Integrand size = 45, antiderivative size = 25 \[ \int e^{-x+2 e^{-x} \left (-4+20 x+4 x \log \left (-\frac {x}{4}\right )\right )} \left (14-10 x+(2-2 x) \log \left (-\frac {x}{4}\right )\right ) \, dx=\frac {1}{4} e^{8 e^{-x} \left (-1+x \left (5+\log \left (-\frac {x}{4}\right )\right )\right )} \] Output:
1/4*exp(4*((5+ln(-1/4*x))*x-1)/exp(x))^2
Time = 6.39 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.56 \[ \int e^{-x+2 e^{-x} \left (-4+20 x+4 x \log \left (-\frac {x}{4}\right )\right )} \left (14-10 x+(2-2 x) \log \left (-\frac {x}{4}\right )\right ) \, dx=4^{-1-8 e^{-x} x} e^{-8 e^{-x} (1-5 x)} (-x)^{8 e^{-x} x} \] Input:
Integrate[E^(-x + (2*(-4 + 20*x + 4*x*Log[-1/4*x]))/E^x)*(14 - 10*x + (2 - 2*x)*Log[-1/4*x]),x]
Output:
(4^(-1 - (8*x)/E^x)*(-x)^((8*x)/E^x))/E^((8*(1 - 5*x))/E^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.
\(\displaystyle \int e^{2 e^{-x} \left (20 x+4 x \log \left (-\frac {x}{4}\right )-4\right )-x} \left (-10 x+(2-2 x) \log \left (-\frac {x}{4}\right )+14\right ) \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \left (-10 x+(2-2 x) \log \left (-\frac {x}{4}\right )+14\right ) \exp \left (-e^{-x} \left (e^x x-40 x-8 x \log \left (-\frac {x}{4}\right )+8\right )\right )dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (-10 x \exp \left (-e^{-x} \left (e^x x-40 x-8 x \log \left (-\frac {x}{4}\right )+8\right )\right )+14 \exp \left (-e^{-x} \left (e^x x-40 x-8 x \log \left (-\frac {x}{4}\right )+8\right )\right )-2 (x-1) \log \left (-\frac {x}{4}\right ) \exp \left (-e^{-x} \left (e^x x-40 x-8 x \log \left (-\frac {x}{4}\right )+8\right )\right )\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 14 \int \exp \left (-e^{-x} \left (e^x x-8 \log \left (-\frac {x}{4}\right ) x-40 x+8\right )\right )dx-10 \int \exp \left (-e^{-x} \left (e^x x-8 \log \left (-\frac {x}{4}\right ) x-40 x+8\right )\right ) xdx+2 \int \exp \left (-e^{-x} \left (e^x x-8 \log \left (-\frac {x}{4}\right ) x-40 x+8\right )\right ) \log \left (-\frac {x}{4}\right )dx-2 \int \exp \left (-e^{-x} \left (e^x x-8 \log \left (-\frac {x}{4}\right ) x-40 x+8\right )\right ) x \log \left (-\frac {x}{4}\right )dx\) |
Input:
Int[E^(-x + (2*(-4 + 20*x + 4*x*Log[-1/4*x]))/E^x)*(14 - 10*x + (2 - 2*x)* Log[-1/4*x]),x]
Output:
$Aborted
Time = 0.21 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84
method | result | size |
risch | \(\frac {{\mathrm e}^{8 \left (x \ln \left (-\frac {x}{4}\right )+5 x -1\right ) {\mathrm e}^{-x}}}{4}\) | \(21\) |
parallelrisch | \(\frac {{\mathrm e}^{8 \left (x \ln \left (-\frac {x}{4}\right )+5 x -1\right ) {\mathrm e}^{-x}}}{4}\) | \(23\) |
Input:
int(((2-2*x)*ln(-1/4*x)-10*x+14)*exp((4*x*ln(-1/4*x)+20*x-4)/exp(x))^2/exp (x),x,method=_RETURNVERBOSE)
Output:
1/4*exp(8*(x*ln(-1/4*x)+5*x-1)*exp(-x))
Time = 0.08 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.04 \[ \int e^{-x+2 e^{-x} \left (-4+20 x+4 x \log \left (-\frac {x}{4}\right )\right )} \left (14-10 x+(2-2 x) \log \left (-\frac {x}{4}\right )\right ) \, dx=\frac {1}{4} \, e^{\left (8 \, x e^{\left (-x\right )} \log \left (-\frac {1}{4} \, x\right ) + 8 \, {\left (5 \, x - 1\right )} e^{\left (-x\right )}\right )} \] Input:
integrate(((2-2*x)*log(-1/4*x)-10*x+14)*exp((4*x*log(-1/4*x)+20*x-4)/exp(x ))^2/exp(x),x, algorithm="fricas")
Output:
1/4*e^(8*x*e^(-x)*log(-1/4*x) + 8*(5*x - 1)*e^(-x))
Time = 0.28 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80 \[ \int e^{-x+2 e^{-x} \left (-4+20 x+4 x \log \left (-\frac {x}{4}\right )\right )} \left (14-10 x+(2-2 x) \log \left (-\frac {x}{4}\right )\right ) \, dx=\frac {e^{2 \cdot \left (4 x \log {\left (- \frac {x}{4} \right )} + 20 x - 4\right ) e^{- x}}}{4} \] Input:
integrate(((2-2*x)*ln(-1/4*x)-10*x+14)*exp((4*x*ln(-1/4*x)+20*x-4)/exp(x)) **2/exp(x),x)
Output:
exp(2*(4*x*log(-x/4) + 20*x - 4)*exp(-x))/4
Time = 0.20 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.48 \[ \int e^{-x+2 e^{-x} \left (-4+20 x+4 x \log \left (-\frac {x}{4}\right )\right )} \left (14-10 x+(2-2 x) \log \left (-\frac {x}{4}\right )\right ) \, dx=\frac {1}{4} \, e^{\left (-16 \, x e^{\left (-x\right )} \log \left (2\right ) + 8 \, x e^{\left (-x\right )} \log \left (-x\right ) + 40 \, x e^{\left (-x\right )} - 8 \, e^{\left (-x\right )}\right )} \] Input:
integrate(((2-2*x)*log(-1/4*x)-10*x+14)*exp((4*x*log(-1/4*x)+20*x-4)/exp(x ))^2/exp(x),x, algorithm="maxima")
Output:
1/4*e^(-16*x*e^(-x)*log(2) + 8*x*e^(-x)*log(-x) + 40*x*e^(-x) - 8*e^(-x))
\[ \int e^{-x+2 e^{-x} \left (-4+20 x+4 x \log \left (-\frac {x}{4}\right )\right )} \left (14-10 x+(2-2 x) \log \left (-\frac {x}{4}\right )\right ) \, dx=\int { -2 \, {\left ({\left (x - 1\right )} \log \left (-\frac {1}{4} \, x\right ) + 5 \, x - 7\right )} e^{\left (8 \, {\left (x \log \left (-\frac {1}{4} \, x\right ) + 5 \, x - 1\right )} e^{\left (-x\right )} - x\right )} \,d x } \] Input:
integrate(((2-2*x)*log(-1/4*x)-10*x+14)*exp((4*x*log(-1/4*x)+20*x-4)/exp(x ))^2/exp(x),x, algorithm="giac")
Output:
integrate(-2*((x - 1)*log(-1/4*x) + 5*x - 7)*e^(8*(x*log(-1/4*x) + 5*x - 1 )*e^(-x) - x), x)
Time = 2.34 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.12 \[ \int e^{-x+2 e^{-x} \left (-4+20 x+4 x \log \left (-\frac {x}{4}\right )\right )} \left (14-10 x+(2-2 x) \log \left (-\frac {x}{4}\right )\right ) \, dx=\frac {{\mathrm {e}}^{-8\,{\mathrm {e}}^{-x}}\,{\mathrm {e}}^{40\,x\,{\mathrm {e}}^{-x}}\,{\left (-\frac {x}{4}\right )}^{8\,x\,{\mathrm {e}}^{-x}}}{4} \] Input:
int(-exp(-x)*exp(2*exp(-x)*(20*x + 4*x*log(-x/4) - 4))*(10*x + log(-x/4)*( 2*x - 2) - 14),x)
Output:
(exp(-8*exp(-x))*exp(40*x*exp(-x))*(-x/4)^(8*x*exp(-x)))/4
Time = 0.24 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.28 \[ \int e^{-x+2 e^{-x} \left (-4+20 x+4 x \log \left (-\frac {x}{4}\right )\right )} \left (14-10 x+(2-2 x) \log \left (-\frac {x}{4}\right )\right ) \, dx=\frac {e^{\frac {8 \,\mathrm {log}\left (-\frac {x}{4}\right ) x +40 x}{e^{x}}}}{4 e^{\frac {8}{e^{x}}}} \] Input:
int(((2-2*x)*log(-1/4*x)-10*x+14)*exp((4*x*log(-1/4*x)+20*x-4)/exp(x))^2/e xp(x),x)
Output:
e**((8*log(( - x)/4)*x + 40*x)/e**x)/(4*e**(8/e**x))