Integrand size = 93, antiderivative size = 29 \begin {dmath*} \int \frac {36 e^{2 e^x} x^2+e^{e^x} \left (-3 e^{2 x} x^2+e^x \left (-6 x+3 x^2\right )\right )}{4 e^{2 x}+e^{e^x+x} \left (-96 x+4 x^2\right )+e^{2 e^x} \left (576 x^2-48 x^3+x^4\right )} \, dx=\frac {x}{16-x+\frac {1}{3} \left (-\frac {4 e^{-e^x+x}}{x}+x\right )} \end {dmath*}
Time = 0.95 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.10 \begin {dmath*} \int \frac {36 e^{2 e^x} x^2+e^{e^x} \left (-3 e^{2 x} x^2+e^x \left (-6 x+3 x^2\right )\right )}{4 e^{2 x}+e^{e^x+x} \left (-96 x+4 x^2\right )+e^{2 e^x} \left (576 x^2-48 x^3+x^4\right )} \, dx=\frac {3 \left (e^x-12 e^{e^x} x\right )}{2 e^x+e^{e^x} (-24+x) x} \end {dmath*}
Integrate[(36*E^(2*E^x)*x^2 + E^E^x*(-3*E^(2*x)*x^2 + E^x*(-6*x + 3*x^2))) /(4*E^(2*x) + E^(E^x + x)*(-96*x + 4*x^2) + E^(2*E^x)*(576*x^2 - 48*x^3 + x^4)),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.
\(\displaystyle \int \frac {36 e^{2 e^x} x^2+e^{e^x} \left (e^x \left (3 x^2-6 x\right )-3 e^{2 x} x^2\right )}{e^{x+e^x} \left (4 x^2-96 x\right )+e^{2 e^x} \left (x^4-48 x^3+576 x^2\right )+4 e^{2 x}} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {3 e^{e^x} x \left (e^x (x-2)+12 e^{e^x} x-e^{2 x} x\right )}{\left (e^{e^x} (x-24) x+2 e^x\right )^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 3 \int -\frac {e^{e^x} x \left (e^x (2-x)-12 e^{e^x} x+e^{2 x} x\right )}{\left (2 e^x-e^{e^x} (24-x) x\right )^2}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -3 \int \frac {e^{e^x} x \left (e^x (2-x)-12 e^{e^x} x+e^{2 x} x\right )}{\left (2 e^x-e^{e^x} (24-x) x\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -3 \int \left (\frac {1}{4} e^{e^x} x^2+\frac {e^{2 e^x} \left (e^{e^x} x^4-48 e^{e^x} x^3+576 e^{e^x} x^2+2 x^2-52 x+48\right ) x^2}{4 \left (e^{e^x} x^2-24 e^{e^x} x+2 e^x\right )^2}-\frac {e^{e^x} \left (e^{e^x} x^3-24 e^{e^x} x^2+x-2\right ) x}{2 \left (e^{e^x} x^2-24 e^{e^x} x+2 e^x\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -3 \left (\frac {1}{4} \int e^{e^x} x^2dx+12 \int \frac {e^{2 e^x} x^2}{\left (e^{e^x} x^2-24 e^{e^x} x+2 e^x\right )^2}dx+\int \frac {e^{e^x} x}{e^{e^x} x^2-24 e^{e^x} x+2 e^x}dx-\frac {1}{2} \int \frac {e^{e^x} x^2}{e^{e^x} x^2-24 e^{e^x} x+2 e^x}dx+\frac {1}{4} \int \frac {e^{3 e^x} x^6}{\left (e^{e^x} x^2-24 e^{e^x} x+2 e^x\right )^2}dx-12 \int \frac {e^{3 e^x} x^5}{\left (e^{e^x} x^2-24 e^{e^x} x+2 e^x\right )^2}dx+\frac {1}{2} \int \frac {e^{2 e^x} x^4}{\left (e^{e^x} x^2-24 e^{e^x} x+2 e^x\right )^2}dx+144 \int \frac {e^{3 e^x} x^4}{\left (e^{e^x} x^2-24 e^{e^x} x+2 e^x\right )^2}dx-\frac {1}{2} \int \frac {e^{2 e^x} x^4}{e^{e^x} x^2-24 e^{e^x} x+2 e^x}dx-13 \int \frac {e^{2 e^x} x^3}{\left (e^{e^x} x^2-24 e^{e^x} x+2 e^x\right )^2}dx+12 \int \frac {e^{2 e^x} x^3}{e^{e^x} x^2-24 e^{e^x} x+2 e^x}dx\right )\) |
Int[(36*E^(2*E^x)*x^2 + E^E^x*(-3*E^(2*x)*x^2 + E^x*(-6*x + 3*x^2)))/(4*E^ (2*x) + E^(E^x + x)*(-96*x + 4*x^2) + E^(2*E^x)*(576*x^2 - 48*x^3 + x^4)), x]
3.11.93.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 0.23 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.14
method | result | size |
parallelrisch | \(\frac {-36 x \,{\mathrm e}^{{\mathrm e}^{x}}+3 \,{\mathrm e}^{x}}{{\mathrm e}^{{\mathrm e}^{x}} x^{2}-24 x \,{\mathrm e}^{{\mathrm e}^{x}}+2 \,{\mathrm e}^{x}}\) | \(33\) |
risch | \(-\frac {36}{x -24}+\frac {3 x \,{\mathrm e}^{x}}{\left (x -24\right ) \left ({\mathrm e}^{{\mathrm e}^{x}} x^{2}-24 x \,{\mathrm e}^{{\mathrm e}^{x}}+2 \,{\mathrm e}^{x}\right )}\) | \(39\) |
int((36*x^2*exp(exp(x))^2+(-3*exp(x)^2*x^2+(3*x^2-6*x)*exp(x))*exp(exp(x)) )/((x^4-48*x^3+576*x^2)*exp(exp(x))^2+(4*x^2-96*x)*exp(x)*exp(exp(x))+4*ex p(x)^2),x,method=_RETURNVERBOSE)
Time = 0.27 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.34 \begin {dmath*} \int \frac {36 e^{2 e^x} x^2+e^{e^x} \left (-3 e^{2 x} x^2+e^x \left (-6 x+3 x^2\right )\right )}{4 e^{2 x}+e^{e^x+x} \left (-96 x+4 x^2\right )+e^{2 e^x} \left (576 x^2-48 x^3+x^4\right )} \, dx=-\frac {3 \, {\left (12 \, x e^{\left (x + e^{x}\right )} - e^{\left (2 \, x\right )}\right )}}{{\left (x^{2} - 24 \, x\right )} e^{\left (x + e^{x}\right )} + 2 \, e^{\left (2 \, x\right )}} \end {dmath*}
integrate((36*x^2*exp(exp(x))^2+(-3*exp(x)^2*x^2+(3*x^2-6*x)*exp(x))*exp(e xp(x)))/((x^4-48*x^3+576*x^2)*exp(exp(x))^2+(4*x^2-96*x)*exp(x)*exp(exp(x) )+4*exp(x)^2),x, algorithm=\
Time = 0.17 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.34 \begin {dmath*} \int \frac {36 e^{2 e^x} x^2+e^{e^x} \left (-3 e^{2 x} x^2+e^x \left (-6 x+3 x^2\right )\right )}{4 e^{2 x}+e^{e^x+x} \left (-96 x+4 x^2\right )+e^{2 e^x} \left (576 x^2-48 x^3+x^4\right )} \, dx=\frac {3 x e^{x}}{2 x e^{x} + \left (x^{3} - 48 x^{2} + 576 x\right ) e^{e^{x}} - 48 e^{x}} - \frac {36}{x - 24} \end {dmath*}
integrate((36*x**2*exp(exp(x))**2+(-3*exp(x)**2*x**2+(3*x**2-6*x)*exp(x))* exp(exp(x)))/((x**4-48*x**3+576*x**2)*exp(exp(x))**2+(4*x**2-96*x)*exp(x)* exp(exp(x))+4*exp(x)**2),x)
Time = 0.25 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \begin {dmath*} \int \frac {36 e^{2 e^x} x^2+e^{e^x} \left (-3 e^{2 x} x^2+e^x \left (-6 x+3 x^2\right )\right )}{4 e^{2 x}+e^{e^x+x} \left (-96 x+4 x^2\right )+e^{2 e^x} \left (576 x^2-48 x^3+x^4\right )} \, dx=-\frac {3 \, {\left (12 \, x e^{\left (e^{x}\right )} - e^{x}\right )}}{{\left (x^{2} - 24 \, x\right )} e^{\left (e^{x}\right )} + 2 \, e^{x}} \end {dmath*}
integrate((36*x^2*exp(exp(x))^2+(-3*exp(x)^2*x^2+(3*x^2-6*x)*exp(x))*exp(e xp(x)))/((x^4-48*x^3+576*x^2)*exp(exp(x))^2+(4*x^2-96*x)*exp(x)*exp(exp(x) )+4*exp(x)^2),x, algorithm=\
Time = 0.30 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.14 \begin {dmath*} \int \frac {36 e^{2 e^x} x^2+e^{e^x} \left (-3 e^{2 x} x^2+e^x \left (-6 x+3 x^2\right )\right )}{4 e^{2 x}+e^{e^x+x} \left (-96 x+4 x^2\right )+e^{2 e^x} \left (576 x^2-48 x^3+x^4\right )} \, dx=-\frac {3 \, {\left (12 \, x e^{\left (e^{x}\right )} - e^{x}\right )}}{x^{2} e^{\left (e^{x}\right )} - 24 \, x e^{\left (e^{x}\right )} + 2 \, e^{x}} \end {dmath*}
integrate((36*x^2*exp(exp(x))^2+(-3*exp(x)^2*x^2+(3*x^2-6*x)*exp(x))*exp(e xp(x)))/((x^4-48*x^3+576*x^2)*exp(exp(x))^2+(4*x^2-96*x)*exp(x)*exp(exp(x) )+4*exp(x)^2),x, algorithm=\
Timed out. \begin {dmath*} \int \frac {36 e^{2 e^x} x^2+e^{e^x} \left (-3 e^{2 x} x^2+e^x \left (-6 x+3 x^2\right )\right )}{4 e^{2 x}+e^{e^x+x} \left (-96 x+4 x^2\right )+e^{2 e^x} \left (576 x^2-48 x^3+x^4\right )} \, dx=-\int \frac {{\mathrm {e}}^{{\mathrm {e}}^x}\,\left (3\,x^2\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^x\,\left (6\,x-3\,x^2\right )\right )-36\,x^2\,{\mathrm {e}}^{2\,{\mathrm {e}}^x}}{4\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{2\,{\mathrm {e}}^x}\,\left (x^4-48\,x^3+576\,x^2\right )-{\mathrm {e}}^{x+{\mathrm {e}}^x}\,\left (96\,x-4\,x^2\right )} \,d x \end {dmath*}
int(-(exp(exp(x))*(3*x^2*exp(2*x) + exp(x)*(6*x - 3*x^2)) - 36*x^2*exp(2*e xp(x)))/(4*exp(2*x) + exp(2*exp(x))*(576*x^2 - 48*x^3 + x^4) - exp(exp(x)) *exp(x)*(96*x - 4*x^2)),x)