\(\int \frac {3528+e^{2 e^{2 x}} (168-48 x)-1008 x+e^{e^{2 x}} (1554-444 x+e^{2 x} (180-420 x+60 x^2))}{3969-18522 x+24255 x^2-6174 x^3+441 x^4+e^{2 e^{2 x}} (144-672 x+880 x^2-224 x^3+16 x^4)+e^{e^{2 x}} (1512-7056 x+9240 x^2-2352 x^3+168 x^4)} \, dx\) [7618]

Optimal result

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

[Out]

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

Rubi [F]

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

[In]

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]

[Out]

(-120*Defer[Int][E^(E^(2*x) + 2*x)/((21 + 4*E^E^(2*x))^2*(7 + Sqrt[37] - 2*x)), x])/Sqrt[37] - (120*Defer[Int]
[E^(E^(2*x) + 2*x)/((21 + 4*E^E^(2*x))^2*(-7 + Sqrt[37] + 2*x)), x])/Sqrt[37] + 3528*Defer[Int][1/((21 + 4*E^E
^(2*x))^2*(3 - 7*x + x^2)^2), x] + 1554*Defer[Int][E^E^(2*x)/((21 + 4*E^E^(2*x))^2*(3 - 7*x + x^2)^2), x] + 16
8*Defer[Int][E^(2*E^(2*x))/((21 + 4*E^E^(2*x))^2*(3 - 7*x + x^2)^2), x] - 1008*Defer[Int][x/((21 + 4*E^E^(2*x)
)^2*(3 - 7*x + x^2)^2), x] - 444*Defer[Int][(E^E^(2*x)*x)/((21 + 4*E^E^(2*x))^2*(3 - 7*x + x^2)^2), x] - 48*De
fer[Int][(E^(2*E^(2*x))*x)/((21 + 4*E^E^(2*x))^2*(3 - 7*x + x^2)^2), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {6 \left (-84 (-7+2 x)-37 e^{e^{2 x}} (-7+2 x)-4 e^{2 e^{2 x}} (-7+2 x)+10 e^{e^{2 x}+2 x} \left (3-7 x+x^2\right )\right )}{\left (21+4 e^{e^{2 x}}\right )^2 \left (3-7 x+x^2\right )^2} \, dx \\ & = 6 \int \frac {-84 (-7+2 x)-37 e^{e^{2 x}} (-7+2 x)-4 e^{2 e^{2 x}} (-7+2 x)+10 e^{e^{2 x}+2 x} \left (3-7 x+x^2\right )}{\left (21+4 e^{e^{2 x}}\right )^2 \left (3-7 x+x^2\right )^2} \, dx \\ & = 6 \int \left (-\frac {84 (-7+2 x)}{\left (21+4 e^{e^{2 x}}\right )^2 \left (3-7 x+x^2\right )^2}-\frac {37 e^{e^{2 x}} (-7+2 x)}{\left (21+4 e^{e^{2 x}}\right )^2 \left (3-7 x+x^2\right )^2}-\frac {4 e^{2 e^{2 x}} (-7+2 x)}{\left (21+4 e^{e^{2 x}}\right )^2 \left (3-7 x+x^2\right )^2}+\frac {10 e^{e^{2 x}+2 x}}{\left (21+4 e^{e^{2 x}}\right )^2 \left (3-7 x+x^2\right )}\right ) \, dx \\ & = -\left (24 \int \frac {e^{2 e^{2 x}} (-7+2 x)}{\left (21+4 e^{e^{2 x}}\right )^2 \left (3-7 x+x^2\right )^2} \, dx\right )+60 \int \frac {e^{e^{2 x}+2 x}}{\left (21+4 e^{e^{2 x}}\right )^2 \left (3-7 x+x^2\right )} \, dx-222 \int \frac {e^{e^{2 x}} (-7+2 x)}{\left (21+4 e^{e^{2 x}}\right )^2 \left (3-7 x+x^2\right )^2} \, dx-504 \int \frac {-7+2 x}{\left (21+4 e^{e^{2 x}}\right )^2 \left (3-7 x+x^2\right )^2} \, dx \\ & = -\left (24 \int \left (-\frac {7 e^{2 e^{2 x}}}{\left (21+4 e^{e^{2 x}}\right )^2 \left (3-7 x+x^2\right )^2}+\frac {2 e^{2 e^{2 x}} x}{\left (21+4 e^{e^{2 x}}\right )^2 \left (3-7 x+x^2\right )^2}\right ) \, dx\right )+60 \int \left (-\frac {2 e^{e^{2 x}+2 x}}{\sqrt {37} \left (21+4 e^{e^{2 x}}\right )^2 \left (7+\sqrt {37}-2 x\right )}-\frac {2 e^{e^{2 x}+2 x}}{\sqrt {37} \left (21+4 e^{e^{2 x}}\right )^2 \left (-7+\sqrt {37}+2 x\right )}\right ) \, dx-222 \int \left (-\frac {7 e^{e^{2 x}}}{\left (21+4 e^{e^{2 x}}\right )^2 \left (3-7 x+x^2\right )^2}+\frac {2 e^{e^{2 x}} x}{\left (21+4 e^{e^{2 x}}\right )^2 \left (3-7 x+x^2\right )^2}\right ) \, dx-504 \int \left (-\frac {7}{\left (21+4 e^{e^{2 x}}\right )^2 \left (3-7 x+x^2\right )^2}+\frac {2 x}{\left (21+4 e^{e^{2 x}}\right )^2 \left (3-7 x+x^2\right )^2}\right ) \, dx \\ & = -\left (48 \int \frac {e^{2 e^{2 x}} x}{\left (21+4 e^{e^{2 x}}\right )^2 \left (3-7 x+x^2\right )^2} \, dx\right )+168 \int \frac {e^{2 e^{2 x}}}{\left (21+4 e^{e^{2 x}}\right )^2 \left (3-7 x+x^2\right )^2} \, dx-444 \int \frac {e^{e^{2 x}} x}{\left (21+4 e^{e^{2 x}}\right )^2 \left (3-7 x+x^2\right )^2} \, dx-1008 \int \frac {x}{\left (21+4 e^{e^{2 x}}\right )^2 \left (3-7 x+x^2\right )^2} \, dx+1554 \int \frac {e^{e^{2 x}}}{\left (21+4 e^{e^{2 x}}\right )^2 \left (3-7 x+x^2\right )^2} \, dx+3528 \int \frac {1}{\left (21+4 e^{e^{2 x}}\right )^2 \left (3-7 x+x^2\right )^2} \, dx-\frac {120 \int \frac {e^{e^{2 x}+2 x}}{\left (21+4 e^{e^{2 x}}\right )^2 \left (7+\sqrt {37}-2 x\right )} \, dx}{\sqrt {37}}-\frac {120 \int \frac {e^{e^{2 x}+2 x}}{\left (21+4 e^{e^{2 x}}\right )^2 \left (-7+\sqrt {37}+2 x\right )} \, dx}{\sqrt {37}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.81 \[ \int \frac {3528+e^{2 e^{2 x}} (168-48 x)-1008 x+e^{e^{2 x}} \left (1554-444 x+e^{2 x} \left (180-420 x+60 x^2\right )\right )}{3969-18522 x+24255 x^2-6174 x^3+441 x^4+e^{2 e^{2 x}} \left (144-672 x+880 x^2-224 x^3+16 x^4\right )+e^{e^{2 x}} \left (1512-7056 x+9240 x^2-2352 x^3+168 x^4\right )} \, dx=\frac {6 \left (4+e^{e^{2 x}}\right )}{\left (21+4 e^{e^{2 x}}\right ) \left (3-7 x+x^2\right )} \]

[In]

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 - 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]

[Out]

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

Maple [A] (verified)

Time = 0.38 (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	\(\frac {3}{2 \left (x^{2}-7 x +3\right )}-\frac {15}{2 \left (x^{2}-7 x +3\right ) \left (4 \,{\mathrm e}^{{\mathrm e}^{2 x}}+21\right )}\)	\(37\)

[In]

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)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.86 \[ \int \frac {3528+e^{2 e^{2 x}} (168-48 x)-1008 x+e^{e^{2 x}} \left (1554-444 x+e^{2 x} \left (180-420 x+60 x^2\right )\right )}{3969-18522 x+24255 x^2-6174 x^3+441 x^4+e^{2 e^{2 x}} \left (144-672 x+880 x^2-224 x^3+16 x^4\right )+e^{e^{2 x}} \left (1512-7056 x+9240 x^2-2352 x^3+168 x^4\right )} \, dx=\frac {6 \, {\left (e^{\left (e^{\left (2 \, x\right )}\right )} + 4\right )}}{21 \, x^{2} + 4 \, {\left (x^{2} - 7 \, x + 3\right )} e^{\left (e^{\left (2 \, x\right )}\right )} - 147 \, x + 63} \]

[In]

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)/((1
6*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, algorithm="fricas")

[Out]

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 = 0.14 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.93 \[ \int \frac {3528+e^{2 e^{2 x}} (168-48 x)-1008 x+e^{e^{2 x}} \left (1554-444 x+e^{2 x} \left (180-420 x+60 x^2\right )\right )}{3969-18522 x+24255 x^2-6174 x^3+441 x^4+e^{2 e^{2 x}} \left (144-672 x+880 x^2-224 x^3+16 x^4\right )+e^{e^{2 x}} \left (1512-7056 x+9240 x^2-2352 x^3+168 x^4\right )} \, dx=- \frac {15}{42 x^{2} - 294 x + \left (8 x^{2} - 56 x + 24\right ) e^{e^{2 x}} + 126} + \frac {3}{2 x^{2} - 14 x + 6} \]

[In]

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(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)

[Out]

-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)

none

Time = 0.28 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.86 \[ \int \frac {3528+e^{2 e^{2 x}} (168-48 x)-1008 x+e^{e^{2 x}} \left (1554-444 x+e^{2 x} \left (180-420 x+60 x^2\right )\right )}{3969-18522 x+24255 x^2-6174 x^3+441 x^4+e^{2 e^{2 x}} \left (144-672 x+880 x^2-224 x^3+16 x^4\right )+e^{e^{2 x}} \left (1512-7056 x+9240 x^2-2352 x^3+168 x^4\right )} \, dx=\frac {6 \, {\left (e^{\left (e^{\left (2 \, x\right )}\right )} + 4\right )}}{21 \, x^{2} + 4 \, {\left (x^{2} - 7 \, x + 3\right )} e^{\left (e^{\left (2 \, x\right )}\right )} - 147 \, x + 63} \]

[In]

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)/((1
6*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, algorithm="maxima")

[Out]

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)

none

Time = 0.30 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.10 \[ \int \frac {3528+e^{2 e^{2 x}} (168-48 x)-1008 x+e^{e^{2 x}} \left (1554-444 x+e^{2 x} \left (180-420 x+60 x^2\right )\right )}{3969-18522 x+24255 x^2-6174 x^3+441 x^4+e^{2 e^{2 x}} \left (144-672 x+880 x^2-224 x^3+16 x^4\right )+e^{e^{2 x}} \left (1512-7056 x+9240 x^2-2352 x^3+168 x^4\right )} \, dx=\frac {6 \, {\left (e^{\left (e^{\left (2 \, x\right )}\right )} + 4\right )}}{4 \, x^{2} e^{\left (e^{\left (2 \, x\right )}\right )} + 21 \, x^{2} - 28 \, x e^{\left (e^{\left (2 \, x\right )}\right )} - 147 \, x + 12 \, e^{\left (e^{\left (2 \, x\right )}\right )} + 63} \]

[In]

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)/((1
6*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, algorithm="giac")

[Out]

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 = 13.40 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.71 \[ \int \frac {3528+e^{2 e^{2 x}} (168-48 x)-1008 x+e^{e^{2 x}} \left (1554-444 x+e^{2 x} \left (180-420 x+60 x^2\right )\right )}{3969-18522 x+24255 x^2-6174 x^3+441 x^4+e^{2 e^{2 x}} \left (144-672 x+880 x^2-224 x^3+16 x^4\right )+e^{e^{2 x}} \left (1512-7056 x+9240 x^2-2352 x^3+168 x^4\right )} \, dx=\frac {6\,\left ({\mathrm {e}}^{{\mathrm {e}}^{2\,x}}+4\right )}{\left (4\,{\mathrm {e}}^{{\mathrm {e}}^{2\,x}}+21\right )\,\left (x^2-7\,x+3\right )} \]

[In]

int(-(1008*x - exp(exp(2*x))*(exp(2*x)*(60*x^2 - 420*x + 180) - 444*x + 1554) + 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)

[Out]

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