Integrand size = 83, antiderivative size = 28 \[ \int \frac {e^{-9-12 e^{\frac {x+4 \log (4)}{\log (4)}}-4 e^{\frac {2 (x+4 \log (4))}{\log (4)}}+x} \left (-24 e^{\frac {x+4 \log (4)}{\log (4)}} x-16 e^{\frac {2 (x+4 \log (4))}{\log (4)}} x+(2+2 x) \log (4)\right )}{\log (4)} \, dx=2 e^{x-\left (3+\log \left (e^{2 e^{4+\frac {x}{\log (4)}}}\right )\right )^2} x \] Output:
2*x*exp(x-(3+ln(exp(exp(1/2*x/ln(2)+4))^2))^2)
Time = 0.51 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.18 \[ \int \frac {e^{-9-12 e^{\frac {x+4 \log (4)}{\log (4)}}-4 e^{\frac {2 (x+4 \log (4))}{\log (4)}}+x} \left (-24 e^{\frac {x+4 \log (4)}{\log (4)}} x-16 e^{\frac {2 (x+4 \log (4))}{\log (4)}} x+(2+2 x) \log (4)\right )}{\log (4)} \, dx=2 e^{-9-12 e^{4+\frac {x}{\log (4)}}-4 e^{8+\frac {2 x}{\log (4)}}+x} x \] Input:
Integrate[(E^(-9 - 12*E^((x + 4*Log[4])/Log[4]) - 4*E^((2*(x + 4*Log[4]))/ Log[4]) + x)*(-24*E^((x + 4*Log[4])/Log[4])*x - 16*E^((2*(x + 4*Log[4]))/L og[4])*x + (2 + 2*x)*Log[4]))/Log[4],x]
Output:
2*E^(-9 - 12*E^(4 + x/Log[4]) - 4*E^(8 + (2*x)/Log[4]) + x)*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.
| \(\displaystyle \int \frac {\left (-24 x e^{\frac {x+4 \log (4)}{\log (4)}}-16 x e^{\frac {2 (x+4 \log (4))}{\log (4)}}+(2 x+2) \log (4)\right ) \exp \left (x-12 e^{\frac {x+4 \log (4)}{\log (4)}}-4 e^{\frac {2 (x+4 \log (4))}{\log (4)}}-9\right )}{\log (4)} \, dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int -2 e^{x-12 e^{\frac {x}{\log (4)}+4}-4 e^{\frac {2 x}{\log (4)}+8}-9} \left (12 e^{\frac {x}{\log (4)}+4} x+8 e^{\frac {2 x}{\log (4)}+8} x-(x+1) \log (4)\right )dx}{\log (4)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 \int e^{x-12 e^{\frac {x}{\log (4)}+4}-4 e^{\frac {2 x}{\log (4)}+8}-9} \left (12 e^{\frac {x}{\log (4)}+4} x+8 e^{\frac {2 x}{\log (4)}+8} x-(x+1) \log (4)\right )dx}{\log (4)}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {2 \int \left (12 \exp \left (\frac {x}{\log (4)}+x-12 e^{\frac {x}{\log (4)}+4}-4 e^{\frac {2 x}{\log (4)}+8}-5\right ) x+8 \exp \left (\frac {2 x}{\log (4)}+x-12 e^{\frac {x}{\log (4)}+4}-4 e^{\frac {2 x}{\log (4)}+8}-1\right ) x-e^{x-12 e^{\frac {x}{\log (4)}+4}-4 e^{\frac {2 x}{\log (4)}+8}-9} (x+1) \log (4)\right )dx}{\log (4)}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 \left (12 \int \exp \left (\left (1+\frac {1}{\log (4)}\right ) x-12 e^{\frac {x}{\log (4)}+4}-4 e^{\frac {2 x}{\log (4)}+8}-5\right ) xdx+8 \int \exp \left (\left (1+\frac {2}{\log (4)}\right ) x-12 e^{\frac {x}{\log (4)}+4}-4 e^{\frac {2 x}{\log (4)}+8}-1\right ) xdx-\log (4) \int e^{x-12 e^{\frac {x}{\log (4)}+4}-4 e^{\frac {2 x}{\log (4)}+8}-9}dx-\log (4) \int e^{x-12 e^{\frac {x}{\log (4)}+4}-4 e^{\frac {2 x}{\log (4)}+8}-9} xdx\right )}{\log (4)}\) |
Input:
Int[(E^(-9 - 12*E^((x + 4*Log[4])/Log[4]) - 4*E^((2*(x + 4*Log[4]))/Log[4] ) + x)*(-24*E^((x + 4*Log[4])/Log[4])*x - 16*E^((2*(x + 4*Log[4]))/Log[4]) *x + (2 + 2*x)*Log[4]))/Log[4],x]
Output:
$Aborted
Time = 0.59 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.32
| method | result | size |
| risch | \(2 x \,{\mathrm e}^{-4 \,{\mathrm e}^{\frac {8 \ln \left (2\right )+x}{\ln \left (2\right )}}-12 \,{\mathrm e}^{\frac {8 \ln \left (2\right )+x}{2 \ln \left (2\right )}}+x -9}\) | \(37\) |
| norman | \(2 x \,{\mathrm e}^{-4 \,{\mathrm e}^{\frac {8 \ln \left (2\right )+x}{\ln \left (2\right )}}-12 \,{\mathrm e}^{\frac {8 \ln \left (2\right )+x}{2 \ln \left (2\right )}}+x -9}\) | \(40\) |
| parallelrisch | \(2 x \,{\mathrm e}^{-4 \,{\mathrm e}^{\frac {8 \ln \left (2\right )+x}{\ln \left (2\right )}}-12 \,{\mathrm e}^{\frac {8 \ln \left (2\right )+x}{2 \ln \left (2\right )}}+x -9}\) | \(40\) |
Input:
int(1/2*(-16*x*exp(1/2*(8*ln(2)+x)/ln(2))^2-24*x*exp(1/2*(8*ln(2)+x)/ln(2) )+2*(2+2*x)*ln(2))*exp(-4*exp(1/2*(8*ln(2)+x)/ln(2))^2-12*exp(1/2*(8*ln(2) +x)/ln(2))+x-9)/ln(2),x,method=_RETURNVERBOSE)
Output:
2*x*exp(-4*exp((8*ln(2)+x)/ln(2))-12*exp(1/2*(8*ln(2)+x)/ln(2))+x-9)
Time = 0.08 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.29 \[ \int \frac {e^{-9-12 e^{\frac {x+4 \log (4)}{\log (4)}}-4 e^{\frac {2 (x+4 \log (4))}{\log (4)}}+x} \left (-24 e^{\frac {x+4 \log (4)}{\log (4)}} x-16 e^{\frac {2 (x+4 \log (4))}{\log (4)}} x+(2+2 x) \log (4)\right )}{\log (4)} \, dx=2 \, x e^{\left (x - 4 \, e^{\left (\frac {x + 8 \, \log \left (2\right )}{\log \left (2\right )}\right )} - 12 \, e^{\left (\frac {x + 8 \, \log \left (2\right )}{2 \, \log \left (2\right )}\right )} - 9\right )} \] Input:
integrate(1/2*(-16*x*exp(1/2*(8*log(2)+x)/log(2))^2-24*x*exp(1/2*(8*log(2) +x)/log(2))+2*(2+2*x)*log(2))*exp(-4*exp(1/2*(8*log(2)+x)/log(2))^2-12*exp (1/2*(8*log(2)+x)/log(2))+x-9)/log(2),x, algorithm="fricas")
Output:
2*x*e^(x - 4*e^((x + 8*log(2))/log(2)) - 12*e^(1/2*(x + 8*log(2))/log(2)) - 9)
Time = 0.19 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.29 \[ \int \frac {e^{-9-12 e^{\frac {x+4 \log (4)}{\log (4)}}-4 e^{\frac {2 (x+4 \log (4))}{\log (4)}}+x} \left (-24 e^{\frac {x+4 \log (4)}{\log (4)}} x-16 e^{\frac {2 (x+4 \log (4))}{\log (4)}} x+(2+2 x) \log (4)\right )}{\log (4)} \, dx=2 x e^{x - 4 e^{\frac {2 \left (\frac {x}{2} + 4 \log {\left (2 \right )}\right )}{\log {\left (2 \right )}}} - 12 e^{\frac {\frac {x}{2} + 4 \log {\left (2 \right )}}{\log {\left (2 \right )}}} - 9} \] Input:
integrate(1/2*(-16*x*exp(1/2*(8*ln(2)+x)/ln(2))**2-24*x*exp(1/2*(8*ln(2)+x )/ln(2))+2*(2+2*x)*ln(2))*exp(-4*exp(1/2*(8*ln(2)+x)/ln(2))**2-12*exp(1/2* (8*ln(2)+x)/ln(2))+x-9)/ln(2),x)
Output:
2*x*exp(x - 4*exp(2*(x/2 + 4*log(2))/log(2)) - 12*exp((x/2 + 4*log(2))/log (2)) - 9)
Time = 0.27 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {e^{-9-12 e^{\frac {x+4 \log (4)}{\log (4)}}-4 e^{\frac {2 (x+4 \log (4))}{\log (4)}}+x} \left (-24 e^{\frac {x+4 \log (4)}{\log (4)}} x-16 e^{\frac {2 (x+4 \log (4))}{\log (4)}} x+(2+2 x) \log (4)\right )}{\log (4)} \, dx=2 \, x e^{\left (x - 4 \, e^{\left (\frac {x}{\log \left (2\right )} + 8\right )} - 12 \, e^{\left (\frac {x}{2 \, \log \left (2\right )} + 4\right )} - 9\right )} \] Input:
integrate(1/2*(-16*x*exp(1/2*(8*log(2)+x)/log(2))^2-24*x*exp(1/2*(8*log(2) +x)/log(2))+2*(2+2*x)*log(2))*exp(-4*exp(1/2*(8*log(2)+x)/log(2))^2-12*exp (1/2*(8*log(2)+x)/log(2))+x-9)/log(2),x, algorithm="maxima")
Output:
2*x*e^(x - 4*e^(x/log(2) + 8) - 12*e^(1/2*x/log(2) + 4) - 9)
\[ \int \frac {e^{-9-12 e^{\frac {x+4 \log (4)}{\log (4)}}-4 e^{\frac {2 (x+4 \log (4))}{\log (4)}}+x} \left (-24 e^{\frac {x+4 \log (4)}{\log (4)}} x-16 e^{\frac {2 (x+4 \log (4))}{\log (4)}} x+(2+2 x) \log (4)\right )}{\log (4)} \, dx=\int { -\frac {2 \, {\left (4 \, x e^{\left (\frac {x + 8 \, \log \left (2\right )}{\log \left (2\right )}\right )} + 6 \, x e^{\left (\frac {x + 8 \, \log \left (2\right )}{2 \, \log \left (2\right )}\right )} - {\left (x + 1\right )} \log \left (2\right )\right )} e^{\left (x - 4 \, e^{\left (\frac {x + 8 \, \log \left (2\right )}{\log \left (2\right )}\right )} - 12 \, e^{\left (\frac {x + 8 \, \log \left (2\right )}{2 \, \log \left (2\right )}\right )} - 9\right )}}{\log \left (2\right )} \,d x } \] Input:
integrate(1/2*(-16*x*exp(1/2*(8*log(2)+x)/log(2))^2-24*x*exp(1/2*(8*log(2) +x)/log(2))+2*(2+2*x)*log(2))*exp(-4*exp(1/2*(8*log(2)+x)/log(2))^2-12*exp (1/2*(8*log(2)+x)/log(2))+x-9)/log(2),x, algorithm="giac")
Output:
integrate(-2*(4*x*e^((x + 8*log(2))/log(2)) + 6*x*e^(1/2*(x + 8*log(2))/lo g(2)) - (x + 1)*log(2))*e^(x - 4*e^((x + 8*log(2))/log(2)) - 12*e^(1/2*(x + 8*log(2))/log(2)) - 9)/log(2), x)
Time = 0.12 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.14 \[ \int \frac {e^{-9-12 e^{\frac {x+4 \log (4)}{\log (4)}}-4 e^{\frac {2 (x+4 \log (4))}{\log (4)}}+x} \left (-24 e^{\frac {x+4 \log (4)}{\log (4)}} x-16 e^{\frac {2 (x+4 \log (4))}{\log (4)}} x+(2+2 x) \log (4)\right )}{\log (4)} \, dx=2\,x\,{\mathrm {e}}^{-4\,{\mathrm {e}}^{\frac {x}{\ln \left (2\right )}}\,{\mathrm {e}}^8}\,{\mathrm {e}}^{-12\,{\mathrm {e}}^{\frac {x}{2\,\ln \left (2\right )}}\,{\mathrm {e}}^4}\,{\mathrm {e}}^{-9}\,{\mathrm {e}}^x \] Input:
int(-(exp(x - 12*exp((x/2 + 4*log(2))/log(2)) - 4*exp((2*(x/2 + 4*log(2))) /log(2)) - 9)*(24*x*exp((x/2 + 4*log(2))/log(2)) - 2*log(2)*(2*x + 2) + 16 *x*exp((2*(x/2 + 4*log(2)))/log(2))))/(2*log(2)),x)
Output:
2*x*exp(-4*exp(x/log(2))*exp(8))*exp(-12*exp(x/(2*log(2)))*exp(4))*exp(-9) *exp(x)
\[ \int \frac {e^{-9-12 e^{\frac {x+4 \log (4)}{\log (4)}}-4 e^{\frac {2 (x+4 \log (4))}{\log (4)}}+x} \left (-24 e^{\frac {x+4 \log (4)}{\log (4)}} x-16 e^{\frac {2 (x+4 \log (4))}{\log (4)}} x+(2+2 x) \log (4)\right )}{\log (4)} \, dx=\frac {2 \left (\int \frac {e^{x}}{e^{4 e^{\frac {x}{\mathrm {log}\left (2\right )}} e^{8}+12 e^{\frac {x}{2 \,\mathrm {log}\left (2\right )}} e^{4}}}d x \right ) \mathrm {log}\left (2\right )-8 \left (\int \frac {e^{\frac {\mathrm {log}\left (2\right ) x +x}{\mathrm {log}\left (2\right )}} x}{e^{4 e^{\frac {x}{\mathrm {log}\left (2\right )}} e^{8}+12 e^{\frac {x}{2 \,\mathrm {log}\left (2\right )}} e^{4}}}d x \right ) e^{8}-12 \left (\int \frac {e^{\frac {2 \,\mathrm {log}\left (2\right ) x +x}{2 \,\mathrm {log}\left (2\right )}} x}{e^{4 e^{\frac {x}{\mathrm {log}\left (2\right )}} e^{8}+12 e^{\frac {x}{2 \,\mathrm {log}\left (2\right )}} e^{4}}}d x \right ) e^{4}+2 \left (\int \frac {e^{x} x}{e^{4 e^{\frac {x}{\mathrm {log}\left (2\right )}} e^{8}+12 e^{\frac {x}{2 \,\mathrm {log}\left (2\right )}} e^{4}}}d x \right ) \mathrm {log}\left (2\right )}{\mathrm {log}\left (2\right ) e^{9}} \] Input:
int(1/2*(-16*x*exp(1/2*(8*log(2)+x)/log(2))^2-24*x*exp(1/2*(8*log(2)+x)/lo g(2))+2*(2+2*x)*log(2))*exp(-4*exp(1/2*(8*log(2)+x)/log(2))^2-12*exp(1/2*( 8*log(2)+x)/log(2))+x-9)/log(2),x)
Output:
(2*(int(e**x/e**(4*e**(x/log(2))*e**8 + 12*e**(x/(2*log(2)))*e**4),x)*log( 2) - 4*int((e**((log(2)*x + x)/log(2))*x)/e**(4*e**(x/log(2))*e**8 + 12*e* *(x/(2*log(2)))*e**4),x)*e**8 - 6*int((e**((2*log(2)*x + x)/(2*log(2)))*x) /e**(4*e**(x/log(2))*e**8 + 12*e**(x/(2*log(2)))*e**4),x)*e**4 + int((e**x *x)/e**(4*e**(x/log(2))*e**8 + 12*e**(x/(2*log(2)))*e**4),x)*log(2)))/(log (2)*e**9)