3.26.38 \(\int (e^2+e^{e^{2 x}} (-e^2-2 e^{2+2 x} x)) \, dx\)

Optimal. Leaf size=22 \[ e^2 \left (x-\left (e^{e^{2 x}}+\frac {15}{x}\right ) x\right ) \]

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Rubi [A]  time = 0.02, antiderivative size = 18, normalized size of antiderivative = 0.82, number of steps used = 2, number of rules used = 1, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {2288} \begin {gather*} e^2 x-e^{e^{2 x}+2} x \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

E^2*x - E^(2 + E^(2*x))*x

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} &=e^2 x+\int e^{e^{2 x}} \left (-e^2-2 e^{2+2 x} x\right ) \, dx\\ &=e^2 x-e^{2+e^{2 x}} x\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 18, normalized size = 0.82 \begin {gather*} -e^2 \left (-x+e^{e^{2 x}} x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(E^2*(-x + E^E^(2*x)*x))

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fricas [A]  time = 1.08, size = 15, normalized size = 0.68 \begin {gather*} x e^{2} - x e^{\left (e^{\left (2 \, x\right )} + 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*e^2 - x*e^(e^(2*x) + 2)

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giac [A]  time = 0.40, size = 15, normalized size = 0.68 \begin {gather*} x e^{2} - x e^{\left (e^{\left (2 \, x\right )} + 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*e^2 - x*e^(e^(2*x) + 2)

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maple [A]  time = 0.07, size = 16, normalized size = 0.73

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*x*exp(2)*exp(2*x)-exp(2))*exp(exp(2*x))+exp(2),x,method=_RETURNVERBOSE)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.06, size = 12, normalized size = 0.55 \begin {gather*} -x\,{\mathrm {e}}^2\,\left ({\mathrm {e}}^{{\mathrm {e}}^{2\,x}}-1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x*exp(2)*(exp(exp(2*x)) - 1)

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sympy [A]  time = 0.15, size = 15, normalized size = 0.68 \begin {gather*} - x e^{2} e^{e^{2 x}} + x e^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x*exp(2)*exp(exp(2*x)) + x*exp(2)

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