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

Optimal. Leaf size=15 \[ \frac {1}{e^{e^{-e+x}}+x^2} \]

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Rubi [A]  time = 0.28, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 52, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {6741, 6686} \begin {gather*} \frac {1}{x^2+e^{e^{x-e}}} \end {gather*}

Antiderivative was successfully verified.

[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 6686

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 6741

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

Rubi steps

\begin {gather*} \begin {aligned} \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 {aligned} \end {gather*}

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Mathematica [A]  time = 0.06, size = 15, normalized size = 1.00 \begin {gather*} \frac {1}{e^{e^{-e+x}}+x^2} \end {gather*}

Antiderivative was successfully verified.

[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)

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fricas [A]  time = 0.90, size = 14, normalized size = 0.93 \begin {gather*} \frac {1}{x^{2} + e^{\left (e^{\left (x - e\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)))

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giac [A]  time = 0.37, size = 14, normalized size = 0.93 \begin {gather*} \frac {1}{x^{2} + e^{\left (e^{\left (x - e\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)))

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maple [A]  time = 0.15, size = 15, normalized size = 1.00

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

Verification of antiderivative is not currently implemented for this CAS.

[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))))

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maxima [A]  time = 0.48, size = 14, normalized size = 0.93 \begin {gather*} \frac {1}{x^{2} + e^{\left (e^{\left (x - e\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)))

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mupad [B]  time = 0.16, size = 15, normalized size = 1.00 \begin {gather*} \frac {1}{{\mathrm {e}}^{{\mathrm {e}}^{-\mathrm {e}}\,{\mathrm {e}}^x}+x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [A]  time = 0.12, size = 12, normalized size = 0.80 \begin {gather*} \frac {1}{x^{2} + e^{e^{x - e}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)))

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