\(\int \frac {2 x+e^{e^{48+2 e^x+6 x} x} (2 x+e^{48+2 e^x+6 x} (-2 x^2-12 x^3-4 e^x x^3))}{1+3 e^{e^{48+2 e^x+6 x} x}+3 e^{2 e^{48+2 e^x+6 x} x}+e^{3 e^{48+2 e^x+6 x} x}} \, dx\) [5589]

Optimal result

Integrand size = 116, antiderivative size = 25 \[ \int \frac {2 x+e^{e^{48+2 e^x+6 x} x} \left (2 x+e^{48+2 e^x+6 x} \left (-2 x^2-12 x^3-4 e^x x^3\right )\right )}{1+3 e^{e^{48+2 e^x+6 x} x}+3 e^{2 e^{48+2 e^x+6 x} x}+e^{3 e^{48+2 e^x+6 x} x}} \, dx=\frac {x^2}{\left (1+e^{e^{2 e^x+6 (8+x)} x}\right )^2} \]

[Out]

x^2/(1+exp(x*exp(exp(x)+3*x+24)^2))^2

Rubi [F]

\[ \int \frac {2 x+e^{e^{48+2 e^x+6 x} x} \left (2 x+e^{48+2 e^x+6 x} \left (-2 x^2-12 x^3-4 e^x x^3\right )\right )}{1+3 e^{e^{48+2 e^x+6 x} x}+3 e^{2 e^{48+2 e^x+6 x} x}+e^{3 e^{48+2 e^x+6 x} x}} \, dx=\int \frac {2 x+e^{e^{48+2 e^x+6 x} x} \left (2 x+e^{48+2 e^x+6 x} \left (-2 x^2-12 x^3-4 e^x x^3\right )\right )}{1+3 e^{e^{48+2 e^x+6 x} x}+3 e^{2 e^{48+2 e^x+6 x} x}+e^{3 e^{48+2 e^x+6 x} x}} \, dx \]

[In]

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]

[Out]

2*Defer[Int][x/(1 + E^(E^(48 + 2*E^x + 6*x)*x))^2, x] + 2*Defer[Int][(E^(2*(24 + E^x + 3*x))*x^2)/(1 + E^(E^(4
8 + 2*E^x + 6*x)*x))^3, x] - 2*Defer[Int][(E^(2*(24 + E^x + 3*x))*x^2)/(1 + E^(E^(48 + 2*E^x + 6*x)*x))^2, x]
+ 12*Defer[Int][(E^(2*(24 + E^x + 3*x))*x^3)/(1 + E^(E^(48 + 2*E^x + 6*x)*x))^3, x] + 4*Defer[Int][(E^(x + 2*(
24 + E^x + 3*x))*x^3)/(1 + E^(E^(48 + 2*E^x + 6*x)*x))^3, x] - 12*Defer[Int][(E^(2*(24 + E^x + 3*x))*x^3)/(1 +
E^(E^(48 + 2*E^x + 6*x)*x))^2, x] - 4*Defer[Int][(E^(48 + 2*E^x + 7*x)*x^3)/(1 + E^(E^(48 + 2*E^x + 6*x)*x))^
2, x]

Rubi steps \begin{align*} \text {integral}& = \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 )}{\left (1+e^{e^{48+2 e^x+6 x} x}\right )^3} \, dx \\ & = \int \left (\frac {2 e^{2 \left (24+e^x+3 x\right )} x^2 \left (1+6 x+2 e^x x\right )}{\left (1+e^{e^{48+2 e^x+6 x} x}\right )^3}-\frac {2 x \left (-1+e^{48+2 e^x+6 x} x+6 e^{48+2 e^x+6 x} x^2+2 e^{48+2 e^x+7 x} x^2\right )}{\left (1+e^{e^{48+2 e^x+6 x} x}\right )^2}\right ) \, dx \\ & = 2 \int \frac {e^{2 \left (24+e^x+3 x\right )} x^2 \left (1+6 x+2 e^x x\right )}{\left (1+e^{e^{48+2 e^x+6 x} x}\right )^3} \, dx-2 \int \frac {x \left (-1+e^{48+2 e^x+6 x} x+6 e^{48+2 e^x+6 x} x^2+2 e^{48+2 e^x+7 x} x^2\right )}{\left (1+e^{e^{48+2 e^x+6 x} x}\right )^2} \, dx \\ & = 2 \int \left (\frac {e^{2 \left (24+e^x+3 x\right )} x^2}{\left (1+e^{e^{48+2 e^x+6 x} x}\right )^3}+\frac {6 e^{2 \left (24+e^x+3 x\right )} x^3}{\left (1+e^{e^{48+2 e^x+6 x} x}\right )^3}+\frac {2 e^{x+2 \left (24+e^x+3 x\right )} x^3}{\left (1+e^{e^{48+2 e^x+6 x} x}\right )^3}\right ) \, dx-2 \int \left (-\frac {x}{\left (1+e^{e^{48+2 e^x+6 x} x}\right )^2}+\frac {e^{2 \left (24+e^x+3 x\right )} x^2}{\left (1+e^{e^{48+2 e^x+6 x} x}\right )^2}+\frac {6 e^{2 \left (24+e^x+3 x\right )} x^3}{\left (1+e^{e^{48+2 e^x+6 x} x}\right )^2}+\frac {2 e^{48+2 e^x+7 x} x^3}{\left (1+e^{e^{48+2 e^x+6 x} x}\right )^2}\right ) \, dx \\ & = 2 \int \frac {x}{\left (1+e^{e^{48+2 e^x+6 x} x}\right )^2} \, dx+2 \int \frac {e^{2 \left (24+e^x+3 x\right )} x^2}{\left (1+e^{e^{48+2 e^x+6 x} x}\right )^3} \, dx-2 \int \frac {e^{2 \left (24+e^x+3 x\right )} x^2}{\left (1+e^{e^{48+2 e^x+6 x} x}\right )^2} \, dx+4 \int \frac {e^{x+2 \left (24+e^x+3 x\right )} x^3}{\left (1+e^{e^{48+2 e^x+6 x} x}\right )^3} \, dx-4 \int \frac {e^{48+2 e^x+7 x} x^3}{\left (1+e^{e^{48+2 e^x+6 x} x}\right )^2} \, dx+12 \int \frac {e^{2 \left (24+e^x+3 x\right )} x^3}{\left (1+e^{e^{48+2 e^x+6 x} x}\right )^3} \, dx-12 \int \frac {e^{2 \left (24+e^x+3 x\right )} x^3}{\left (1+e^{e^{48+2 e^x+6 x} x}\right )^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {2 x+e^{e^{48+2 e^x+6 x} x} \left (2 x+e^{48+2 e^x+6 x} \left (-2 x^2-12 x^3-4 e^x x^3\right )\right )}{1+3 e^{e^{48+2 e^x+6 x} x}+3 e^{2 e^{48+2 e^x+6 x} x}+e^{3 e^{48+2 e^x+6 x} x}} \, dx=\frac {x^2}{\left (1+e^{e^{48+2 e^x+6 x} x}\right )^2} \]

[In]

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]

[Out]

x^2/(1 + E^(E^(48 + 2*E^x + 6*x)*x))^2

Maple [A] (verified)

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	\(\frac {x^{2}}{{\mathrm e}^{2 x \,{\mathrm e}^{2 \,{\mathrm e}^{x}+6 x +48}}+2 \,{\mathrm e}^{x \,{\mathrm e}^{2 \,{\mathrm e}^{x}+6 x +48}}+1}\)	\(39\)

[In]

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)

[Out]

x^2/(1+exp(x*exp(2*exp(x)+6*x+48)))^2

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.48 \[ \int \frac {2 x+e^{e^{48+2 e^x+6 x} x} \left (2 x+e^{48+2 e^x+6 x} \left (-2 x^2-12 x^3-4 e^x x^3\right )\right )}{1+3 e^{e^{48+2 e^x+6 x} x}+3 e^{2 e^{48+2 e^x+6 x} x}+e^{3 e^{48+2 e^x+6 x} x}} \, dx=\frac {x^{2}}{e^{\left (2 \, x e^{\left (6 \, x + 2 \, e^{x} + 48\right )}\right )} + 2 \, e^{\left (x e^{\left (6 \, x + 2 \, e^{x} + 48\right )}\right )} + 1} \]

[In]

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*exp(exp(x)+3*x+24)^2)^2+3*exp(x*exp(exp(x)+3*x+24)^2)+1),x, algorithm="fricas")

[Out]

x^2/(e^(2*x*e^(6*x + 2*e^x + 48)) + 2*e^(x*e^(6*x + 2*e^x + 48)) + 1)

Sympy [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.48 \[ \int \frac {2 x+e^{e^{48+2 e^x+6 x} x} \left (2 x+e^{48+2 e^x+6 x} \left (-2 x^2-12 x^3-4 e^x x^3\right )\right )}{1+3 e^{e^{48+2 e^x+6 x} x}+3 e^{2 e^{48+2 e^x+6 x} x}+e^{3 e^{48+2 e^x+6 x} x}} \, dx=\frac {x^{2}}{e^{2 x e^{6 x + 2 e^{x} + 48}} + 2 e^{x e^{6 x + 2 e^{x} + 48}} + 1} \]

[In]

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*exp(exp(x)+3*x+24)**2)**2+3*exp(x*exp(exp(x)+3*x+24)**2)+1),x)

[Out]

x**2/(exp(2*x*exp(6*x + 2*exp(x) + 48)) + 2*exp(x*exp(6*x + 2*exp(x) + 48)) + 1)

Maxima [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.48 \[ \int \frac {2 x+e^{e^{48+2 e^x+6 x} x} \left (2 x+e^{48+2 e^x+6 x} \left (-2 x^2-12 x^3-4 e^x x^3\right )\right )}{1+3 e^{e^{48+2 e^x+6 x} x}+3 e^{2 e^{48+2 e^x+6 x} x}+e^{3 e^{48+2 e^x+6 x} x}} \, dx=\frac {x^{2}}{e^{\left (2 \, x e^{\left (6 \, x + 2 \, e^{x} + 48\right )}\right )} + 2 \, e^{\left (x e^{\left (6 \, x + 2 \, e^{x} + 48\right )}\right )} + 1} \]

[In]

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*exp(exp(x)+3*x+24)^2)^2+3*exp(x*exp(exp(x)+3*x+24)^2)+1),x, algorithm="maxima")

[Out]

x^2/(e^(2*x*e^(6*x + 2*e^x + 48)) + 2*e^(x*e^(6*x + 2*e^x + 48)) + 1)

Giac [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.48 \[ \int \frac {2 x+e^{e^{48+2 e^x+6 x} x} \left (2 x+e^{48+2 e^x+6 x} \left (-2 x^2-12 x^3-4 e^x x^3\right )\right )}{1+3 e^{e^{48+2 e^x+6 x} x}+3 e^{2 e^{48+2 e^x+6 x} x}+e^{3 e^{48+2 e^x+6 x} x}} \, dx=\frac {x^{2}}{e^{\left (2 \, x e^{\left (6 \, x + 2 \, e^{x} + 48\right )}\right )} + 2 \, e^{\left (x e^{\left (6 \, x + 2 \, e^{x} + 48\right )}\right )} + 1} \]

[In]

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*exp(exp(x)+3*x+24)^2)^2+3*exp(x*exp(exp(x)+3*x+24)^2)+1),x, algorithm="giac")

[Out]

x^2/(e^(2*x*e^(6*x + 2*e^x + 48)) + 2*e^(x*e^(6*x + 2*e^x + 48)) + 1)

Mupad [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.48 \[ \int \frac {2 x+e^{e^{48+2 e^x+6 x} x} \left (2 x+e^{48+2 e^x+6 x} \left (-2 x^2-12 x^3-4 e^x x^3\right )\right )}{1+3 e^{e^{48+2 e^x+6 x} x}+3 e^{2 e^{48+2 e^x+6 x} x}+e^{3 e^{48+2 e^x+6 x} x}} \, dx=\frac {x^2\,{\mathrm {e}}^{-x\,{\mathrm {e}}^{6\,x}\,{\mathrm {e}}^{48}\,{\mathrm {e}}^{2\,{\mathrm {e}}^x}}}{4\,{\mathrm {cosh}\left (\frac {x\,{\mathrm {e}}^{6\,x}\,{\mathrm {e}}^{48}\,{\mathrm {e}}^{2\,{\mathrm {e}}^x}}{2}\right )}^2} \]

[In]

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)

[Out]

(x^2*exp(-x*exp(6*x)*exp(48)*exp(2*exp(x))))/(4*cosh((x*exp(6*x)*exp(48)*exp(2*exp(x)))/2)^2)