\(\int \frac {3528+e^{2 e^{2 x}} (168-48 x)-1008 x+e^{e^{2 x}} (1554-444 x+e^{2 x} (180-420 x+60 x^2))}{3969-18522 x+24255 x^2-6174 x^3+441 x^4+e^{2 e^{2 x}} (144-672 x+880 x^2-224 x^3+16 x^4)+e^{e^{2 x}} (1512-7056 x+9240 x^2-2352 x^3+168 x^4)} \, dx\) [1618]

Optimal result

Integrand size = 130, antiderivative size = 42 \[ \int \frac {3528+e^{2 e^{2 x}} (168-48 x)-1008 x+e^{e^{2 x}} \left (1554-444 x+e^{2 x} \left (180-420 x+60 x^2\right )\right )}{3969-18522 x+24255 x^2-6174 x^3+441 x^4+e^{2 e^{2 x}} \left (144-672 x+880 x^2-224 x^3+16 x^4\right )+e^{e^{2 x}} \left (1512-7056 x+9240 x^2-2352 x^3+168 x^4\right )} \, dx=\frac {2}{\left (4+\frac {5}{4+e^{e^{2 x}}}\right ) \left (-2 x+\frac {1}{3} \left (-x+x \left (\frac {3}{x}+x\right )\right )\right )} \] Output:

2/(4+5/(4+exp(exp(x)^2)))/(1/3*(x+3/x)*x-7/3*x)
 

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.81 \[ \int \frac {3528+e^{2 e^{2 x}} (168-48 x)-1008 x+e^{e^{2 x}} \left (1554-444 x+e^{2 x} \left (180-420 x+60 x^2\right )\right )}{3969-18522 x+24255 x^2-6174 x^3+441 x^4+e^{2 e^{2 x}} \left (144-672 x+880 x^2-224 x^3+16 x^4\right )+e^{e^{2 x}} \left (1512-7056 x+9240 x^2-2352 x^3+168 x^4\right )} \, dx=\frac {6 \left (4+e^{e^{2 x}}\right )}{\left (21+4 e^{e^{2 x}}\right ) \left (3-7 x+x^2\right )} \] Input:

Integrate[(3528 + E^(2*E^(2*x))*(168 - 48*x) - 1008*x + E^E^(2*x)*(1554 - 
444*x + E^(2*x)*(180 - 420*x + 60*x^2)))/(3969 - 18522*x + 24255*x^2 - 617 
4*x^3 + 441*x^4 + E^(2*E^(2*x))*(144 - 672*x + 880*x^2 - 224*x^3 + 16*x^4) 
 + E^E^(2*x)*(1512 - 7056*x + 9240*x^2 - 2352*x^3 + 168*x^4)),x]
 

Output:

(6*(4 + E^E^(2*x)))/((21 + 4*E^E^(2*x))*(3 - 7*x + x^2))
 

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.

\(\displaystyle \int \frac {e^{e^{2 x}} \left (e^{2 x} \left (60 x^2-420 x+180\right )-444 x+1554\right )+e^{2 e^{2 x}} (168-48 x)-1008 x+3528}{441 x^4-6174 x^3+24255 x^2+e^{2 e^{2 x}} \left (16 x^4-224 x^3+880 x^2-672 x+144\right )+e^{e^{2 x}} \left (168 x^4-2352 x^3+9240 x^2-7056 x+1512\right )-18522 x+3969} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {e^{e^{2 x}} \left (e^{2 x} \left (60 x^2-420 x+180\right )-444 x+1554\right )+e^{2 e^{2 x}} (168-48 x)-1008 x+3528}{\left (4 e^{e^{2 x}}+21\right )^2 \left (x^2-7 x+3\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {444 e^{e^{2 x}} x}{\left (4 e^{e^{2 x}}+21\right )^2 \left (x^2-7 x+3\right )^2}-\frac {1008 x}{\left (4 e^{e^{2 x}}+21\right )^2 \left (x^2-7 x+3\right )^2}+\frac {60 e^{2 x+e^{2 x}}}{\left (4 e^{e^{2 x}}+21\right )^2 \left (x^2-7 x+3\right )}-\frac {24 e^{2 e^{2 x}} (2 x-7)}{\left (4 e^{e^{2 x}}+21\right )^2 \left (x^2-7 x+3\right )^2}+\frac {1554 e^{e^{2 x}}}{\left (4 e^{e^{2 x}}+21\right )^2 \left (x^2-7 x+3\right )^2}+\frac {3528}{\left (4 e^{e^{2 x}}+21\right )^2 \left (x^2-7 x+3\right )^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 3528 \int \frac {1}{\left (21+4 e^{e^{2 x}}\right )^2 \left (x^2-7 x+3\right )^2}dx+1554 \int \frac {e^{e^{2 x}}}{\left (21+4 e^{e^{2 x}}\right )^2 \left (x^2-7 x+3\right )^2}dx+168 \int \frac {e^{2 e^{2 x}}}{\left (21+4 e^{e^{2 x}}\right )^2 \left (x^2-7 x+3\right )^2}dx-1008 \int \frac {x}{\left (21+4 e^{e^{2 x}}\right )^2 \left (x^2-7 x+3\right )^2}dx-444 \int \frac {e^{e^{2 x}} x}{\left (21+4 e^{e^{2 x}}\right )^2 \left (x^2-7 x+3\right )^2}dx-48 \int \frac {e^{2 e^{2 x}} x}{\left (21+4 e^{e^{2 x}}\right )^2 \left (x^2-7 x+3\right )^2}dx-\frac {120 \int \frac {e^{2 x+e^{2 x}}}{\left (21+4 e^{e^{2 x}}\right )^2 \left (-2 x+\sqrt {37}+7\right )}dx}{\sqrt {37}}-\frac {120 \int \frac {e^{2 x+e^{2 x}}}{\left (21+4 e^{e^{2 x}}\right )^2 \left (2 x+\sqrt {37}-7\right )}dx}{\sqrt {37}}\)

Input:

Int[(3528 + E^(2*E^(2*x))*(168 - 48*x) - 1008*x + E^E^(2*x)*(1554 - 444*x 
+ E^(2*x)*(180 - 420*x + 60*x^2)))/(3969 - 18522*x + 24255*x^2 - 6174*x^3 
+ 441*x^4 + E^(2*E^(2*x))*(144 - 672*x + 880*x^2 - 224*x^3 + 16*x^4) + E^E 
^(2*x)*(1512 - 7056*x + 9240*x^2 - 2352*x^3 + 168*x^4)),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.49 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.79

method	result	size
parallelrisch	\(\frac {96+24 \,{\mathrm e}^{{\mathrm e}^{2 x}}}{4 \left (4 \,{\mathrm e}^{{\mathrm e}^{2 x}}+21\right ) \left (x^{2}-7 x +3\right )}\)	\(33\)
risch	\(\frac {3}{2 \left (x^{2}-7 x +3\right )}-\frac {15}{2 \left (x^{2}-7 x +3\right ) \left (4 \,{\mathrm e}^{{\mathrm e}^{2 x}}+21\right )}\)	\(37\)

Input:

int(((-48*x+168)*exp(exp(x)^2)^2+((60*x^2-420*x+180)*exp(x)^2-444*x+1554)* 
exp(exp(x)^2)-1008*x+3528)/((16*x^4-224*x^3+880*x^2-672*x+144)*exp(exp(x)^ 
2)^2+(168*x^4-2352*x^3+9240*x^2-7056*x+1512)*exp(exp(x)^2)+441*x^4-6174*x^ 
3+24255*x^2-18522*x+3969),x,method=_RETURNVERBOSE)
 

Output:

1/4*(96+24*exp(exp(x)^2))/(4*exp(exp(x)^2)+21)/(x^2-7*x+3)
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.86 \[ \int \frac {3528+e^{2 e^{2 x}} (168-48 x)-1008 x+e^{e^{2 x}} \left (1554-444 x+e^{2 x} \left (180-420 x+60 x^2\right )\right )}{3969-18522 x+24255 x^2-6174 x^3+441 x^4+e^{2 e^{2 x}} \left (144-672 x+880 x^2-224 x^3+16 x^4\right )+e^{e^{2 x}} \left (1512-7056 x+9240 x^2-2352 x^3+168 x^4\right )} \, dx=\frac {6 \, {\left (e^{\left (e^{\left (2 \, x\right )}\right )} + 4\right )}}{21 \, x^{2} + 4 \, {\left (x^{2} - 7 \, x + 3\right )} e^{\left (e^{\left (2 \, x\right )}\right )} - 147 \, x + 63} \] Input:

integrate(((-48*x+168)*exp(exp(x)^2)^2+((60*x^2-420*x+180)*exp(x)^2-444*x+ 
1554)*exp(exp(x)^2)-1008*x+3528)/((16*x^4-224*x^3+880*x^2-672*x+144)*exp(e 
xp(x)^2)^2+(168*x^4-2352*x^3+9240*x^2-7056*x+1512)*exp(exp(x)^2)+441*x^4-6 
174*x^3+24255*x^2-18522*x+3969),x, algorithm="fricas")
 

Output:

6*(e^(e^(2*x)) + 4)/(21*x^2 + 4*(x^2 - 7*x + 3)*e^(e^(2*x)) - 147*x + 63)
 

Sympy [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.93 \[ \int \frac {3528+e^{2 e^{2 x}} (168-48 x)-1008 x+e^{e^{2 x}} \left (1554-444 x+e^{2 x} \left (180-420 x+60 x^2\right )\right )}{3969-18522 x+24255 x^2-6174 x^3+441 x^4+e^{2 e^{2 x}} \left (144-672 x+880 x^2-224 x^3+16 x^4\right )+e^{e^{2 x}} \left (1512-7056 x+9240 x^2-2352 x^3+168 x^4\right )} \, dx=- \frac {15}{42 x^{2} - 294 x + \left (8 x^{2} - 56 x + 24\right ) e^{e^{2 x}} + 126} + \frac {3}{2 x^{2} - 14 x + 6} \] Input:

integrate(((-48*x+168)*exp(exp(x)**2)**2+((60*x**2-420*x+180)*exp(x)**2-44 
4*x+1554)*exp(exp(x)**2)-1008*x+3528)/((16*x**4-224*x**3+880*x**2-672*x+14 
4)*exp(exp(x)**2)**2+(168*x**4-2352*x**3+9240*x**2-7056*x+1512)*exp(exp(x) 
**2)+441*x**4-6174*x**3+24255*x**2-18522*x+3969),x)
 

Output:

-15/(42*x**2 - 294*x + (8*x**2 - 56*x + 24)*exp(exp(2*x)) + 126) + 3/(2*x* 
*2 - 14*x + 6)
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.86 \[ \int \frac {3528+e^{2 e^{2 x}} (168-48 x)-1008 x+e^{e^{2 x}} \left (1554-444 x+e^{2 x} \left (180-420 x+60 x^2\right )\right )}{3969-18522 x+24255 x^2-6174 x^3+441 x^4+e^{2 e^{2 x}} \left (144-672 x+880 x^2-224 x^3+16 x^4\right )+e^{e^{2 x}} \left (1512-7056 x+9240 x^2-2352 x^3+168 x^4\right )} \, dx=\frac {6 \, {\left (e^{\left (e^{\left (2 \, x\right )}\right )} + 4\right )}}{21 \, x^{2} + 4 \, {\left (x^{2} - 7 \, x + 3\right )} e^{\left (e^{\left (2 \, x\right )}\right )} - 147 \, x + 63} \] Input:

integrate(((-48*x+168)*exp(exp(x)^2)^2+((60*x^2-420*x+180)*exp(x)^2-444*x+ 
1554)*exp(exp(x)^2)-1008*x+3528)/((16*x^4-224*x^3+880*x^2-672*x+144)*exp(e 
xp(x)^2)^2+(168*x^4-2352*x^3+9240*x^2-7056*x+1512)*exp(exp(x)^2)+441*x^4-6 
174*x^3+24255*x^2-18522*x+3969),x, algorithm="maxima")
 

Output:

6*(e^(e^(2*x)) + 4)/(21*x^2 + 4*(x^2 - 7*x + 3)*e^(e^(2*x)) - 147*x + 63)
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.10 \[ \int \frac {3528+e^{2 e^{2 x}} (168-48 x)-1008 x+e^{e^{2 x}} \left (1554-444 x+e^{2 x} \left (180-420 x+60 x^2\right )\right )}{3969-18522 x+24255 x^2-6174 x^3+441 x^4+e^{2 e^{2 x}} \left (144-672 x+880 x^2-224 x^3+16 x^4\right )+e^{e^{2 x}} \left (1512-7056 x+9240 x^2-2352 x^3+168 x^4\right )} \, dx=\frac {6 \, {\left (e^{\left (e^{\left (2 \, x\right )}\right )} + 4\right )}}{4 \, x^{2} e^{\left (e^{\left (2 \, x\right )}\right )} + 21 \, x^{2} - 28 \, x e^{\left (e^{\left (2 \, x\right )}\right )} - 147 \, x + 12 \, e^{\left (e^{\left (2 \, x\right )}\right )} + 63} \] Input:

integrate(((-48*x+168)*exp(exp(x)^2)^2+((60*x^2-420*x+180)*exp(x)^2-444*x+ 
1554)*exp(exp(x)^2)-1008*x+3528)/((16*x^4-224*x^3+880*x^2-672*x+144)*exp(e 
xp(x)^2)^2+(168*x^4-2352*x^3+9240*x^2-7056*x+1512)*exp(exp(x)^2)+441*x^4-6 
174*x^3+24255*x^2-18522*x+3969),x, algorithm="giac")
 

Output:

6*(e^(e^(2*x)) + 4)/(4*x^2*e^(e^(2*x)) + 21*x^2 - 28*x*e^(e^(2*x)) - 147*x 
 + 12*e^(e^(2*x)) + 63)
 

Mupad [B] (verification not implemented)

Time = 3.05 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.71 \[ \int \frac {3528+e^{2 e^{2 x}} (168-48 x)-1008 x+e^{e^{2 x}} \left (1554-444 x+e^{2 x} \left (180-420 x+60 x^2\right )\right )}{3969-18522 x+24255 x^2-6174 x^3+441 x^4+e^{2 e^{2 x}} \left (144-672 x+880 x^2-224 x^3+16 x^4\right )+e^{e^{2 x}} \left (1512-7056 x+9240 x^2-2352 x^3+168 x^4\right )} \, dx=\frac {6\,\left ({\mathrm {e}}^{{\mathrm {e}}^{2\,x}}+4\right )}{\left (4\,{\mathrm {e}}^{{\mathrm {e}}^{2\,x}}+21\right )\,\left (x^2-7\,x+3\right )} \] Input:

int(-(1008*x - exp(exp(2*x))*(exp(2*x)*(60*x^2 - 420*x + 180) - 444*x + 15 
54) + exp(2*exp(2*x))*(48*x - 168) - 3528)/(exp(2*exp(2*x))*(880*x^2 - 672 
*x - 224*x^3 + 16*x^4 + 144) - 18522*x + 24255*x^2 - 6174*x^3 + 441*x^4 + 
exp(exp(2*x))*(9240*x^2 - 7056*x - 2352*x^3 + 168*x^4 + 1512) + 3969),x)
 

Output:

(6*(exp(exp(2*x)) + 4))/((4*exp(exp(2*x)) + 21)*(x^2 - 7*x + 3))
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.29 \[ \int \frac {3528+e^{2 e^{2 x}} (168-48 x)-1008 x+e^{e^{2 x}} \left (1554-444 x+e^{2 x} \left (180-420 x+60 x^2\right )\right )}{3969-18522 x+24255 x^2-6174 x^3+441 x^4+e^{2 e^{2 x}} \left (144-672 x+880 x^2-224 x^3+16 x^4\right )+e^{e^{2 x}} \left (1512-7056 x+9240 x^2-2352 x^3+168 x^4\right )} \, dx=\frac {6 e^{e^{2 x}}+24}{4 e^{e^{2 x}} x^{2}-28 e^{e^{2 x}} x +12 e^{e^{2 x}}+21 x^{2}-147 x +63} \] Input:

int(((-48*x+168)*exp(exp(x)^2)^2+((60*x^2-420*x+180)*exp(x)^2-444*x+1554)* 
exp(exp(x)^2)-1008*x+3528)/((16*x^4-224*x^3+880*x^2-672*x+144)*exp(exp(x)^ 
2)^2+(168*x^4-2352*x^3+9240*x^2-7056*x+1512)*exp(exp(x)^2)+441*x^4-6174*x^ 
3+24255*x^2-18522*x+3969),x)
 

Output:

(6*(e**(e**(2*x)) + 4))/(4*e**(e**(2*x))*x**2 - 28*e**(e**(2*x))*x + 12*e* 
*(e**(2*x)) + 21*x**2 - 147*x + 63)