\(\int \frac {-e^{-e+e^{-e+x}+x}-2 x}{e^{2 e^{-e+x}}+2 e^{e^{-e+x}} x^2+x^4} \, dx\) [1339]

Optimal result

Integrand size = 52, antiderivative size = 15 \[ \int \frac {-e^{-e+e^{-e+x}+x}-2 x}{e^{2 e^{-e+x}}+2 e^{e^{-e+x}} x^2+x^4} \, dx=\frac {1}{e^{e^{-e+x}}+x^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {6873, 6818} \[ \int \frac {-e^{-e+e^{-e+x}+x}-2 x}{e^{2 e^{-e+x}}+2 e^{e^{-e+x}} x^2+x^4} \, dx=\frac {1}{x^2+e^{e^{x-e}}} \]

[In]

Int[(-E^(-E + E^(-E + x) + x) - 2*x)/(E^(2*E^(-E + x)) + 2*E^E^(-E + x)*x^2 + x^4),x]

[Out]

(E^E^(-E + x) + x^2)^(-1)

Rule 6818

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*(y^(m + 1)/(m + 1)), x] /;  !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

Rule 6873

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]

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

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {-e^{-e+e^{-e+x}+x}-2 x}{e^{2 e^{-e+x}}+2 e^{e^{-e+x}} x^2+x^4} \, dx=\frac {1}{e^{e^{-e+x}}+x^2} \]

[In]

Integrate[(-E^(-E + E^(-E + x) + x) - 2*x)/(E^(2*E^(-E + x)) + 2*E^E^(-E + x)*x^2 + x^4),x]

[Out]

(E^E^(-E + x) + x^2)^(-1)

Maple [A] (verified)

Time = 0.16 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00

method	result	size
norman	\(\frac {1}{x^{2}+{\mathrm e}^{{\mathrm e}^{x -{\mathrm e}}}}\)	\(15\)
risch	\(\frac {1}{x^{2}+{\mathrm e}^{{\mathrm e}^{x -{\mathrm e}}}}\)	\(15\)
parallelrisch	\(\frac {1}{x^{2}+{\mathrm e}^{{\mathrm e}^{x -{\mathrm e}}}}\)	\(15\)

[In]

int((-exp(x-exp(1))*exp(exp(x-exp(1)))-2*x)/(exp(exp(x-exp(1)))^2+2*x^2*exp(exp(x-exp(1)))+x^4),x,method=_RETU
RNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.93 \[ \int \frac {-e^{-e+e^{-e+x}+x}-2 x}{e^{2 e^{-e+x}}+2 e^{e^{-e+x}} x^2+x^4} \, dx=\frac {1}{x^{2} + e^{\left (e^{\left (x - e\right )}\right )}} \]

[In]

integrate((-exp(x-exp(1))*exp(exp(x-exp(1)))-2*x)/(exp(exp(x-exp(1)))^2+2*x^2*exp(exp(x-exp(1)))+x^4),x, algor
ithm="fricas")

[Out]

1/(x^2 + e^(e^(x - e)))

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.80 \[ \int \frac {-e^{-e+e^{-e+x}+x}-2 x}{e^{2 e^{-e+x}}+2 e^{e^{-e+x}} x^2+x^4} \, dx=\frac {1}{x^{2} + e^{e^{x - e}}} \]

[In]

integrate((-exp(x-exp(1))*exp(exp(x-exp(1)))-2*x)/(exp(exp(x-exp(1)))**2+2*x**2*exp(exp(x-exp(1)))+x**4),x)

[Out]

1/(x**2 + exp(exp(x - E)))

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.93 \[ \int \frac {-e^{-e+e^{-e+x}+x}-2 x}{e^{2 e^{-e+x}}+2 e^{e^{-e+x}} x^2+x^4} \, dx=\frac {1}{x^{2} + e^{\left (e^{\left (x - e\right )}\right )}} \]

[In]

integrate((-exp(x-exp(1))*exp(exp(x-exp(1)))-2*x)/(exp(exp(x-exp(1)))^2+2*x^2*exp(exp(x-exp(1)))+x^4),x, algor
ithm="maxima")

[Out]

1/(x^2 + e^(e^(x - e)))

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.93 \[ \int \frac {-e^{-e+e^{-e+x}+x}-2 x}{e^{2 e^{-e+x}}+2 e^{e^{-e+x}} x^2+x^4} \, dx=\frac {1}{x^{2} + e^{\left (e^{\left (x - e\right )}\right )}} \]

[In]

integrate((-exp(x-exp(1))*exp(exp(x-exp(1)))-2*x)/(exp(exp(x-exp(1)))^2+2*x^2*exp(exp(x-exp(1)))+x^4),x, algor
ithm="giac")

[Out]

1/(x^2 + e^(e^(x - e)))

Mupad [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {-e^{-e+e^{-e+x}+x}-2 x}{e^{2 e^{-e+x}}+2 e^{e^{-e+x}} x^2+x^4} \, dx=\frac {1}{{\mathrm {e}}^{{\mathrm {e}}^{-\mathrm {e}}\,{\mathrm {e}}^x}+x^2} \]

[In]

int(-(2*x + exp(x - exp(1))*exp(exp(x - exp(1))))/(exp(2*exp(x - exp(1))) + 2*x^2*exp(exp(x - exp(1))) + x^4),
x)

[Out]

1/(exp(exp(-exp(1))*exp(x)) + x^2)