Integrand size = 111, antiderivative size = 24 \[ \int \frac {3 e^{e^2} x^2+e^{2 e^{-e^2} x} \left (e^{e^2} x^3-4 x^4\right )}{25 e^{e^2}+75 e^{e^2+2 e^{-e^2} x} x+75 e^{e^2+4 e^{-e^2} x} x^2+25 e^{e^2+6 e^{-e^2} x} x^3} \, dx=\frac {x}{\left (5 e^{2 e^{-e^2} x}+\frac {5}{x}\right )^2} \] Output:
x/(5/x+5*exp(2*x/exp(exp(2))))^2
Time = 1.69 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.04 \[ \int \frac {3 e^{e^2} x^2+e^{2 e^{-e^2} x} \left (e^{e^2} x^3-4 x^4\right )}{25 e^{e^2}+75 e^{e^2+2 e^{-e^2} x} x+75 e^{e^2+4 e^{-e^2} x} x^2+25 e^{e^2+6 e^{-e^2} x} x^3} \, dx=\frac {x^3}{25 \left (1+e^{2 e^{-e^2} x} x\right )^2} \] Input:
Integrate[(3*E^E^2*x^2 + E^((2*x)/E^E^2)*(E^E^2*x^3 - 4*x^4))/(25*E^E^2 + 75*E^(E^2 + (2*x)/E^E^2)*x + 75*E^(E^2 + (4*x)/E^E^2)*x^2 + 25*E^(E^2 + (6 *x)/E^E^2)*x^3),x]
Output:
x^3/(25*(1 + E^((2*x)/E^E^2)*x)^2)
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.
\(\displaystyle \int \frac {3 e^{e^2} x^2+e^{2 e^{-e^2} x} \left (e^{e^2} x^3-4 x^4\right )}{25 e^{6 e^{-e^2} x+e^2} x^3+75 e^{4 e^{-e^2} x+e^2} x^2+75 e^{2 e^{-e^2} x+e^2} x+25 e^{e^2}} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {e^{-e^2} \left (3 e^{e^2} x^2+e^{2 e^{-e^2} x} \left (e^{e^2} x^3-4 x^4\right )\right )}{25 \left (e^{2 e^{-e^2} x} x+1\right )^3}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{25} e^{-e^2} \int \frac {3 e^{e^2} x^2+e^{2 e^{-e^2} x} \left (e^{e^2} x^3-4 x^4\right )}{\left (e^{2 e^{-e^2} x} x+1\right )^3}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \frac {1}{25} e^{-e^2} \int \left (\frac {\left (e^{e^2}-4 x\right ) x^2}{\left (e^{2 e^{-e^2} x} x+1\right )^2}+\frac {2 \left (2 x+e^{e^2}\right ) x^2}{\left (e^{2 e^{-e^2} x} x+1\right )^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{25} e^{-e^2} \left (4 \int \frac {x^3}{\left (e^{2 e^{-e^2} x} x+1\right )^3}dx-4 \int \frac {x^3}{\left (e^{2 e^{-e^2} x} x+1\right )^2}dx+2 e^{e^2} \int \frac {x^2}{\left (e^{2 e^{-e^2} x} x+1\right )^3}dx+e^{e^2} \int \frac {x^2}{\left (e^{2 e^{-e^2} x} x+1\right )^2}dx\right )\) |
Input:
Int[(3*E^E^2*x^2 + E^((2*x)/E^E^2)*(E^E^2*x^3 - 4*x^4))/(25*E^E^2 + 75*E^( E^2 + (2*x)/E^E^2)*x + 75*E^(E^2 + (4*x)/E^E^2)*x^2 + 25*E^(E^2 + (6*x)/E^ E^2)*x^3),x]
Output:
$Aborted
Time = 1.66 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.88
method | result | size |
norman | \(\frac {x^{3}}{25 \left (x \,{\mathrm e}^{2 x \,{\mathrm e}^{-{\mathrm e}^{2}}}+1\right )^{2}}\) | \(21\) |
risch | \(\frac {x^{3}}{25 \left (x \,{\mathrm e}^{2 x \,{\mathrm e}^{-{\mathrm e}^{2}}}+1\right )^{2}}\) | \(21\) |
parallelrisch | \(\frac {x^{3}}{25 x^{2} {\mathrm e}^{4 x \,{\mathrm e}^{-{\mathrm e}^{2}}}+50 x \,{\mathrm e}^{2 x \,{\mathrm e}^{-{\mathrm e}^{2}}}+25}\) | \(37\) |
Input:
int(((x^3*exp(exp(2))-4*x^4)*exp(2*x/exp(exp(2)))+3*x^2*exp(exp(2)))/(25*x ^3*exp(exp(2))*exp(2*x/exp(exp(2)))^3+75*x^2*exp(exp(2))*exp(2*x/exp(exp(2 )))^2+75*x*exp(exp(2))*exp(2*x/exp(exp(2)))+25*exp(exp(2))),x,method=_RETU RNVERBOSE)
Output:
1/25*x^3/(x*exp(2*x/exp(exp(2)))+1)^2
Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (18) = 36\).
Time = 0.09 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.54 \[ \int \frac {3 e^{e^2} x^2+e^{2 e^{-e^2} x} \left (e^{e^2} x^3-4 x^4\right )}{25 e^{e^2}+75 e^{e^2+2 e^{-e^2} x} x+75 e^{e^2+4 e^{-e^2} x} x^2+25 e^{e^2+6 e^{-e^2} x} x^3} \, dx=\frac {x^{3} e^{\left (2 \, e^{2}\right )}}{25 \, {\left (x^{2} e^{\left (2 \, {\left (2 \, x + e^{\left (e^{2} + 2\right )}\right )} e^{\left (-e^{2}\right )}\right )} + 2 \, x e^{\left ({\left (2 \, x + e^{\left (e^{2} + 2\right )}\right )} e^{\left (-e^{2}\right )} + e^{2}\right )} + e^{\left (2 \, e^{2}\right )}\right )}} \] Input:
integrate(((x^3*exp(exp(2))-4*x^4)*exp(2*x/exp(exp(2)))+3*x^2*exp(exp(2))) /(25*x^3*exp(exp(2))*exp(2*x/exp(exp(2)))^3+75*x^2*exp(exp(2))*exp(2*x/exp (exp(2)))^2+75*x*exp(exp(2))*exp(2*x/exp(exp(2)))+25*exp(exp(2))),x, algor ithm="fricas")
Output:
1/25*x^3*e^(2*e^2)/(x^2*e^(2*(2*x + e^(e^2 + 2))*e^(-e^2)) + 2*x*e^((2*x + e^(e^2 + 2))*e^(-e^2) + e^2) + e^(2*e^2))
Time = 0.08 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.33 \[ \int \frac {3 e^{e^2} x^2+e^{2 e^{-e^2} x} \left (e^{e^2} x^3-4 x^4\right )}{25 e^{e^2}+75 e^{e^2+2 e^{-e^2} x} x+75 e^{e^2+4 e^{-e^2} x} x^2+25 e^{e^2+6 e^{-e^2} x} x^3} \, dx=\frac {x^{3}}{25 x^{2} e^{\frac {4 x}{e^{e^{2}}}} + 50 x e^{\frac {2 x}{e^{e^{2}}}} + 25} \] Input:
integrate(((x**3*exp(exp(2))-4*x**4)*exp(2*x/exp(exp(2)))+3*x**2*exp(exp(2 )))/(25*x**3*exp(exp(2))*exp(2*x/exp(exp(2)))**3+75*x**2*exp(exp(2))*exp(2 *x/exp(exp(2)))**2+75*x*exp(exp(2))*exp(2*x/exp(exp(2)))+25*exp(exp(2))),x )
Output:
x**3/(25*x**2*exp(4*x*exp(-exp(2))) + 50*x*exp(2*x*exp(-exp(2))) + 25)
Time = 0.08 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.42 \[ \int \frac {3 e^{e^2} x^2+e^{2 e^{-e^2} x} \left (e^{e^2} x^3-4 x^4\right )}{25 e^{e^2}+75 e^{e^2+2 e^{-e^2} x} x+75 e^{e^2+4 e^{-e^2} x} x^2+25 e^{e^2+6 e^{-e^2} x} x^3} \, dx=\frac {x^{3}}{25 \, {\left (x^{2} e^{\left (4 \, x e^{\left (-e^{2}\right )}\right )} + 2 \, x e^{\left (2 \, x e^{\left (-e^{2}\right )}\right )} + 1\right )}} \] Input:
integrate(((x^3*exp(exp(2))-4*x^4)*exp(2*x/exp(exp(2)))+3*x^2*exp(exp(2))) /(25*x^3*exp(exp(2))*exp(2*x/exp(exp(2)))^3+75*x^2*exp(exp(2))*exp(2*x/exp (exp(2)))^2+75*x*exp(exp(2))*exp(2*x/exp(exp(2)))+25*exp(exp(2))),x, algor ithm="maxima")
Output:
1/25*x^3/(x^2*e^(4*x*e^(-e^2)) + 2*x*e^(2*x*e^(-e^2)) + 1)
Time = 0.11 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.42 \[ \int \frac {3 e^{e^2} x^2+e^{2 e^{-e^2} x} \left (e^{e^2} x^3-4 x^4\right )}{25 e^{e^2}+75 e^{e^2+2 e^{-e^2} x} x+75 e^{e^2+4 e^{-e^2} x} x^2+25 e^{e^2+6 e^{-e^2} x} x^3} \, dx=\frac {x^{3}}{25 \, {\left (x^{2} e^{\left (4 \, x e^{\left (-e^{2}\right )}\right )} + 2 \, x e^{\left (2 \, x e^{\left (-e^{2}\right )}\right )} + 1\right )}} \] Input:
integrate(((x^3*exp(exp(2))-4*x^4)*exp(2*x/exp(exp(2)))+3*x^2*exp(exp(2))) /(25*x^3*exp(exp(2))*exp(2*x/exp(exp(2)))^3+75*x^2*exp(exp(2))*exp(2*x/exp (exp(2)))^2+75*x*exp(exp(2))*exp(2*x/exp(exp(2)))+25*exp(exp(2))),x, algor ithm="giac")
Output:
1/25*x^3/(x^2*e^(4*x*e^(-e^2)) + 2*x*e^(2*x*e^(-e^2)) + 1)
Time = 0.17 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.42 \[ \int \frac {3 e^{e^2} x^2+e^{2 e^{-e^2} x} \left (e^{e^2} x^3-4 x^4\right )}{25 e^{e^2}+75 e^{e^2+2 e^{-e^2} x} x+75 e^{e^2+4 e^{-e^2} x} x^2+25 e^{e^2+6 e^{-e^2} x} x^3} \, dx=\frac {x^3}{50\,x\,{\mathrm {e}}^{2\,x\,{\mathrm {e}}^{-{\mathrm {e}}^2}}+25\,x^2\,{\mathrm {e}}^{4\,x\,{\mathrm {e}}^{-{\mathrm {e}}^2}}+25} \] Input:
int((3*x^2*exp(exp(2)) + exp(2*x*exp(-exp(2)))*(x^3*exp(exp(2)) - 4*x^4))/ (25*exp(exp(2)) + 75*x*exp(2*x*exp(-exp(2)))*exp(exp(2)) + 75*x^2*exp(4*x* exp(-exp(2)))*exp(exp(2)) + 25*x^3*exp(6*x*exp(-exp(2)))*exp(exp(2))),x)
Output:
x^3/(50*x*exp(2*x*exp(-exp(2))) + 25*x^2*exp(4*x*exp(-exp(2))) + 25)
Time = 0.16 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.67 \[ \int \frac {3 e^{e^2} x^2+e^{2 e^{-e^2} x} \left (e^{e^2} x^3-4 x^4\right )}{25 e^{e^2}+75 e^{e^2+2 e^{-e^2} x} x+75 e^{e^2+4 e^{-e^2} x} x^2+25 e^{e^2+6 e^{-e^2} x} x^3} \, dx=\frac {x^{3}}{25 e^{\frac {4 x}{e^{e^{2}}}} x^{2}+50 e^{\frac {2 x}{e^{e^{2}}}} x +25} \] Input:
int(((x^3*exp(exp(2))-4*x^4)*exp(2*x/exp(exp(2)))+3*x^2*exp(exp(2)))/(25*x ^3*exp(exp(2))*exp(2*x/exp(exp(2)))^3+75*x^2*exp(exp(2))*exp(2*x/exp(exp(2 )))^2+75*x*exp(exp(2))*exp(2*x/exp(exp(2)))+25*exp(exp(2))),x)
Output:
x**3/(25*(e**((4*x)/e**(e**2))*x**2 + 2*e**((2*x)/e**(e**2))*x + 1))