\(\int \frac {1}{3} e^{-e^x+\frac {1}{3} e^{-e^x} (12 x^3+100 x^4+(-12 x^2-100 x^3) \log (4))} (36 x^2+400 x^3+(-24 x-300 x^2) \log (4)+e^x (-12 x^3-100 x^4+(12 x^2+100 x^3) \log (4))) \, dx\) [105]

Optimal result

Integrand size = 100, antiderivative size = 27 \[ \int \frac {1}{3} e^{-e^x+\frac {1}{3} e^{-e^x} \left (12 x^3+100 x^4+\left (-12 x^2-100 x^3\right ) \log (4)\right )} \left (36 x^2+400 x^3+\left (-24 x-300 x^2\right ) \log (4)+e^x \left (-12 x^3-100 x^4+\left (12 x^2+100 x^3\right ) \log (4)\right )\right ) \, dx=e^{\frac {4}{3} e^{-e^x} x^2 (3+25 x) (x-\log (4))} \] Output:

exp(4/3/exp(exp(x))*x^2*(x-2*ln(2))*(3+25*x))
 

Mathematica [F]

\[ \int \frac {1}{3} e^{-e^x+\frac {1}{3} e^{-e^x} \left (12 x^3+100 x^4+\left (-12 x^2-100 x^3\right ) \log (4)\right )} \left (36 x^2+400 x^3+\left (-24 x-300 x^2\right ) \log (4)+e^x \left (-12 x^3-100 x^4+\left (12 x^2+100 x^3\right ) \log (4)\right )\right ) \, dx=\int \frac {1}{3} e^{-e^x+\frac {1}{3} e^{-e^x} \left (12 x^3+100 x^4+\left (-12 x^2-100 x^3\right ) \log (4)\right )} \left (36 x^2+400 x^3+\left (-24 x-300 x^2\right ) \log (4)+e^x \left (-12 x^3-100 x^4+\left (12 x^2+100 x^3\right ) \log (4)\right )\right ) \, dx \] Input:

Integrate[(E^(-E^x + (12*x^3 + 100*x^4 + (-12*x^2 - 100*x^3)*Log[4])/(3*E^ 
E^x))*(36*x^2 + 400*x^3 + (-24*x - 300*x^2)*Log[4] + E^x*(-12*x^3 - 100*x^ 
4 + (12*x^2 + 100*x^3)*Log[4])))/3,x]
 

Output:

Integrate[E^(-E^x + (12*x^3 + 100*x^4 + (-12*x^2 - 100*x^3)*Log[4])/(3*E^E 
^x))*(36*x^2 + 400*x^3 + (-24*x - 300*x^2)*Log[4] + E^x*(-12*x^3 - 100*x^4 
 + (12*x^2 + 100*x^3)*Log[4])), x]/3
 

Rubi [F]

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.

\(\displaystyle \int \frac {1}{3} \left (400 x^3+36 x^2+\left (-300 x^2-24 x\right ) \log (4)+e^x \left (-100 x^4-12 x^3+\left (100 x^3+12 x^2\right ) \log (4)\right )\right ) \exp \left (\frac {1}{3} e^{-e^x} \left (100 x^4+12 x^3+\left (-100 x^3-12 x^2\right ) \log (4)\right )-e^x\right ) \, dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{3} \int 4 \exp \left (\frac {4}{3} e^{-e^x} \left (25 x^4+3 x^3-\left (25 x^3+3 x^2\right ) \log (4)\right )-e^x\right ) \left (100 x^3+9 x^2-e^x \left (25 x^4+3 x^3-\left (25 x^3+3 x^2\right ) \log (4)\right )-3 \left (25 x^2+2 x\right ) \log (4)\right )dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {4}{3} \int \exp \left (\frac {4}{3} e^{-e^x} \left (25 x^4+3 x^3-\left (25 x^3+3 x^2\right ) \log (4)\right )-e^x\right ) \left (100 x^3+9 x^2-e^x \left (25 x^4+3 x^3-\left (25 x^3+3 x^2\right ) \log (4)\right )-3 \left (25 x^2+2 x\right ) \log (4)\right )dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \frac {4}{3} \int \exp \left (-\frac {1}{3} e^{-e^x} \left (-100 x^4-12 \left (1-\frac {25 \log (4)}{3}\right ) x^3+12 \log (4) x^2+3 e^{x+e^x}\right )\right ) \left (100 x^3+9 x^2-e^x \left (25 x^4+3 x^3-\left (25 x^3+3 x^2\right ) \log (4)\right )-3 \left (25 x^2+2 x\right ) \log (4)\right )dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {4}{3} \int \left (100 \exp \left (-\frac {1}{3} e^{-e^x} \left (-100 x^4-12 \left (1-\frac {25 \log (4)}{3}\right ) x^3+12 \log (4) x^2+3 e^{x+e^x}\right )\right ) x^3+9 \exp \left (-\frac {1}{3} e^{-e^x} \left (-100 x^4-12 \left (1-\frac {25 \log (4)}{3}\right ) x^3+12 \log (4) x^2+3 e^{x+e^x}\right )\right ) x^2-\exp \left (x-\frac {1}{3} e^{-e^x} \left (-100 x^4-12 \left (1-\frac {25 \log (4)}{3}\right ) x^3+12 \log (4) x^2+3 e^{x+e^x}\right )\right ) (25 x+3) (x-\log (4)) x^2-3 \exp \left (-\frac {1}{3} e^{-e^x} \left (-100 x^4-12 \left (1-\frac {25 \log (4)}{3}\right ) x^3+12 \log (4) x^2+3 e^{x+e^x}\right )\right ) (25 x+2) \log (4) x\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {4}{3} \left (-6 \log (4) \int \exp \left (-\frac {1}{3} e^{-e^x} \left (-100 x^4-12 \left (1-\frac {25 \log (4)}{3}\right ) x^3+12 \log (4) x^2+3 e^{x+e^x}\right )\right ) xdx-75 \log (4) \int \exp \left (-\frac {1}{3} e^{-e^x} \left (-100 x^4-12 \left (1-\frac {25 \log (4)}{3}\right ) x^3+12 \log (4) x^2+3 e^{x+e^x}\right )\right ) x^2dx+9 \int \exp \left (-\frac {1}{3} e^{-e^x} \left (-100 x^4-12 \left (1-\frac {25 \log (4)}{3}\right ) x^3+12 \log (4) x^2+3 e^{x+e^x}\right )\right ) x^2dx+3 \log (4) \int \exp \left (x-\frac {1}{3} e^{-e^x} \left (-100 x^4-12 \left (1-\frac {25 \log (4)}{3}\right ) x^3+12 \log (4) x^2+3 e^{x+e^x}\right )\right ) x^2dx+100 \int \exp \left (-\frac {1}{3} e^{-e^x} \left (-100 x^4-12 \left (1-\frac {25 \log (4)}{3}\right ) x^3+12 \log (4) x^2+3 e^{x+e^x}\right )\right ) x^3dx-(3-25 \log (4)) \int \exp \left (x-\frac {1}{3} e^{-e^x} \left (-100 x^4-12 \left (1-\frac {25 \log (4)}{3}\right ) x^3+12 \log (4) x^2+3 e^{x+e^x}\right )\right ) x^3dx-25 \int \exp \left (x-\frac {1}{3} e^{-e^x} \left (-100 x^4-12 \left (1-\frac {25 \log (4)}{3}\right ) x^3+12 \log (4) x^2+3 e^{x+e^x}\right )\right ) x^4dx\right )\)

Input:

Int[(E^(-E^x + (12*x^3 + 100*x^4 + (-12*x^2 - 100*x^3)*Log[4])/(3*E^E^x))* 
(36*x^2 + 400*x^3 + (-24*x - 300*x^2)*Log[4] + E^x*(-12*x^3 - 100*x^4 + (1 
2*x^2 + 100*x^3)*Log[4])))/3,x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 4.94 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93

method	result	size
risch	\({\mathrm e}^{-\frac {4 x^{2} \left (3+25 x \right ) \left (2 \ln \left (2\right )-x \right ) {\mathrm e}^{-{\mathrm e}^{x}}}{3}}\)	\(25\)
parallelrisch	\({\mathrm e}^{-\frac {4 x^{2} \left (50 x \ln \left (2\right )-25 x^{2}+6 \ln \left (2\right )-3 x \right ) {\mathrm e}^{-{\mathrm e}^{x}}}{3}}\)	\(30\)

Input:

int(1/3*((2*(100*x^3+12*x^2)*ln(2)-100*x^4-12*x^3)*exp(x)+2*(-300*x^2-24*x 
)*ln(2)+400*x^3+36*x^2)*exp(1/3*(2*(-100*x^3-12*x^2)*ln(2)+100*x^4+12*x^3) 
/exp(exp(x)))/exp(exp(x)),x,method=_RETURNVERBOSE)
 

Output:

exp(-4/3*x^2*(3+25*x)*(2*ln(2)-x)*exp(-exp(x)))
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.26 \[ \int \frac {1}{3} e^{-e^x+\frac {1}{3} e^{-e^x} \left (12 x^3+100 x^4+\left (-12 x^2-100 x^3\right ) \log (4)\right )} \left (36 x^2+400 x^3+\left (-24 x-300 x^2\right ) \log (4)+e^x \left (-12 x^3-100 x^4+\left (12 x^2+100 x^3\right ) \log (4)\right )\right ) \, dx=e^{\left (\frac {4}{3} \, {\left (25 \, x^{4} + 3 \, x^{3} - 2 \, {\left (25 \, x^{3} + 3 \, x^{2}\right )} \log \left (2\right )\right )} e^{\left (-e^{x}\right )}\right )} \] Input:

integrate(1/3*((2*(100*x^3+12*x^2)*log(2)-100*x^4-12*x^3)*exp(x)+2*(-300*x 
^2-24*x)*log(2)+400*x^3+36*x^2)*exp(1/3*(2*(-100*x^3-12*x^2)*log(2)+100*x^ 
4+12*x^3)/exp(exp(x)))/exp(exp(x)),x, algorithm="fricas")
 

Output:

e^(4/3*(25*x^4 + 3*x^3 - 2*(25*x^3 + 3*x^2)*log(2))*e^(-e^x))
 

Sympy [A] (verification not implemented)

Time = 0.38 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.26 \[ \int \frac {1}{3} e^{-e^x+\frac {1}{3} e^{-e^x} \left (12 x^3+100 x^4+\left (-12 x^2-100 x^3\right ) \log (4)\right )} \left (36 x^2+400 x^3+\left (-24 x-300 x^2\right ) \log (4)+e^x \left (-12 x^3-100 x^4+\left (12 x^2+100 x^3\right ) \log (4)\right )\right ) \, dx=e^{\left (\frac {100 x^{4}}{3} + 4 x^{3} + \frac {\left (- 200 x^{3} - 24 x^{2}\right ) \log {\left (2 \right )}}{3}\right ) e^{- e^{x}}} \] Input:

integrate(1/3*((2*(100*x**3+12*x**2)*ln(2)-100*x**4-12*x**3)*exp(x)+2*(-30 
0*x**2-24*x)*ln(2)+400*x**3+36*x**2)*exp(1/3*(2*(-100*x**3-12*x**2)*ln(2)+ 
100*x**4+12*x**3)/exp(exp(x)))/exp(exp(x)),x)
 

Output:

exp((100*x**4/3 + 4*x**3 + (-200*x**3 - 24*x**2)*log(2)/3)*exp(-exp(x)))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (22) = 44\).

Time = 0.44 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.70 \[ \int \frac {1}{3} e^{-e^x+\frac {1}{3} e^{-e^x} \left (12 x^3+100 x^4+\left (-12 x^2-100 x^3\right ) \log (4)\right )} \left (36 x^2+400 x^3+\left (-24 x-300 x^2\right ) \log (4)+e^x \left (-12 x^3-100 x^4+\left (12 x^2+100 x^3\right ) \log (4)\right )\right ) \, dx=e^{\left (\frac {100}{3} \, x^{4} e^{\left (-e^{x}\right )} - \frac {200}{3} \, x^{3} e^{\left (-e^{x}\right )} \log \left (2\right ) + 4 \, x^{3} e^{\left (-e^{x}\right )} - 8 \, x^{2} e^{\left (-e^{x}\right )} \log \left (2\right )\right )} \] Input:

integrate(1/3*((2*(100*x^3+12*x^2)*log(2)-100*x^4-12*x^3)*exp(x)+2*(-300*x 
^2-24*x)*log(2)+400*x^3+36*x^2)*exp(1/3*(2*(-100*x^3-12*x^2)*log(2)+100*x^ 
4+12*x^3)/exp(exp(x)))/exp(exp(x)),x, algorithm="maxima")
 

Output:

e^(100/3*x^4*e^(-e^x) - 200/3*x^3*e^(-e^x)*log(2) + 4*x^3*e^(-e^x) - 8*x^2 
*e^(-e^x)*log(2))
 

Giac [F]

\[ \int \frac {1}{3} e^{-e^x+\frac {1}{3} e^{-e^x} \left (12 x^3+100 x^4+\left (-12 x^2-100 x^3\right ) \log (4)\right )} \left (36 x^2+400 x^3+\left (-24 x-300 x^2\right ) \log (4)+e^x \left (-12 x^3-100 x^4+\left (12 x^2+100 x^3\right ) \log (4)\right )\right ) \, dx=\int { \frac {4}{3} \, {\left (100 \, x^{3} + 9 \, x^{2} - {\left (25 \, x^{4} + 3 \, x^{3} - 2 \, {\left (25 \, x^{3} + 3 \, x^{2}\right )} \log \left (2\right )\right )} e^{x} - 6 \, {\left (25 \, x^{2} + 2 \, x\right )} \log \left (2\right )\right )} e^{\left (\frac {4}{3} \, {\left (25 \, x^{4} + 3 \, x^{3} - 2 \, {\left (25 \, x^{3} + 3 \, x^{2}\right )} \log \left (2\right )\right )} e^{\left (-e^{x}\right )} - e^{x}\right )} \,d x } \] Input:

integrate(1/3*((2*(100*x^3+12*x^2)*log(2)-100*x^4-12*x^3)*exp(x)+2*(-300*x 
^2-24*x)*log(2)+400*x^3+36*x^2)*exp(1/3*(2*(-100*x^3-12*x^2)*log(2)+100*x^ 
4+12*x^3)/exp(exp(x)))/exp(exp(x)),x, algorithm="giac")
 

Output:

integrate(4/3*(100*x^3 + 9*x^2 - (25*x^4 + 3*x^3 - 2*(25*x^3 + 3*x^2)*log( 
2))*e^x - 6*(25*x^2 + 2*x)*log(2))*e^(4/3*(25*x^4 + 3*x^3 - 2*(25*x^3 + 3* 
x^2)*log(2))*e^(-e^x) - e^x), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{3} e^{-e^x+\frac {1}{3} e^{-e^x} \left (12 x^3+100 x^4+\left (-12 x^2-100 x^3\right ) \log (4)\right )} \left (36 x^2+400 x^3+\left (-24 x-300 x^2\right ) \log (4)+e^x \left (-12 x^3-100 x^4+\left (12 x^2+100 x^3\right ) \log (4)\right )\right ) \, dx=\int -\frac {{\mathrm {e}}^{-{\mathrm {e}}^x}\,{\mathrm {e}}^{{\mathrm {e}}^{-{\mathrm {e}}^x}\,\left (4\,x^3-\frac {2\,\ln \left (2\right )\,\left (100\,x^3+12\,x^2\right )}{3}+\frac {100\,x^4}{3}\right )}\,\left (2\,\ln \left (2\right )\,\left (300\,x^2+24\,x\right )+{\mathrm {e}}^x\,\left (12\,x^3-2\,\ln \left (2\right )\,\left (100\,x^3+12\,x^2\right )+100\,x^4\right )-36\,x^2-400\,x^3\right )}{3} \,d x \] Input:

int(-(exp(-exp(x))*exp(exp(-exp(x))*(4*x^3 - (2*log(2)*(12*x^2 + 100*x^3)) 
/3 + (100*x^4)/3))*(2*log(2)*(24*x + 300*x^2) + exp(x)*(12*x^3 - 2*log(2)* 
(12*x^2 + 100*x^3) + 100*x^4) - 36*x^2 - 400*x^3))/3,x)
 

Output:

int(-(exp(-exp(x))*exp(exp(-exp(x))*(4*x^3 - (2*log(2)*(12*x^2 + 100*x^3)) 
/3 + (100*x^4)/3))*(2*log(2)*(24*x + 300*x^2) + exp(x)*(12*x^3 - 2*log(2)* 
(12*x^2 + 100*x^3) + 100*x^4) - 36*x^2 - 400*x^3))/3, x)
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.89 \[ \int \frac {1}{3} e^{-e^x+\frac {1}{3} e^{-e^x} \left (12 x^3+100 x^4+\left (-12 x^2-100 x^3\right ) \log (4)\right )} \left (36 x^2+400 x^3+\left (-24 x-300 x^2\right ) \log (4)+e^x \left (-12 x^3-100 x^4+\left (12 x^2+100 x^3\right ) \log (4)\right )\right ) \, dx=\frac {e^{\frac {100 x^{4}+12 x^{3}}{3 e^{e^{x}}}}}{e^{\frac {200 \,\mathrm {log}\left (2\right ) x^{3}+24 \,\mathrm {log}\left (2\right ) x^{2}}{3 e^{e^{x}}}}} \] Input:

int(1/3*((2*(100*x^3+12*x^2)*log(2)-100*x^4-12*x^3)*exp(x)+2*(-300*x^2-24* 
x)*log(2)+400*x^3+36*x^2)*exp(1/3*(2*(-100*x^3-12*x^2)*log(2)+100*x^4+12*x 
^3)/exp(exp(x)))/exp(exp(x)),x)
 

Output:

e**((100*x**4 + 12*x**3)/(3*e**(e**x)))/e**((200*log(2)*x**3 + 24*log(2)*x 
**2)/(3*e**(e**x)))