3.84.3 \(\int e^{e^{e^{e^x} (-2+x^2)}+x} (1+x+e^{e^x+e^{e^x} (-2+x^2)} (2 x^2+e^x (-2 x+x^3))) \, dx\)

Optimal. Leaf size=19 \[ e^{e^{e^{e^x} \left (-2+x^2\right )}+x} x \]

________________________________________________________________________________________

Rubi [B]  time = 0.68, antiderivative size = 110, normalized size of antiderivative = 5.79, number of steps used = 1, number of rules used = 1, integrand size = 56, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.018, Rules used = {2288} \begin {gather*} \frac {e^{e^{-e^{e^x} \left (2-x^2\right )}+x} \left (e^{e^x-e^{e^x} \left (2-x^2\right )} \left (2 x^2-e^x \left (2 x-x^3\right )\right )+x\right )}{e^{-e^{e^x} \left (2-x^2\right )} \left (2 e^{e^x} x-e^{x+e^x} \left (2-x^2\right )\right )+1} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(E^(E^E^x*(-2 + x^2)) + x)*(1 + x + E^(E^x + E^E^x*(-2 + x^2))*(2*x^2 + E^x*(-2*x + x^3))),x]

[Out]

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

Rule 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {e^{e^{-e^{e^x} \left (2-x^2\right )}+x} \left (x+e^{e^x-e^{e^x} \left (2-x^2\right )} \left (2 x^2-e^x \left (2 x-x^3\right )\right )\right )}{1+e^{-e^{e^x} \left (2-x^2\right )} \left (2 e^{e^x} x-e^{e^x+x} \left (2-x^2\right )\right )}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.62, size = 19, normalized size = 1.00 \begin {gather*} e^{e^{e^{e^x} \left (-2+x^2\right )}+x} x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(E^(E^E^x*(-2 + x^2)) + x)*(1 + x + E^(E^x + E^E^x*(-2 + x^2))*(2*x^2 + E^x*(-2*x + x^3))),x]

[Out]

E^(E^(E^E^x*(-2 + x^2)) + x)*x

________________________________________________________________________________________

fricas [A]  time = 0.63, size = 28, normalized size = 1.47 \begin {gather*} x e^{\left ({\left (x e^{\left (e^{x}\right )} + e^{\left ({\left (x^{2} - 2\right )} e^{\left (e^{x}\right )} + e^{x}\right )}\right )} e^{\left (-e^{x}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int {\left ({\left (2 \, x^{2} + {\left (x^{3} - 2 \, x\right )} e^{x}\right )} e^{\left ({\left (x^{2} - 2\right )} e^{\left (e^{x}\right )} + e^{x}\right )} + x + 1\right )} e^{\left (x + e^{\left ({\left (x^{2} - 2\right )} e^{\left (e^{x}\right )}\right )}\right )}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(((2*x^2 + (x^3 - 2*x)*e^x)*e^((x^2 - 2)*e^(e^x) + e^x) + x + 1)*e^(x + e^((x^2 - 2)*e^(e^x))), x)

________________________________________________________________________________________

maple [A]  time = 0.11, size = 16, normalized size = 0.84

method	result	size
risch	\(x \,{\mathrm e}^{{\mathrm e}^{\left (x^{2}-2\right ) {\mathrm e}^{{\mathrm e}^{x}}}+x}\)	\(16\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((x^3-2*x)*exp(x)+2*x^2)*exp(exp(x))*exp((x^2-2)*exp(exp(x)))+x+1)*exp(exp((x^2-2)*exp(exp(x)))+x),x,meth
od=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.55, size = 19, normalized size = 1.00 \begin {gather*} x e^{\left (x + e^{\left (x^{2} e^{\left (e^{x}\right )} - 2 \, e^{\left (e^{x}\right )}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*e^(x + e^(x^2*e^(e^x) - 2*e^(e^x)))

________________________________________________________________________________________

mupad [B]  time = 5.20, size = 20, normalized size = 1.05 \begin {gather*} x\,{\mathrm {e}}^{{\mathrm {e}}^{-2\,{\mathrm {e}}^{{\mathrm {e}}^x}}\,{\mathrm {e}}^{x^2\,{\mathrm {e}}^{{\mathrm {e}}^x}}}\,{\mathrm {e}}^x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x + exp(exp(exp(x))*(x^2 - 2)))*(x - exp(exp(exp(x))*(x^2 - 2))*exp(exp(x))*(exp(x)*(2*x - x^3) - 2*x^
2) + 1),x)

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 12.46, size = 15, normalized size = 0.79 \begin {gather*} x e^{x + e^{\left (x^{2} - 2\right ) e^{e^{x}}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x**3-2*x)*exp(x)+2*x**2)*exp(exp(x))*exp((x**2-2)*exp(exp(x)))+x+1)*exp(exp((x**2-2)*exp(exp(x)))
+x),x)

[Out]

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

________________________________________________________________________________________