\(\int \frac {e^{-e^2} (e^{e^2} (36 x^3+12 x^5+x^7)+e^{\frac {e^{-e^2} (-4-x^2+6 x^3+x^5)}{6 x^2+x^4}} (48+16 x^2+36 x^3+2 x^4+12 x^5+x^7))}{36 x^3+12 x^5+x^7} \, dx\) [7415]

Optimal result

Integrand size = 107, antiderivative size = 32 \[ \int \frac {e^{-e^2} \left (e^{e^2} \left (36 x^3+12 x^5+x^7\right )+e^{\frac {e^{-e^2} \left (-4-x^2+6 x^3+x^5\right )}{6 x^2+x^4}} \left (48+16 x^2+36 x^3+2 x^4+12 x^5+x^7\right )\right )}{36 x^3+12 x^5+x^7} \, dx=e^{e^{-e^2} \left (x-\frac {1}{x^2 \left (1+\frac {2}{4+x^2}\right )}\right )}+x \]

[Out]

exp((x-1/x^2/(2/(x^2+4)+1))/exp(exp(2)))+x

Rubi [F]

\[ \int \frac {e^{-e^2} \left (e^{e^2} \left (36 x^3+12 x^5+x^7\right )+e^{\frac {e^{-e^2} \left (-4-x^2+6 x^3+x^5\right )}{6 x^2+x^4}} \left (48+16 x^2+36 x^3+2 x^4+12 x^5+x^7\right )\right )}{36 x^3+12 x^5+x^7} \, dx=\int \frac {e^{-e^2} \left (e^{e^2} \left (36 x^3+12 x^5+x^7\right )+\exp \left (\frac {e^{-e^2} \left (-4-x^2+6 x^3+x^5\right )}{6 x^2+x^4}\right ) \left (48+16 x^2+36 x^3+2 x^4+12 x^5+x^7\right )\right )}{36 x^3+12 x^5+x^7} \, dx \]

[In]

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]

[Out]

x + Defer[Int][E^((-4 - x^2 + 6*x^3 + x^5)/(E^E^2*x^2*(6 + x^2))), x]/E^E^2 + (4*Defer[Int][E^((-4 - x^2 + 6*x
^3 + x^5)/(E^E^2*x^2*(6 + x^2)))/x^3, x])/(3*E^E^2) + (2*Defer[Int][(E^((-4 - x^2 + 6*x^3 + x^5)/(E^E^2*x^2*(6
+ x^2)))*x)/(6 + x^2)^2, x])/(3*E^E^2)

Rubi steps \begin{align*} \text {integral}& = e^{-e^2} \int \frac {e^{e^2} \left (36 x^3+12 x^5+x^7\right )+\exp \left (\frac {e^{-e^2} \left (-4-x^2+6 x^3+x^5\right )}{6 x^2+x^4}\right ) \left (48+16 x^2+36 x^3+2 x^4+12 x^5+x^7\right )}{36 x^3+12 x^5+x^7} \, dx \\ & = e^{-e^2} \int \frac {e^{e^2} \left (36 x^3+12 x^5+x^7\right )+\exp \left (\frac {e^{-e^2} \left (-4-x^2+6 x^3+x^5\right )}{6 x^2+x^4}\right ) \left (48+16 x^2+36 x^3+2 x^4+12 x^5+x^7\right )}{x^3 \left (36+12 x^2+x^4\right )} \, dx \\ & = e^{-e^2} \int \frac {e^{e^2} \left (36 x^3+12 x^5+x^7\right )+\exp \left (\frac {e^{-e^2} \left (-4-x^2+6 x^3+x^5\right )}{6 x^2+x^4}\right ) \left (48+16 x^2+36 x^3+2 x^4+12 x^5+x^7\right )}{x^3 \left (6+x^2\right )^2} \, dx \\ & = e^{-e^2} \int \left (e^{e^2}+\frac {\exp \left (\frac {e^{-e^2} \left (-4-x^2+6 x^3+x^5\right )}{x^2 \left (6+x^2\right )}\right ) \left (48+16 x^2+36 x^3+2 x^4+12 x^5+x^7\right )}{x^3 \left (6+x^2\right )^2}\right ) \, dx \\ & = x+e^{-e^2} \int \frac {\exp \left (\frac {e^{-e^2} \left (-4-x^2+6 x^3+x^5\right )}{x^2 \left (6+x^2\right )}\right ) \left (48+16 x^2+36 x^3+2 x^4+12 x^5+x^7\right )}{x^3 \left (6+x^2\right )^2} \, dx \\ & = x+e^{-e^2} \int \left (\exp \left (\frac {e^{-e^2} \left (-4-x^2+6 x^3+x^5\right )}{x^2 \left (6+x^2\right )}\right )+\frac {4 \exp \left (\frac {e^{-e^2} \left (-4-x^2+6 x^3+x^5\right )}{x^2 \left (6+x^2\right )}\right )}{3 x^3}+\frac {2 \exp \left (\frac {e^{-e^2} \left (-4-x^2+6 x^3+x^5\right )}{x^2 \left (6+x^2\right )}\right ) x}{3 \left (6+x^2\right )^2}\right ) \, dx \\ & = x+\frac {1}{3} \left (2 e^{-e^2}\right ) \int \frac {\exp \left (\frac {e^{-e^2} \left (-4-x^2+6 x^3+x^5\right )}{x^2 \left (6+x^2\right )}\right ) x}{\left (6+x^2\right )^2} \, dx+e^{-e^2} \int \exp \left (\frac {e^{-e^2} \left (-4-x^2+6 x^3+x^5\right )}{x^2 \left (6+x^2\right )}\right ) \, dx+\frac {1}{3} \left (4 e^{-e^2}\right ) \int \frac {\exp \left (\frac {e^{-e^2} \left (-4-x^2+6 x^3+x^5\right )}{x^2 \left (6+x^2\right )}\right )}{x^3} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 5.24 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.44 \[ \int \frac {e^{-e^2} \left (e^{e^2} \left (36 x^3+12 x^5+x^7\right )+e^{\frac {e^{-e^2} \left (-4-x^2+6 x^3+x^5\right )}{6 x^2+x^4}} \left (48+16 x^2+36 x^3+2 x^4+12 x^5+x^7\right )\right )}{36 x^3+12 x^5+x^7} \, dx=e^{-\frac {2 e^{-e^2}}{3 x^2}+e^{-e^2} x-\frac {e^{-e^2}}{3 \left (6+x^2\right )}}+x \]

[In]

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 + 3
6*x^3 + 2*x^4 + 12*x^5 + x^7))/(E^E^2*(36*x^3 + 12*x^5 + x^7)),x]

[Out]

E^(-2/(3*E^E^2*x^2) + x/E^E^2 - 1/(3*E^E^2*(6 + x^2))) + x

Maple [A] (verified)

Time = 0.94 (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	\(\frac {x^{5}+x^{4} {\mathrm e}^{\frac {\left (x^{5}+6 x^{3}-x^{2}-4\right ) {\mathrm e}^{-{\mathrm e}^{2}}}{x^{4}+6 x^{2}}}+6 x^{3}+6 x^{2} {\mathrm e}^{\frac {\left (x^{5}+6 x^{3}-x^{2}-4\right ) {\mathrm e}^{-{\mathrm e}^{2}}}{x^{4}+6 x^{2}}}}{x^{2} \left (x^{2}+6\right )}\)	\(96\)

[In]

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)*ex
p(exp(2)))/(x^7+12*x^5+36*x^3)/exp(exp(2)),x,method=_RETURNVERBOSE)

[Out]

x+exp((x^5+6*x^3-x^2-4)*exp(-exp(2))/x^2/(x^2+6))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.09 \[ \int \frac {e^{-e^2} \left (e^{e^2} \left (36 x^3+12 x^5+x^7\right )+e^{\frac {e^{-e^2} \left (-4-x^2+6 x^3+x^5\right )}{6 x^2+x^4}} \left (48+16 x^2+36 x^3+2 x^4+12 x^5+x^7\right )\right )}{36 x^3+12 x^5+x^7} \, dx=x + e^{\left (\frac {{\left (x^{5} + 6 \, x^{3} - x^{2} - 4\right )} e^{\left (-e^{2}\right )}}{x^{4} + 6 \, x^{2}}\right )} \]

[In]

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, algorithm="fricas")

[Out]

x + e^((x^5 + 6*x^3 - x^2 - 4)*e^(-e^2)/(x^4 + 6*x^2))

Sympy [A] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.91 \[ \int \frac {e^{-e^2} \left (e^{e^2} \left (36 x^3+12 x^5+x^7\right )+e^{\frac {e^{-e^2} \left (-4-x^2+6 x^3+x^5\right )}{6 x^2+x^4}} \left (48+16 x^2+36 x^3+2 x^4+12 x^5+x^7\right )\right )}{36 x^3+12 x^5+x^7} \, dx=x + e^{\frac {x^{5} + 6 x^{3} - x^{2} - 4}{\left (x^{4} + 6 x^{2}\right ) e^{e^{2}}}} \]

[In]

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)

[Out]

x + exp((x**5 + 6*x**3 - x**2 - 4)*exp(-exp(2))/(x**4 + 6*x**2))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 132 vs. \(2 (29) = 58\).

Time = 0.49 (sec) , antiderivative size = 132, normalized size of antiderivative = 4.12 \[ \int \frac {e^{-e^2} \left (e^{e^2} \left (36 x^3+12 x^5+x^7\right )+e^{\frac {e^{-e^2} \left (-4-x^2+6 x^3+x^5\right )}{6 x^2+x^4}} \left (48+16 x^2+36 x^3+2 x^4+12 x^5+x^7\right )\right )}{36 x^3+12 x^5+x^7} \, dx=-\frac {1}{2} \, {\left ({\left (3 \, \sqrt {6} \arctan \left (\frac {1}{6} \, \sqrt {6} x\right ) - 2 \, x - \frac {6 \, x}{x^{2} + 6}\right )} e^{\left (e^{2}\right )} - {\left (\sqrt {6} \arctan \left (\frac {1}{6} \, \sqrt {6} x\right ) + \frac {6 \, x}{x^{2} + 6}\right )} e^{\left (e^{2}\right )} - 2 \, {\left (\sqrt {6} \arctan \left (\frac {1}{6} \, \sqrt {6} x\right ) - \frac {6 \, x}{x^{2} + 6}\right )} e^{\left (e^{2}\right )} - 2 \, e^{\left (x e^{\left (-e^{2}\right )} - \frac {1}{3 \, {\left (x^{2} e^{\left (e^{2}\right )} + 6 \, e^{\left (e^{2}\right )}\right )}} - \frac {2 \, e^{\left (-e^{2}\right )}}{3 \, x^{2}} + e^{2}\right )}\right )} e^{\left (-e^{2}\right )} \]

[In]

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, algorithm="maxima")

[Out]

-1/2*((3*sqrt(6)*arctan(1/6*sqrt(6)*x) - 2*x - 6*x/(x^2 + 6))*e^(e^2) - (sqrt(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)

Giac [A] (verification not implemented)

none

Time = 0.41 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.62 \[ \int \frac {e^{-e^2} \left (e^{e^2} \left (36 x^3+12 x^5+x^7\right )+e^{\frac {e^{-e^2} \left (-4-x^2+6 x^3+x^5\right )}{6 x^2+x^4}} \left (48+16 x^2+36 x^3+2 x^4+12 x^5+x^7\right )\right )}{36 x^3+12 x^5+x^7} \, dx=x + e^{\left (\frac {x^{5} e^{\left (-e^{2}\right )} + 6 \, x^{3} e^{\left (-e^{2}\right )} - x^{2} e^{\left (-e^{2}\right )} - 4 \, e^{\left (-e^{2}\right )}}{x^{4} + 6 \, x^{2}}\right )} \]

[In]

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, algorithm="giac")

[Out]

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))

Mupad [B] (verification not implemented)

Time = 12.30 (sec) , antiderivative size = 87, normalized size of antiderivative = 2.72 \[ \int \frac {e^{-e^2} \left (e^{e^2} \left (36 x^3+12 x^5+x^7\right )+e^{\frac {e^{-e^2} \left (-4-x^2+6 x^3+x^5\right )}{6 x^2+x^4}} \left (48+16 x^2+36 x^3+2 x^4+12 x^5+x^7\right )\right )}{36 x^3+12 x^5+x^7} \, dx=x+{\mathrm {e}}^{-\frac {x^2\,{\mathrm {e}}^{-{\mathrm {e}}^2}}{x^4+6\,x^2}}\,{\mathrm {e}}^{\frac {x^5\,{\mathrm {e}}^{-{\mathrm {e}}^2}}{x^4+6\,x^2}}\,{\mathrm {e}}^{\frac {6\,x^3\,{\mathrm {e}}^{-{\mathrm {e}}^2}}{x^4+6\,x^2}}\,{\mathrm {e}}^{-\frac {4\,{\mathrm {e}}^{-{\mathrm {e}}^2}}{x^4+6\,x^2}} \]

[In]

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)

[Out]

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))