\(\int e^{58 x+60 x^2-2 e^{e^{-2 x}} (x+x^2)} (e^{2 x} (2 x+60 x^2+120 x^3)+e^{e^{-2 x}} (4 x^3+4 x^4+e^{2 x} (-2 x^2-4 x^3))) \, dx\) [5715]

Optimal result

Integrand size = 83, antiderivative size = 21 \[ \int e^{58 x+60 x^2-2 e^{e^{-2 x}} \left (x+x^2\right )} \left (e^{2 x} \left (2 x+60 x^2+120 x^3\right )+e^{e^{-2 x}} \left (4 x^3+4 x^4+e^{2 x} \left (-2 x^2-4 x^3\right )\right )\right ) \, dx=e^{-2 \left (-30+e^{e^{-2 x}}\right ) x (1+x)} x^2 \]

[Out]

x^2/exp((1+x)*(exp(1/exp(x)^2)-30)*x)^2

Rubi [F]

\[ \int e^{58 x+60 x^2-2 e^{e^{-2 x}} \left (x+x^2\right )} \left (e^{2 x} \left (2 x+60 x^2+120 x^3\right )+e^{e^{-2 x}} \left (4 x^3+4 x^4+e^{2 x} \left (-2 x^2-4 x^3\right )\right )\right ) \, dx=\int e^{58 x+60 x^2-2 e^{e^{-2 x}} \left (x+x^2\right )} \left (e^{2 x} \left (2 x+60 x^2+120 x^3\right )+e^{e^{-2 x}} \left (4 x^3+4 x^4+e^{2 x} \left (-2 x^2-4 x^3\right )\right )\right ) \, dx \]

[In]

Int[E^(58*x + 60*x^2 - 2*E^E^(-2*x)*(x + x^2))*(E^(2*x)*(2*x + 60*x^2 + 120*x^3) + E^E^(-2*x)*(4*x^3 + 4*x^4 +
E^(2*x)*(-2*x^2 - 4*x^3))),x]

[Out]

2*Defer[Int][x/E^(2*(-30 + E^E^(-2*x))*x*(1 + x)), x] + 60*Defer[Int][x^2/E^(2*(-30 + E^E^(-2*x))*x*(1 + x)),
x] - 2*Defer[Int][E^(E^(-2*x) + 2*x - 2*x*(-29 + E^E^(-2*x) - 30*x + E^E^(-2*x)*x))*x^2, x] + 120*Defer[Int][x
^3/E^(2*(-30 + E^E^(-2*x))*x*(1 + x)), x] + 4*Defer[Int][E^(E^(-2*x) - 2*x*(-29 + E^E^(-2*x) - 30*x + E^E^(-2*
x)*x))*x^3, x] - 4*Defer[Int][E^(E^(-2*x) + 2*x - 2*x*(-29 + E^E^(-2*x) - 30*x + E^E^(-2*x)*x))*x^3, x] + 4*De
fer[Int][E^(E^(-2*x) - 2*x*(-29 + E^E^(-2*x) - 30*x + E^E^(-2*x)*x))*x^4, x]

Rubi steps \begin{align*} \text {integral}& = \int \exp \left (-2 x \left (-29+e^{e^{-2 x}}-30 x+e^{e^{-2 x}} x\right )\right ) \left (e^{2 x} \left (2 x+60 x^2+120 x^3\right )+e^{e^{-2 x}} \left (4 x^3+4 x^4+e^{2 x} \left (-2 x^2-4 x^3\right )\right )\right ) \, dx \\ & = \int \left (2 \exp \left (e^{-2 x}-2 x \left (-29+e^{e^{-2 x}}-30 x+e^{e^{-2 x}} x\right )\right ) x^2 \left (-e^{2 x}+2 x-2 e^{2 x} x+2 x^2\right )+2 \exp \left (2 x-2 x \left (-29+e^{e^{-2 x}}-30 x+e^{e^{-2 x}} x\right )\right ) x \left (1+30 x+60 x^2\right )\right ) \, dx \\ & = 2 \int \exp \left (e^{-2 x}-2 x \left (-29+e^{e^{-2 x}}-30 x+e^{e^{-2 x}} x\right )\right ) x^2 \left (-e^{2 x}+2 x-2 e^{2 x} x+2 x^2\right ) \, dx+2 \int \exp \left (2 x-2 x \left (-29+e^{e^{-2 x}}-30 x+e^{e^{-2 x}} x\right )\right ) x \left (1+30 x+60 x^2\right ) \, dx \\ & = 2 \int e^{-2 \left (-30+e^{e^{-2 x}}\right ) x (1+x)} x \left (1+30 x+60 x^2\right ) \, dx+2 \int \left (2 \exp \left (e^{-2 x}-2 x \left (-29+e^{e^{-2 x}}-30 x+e^{e^{-2 x}} x\right )\right ) x^3 (1+x)-\exp \left (e^{-2 x}+2 x-2 x \left (-29+e^{e^{-2 x}}-30 x+e^{e^{-2 x}} x\right )\right ) x^2 (1+2 x)\right ) \, dx \\ & = -\left (2 \int \exp \left (e^{-2 x}+2 x-2 x \left (-29+e^{e^{-2 x}}-30 x+e^{e^{-2 x}} x\right )\right ) x^2 (1+2 x) \, dx\right )+2 \int \left (e^{-2 \left (-30+e^{e^{-2 x}}\right ) x (1+x)} x+30 e^{-2 \left (-30+e^{e^{-2 x}}\right ) x (1+x)} x^2+60 e^{-2 \left (-30+e^{e^{-2 x}}\right ) x (1+x)} x^3\right ) \, dx+4 \int \exp \left (e^{-2 x}-2 x \left (-29+e^{e^{-2 x}}-30 x+e^{e^{-2 x}} x\right )\right ) x^3 (1+x) \, dx \\ & = 2 \int e^{-2 \left (-30+e^{e^{-2 x}}\right ) x (1+x)} x \, dx-2 \int \left (\exp \left (e^{-2 x}+2 x-2 x \left (-29+e^{e^{-2 x}}-30 x+e^{e^{-2 x}} x\right )\right ) x^2+2 \exp \left (e^{-2 x}+2 x-2 x \left (-29+e^{e^{-2 x}}-30 x+e^{e^{-2 x}} x\right )\right ) x^3\right ) \, dx+4 \int \left (\exp \left (e^{-2 x}-2 x \left (-29+e^{e^{-2 x}}-30 x+e^{e^{-2 x}} x\right )\right ) x^3+\exp \left (e^{-2 x}-2 x \left (-29+e^{e^{-2 x}}-30 x+e^{e^{-2 x}} x\right )\right ) x^4\right ) \, dx+60 \int e^{-2 \left (-30+e^{e^{-2 x}}\right ) x (1+x)} x^2 \, dx+120 \int e^{-2 \left (-30+e^{e^{-2 x}}\right ) x (1+x)} x^3 \, dx \\ & = 2 \int e^{-2 \left (-30+e^{e^{-2 x}}\right ) x (1+x)} x \, dx-2 \int \exp \left (e^{-2 x}+2 x-2 x \left (-29+e^{e^{-2 x}}-30 x+e^{e^{-2 x}} x\right )\right ) x^2 \, dx+4 \int \exp \left (e^{-2 x}-2 x \left (-29+e^{e^{-2 x}}-30 x+e^{e^{-2 x}} x\right )\right ) x^3 \, dx-4 \int \exp \left (e^{-2 x}+2 x-2 x \left (-29+e^{e^{-2 x}}-30 x+e^{e^{-2 x}} x\right )\right ) x^3 \, dx+4 \int \exp \left (e^{-2 x}-2 x \left (-29+e^{e^{-2 x}}-30 x+e^{e^{-2 x}} x\right )\right ) x^4 \, dx+60 \int e^{-2 \left (-30+e^{e^{-2 x}}\right ) x (1+x)} x^2 \, dx+120 \int e^{-2 \left (-30+e^{e^{-2 x}}\right ) x (1+x)} x^3 \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 5.09 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int e^{58 x+60 x^2-2 e^{e^{-2 x}} \left (x+x^2\right )} \left (e^{2 x} \left (2 x+60 x^2+120 x^3\right )+e^{e^{-2 x}} \left (4 x^3+4 x^4+e^{2 x} \left (-2 x^2-4 x^3\right )\right )\right ) \, dx=e^{-2 \left (-30+e^{e^{-2 x}}\right ) x (1+x)} x^2 \]

[In]

Integrate[E^(58*x + 60*x^2 - 2*E^E^(-2*x)*(x + x^2))*(E^(2*x)*(2*x + 60*x^2 + 120*x^3) + E^E^(-2*x)*(4*x^3 + 4
*x^4 + E^(2*x)*(-2*x^2 - 4*x^3))),x]

[Out]

x^2/E^(2*(-30 + E^E^(-2*x))*x*(1 + x))

Maple [A] (verified)

Time = 0.43 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90

method	result	size
risch	\(x^{2} {\mathrm e}^{-2 x \left (1+x \right ) \left ({\mathrm e}^{{\mathrm e}^{-2 x}}-30\right )}\)	\(19\)
parallelrisch	\(x^{2} {\mathrm e}^{-2 \left (x^{2}+x \right ) {\mathrm e}^{{\mathrm e}^{-2 x}}+60 x^{2}+60 x}\)	\(28\)

[In]

int((((-4*x^3-2*x^2)*exp(x)^2+4*x^4+4*x^3)*exp(1/exp(x)^2)+(120*x^3+60*x^2+2*x)*exp(x)^2)/exp(x)^2/exp((x^2+x)
*exp(1/exp(x)^2)-30*x^2-30*x)^2,x,method=_RETURNVERBOSE)

[Out]

x^2*exp(-2*x*(1+x)*(exp(exp(-2*x))-30))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.24 \[ \int e^{58 x+60 x^2-2 e^{e^{-2 x}} \left (x+x^2\right )} \left (e^{2 x} \left (2 x+60 x^2+120 x^3\right )+e^{e^{-2 x}} \left (4 x^3+4 x^4+e^{2 x} \left (-2 x^2-4 x^3\right )\right )\right ) \, dx=x^{2} e^{\left (60 \, x^{2} - 2 \, {\left (x^{2} + x\right )} e^{\left (e^{\left (-2 \, x\right )}\right )} + 60 \, x\right )} \]

[In]

integrate((((-4*x^3-2*x^2)*exp(x)^2+4*x^4+4*x^3)*exp(1/exp(x)^2)+(120*x^3+60*x^2+2*x)*exp(x)^2)/exp(x)^2/exp((
x^2+x)*exp(1/exp(x)^2)-30*x^2-30*x)^2,x, algorithm="fricas")

[Out]

x^2*e^(60*x^2 - 2*(x^2 + x)*e^(e^(-2*x)) + 60*x)

Sympy [A] (verification not implemented)

Time = 9.75 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.29 \[ \int e^{58 x+60 x^2-2 e^{e^{-2 x}} \left (x+x^2\right )} \left (e^{2 x} \left (2 x+60 x^2+120 x^3\right )+e^{e^{-2 x}} \left (4 x^3+4 x^4+e^{2 x} \left (-2 x^2-4 x^3\right )\right )\right ) \, dx=x^{2} e^{60 x^{2} + 60 x - 2 \left (x^{2} + x\right ) e^{e^{- 2 x}}} \]

[In]

integrate((((-4*x**3-2*x**2)*exp(x)**2+4*x**4+4*x**3)*exp(1/exp(x)**2)+(120*x**3+60*x**2+2*x)*exp(x)**2)/exp(x
)**2/exp((x**2+x)*exp(1/exp(x)**2)-30*x**2-30*x)**2,x)

[Out]

x**2*exp(60*x**2 + 60*x - 2*(x**2 + x)*exp(exp(-2*x)))

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.52 \[ \int e^{58 x+60 x^2-2 e^{e^{-2 x}} \left (x+x^2\right )} \left (e^{2 x} \left (2 x+60 x^2+120 x^3\right )+e^{e^{-2 x}} \left (4 x^3+4 x^4+e^{2 x} \left (-2 x^2-4 x^3\right )\right )\right ) \, dx=x^{2} e^{\left (-2 \, x^{2} e^{\left (e^{\left (-2 \, x\right )}\right )} + 60 \, x^{2} - 2 \, x e^{\left (e^{\left (-2 \, x\right )}\right )} + 60 \, x\right )} \]

[In]

integrate((((-4*x^3-2*x^2)*exp(x)^2+4*x^4+4*x^3)*exp(1/exp(x)^2)+(120*x^3+60*x^2+2*x)*exp(x)^2)/exp(x)^2/exp((
x^2+x)*exp(1/exp(x)^2)-30*x^2-30*x)^2,x, algorithm="maxima")

[Out]

x^2*e^(-2*x^2*e^(e^(-2*x)) + 60*x^2 - 2*x*e^(e^(-2*x)) + 60*x)

Giac [F]

\[ \int e^{58 x+60 x^2-2 e^{e^{-2 x}} \left (x+x^2\right )} \left (e^{2 x} \left (2 x+60 x^2+120 x^3\right )+e^{e^{-2 x}} \left (4 x^3+4 x^4+e^{2 x} \left (-2 x^2-4 x^3\right )\right )\right ) \, dx=\int { 2 \, {\left ({\left (60 \, x^{3} + 30 \, x^{2} + x\right )} e^{\left (2 \, x\right )} + {\left (2 \, x^{4} + 2 \, x^{3} - {\left (2 \, x^{3} + x^{2}\right )} e^{\left (2 \, x\right )}\right )} e^{\left (e^{\left (-2 \, x\right )}\right )}\right )} e^{\left (60 \, x^{2} - 2 \, {\left (x^{2} + x\right )} e^{\left (e^{\left (-2 \, x\right )}\right )} + 58 \, x\right )} \,d x } \]

[In]

integrate((((-4*x^3-2*x^2)*exp(x)^2+4*x^4+4*x^3)*exp(1/exp(x)^2)+(120*x^3+60*x^2+2*x)*exp(x)^2)/exp(x)^2/exp((
x^2+x)*exp(1/exp(x)^2)-30*x^2-30*x)^2,x, algorithm="giac")

[Out]

integrate(2*((60*x^3 + 30*x^2 + x)*e^(2*x) + (2*x^4 + 2*x^3 - (2*x^3 + x^2)*e^(2*x))*e^(e^(-2*x)))*e^(60*x^2 -
2*(x^2 + x)*e^(e^(-2*x)) + 58*x), x)

Mupad [B] (verification not implemented)

Time = 12.64 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.62 \[ \int e^{58 x+60 x^2-2 e^{e^{-2 x}} \left (x+x^2\right )} \left (e^{2 x} \left (2 x+60 x^2+120 x^3\right )+e^{e^{-2 x}} \left (4 x^3+4 x^4+e^{2 x} \left (-2 x^2-4 x^3\right )\right )\right ) \, dx=x^2\,{\mathrm {e}}^{60\,x}\,{\mathrm {e}}^{-2\,x^2\,{\mathrm {e}}^{{\mathrm {e}}^{-2\,x}}}\,{\mathrm {e}}^{60\,x^2}\,{\mathrm {e}}^{-2\,x\,{\mathrm {e}}^{{\mathrm {e}}^{-2\,x}}} \]

[In]

int(exp(-2*x)*exp(60*x - 2*exp(exp(-2*x))*(x + x^2) + 60*x^2)*(exp(exp(-2*x))*(4*x^3 - exp(2*x)*(2*x^2 + 4*x^3
) + 4*x^4) + exp(2*x)*(2*x + 60*x^2 + 120*x^3)),x)

[Out]

x^2*exp(60*x)*exp(-2*x^2*exp(exp(-2*x)))*exp(60*x^2)*exp(-2*x*exp(exp(-2*x)))