Integrand size = 116, antiderivative size = 25 \[ \int \frac {2 x+e^{e^{48+2 e^x+6 x} x} \left (2 x+e^{48+2 e^x+6 x} \left (-2 x^2-12 x^3-4 e^x x^3\right )\right )}{1+3 e^{e^{48+2 e^x+6 x} x}+3 e^{2 e^{48+2 e^x+6 x} x}+e^{3 e^{48+2 e^x+6 x} x}} \, dx=\frac {x^2}{\left (1+e^{e^{2 e^x+6 (8+x)} x}\right )^2} \]
Time = 0.19 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {2 x+e^{e^{48+2 e^x+6 x} x} \left (2 x+e^{48+2 e^x+6 x} \left (-2 x^2-12 x^3-4 e^x x^3\right )\right )}{1+3 e^{e^{48+2 e^x+6 x} x}+3 e^{2 e^{48+2 e^x+6 x} x}+e^{3 e^{48+2 e^x+6 x} x}} \, dx=\frac {x^2}{\left (1+e^{e^{48+2 e^x+6 x} x}\right )^2} \]
Integrate[(2*x + E^(E^(48 + 2*E^x + 6*x)*x)*(2*x + E^(48 + 2*E^x + 6*x)*(- 2*x^2 - 12*x^3 - 4*E^x*x^3)))/(1 + 3*E^(E^(48 + 2*E^x + 6*x)*x) + 3*E^(2*E ^(48 + 2*E^x + 6*x)*x) + E^(3*E^(48 + 2*E^x + 6*x)*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.
\(\displaystyle \int \frac {e^{e^{6 x+2 e^x+48} x} \left (e^{6 x+2 e^x+48} \left (-4 e^x x^3-12 x^3-2 x^2\right )+2 x\right )+2 x}{3 e^{e^{6 x+2 e^x+48} x}+3 e^{2 e^{6 x+2 e^x+48} x}+e^{3 e^{6 x+2 e^x+48} x}+1} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {e^{e^{6 x+2 e^x+48} x} \left (e^{6 x+2 e^x+48} \left (-4 e^x x^3-12 x^3-2 x^2\right )+2 x\right )+2 x}{\left (e^{e^{6 x+2 e^x+48} x}+1\right )^3}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {2 e^{2 \left (3 x+e^x+24\right )} x^2 \left (2 e^x x+6 x+1\right )}{\left (e^{e^{6 x+2 e^x+48} x}+1\right )^3}-\frac {2 x \left (6 e^{6 x+2 e^x+48} x^2+2 e^{7 x+2 e^x+48} x^2+e^{6 x+2 e^x+48} x-1\right )}{\left (e^{e^{6 x+2 e^x+48} x}+1\right )^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 12 \int \frac {e^{2 \left (3 x+e^x+24\right )} x^3}{\left (1+e^{e^{6 x+2 e^x+48} x}\right )^3}dx+4 \int \frac {e^{x+2 \left (3 x+e^x+24\right )} x^3}{\left (1+e^{e^{6 x+2 e^x+48} x}\right )^3}dx-12 \int \frac {e^{2 \left (3 x+e^x+24\right )} x^3}{\left (1+e^{e^{6 x+2 e^x+48} x}\right )^2}dx-4 \int \frac {e^{7 x+2 e^x+48} x^3}{\left (1+e^{e^{6 x+2 e^x+48} x}\right )^2}dx+2 \int \frac {e^{2 \left (3 x+e^x+24\right )} x^2}{\left (1+e^{e^{6 x+2 e^x+48} x}\right )^3}dx-2 \int \frac {e^{2 \left (3 x+e^x+24\right )} x^2}{\left (1+e^{e^{6 x+2 e^x+48} x}\right )^2}dx+2 \int \frac {x}{\left (1+e^{e^{6 x+2 e^x+48} x}\right )^2}dx\) |
Int[(2*x + E^(E^(48 + 2*E^x + 6*x)*x)*(2*x + E^(48 + 2*E^x + 6*x)*(-2*x^2 - 12*x^3 - 4*E^x*x^3)))/(1 + 3*E^(E^(48 + 2*E^x + 6*x)*x) + 3*E^(2*E^(48 + 2*E^x + 6*x)*x) + E^(3*E^(48 + 2*E^x + 6*x)*x)),x]
3.26.89.3.1 Defintions of rubi rules used
Time = 1.77 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88
method | result | size |
risch | \(\frac {x^{2}}{\left (1+{\mathrm e}^{x \,{\mathrm e}^{2 \,{\mathrm e}^{x}+6 x +48}}\right )^{2}}\) | \(22\) |
parallelrisch | \(\frac {x^{2}}{{\mathrm e}^{2 x \,{\mathrm e}^{2 \,{\mathrm e}^{x}+6 x +48}}+2 \,{\mathrm e}^{x \,{\mathrm e}^{2 \,{\mathrm e}^{x}+6 x +48}}+1}\) | \(39\) |
int((((-4*exp(x)*x^3-12*x^3-2*x^2)*exp(exp(x)+3*x+24)^2+2*x)*exp(x*exp(exp (x)+3*x+24)^2)+2*x)/(exp(x*exp(exp(x)+3*x+24)^2)^3+3*exp(x*exp(exp(x)+3*x+ 24)^2)^2+3*exp(x*exp(exp(x)+3*x+24)^2)+1),x,method=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.48 \[ \int \frac {2 x+e^{e^{48+2 e^x+6 x} x} \left (2 x+e^{48+2 e^x+6 x} \left (-2 x^2-12 x^3-4 e^x x^3\right )\right )}{1+3 e^{e^{48+2 e^x+6 x} x}+3 e^{2 e^{48+2 e^x+6 x} x}+e^{3 e^{48+2 e^x+6 x} x}} \, dx=\frac {x^{2}}{e^{\left (2 \, x e^{\left (6 \, x + 2 \, e^{x} + 48\right )}\right )} + 2 \, e^{\left (x e^{\left (6 \, x + 2 \, e^{x} + 48\right )}\right )} + 1} \]
integrate((((-4*exp(x)*x^3-12*x^3-2*x^2)*exp(exp(x)+3*x+24)^2+2*x)*exp(x*e xp(exp(x)+3*x+24)^2)+2*x)/(exp(x*exp(exp(x)+3*x+24)^2)^3+3*exp(x*exp(exp(x )+3*x+24)^2)^2+3*exp(x*exp(exp(x)+3*x+24)^2)+1),x, algorithm=\
Time = 0.20 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.48 \[ \int \frac {2 x+e^{e^{48+2 e^x+6 x} x} \left (2 x+e^{48+2 e^x+6 x} \left (-2 x^2-12 x^3-4 e^x x^3\right )\right )}{1+3 e^{e^{48+2 e^x+6 x} x}+3 e^{2 e^{48+2 e^x+6 x} x}+e^{3 e^{48+2 e^x+6 x} x}} \, dx=\frac {x^{2}}{e^{2 x e^{6 x + 2 e^{x} + 48}} + 2 e^{x e^{6 x + 2 e^{x} + 48}} + 1} \]
integrate((((-4*exp(x)*x**3-12*x**3-2*x**2)*exp(exp(x)+3*x+24)**2+2*x)*exp (x*exp(exp(x)+3*x+24)**2)+2*x)/(exp(x*exp(exp(x)+3*x+24)**2)**3+3*exp(x*ex p(exp(x)+3*x+24)**2)**2+3*exp(x*exp(exp(x)+3*x+24)**2)+1),x)
Time = 0.25 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.48 \[ \int \frac {2 x+e^{e^{48+2 e^x+6 x} x} \left (2 x+e^{48+2 e^x+6 x} \left (-2 x^2-12 x^3-4 e^x x^3\right )\right )}{1+3 e^{e^{48+2 e^x+6 x} x}+3 e^{2 e^{48+2 e^x+6 x} x}+e^{3 e^{48+2 e^x+6 x} x}} \, dx=\frac {x^{2}}{e^{\left (2 \, x e^{\left (6 \, x + 2 \, e^{x} + 48\right )}\right )} + 2 \, e^{\left (x e^{\left (6 \, x + 2 \, e^{x} + 48\right )}\right )} + 1} \]
integrate((((-4*exp(x)*x^3-12*x^3-2*x^2)*exp(exp(x)+3*x+24)^2+2*x)*exp(x*e xp(exp(x)+3*x+24)^2)+2*x)/(exp(x*exp(exp(x)+3*x+24)^2)^3+3*exp(x*exp(exp(x )+3*x+24)^2)^2+3*exp(x*exp(exp(x)+3*x+24)^2)+1),x, algorithm=\
Time = 0.33 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.48 \[ \int \frac {2 x+e^{e^{48+2 e^x+6 x} x} \left (2 x+e^{48+2 e^x+6 x} \left (-2 x^2-12 x^3-4 e^x x^3\right )\right )}{1+3 e^{e^{48+2 e^x+6 x} x}+3 e^{2 e^{48+2 e^x+6 x} x}+e^{3 e^{48+2 e^x+6 x} x}} \, dx=\frac {x^{2}}{e^{\left (2 \, x e^{\left (6 \, x + 2 \, e^{x} + 48\right )}\right )} + 2 \, e^{\left (x e^{\left (6 \, x + 2 \, e^{x} + 48\right )}\right )} + 1} \]
integrate((((-4*exp(x)*x^3-12*x^3-2*x^2)*exp(exp(x)+3*x+24)^2+2*x)*exp(x*e xp(exp(x)+3*x+24)^2)+2*x)/(exp(x*exp(exp(x)+3*x+24)^2)^3+3*exp(x*exp(exp(x )+3*x+24)^2)^2+3*exp(x*exp(exp(x)+3*x+24)^2)+1),x, algorithm=\
Time = 0.19 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.48 \[ \int \frac {2 x+e^{e^{48+2 e^x+6 x} x} \left (2 x+e^{48+2 e^x+6 x} \left (-2 x^2-12 x^3-4 e^x x^3\right )\right )}{1+3 e^{e^{48+2 e^x+6 x} x}+3 e^{2 e^{48+2 e^x+6 x} x}+e^{3 e^{48+2 e^x+6 x} x}} \, dx=\frac {x^2\,{\mathrm {e}}^{-x\,{\mathrm {e}}^{6\,x}\,{\mathrm {e}}^{48}\,{\mathrm {e}}^{2\,{\mathrm {e}}^x}}}{4\,{\mathrm {cosh}\left (\frac {x\,{\mathrm {e}}^{6\,x}\,{\mathrm {e}}^{48}\,{\mathrm {e}}^{2\,{\mathrm {e}}^x}}{2}\right )}^2} \]
int((2*x + exp(x*exp(6*x + 2*exp(x) + 48))*(2*x - exp(6*x + 2*exp(x) + 48) *(4*x^3*exp(x) + 2*x^2 + 12*x^3)))/(3*exp(x*exp(6*x + 2*exp(x) + 48)) + 3* exp(2*x*exp(6*x + 2*exp(x) + 48)) + exp(3*x*exp(6*x + 2*exp(x) + 48)) + 1) ,x)