Integrand size = 107, antiderivative size = 32 \[ \int \frac {e^{-e^2} \left (e^{e^2} \left (36 x^3+12 x^5+x^7\right )+e^{\frac {e^{-e^2} \left (-4-x^2+6 x^3+x^5\right )}{6 x^2+x^4}} \left (48+16 x^2+36 x^3+2 x^4+12 x^5+x^7\right )\right )}{36 x^3+12 x^5+x^7} \, dx=e^{e^{-e^2} \left (x-\frac {1}{x^2 \left (1+\frac {2}{4+x^2}\right )}\right )}+x \] Output:
exp((x-1/x^2/(2/(x^2+4)+1))/exp(exp(2)))+x
Time = 5.07 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.44 \[ \int \frac {e^{-e^2} \left (e^{e^2} \left (36 x^3+12 x^5+x^7\right )+e^{\frac {e^{-e^2} \left (-4-x^2+6 x^3+x^5\right )}{6 x^2+x^4}} \left (48+16 x^2+36 x^3+2 x^4+12 x^5+x^7\right )\right )}{36 x^3+12 x^5+x^7} \, dx=e^{-\frac {2 e^{-e^2}}{3 x^2}+e^{-e^2} x-\frac {e^{-e^2}}{3 \left (6+x^2\right )}}+x \] Input:
Integrate[(E^E^2*(36*x^3 + 12*x^5 + x^7) + E^((-4 - x^2 + 6*x^3 + x^5)/(E^ E^2*(6*x^2 + x^4)))*(48 + 16*x^2 + 36*x^3 + 2*x^4 + 12*x^5 + x^7))/(E^E^2* (36*x^3 + 12*x^5 + x^7)),x]
Output:
E^(-2/(3*E^E^2*x^2) + x/E^E^2 - 1/(3*E^E^2*(6 + x^2))) + 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.
\(\displaystyle \int \frac {e^{-e^2} \left (e^{e^2} \left (x^7+12 x^5+36 x^3\right )+e^{\frac {e^{-e^2} \left (x^5+6 x^3-x^2-4\right )}{x^4+6 x^2}} \left (x^7+12 x^5+2 x^4+36 x^3+16 x^2+48\right )\right )}{x^7+12 x^5+36 x^3} \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle e^{-e^2} \int \frac {e^{e^2} \left (x^7+12 x^5+36 x^3\right )+\exp \left (-\frac {e^{-e^2} \left (-x^5-6 x^3+x^2+4\right )}{x^4+6 x^2}\right ) \left (x^7+12 x^5+2 x^4+36 x^3+16 x^2+48\right )}{x^7+12 x^5+36 x^3}dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle e^{-e^2} \int \frac {e^{e^2} \left (x^7+12 x^5+36 x^3\right )+\exp \left (-\frac {e^{-e^2} \left (-x^5-6 x^3+x^2+4\right )}{x^4+6 x^2}\right ) \left (x^7+12 x^5+2 x^4+36 x^3+16 x^2+48\right )}{x^3 \left (x^4+12 x^2+36\right )}dx\) |
\(\Big \downarrow \) 1380 |
\(\displaystyle e^{-e^2} \int \frac {e^{e^2} \left (x^7+12 x^5+36 x^3\right )+\exp \left (-\frac {e^{-e^2} \left (-x^5-6 x^3+x^2+4\right )}{x^4+6 x^2}\right ) \left (x^7+12 x^5+2 x^4+36 x^3+16 x^2+48\right )}{x^3 \left (x^2+6\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle e^{-e^2} \int \left (\frac {\exp \left (\frac {e^{-e^2} \left (x^5+6 x^3-x^2-4\right )}{x^2 \left (x^2+6\right )}\right ) \left (x^7+12 x^5+2 x^4+36 x^3+16 x^2+48\right )}{x^3 \left (x^2+6\right )^2}+e^{e^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle e^{-e^2} \left (\int \exp \left (\frac {e^{-e^2} \left (x^5+6 x^3-x^2-4\right )}{x^2 \left (x^2+6\right )}\right )dx+\frac {4}{3} \int \frac {\exp \left (\frac {e^{-e^2} \left (x^5+6 x^3-x^2-4\right )}{x^2 \left (x^2+6\right )}\right )}{x^3}dx+\frac {2}{3} \int \frac {\exp \left (\frac {e^{-e^2} \left (x^5+6 x^3-x^2-4\right )}{x^2 \left (x^2+6\right )}\right ) x}{\left (x^2+6\right )^2}dx+e^{e^2} x\right )\) |
Input:
Int[(E^E^2*(36*x^3 + 12*x^5 + x^7) + E^((-4 - x^2 + 6*x^3 + x^5)/(E^E^2*(6 *x^2 + x^4)))*(48 + 16*x^2 + 36*x^3 + 2*x^4 + 12*x^5 + x^7))/(E^E^2*(36*x^ 3 + 12*x^5 + x^7)),x]
Output:
$Aborted
Time = 1.40 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.09
method | result | size |
risch | \(x +{\mathrm e}^{\frac {\left (x^{5}+6 x^{3}-x^{2}-4\right ) {\mathrm e}^{-{\mathrm e}^{2}}}{x^{2} \left (x^{2}+6\right )}}\) | \(35\) |
parallelrisch | \({\mathrm e}^{-{\mathrm e}^{2}} \left (x \,{\mathrm e}^{{\mathrm e}^{2}}+{\mathrm e}^{{\mathrm e}^{2}} {\mathrm e}^{\frac {\left (x^{5}+6 x^{3}-x^{2}-4\right ) {\mathrm e}^{-{\mathrm e}^{2}}}{x^{2} \left (x^{2}+6\right )}}\right )\) | \(49\) |
parts | \(x +\frac {x^{4} {\mathrm e}^{\frac {\left (x^{5}+6 x^{3}-x^{2}-4\right ) {\mathrm e}^{-{\mathrm e}^{2}}}{x^{4}+6 x^{2}}}+6 x^{2} {\mathrm e}^{\frac {\left (x^{5}+6 x^{3}-x^{2}-4\right ) {\mathrm e}^{-{\mathrm e}^{2}}}{x^{4}+6 x^{2}}}}{x^{2} \left (x^{2}+6\right )}\) | \(90\) |
norman | \(\frac {x^{5}+x^{4} {\mathrm e}^{\frac {\left (x^{5}+6 x^{3}-x^{2}-4\right ) {\mathrm e}^{-{\mathrm e}^{2}}}{x^{4}+6 x^{2}}}+6 x^{3}+6 x^{2} {\mathrm e}^{\frac {\left (x^{5}+6 x^{3}-x^{2}-4\right ) {\mathrm e}^{-{\mathrm e}^{2}}}{x^{4}+6 x^{2}}}}{x^{2} \left (x^{2}+6\right )}\) | \(96\) |
Input:
int(((x^7+12*x^5+2*x^4+36*x^3+16*x^2+48)*exp((x^5+6*x^3-x^2-4)/(x^4+6*x^2) /exp(exp(2)))+(x^7+12*x^5+36*x^3)*exp(exp(2)))/(x^7+12*x^5+36*x^3)/exp(exp (2)),x,method=_RETURNVERBOSE)
Output:
x+exp((x^5+6*x^3-x^2-4)*exp(-exp(2))/x^2/(x^2+6))
Time = 0.09 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.09 \[ \int \frac {e^{-e^2} \left (e^{e^2} \left (36 x^3+12 x^5+x^7\right )+e^{\frac {e^{-e^2} \left (-4-x^2+6 x^3+x^5\right )}{6 x^2+x^4}} \left (48+16 x^2+36 x^3+2 x^4+12 x^5+x^7\right )\right )}{36 x^3+12 x^5+x^7} \, dx=x + e^{\left (\frac {{\left (x^{5} + 6 \, x^{3} - x^{2} - 4\right )} e^{\left (-e^{2}\right )}}{x^{4} + 6 \, x^{2}}\right )} \] Input:
integrate(((x^7+12*x^5+2*x^4+36*x^3+16*x^2+48)*exp((x^5+6*x^3-x^2-4)/(x^4+ 6*x^2)/exp(exp(2)))+(x^7+12*x^5+36*x^3)*exp(exp(2)))/(x^7+12*x^5+36*x^3)/e xp(exp(2)),x, algorithm="fricas")
Output:
x + e^((x^5 + 6*x^3 - x^2 - 4)*e^(-e^2)/(x^4 + 6*x^2))
Time = 0.27 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.91 \[ \int \frac {e^{-e^2} \left (e^{e^2} \left (36 x^3+12 x^5+x^7\right )+e^{\frac {e^{-e^2} \left (-4-x^2+6 x^3+x^5\right )}{6 x^2+x^4}} \left (48+16 x^2+36 x^3+2 x^4+12 x^5+x^7\right )\right )}{36 x^3+12 x^5+x^7} \, dx=x + e^{\frac {x^{5} + 6 x^{3} - x^{2} - 4}{\left (x^{4} + 6 x^{2}\right ) e^{e^{2}}}} \] Input:
integrate(((x**7+12*x**5+2*x**4+36*x**3+16*x**2+48)*exp((x**5+6*x**3-x**2- 4)/(x**4+6*x**2)/exp(exp(2)))+(x**7+12*x**5+36*x**3)*exp(exp(2)))/(x**7+12 *x**5+36*x**3)/exp(exp(2)),x)
Output:
x + exp((x**5 + 6*x**3 - x**2 - 4)*exp(-exp(2))/(x**4 + 6*x**2))
Leaf count of result is larger than twice the leaf count of optimal. 132 vs. \(2 (29) = 58\).
Time = 0.27 (sec) , antiderivative size = 132, normalized size of antiderivative = 4.12 \[ \int \frac {e^{-e^2} \left (e^{e^2} \left (36 x^3+12 x^5+x^7\right )+e^{\frac {e^{-e^2} \left (-4-x^2+6 x^3+x^5\right )}{6 x^2+x^4}} \left (48+16 x^2+36 x^3+2 x^4+12 x^5+x^7\right )\right )}{36 x^3+12 x^5+x^7} \, dx=-\frac {1}{2} \, {\left ({\left (3 \, \sqrt {6} \arctan \left (\frac {1}{6} \, \sqrt {6} x\right ) - 2 \, x - \frac {6 \, x}{x^{2} + 6}\right )} e^{\left (e^{2}\right )} - {\left (\sqrt {6} \arctan \left (\frac {1}{6} \, \sqrt {6} x\right ) + \frac {6 \, x}{x^{2} + 6}\right )} e^{\left (e^{2}\right )} - 2 \, {\left (\sqrt {6} \arctan \left (\frac {1}{6} \, \sqrt {6} x\right ) - \frac {6 \, x}{x^{2} + 6}\right )} e^{\left (e^{2}\right )} - 2 \, e^{\left (x e^{\left (-e^{2}\right )} - \frac {1}{3 \, {\left (x^{2} e^{\left (e^{2}\right )} + 6 \, e^{\left (e^{2}\right )}\right )}} - \frac {2 \, e^{\left (-e^{2}\right )}}{3 \, x^{2}} + e^{2}\right )}\right )} e^{\left (-e^{2}\right )} \] Input:
integrate(((x^7+12*x^5+2*x^4+36*x^3+16*x^2+48)*exp((x^5+6*x^3-x^2-4)/(x^4+ 6*x^2)/exp(exp(2)))+(x^7+12*x^5+36*x^3)*exp(exp(2)))/(x^7+12*x^5+36*x^3)/e xp(exp(2)),x, algorithm="maxima")
Output:
-1/2*((3*sqrt(6)*arctan(1/6*sqrt(6)*x) - 2*x - 6*x/(x^2 + 6))*e^(e^2) - (s qrt(6)*arctan(1/6*sqrt(6)*x) + 6*x/(x^2 + 6))*e^(e^2) - 2*(sqrt(6)*arctan( 1/6*sqrt(6)*x) - 6*x/(x^2 + 6))*e^(e^2) - 2*e^(x*e^(-e^2) - 1/3/(x^2*e^(e^ 2) + 6*e^(e^2)) - 2/3*e^(-e^2)/x^2 + e^2))*e^(-e^2)
Time = 0.27 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.62 \[ \int \frac {e^{-e^2} \left (e^{e^2} \left (36 x^3+12 x^5+x^7\right )+e^{\frac {e^{-e^2} \left (-4-x^2+6 x^3+x^5\right )}{6 x^2+x^4}} \left (48+16 x^2+36 x^3+2 x^4+12 x^5+x^7\right )\right )}{36 x^3+12 x^5+x^7} \, dx=x + e^{\left (\frac {x^{5} e^{\left (-e^{2}\right )} + 6 \, x^{3} e^{\left (-e^{2}\right )} - x^{2} e^{\left (-e^{2}\right )} - 4 \, e^{\left (-e^{2}\right )}}{x^{4} + 6 \, x^{2}}\right )} \] Input:
integrate(((x^7+12*x^5+2*x^4+36*x^3+16*x^2+48)*exp((x^5+6*x^3-x^2-4)/(x^4+ 6*x^2)/exp(exp(2)))+(x^7+12*x^5+36*x^3)*exp(exp(2)))/(x^7+12*x^5+36*x^3)/e xp(exp(2)),x, algorithm="giac")
Output:
x + e^((x^5*e^(-e^2) + 6*x^3*e^(-e^2) - x^2*e^(-e^2) - 4*e^(-e^2))/(x^4 + 6*x^2))
Time = 3.31 (sec) , antiderivative size = 87, normalized size of antiderivative = 2.72 \[ \int \frac {e^{-e^2} \left (e^{e^2} \left (36 x^3+12 x^5+x^7\right )+e^{\frac {e^{-e^2} \left (-4-x^2+6 x^3+x^5\right )}{6 x^2+x^4}} \left (48+16 x^2+36 x^3+2 x^4+12 x^5+x^7\right )\right )}{36 x^3+12 x^5+x^7} \, dx=x+{\mathrm {e}}^{-\frac {x^2\,{\mathrm {e}}^{-{\mathrm {e}}^2}}{x^4+6\,x^2}}\,{\mathrm {e}}^{\frac {x^5\,{\mathrm {e}}^{-{\mathrm {e}}^2}}{x^4+6\,x^2}}\,{\mathrm {e}}^{\frac {6\,x^3\,{\mathrm {e}}^{-{\mathrm {e}}^2}}{x^4+6\,x^2}}\,{\mathrm {e}}^{-\frac {4\,{\mathrm {e}}^{-{\mathrm {e}}^2}}{x^4+6\,x^2}} \] Input:
int((exp(-exp(2))*(exp(exp(2))*(36*x^3 + 12*x^5 + x^7) + exp(-(exp(-exp(2) )*(x^2 - 6*x^3 - x^5 + 4))/(6*x^2 + x^4))*(16*x^2 + 36*x^3 + 2*x^4 + 12*x^ 5 + x^7 + 48)))/(36*x^3 + 12*x^5 + x^7),x)
Output:
x + exp(-(x^2*exp(-exp(2)))/(6*x^2 + x^4))*exp((x^5*exp(-exp(2)))/(6*x^2 + x^4))*exp((6*x^3*exp(-exp(2)))/(6*x^2 + x^4))*exp(-(4*exp(-exp(2)))/(6*x^ 2 + x^4))
Time = 0.18 (sec) , antiderivative size = 77, normalized size of antiderivative = 2.41 \[ \int \frac {e^{-e^2} \left (e^{e^2} \left (36 x^3+12 x^5+x^7\right )+e^{\frac {e^{-e^2} \left (-4-x^2+6 x^3+x^5\right )}{6 x^2+x^4}} \left (48+16 x^2+36 x^3+2 x^4+12 x^5+x^7\right )\right )}{36 x^3+12 x^5+x^7} \, dx=\frac {e^{\frac {x}{e^{e^{2}}}}+e^{\frac {x^{2}+4}{e^{e^{2}} x^{4}+6 e^{e^{2}} x^{2}}} x}{e^{\frac {x^{2}+4}{e^{e^{2}} x^{4}+6 e^{e^{2}} x^{2}}}} \] Input:
int(((x^7+12*x^5+2*x^4+36*x^3+16*x^2+48)*exp((x^5+6*x^3-x^2-4)/(x^4+6*x^2) /exp(exp(2)))+(x^7+12*x^5+36*x^3)*exp(exp(2)))/(x^7+12*x^5+36*x^3)/exp(exp (2)),x)
Output:
(e**(x/e**(e**2)) + e**((x**2 + 4)/(e**(e**2)*x**4 + 6*e**(e**2)*x**2))*x) /e**((x**2 + 4)/(e**(e**2)*x**4 + 6*e**(e**2)*x**2))