Integrand size = 105, antiderivative size = 26 \[ \int \frac {e^{\frac {2}{100 x+10 e^{1+e^{e^x}} x}} \left (-10-1500 x-15 e^{2+2 e^{e^x}} x+e^{1+e^{e^x}} \left (-1-300 x-e^{e^x+x} x\right )\right )}{500 x^5+100 e^{1+e^{e^x}} x^5+5 e^{2+2 e^{e^x}} x^5} \, dx=\frac {e^{\frac {1}{5 \left (10+e^{1+e^{e^x}}\right ) x}}}{x^3} \]
Time = 0.33 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\frac {2}{100 x+10 e^{1+e^{e^x}} x}} \left (-10-1500 x-15 e^{2+2 e^{e^x}} x+e^{1+e^{e^x}} \left (-1-300 x-e^{e^x+x} x\right )\right )}{500 x^5+100 e^{1+e^{e^x}} x^5+5 e^{2+2 e^{e^x}} x^5} \, dx=\frac {e^{\frac {1}{5 \left (10+e^{1+e^{e^x}}\right ) x}}}{x^3} \]
Integrate[(E^(2/(100*x + 10*E^(1 + E^E^x)*x))*(-10 - 1500*x - 15*E^(2 + 2* E^E^x)*x + E^(1 + E^E^x)*(-1 - 300*x - E^(E^x + x)*x)))/(500*x^5 + 100*E^( 1 + E^E^x)*x^5 + 5*E^(2 + 2*E^E^x)*x^5),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.
\(\displaystyle \int \frac {e^{\frac {2}{10 e^{e^{e^x}+1} x+100 x}} \left (-15 e^{2 e^{e^x}+2} x-1500 x+e^{e^{e^x}+1} \left (-e^{x+e^x} x-300 x-1\right )-10\right )}{100 e^{e^{e^x}+1} x^5+5 e^{2 e^{e^x}+2} x^5+500 x^5} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {e^{\frac {1}{5 \left (e^{e^{e^x}+1}+10\right ) x}} \left (-15 e^{2 e^{e^x}+2} x-1500 x+e^{e^{e^x}+1} \left (-e^{x+e^x} x-300 x-1\right )-10\right )}{5 \left (e^{e^{e^x}+1}+10\right )^2 x^5}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{5} \int -\frac {e^{\frac {1}{5 \left (10+e^{1+e^{e^x}}\right ) x}} \left (15 e^{2+2 e^{e^x}} x+1500 x+e^{1+e^{e^x}} \left (e^{x+e^x} x+300 x+1\right )+10\right )}{\left (10+e^{1+e^{e^x}}\right )^2 x^5}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {1}{5} \int \frac {e^{\frac {1}{5 \left (10+e^{1+e^{e^x}}\right ) x}} \left (15 e^{2+2 e^{e^x}} x+1500 x+e^{1+e^{e^x}} \left (e^{x+e^x} x+300 x+1\right )+10\right )}{\left (10+e^{1+e^{e^x}}\right )^2 x^5}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {1}{5} \int \left (\frac {300 e^{e^{e^x}+1+\frac {1}{5 \left (10+e^{1+e^{e^x}}\right ) x}}}{x^4 \left (10+e^{1+e^{e^x}}\right )^2}+\frac {15 e^{2 \left (1+e^{e^x}\right )+\frac {1}{5 \left (10+e^{1+e^{e^x}}\right ) x}}}{x^4 \left (10+e^{1+e^{e^x}}\right )^2}+\frac {1500 e^{\frac {1}{5 \left (10+e^{1+e^{e^x}}\right ) x}}}{x^4 \left (10+e^{1+e^{e^x}}\right )^2}+\frac {e^{x+e^{e^x}+e^x+1+\frac {1}{5 \left (10+e^{1+e^{e^x}}\right ) x}}}{x^4 \left (10+e^{1+e^{e^x}}\right )^2}+\frac {e^{e^{e^x}+1+\frac {1}{5 \left (10+e^{1+e^{e^x}}\right ) x}}}{x^5 \left (10+e^{1+e^{e^x}}\right )^2}+\frac {10 e^{\frac {1}{5 \left (10+e^{1+e^{e^x}}\right ) x}}}{x^5 \left (10+e^{1+e^{e^x}}\right )^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{5} \left (-\int \frac {e^{e^{e^x}+1+\frac {1}{5 \left (10+e^{1+e^{e^x}}\right ) x}}}{\left (10+e^{1+e^{e^x}}\right )^2 x^5}dx-10 \int \frac {e^{\frac {1}{5 \left (10+e^{1+e^{e^x}}\right ) x}}}{\left (10+e^{1+e^{e^x}}\right )^2 x^5}dx-300 \int \frac {e^{e^{e^x}+1+\frac {1}{5 \left (10+e^{1+e^{e^x}}\right ) x}}}{\left (10+e^{1+e^{e^x}}\right )^2 x^4}dx-15 \int \frac {e^{2 \left (1+e^{e^x}\right )+\frac {1}{5 \left (10+e^{1+e^{e^x}}\right ) x}}}{\left (10+e^{1+e^{e^x}}\right )^2 x^4}dx-1500 \int \frac {e^{\frac {1}{5 \left (10+e^{1+e^{e^x}}\right ) x}}}{\left (10+e^{1+e^{e^x}}\right )^2 x^4}dx-\int \frac {e^{x+e^{e^x}+e^x+1+\frac {1}{5 \left (10+e^{1+e^{e^x}}\right ) x}}}{\left (10+e^{1+e^{e^x}}\right )^2 x^4}dx\right )\) |
Int[(E^(2/(100*x + 10*E^(1 + E^E^x)*x))*(-10 - 1500*x - 15*E^(2 + 2*E^E^x) *x + E^(1 + E^E^x)*(-1 - 300*x - E^(E^x + x)*x)))/(500*x^5 + 100*E^(1 + E^ E^x)*x^5 + 5*E^(2 + 2*E^E^x)*x^5),x]
3.16.5.3.1 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]]
Time = 149.52 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.81
method | result | size |
risch | \(\frac {{\mathrm e}^{\frac {1}{5 \left ({\mathrm e}^{{\mathrm e}^{{\mathrm e}^{x}}+1}+10\right ) x}}}{x^{3}}\) | \(21\) |
parallelrisch | \(\frac {{\mathrm e}^{\frac {1}{5 \left ({\mathrm e}^{{\mathrm e}^{{\mathrm e}^{x}}+1}+10\right ) x}}}{x^{3}}\) | \(23\) |
int((-15*x*exp(exp(exp(x))+1)^2+(-x*exp(x)*exp(exp(x))-300*x-1)*exp(exp(ex p(x))+1)-1500*x-10)*exp(1/(10*x*exp(exp(exp(x))+1)+100*x))^2/(5*x^5*exp(ex p(exp(x))+1)^2+100*x^5*exp(exp(exp(x))+1)+500*x^5),x,method=_RETURNVERBOSE )
Time = 0.28 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12 \[ \int \frac {e^{\frac {2}{100 x+10 e^{1+e^{e^x}} x}} \left (-10-1500 x-15 e^{2+2 e^{e^x}} x+e^{1+e^{e^x}} \left (-1-300 x-e^{e^x+x} x\right )\right )}{500 x^5+100 e^{1+e^{e^x}} x^5+5 e^{2+2 e^{e^x}} x^5} \, dx=\frac {e^{\left (\frac {1}{5 \, {\left (x e^{\left ({\left (e^{\left (x + e^{x}\right )} + e^{x}\right )} e^{\left (-x\right )}\right )} + 10 \, x\right )}}\right )}}{x^{3}} \]
integrate((-15*x*exp(exp(exp(x))+1)^2+(-x*exp(x)*exp(exp(x))-300*x-1)*exp( exp(exp(x))+1)-1500*x-10)*exp(1/(10*x*exp(exp(exp(x))+1)+100*x))^2/(5*x^5* exp(exp(exp(x))+1)^2+100*x^5*exp(exp(exp(x))+1)+500*x^5),x, algorithm=\
Time = 0.88 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.77 \[ \int \frac {e^{\frac {2}{100 x+10 e^{1+e^{e^x}} x}} \left (-10-1500 x-15 e^{2+2 e^{e^x}} x+e^{1+e^{e^x}} \left (-1-300 x-e^{e^x+x} x\right )\right )}{500 x^5+100 e^{1+e^{e^x}} x^5+5 e^{2+2 e^{e^x}} x^5} \, dx=\frac {e^{\frac {2}{10 x e^{e^{e^{x}} + 1} + 100 x}}}{x^{3}} \]
integrate((-15*x*exp(exp(exp(x))+1)**2+(-x*exp(x)*exp(exp(x))-300*x-1)*exp (exp(exp(x))+1)-1500*x-10)*exp(1/(10*x*exp(exp(exp(x))+1)+100*x))**2/(5*x* *5*exp(exp(exp(x))+1)**2+100*x**5*exp(exp(exp(x))+1)+500*x**5),x)
\[ \int \frac {e^{\frac {2}{100 x+10 e^{1+e^{e^x}} x}} \left (-10-1500 x-15 e^{2+2 e^{e^x}} x+e^{1+e^{e^x}} \left (-1-300 x-e^{e^x+x} x\right )\right )}{500 x^5+100 e^{1+e^{e^x}} x^5+5 e^{2+2 e^{e^x}} x^5} \, dx=\int { -\frac {{\left (15 \, x e^{\left (2 \, e^{\left (e^{x}\right )} + 2\right )} + {\left (x e^{\left (x + e^{x}\right )} + 300 \, x + 1\right )} e^{\left (e^{\left (e^{x}\right )} + 1\right )} + 1500 \, x + 10\right )} e^{\left (\frac {1}{5 \, {\left (x e^{\left (e^{\left (e^{x}\right )} + 1\right )} + 10 \, x\right )}}\right )}}{5 \, {\left (x^{5} e^{\left (2 \, e^{\left (e^{x}\right )} + 2\right )} + 20 \, x^{5} e^{\left (e^{\left (e^{x}\right )} + 1\right )} + 100 \, x^{5}\right )}} \,d x } \]
integrate((-15*x*exp(exp(exp(x))+1)^2+(-x*exp(x)*exp(exp(x))-300*x-1)*exp( exp(exp(x))+1)-1500*x-10)*exp(1/(10*x*exp(exp(exp(x))+1)+100*x))^2/(5*x^5* exp(exp(exp(x))+1)^2+100*x^5*exp(exp(exp(x))+1)+500*x^5),x, algorithm=\
-1/5*integrate((15*x*e^(2*e^(e^x) + 2) + (x*e^(x + e^x) + 300*x + 1)*e^(e^ (e^x) + 1) + 1500*x + 10)*e^(1/5/(x*e^(e^(e^x) + 1) + 10*x))/(x^5*e^(2*e^( e^x) + 2) + 20*x^5*e^(e^(e^x) + 1) + 100*x^5), x)
Exception generated. \[ \int \frac {e^{\frac {2}{100 x+10 e^{1+e^{e^x}} x}} \left (-10-1500 x-15 e^{2+2 e^{e^x}} x+e^{1+e^{e^x}} \left (-1-300 x-e^{e^x+x} x\right )\right )}{500 x^5+100 e^{1+e^{e^x}} x^5+5 e^{2+2 e^{e^x}} x^5} \, dx=\text {Exception raised: TypeError} \]
integrate((-15*x*exp(exp(exp(x))+1)^2+(-x*exp(x)*exp(exp(x))-300*x-1)*exp( exp(exp(x))+1)-1500*x-10)*exp(1/(10*x*exp(exp(exp(x))+1)+100*x))^2/(5*x^5* exp(exp(exp(x))+1)^2+100*x^5*exp(exp(exp(x))+1)+500*x^5),x, algorithm=\
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro unding error%%%{4687500000000,[0,8,6,6,15,8]%%%}+%%%{31250000000,[0,8,6,6, 14,8]%%%}
Time = 8.69 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.77 \[ \int \frac {e^{\frac {2}{100 x+10 e^{1+e^{e^x}} x}} \left (-10-1500 x-15 e^{2+2 e^{e^x}} x+e^{1+e^{e^x}} \left (-1-300 x-e^{e^x+x} x\right )\right )}{500 x^5+100 e^{1+e^{e^x}} x^5+5 e^{2+2 e^{e^x}} x^5} \, dx=\frac {{\mathrm {e}}^{\frac {1}{50\,x+5\,x\,\mathrm {e}\,{\mathrm {e}}^{{\mathrm {e}}^{{\mathrm {e}}^x}}}}}{x^3} \]