac {x^{2} {\mathrm e}^{-2 x}}{2}-\frac {x \,{\mathrm e}^{-2 x}}{2}-\frac {{\mathrm e}^{-2 x}}{4}\right )+20 \,{\mathrm e}^{8} \left (-\frac {{\mathrm e}^{-2 x} x^{3}}{2}-\frac {3 x^{2} {\mathrm e}^{-2 x}}{4}-\frac {3 x \,{\mathrm e}^{-2 x}}{4}-\frac {3 \,{\mathrm e}^{-2 x}}{8}\right )-2 \,{\mathrm e}^{8} \left (-\frac {x^{4} {\mathrm e}^{-2 x}}{2}-{\mathrm e}^{-2 x} x^{3}-\frac {3 x^{2} {\mathrm e}^{-2 x}}{2}-\frac {3 x \,{\mathrm e}^{-2 x}}{2}-\frac {3 \,{\mathrm e}^{-2 x}}{4}\right )+\frac {4 \,{\mathrm e}^{8} {\mathrm e}^{x \ln \left (3\right )} {\mathrm e}^{-2 x}}{\ln \left (3\right )-2}-\frac {{\mathrm e}^{8} {\mathrm e}^{2 x \ln \left (3\right )} {\mathrm e}^{-2 x}}{4 \left (\ln \left (3\right )-1\right )}-12 \,{\mathrm e}^{4} \left (-x \,{\mathrm e}^{-x}-{\mathrm e}^{-x}\right )+2 \,{\mathrm e}^{4} \left (-x^{2} {\mathrm e}^{-x}-2 x \,{\mathrm e}^{-x}-2 \,{\mathrm e}^{-x}\right )-\frac {{\mathrm e}^{4} {\mathrm e}^{x \ln \left (3\right )} {\mathrm e}^{-x}}{\ln \left (3\right )-1}+\frac {\left (\frac {20 \,{\mathrm e}^{8} {\mathrm e}^{x \ln \left (3\right )}}{\ln \left (3\right )^{2}-4 \ln \left (3\right )+4}-\frac {20 \,{\mathrm e}^{8} x \,{\mathrm e}^{x \ln \left (3\right )}}{\ln \left (3\right )-2}\right ) {\mathrm e}^{-2 x}}{2}+\frac {\left (\frac {4 \,{\mathrm e}^{8} x^{2} {\mathrm e}^{x \ln \left (3\right )}}{\ln \left (3\right )-2}+\frac {8 \,{\mathrm e}^{8} {\mathrm e}^{x \ln \left (3\right )}}{\left (\ln \left (3\right )^{2}-4 \ln \left (3\right )+4\right ) \left (\ln \left (3\right )-2\right )}-\frac {8 \,{\mathrm e}^{8} x \,{\mathrm e}^{x \ln \left (3\right )}}{\ln \left (3\right )^{2}-4 \ln \left (3\right )+4}\right ) {\mathrm e}^{-2 x}}{2}+\frac {{\mathrm e}^{4} \ln \left (3\right ) {\mathrm e}^{x \ln \left (3\right )} {\mathrm e}^{-x}}{\ln \left (3\right )-1}+\frac {\left (-\frac {8 \,{\mathrm e}^{8} \ln \left (3\right ) {\mathrm e}^{x \ln \left (3\right )}}{\ln \left (3\right )^{2}-4 \ln \left (3\right )+4}+\frac {8 \,{\mathrm e}^{8} \ln \left (3\right ) x \,{\mathrm e}^{x \ln \left (3\right )}}{\ln \left (3\right )-2}\right ) {\mathrm e}^{-2 x}}{2}+\frac {\left (-\frac {4 \,{\mathrm e}^{8} \ln \left (3\right ) {\mathrm e}^{x \ln \left (3\right )}}{\left (\ln \left (3\right )^{2}-4 \ln \left (3\right )+4\right ) \left (\ln \left (3\right )-2\right )}+\frac {4 \,{\mathrm e}^{8} \ln \left (3\right ) x \,{\mathrm e}^{x \ln \left (3\right )}}{\ln \left (3\right )^{2}-4 \ln \left (3\right )+4}-\frac {2 \,{\mathrm e}^{8} \ln \left (3\right ) x^{2} {\mathrm e}^{x \ln \left (3\right )}}{\ln \left (3\right )-2}\right ) {\mathrm e}^{-2 x}}{2}\) | \(569\) |
| meijerg | \(\text {Expression too large to display}\) | \(902\) |
int(1/2*((ln(3)-1)*exp(x*ln(3))^2+((2*ln(3)-2)*exp(x-4)+(-2*x^2+8*x)*ln(3) +4*x^2-20*x+8)*exp(x*ln(3))+(4*x^2-24*x+16)*exp(x-4)-4*x^4+40*x^3-112*x^2+ 64*x)/exp(x-4)^2,x,method=_RETURNVERBOSE)
1/4*(4*x^4-32*x^3-8*x^2*exp(x-4)-4*exp(x*ln(3))*x^2+64*x^2+32*x*exp(x-4)+1 6*exp(x*ln(3))*x+4*exp(x*ln(3))*exp(x-4)+exp(x*ln(3))^2)/exp(x-4)^2
Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (27) = 54\).
Time = 0.26 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.14 \[ \int \frac {1}{2} e^{8-2 x} \left (64 x-112 x^2+40 x^3-4 x^4+e^{-4+x} \left (16-24 x+4 x^2\right )+3^{2 x} (-1+\log (3))+3^x \left (8-20 x+4 x^2+\left (8 x-2 x^2\right ) \log (3)+e^{-4+x} (-2+2 \log (3))\right )\right ) \, dx=\frac {1}{4} \, {\left (4 \, x^{4} - 32 \, x^{3} - 4 \, {\left (x^{2} - 4 \, x - e^{\left (x - 4\right )}\right )} 3^{x} + 64 \, x^{2} - 8 \, {\left (x^{2} - 4 \, x\right )} e^{\left (x - 4\right )} + 3^{2 \, x}\right )} e^{\left (-2 \, x + 8\right )} \]
integrate(1/2*((log(3)-1)*exp(x*log(3))^2+((2*log(3)-2)*exp(x-4)+(-2*x^2+8 *x)*log(3)+4*x^2-20*x+8)*exp(x*log(3))+(4*x^2-24*x+16)*exp(x-4)-4*x^4+40*x ^3-112*x^2+64*x)/exp(x-4)^2,x, algorithm=\
1/4*(4*x^4 - 32*x^3 - 4*(x^2 - 4*x - e^(x - 4))*3^x + 64*x^2 - 8*(x^2 - 4* x)*e^(x - 4) + 3^(2*x))*e^(-2*x + 8)
Leaf count of result is larger than twice the leaf count of optimal. 129 vs. \(2 (22) = 44\).
Time = 12.62 (sec) , antiderivative size = 129, normalized size of antiderivative = 4.61 \[ \int \frac {1}{2} e^{8-2 x} \left (64 x-112 x^2+40 x^3-4 x^4+e^{-4+x} \left (16-24 x+4 x^2\right )+3^{2 x} (-1+\log (3))+3^x \left (8-20 x+4 x^2+\left (8 x-2 x^2\right ) \log (3)+e^{-4+x} (-2+2 \log (3))\right )\right ) \, dx=x^{4} e^{8} e^{- 2 x} - 8 x^{3} e^{8} e^{- 2 x} - 2 x^{2} e^{4} e^{- x} - x^{2} e^{8} e^{- 2 x} e^{x \log {\left (3 \right )}} + 16 x^{2} e^{8} e^{- 2 x} + 8 x e^{4} e^{- x} + 4 x e^{8} e^{- 2 x} e^{x \log {\left (3 \right )}} + e^{4} e^{- x} e^{x \log {\left (3 \right )}} + \frac {e^{8} e^{- 2 x} e^{2 x \log {\left (3 \right )}}}{4} \]
integrate(1/2*((ln(3)-1)*exp(x*ln(3))**2+((2*ln(3)-2)*exp(x-4)+(-2*x**2+8* x)*ln(3)+4*x**2-20*x+8)*exp(x*ln(3))+(4*x**2-24*x+16)*exp(x-4)-4*x**4+40*x **3-112*x**2+64*x)/exp(x-4)**2,x)
x**4*exp(8)*exp(-2*x) - 8*x**3*exp(8)*exp(-2*x) - 2*x**2*exp(4)*exp(-x) - x**2*exp(8)*exp(-2*x)*exp(x*log(3)) + 16*x**2*exp(8)*exp(-2*x) + 8*x*exp(4 )*exp(-x) + 4*x*exp(8)*exp(-2*x)*exp(x*log(3)) + exp(4)*exp(-x)*exp(x*log( 3)) + exp(8)*exp(-2*x)*exp(2*x*log(3))/4
Leaf count of result is larger than twice the leaf count of optimal. 431 vs. \(2 (27) = 54\).
Time = 0.30 (sec) , antiderivative size = 431, normalized size of antiderivative = 15.39 \[ \int \frac {1}{2} e^{8-2 x} \left (64 x-112 x^2+40 x^3-4 x^4+e^{-4+x} \left (16-24 x+4 x^2\right )+3^{2 x} (-1+\log (3))+3^x \left (8-20 x+4 x^2+\left (8 x-2 x^2\right ) \log (3)+e^{-4+x} (-2+2 \log (3))\right )\right ) \, dx =\text {Too large to display} \]
integrate(1/2*((log(3)-1)*exp(x*log(3))^2+((2*log(3)-2)*exp(x-4)+(-2*x^2+8 *x)*log(3)+4*x^2-20*x+8)*exp(x*log(3))+(4*x^2-24*x+16)*exp(x-4)-4*x^4+40*x ^3-112*x^2+64*x)/exp(x-4)^2,x, algorithm=\
-2*(x^2*e^4 + 2*x*e^4 + 2*e^4)*e^(-x) + 12*(x*e^4 + e^4)*e^(-x) + 1/2*(2*x ^4*e^8 + 4*x^3*e^8 + 6*x^2*e^8 + 6*x*e^8 + 3*e^8)*e^(-2*x) - 5/2*(4*x^3*e^ 8 + 6*x^2*e^8 + 6*x*e^8 + 3*e^8)*e^(-2*x) + 14*(2*x^2*e^8 + 2*x*e^8 + e^8) *e^(-2*x) - 8*(2*x*e^8 + e^8)*e^(-2*x) - ((log(3)^2 - 4*log(3) + 4)*x^2*e^ 8 - 2*x*(log(3) - 2)*e^8 + 2*e^8)*e^(x*log(3) - 2*x)*log(3)/(log(3)^3 - 6* log(3)^2 + 12*log(3) - 8) + 4*(x*(log(3) - 2)*e^8 - e^8)*e^(x*log(3) - 2*x )*log(3)/(log(3)^2 - 4*log(3) + 4) + 2*((log(3)^2 - 4*log(3) + 4)*x^2*e^8 - 2*x*(log(3) - 2)*e^8 + 2*e^8)*e^(x*log(3) - 2*x)/(log(3)^3 - 6*log(3)^2 + 12*log(3) - 8) - 10*(x*(log(3) - 2)*e^8 - e^8)*e^(x*log(3) - 2*x)/(log(3 )^2 - 4*log(3) + 4) + 6561/4*e^(2*(x - 4)*(log(3) - 1))*log(3)/(log(3) - 1 ) + 81*e^((x - 4)*(log(3) - 1))*log(3)/(log(3) - 1) - 6561/4*e^(2*(x - 4)* (log(3) - 1))/(log(3) - 1) - 81*e^((x - 4)*(log(3) - 1))/(log(3) - 1) + 32 4*e^((x - 4)*(log(3) - 2))/(log(3) - 2) - 8*e^(-x + 4)
Leaf count of result is larger than twice the leaf count of optimal. 142 vs. \(2 (27) = 54\).
Time = 0.26 (sec) , antiderivative size = 142, normalized size of antiderivative = 5.07 \[ \int \frac {1}{2} e^{8-2 x} \left (64 x-112 x^2+40 x^3-4 x^4+e^{-4+x} \left (16-24 x+4 x^2\right )+3^{2 x} (-1+\log (3))+3^x \left (8-20 x+4 x^2+\left (8 x-2 x^2\right ) \log (3)+e^{-4+x} (-2+2 \log (3))\right )\right ) \, dx=-2 \, {\left (x^{2} - 4 \, x\right )} e^{\left (-x + 4\right )} + {\left (x^{4} - 8 \, x^{3} + 16 \, x^{2}\right )} e^{\left (-2 \, x + 8\right )} - \frac {{\left (x^{2} \log \left (3\right )^{3} - 6 \, x^{2} \log \left (3\right )^{2} - 4 \, x \log \left (3\right )^{3} + 12 \, x^{2} \log \left (3\right ) + 24 \, x \log \left (3\right )^{2} - 8 \, x^{2} - 48 \, x \log \left (3\right ) + 32 \, x\right )} e^{\left (x \log \left (3\right ) - 2 \, x + 8\right )}}{\log \left (3\right )^{3} - 6 \, \log \left (3\right )^{2} + 12 \, \log \left (3\right ) - 8} + \frac {1}{4} \, e^{\left (2 \, x \log \left (3\right ) - 2 \, x + 8\right )} + e^{\left (x \log \left (3\right ) - x + 4\right )} \]
integrate(1/2*((log(3)-1)*exp(x*log(3))^2+((2*log(3)-2)*exp(x-4)+(-2*x^2+8 *x)*log(3)+4*x^2-20*x+8)*exp(x*log(3))+(4*x^2-24*x+16)*exp(x-4)-4*x^4+40*x ^3-112*x^2+64*x)/exp(x-4)^2,x, algorithm=\
-2*(x^2 - 4*x)*e^(-x + 4) + (x^4 - 8*x^3 + 16*x^2)*e^(-2*x + 8) - (x^2*log (3)^3 - 6*x^2*log(3)^2 - 4*x*log(3)^3 + 12*x^2*log(3) + 24*x*log(3)^2 - 8* x^2 - 48*x*log(3) + 32*x)*e^(x*log(3) - 2*x + 8)/(log(3)^3 - 6*log(3)^2 + 12*log(3) - 8) + 1/4*e^(2*x*log(3) - 2*x + 8) + e^(x*log(3) - x + 4)
Time = 9.40 (sec) , antiderivative size = 140, normalized size of antiderivative = 5.00 \[ \int \frac {1}{2} e^{8-2 x} \left (64 x-112 x^2+40 x^3-4 x^4+e^{-4+x} \left (16-24 x+4 x^2\right )+3^{2 x} (-1+\log (3))+3^x \left (8-20 x+4 x^2+\left (8 x-2 x^2\right ) \log (3)+e^{-4+x} (-2+2 \log (3))\right )\right ) \, dx={\mathrm {e}}^{8-2\,x}\,\left (28\,x^2+28\,x+14\right )-{\mathrm {e}}^{8-2\,x}\,\left (16\,x+8\right )+{\mathrm {e}}^{8-2\,x}\,\left (x^4+2\,x^3+3\,x^2+3\,x+\frac {3}{2}\right )-{\mathrm {e}}^{8-2\,x}\,\left (10\,x^3+15\,x^2+15\,x+\frac {15}{2}\right )-2\,x\,{\mathrm {e}}^{4-x}\,\left (x-4\right )+3^x\,{\mathrm {e}}^{4-2\,x}\,\left ({\mathrm {e}}^x+4\,x\,{\mathrm {e}}^4-x^2\,{\mathrm {e}}^4\right )+\frac {3^{2\,x}\,{\mathrm {e}}^{8-2\,x}\,\left (\ln \left (3\right )-1\right )}{2\,\left (\ln \left (9\right )-2\right )} \]
int(exp(8 - 2*x)*(32*x + (exp(x - 4)*(4*x^2 - 24*x + 16))/2 + (exp(x*log(3 ))*(log(3)*(8*x - 2*x^2) - 20*x + exp(x - 4)*(2*log(3) - 2) + 4*x^2 + 8))/ 2 + (exp(2*x*log(3))*(log(3) - 1))/2 - 56*x^2 + 20*x^3 - 2*x^4),x)