\(\int (e^2+e^{e^{2 x}} (-e^2-2 e^{2+2 x} x)) \, dx\) [2538]

Optimal result

Integrand size = 28, antiderivative size = 22 \[ \int \left (e^2+e^{e^{2 x}} \left (-e^2-2 e^{2+2 x} x\right )\right ) \, dx=e^2 \left (x-\left (e^{e^{2 x}}+\frac {15}{x}\right ) x\right ) \]

[Out]

(x-(exp(exp(2*x))+15/x)*x)*exp(2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {2326} \[ \int \left (e^2+e^{e^{2 x}} \left (-e^2-2 e^{2+2 x} x\right )\right ) \, dx=e^2 x-e^{e^{2 x}+2} x \]

[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 2326

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{align*} \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{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82 \[ \int \left (e^2+e^{e^{2 x}} \left (-e^2-2 e^{2+2 x} x\right )\right ) \, dx=-e^2 \left (-x+e^{e^{2 x}} x\right ) \]

[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))

Maple [A] (verified)

Time = 0.21 (sec) , antiderivative size = 16, normalized size of antiderivative = 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	\(-x \,{\mathrm e}^{2+{\mathrm e}^{2 x}}+{\mathrm e}^{2} x\)	\(16\)
parallelrisch	\(-{\mathrm e}^{2} x \,{\mathrm e}^{{\mathrm e}^{2 x}}+{\mathrm e}^{2} x\)	\(16\)

[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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.68 \[ \int \left (e^2+e^{e^{2 x}} \left (-e^2-2 e^{2+2 x} x\right )\right ) \, dx=x e^{2} - x e^{\left (e^{\left (2 \, x\right )} + 2\right )} \]

[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)

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.68 \[ \int \left (e^2+e^{e^{2 x}} \left (-e^2-2 e^{2+2 x} x\right )\right ) \, dx=- x e^{2} e^{e^{2 x}} + x e^{2} \]

[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)

Maxima [F]

\[ \int \left (e^2+e^{e^{2 x}} \left (-e^2-2 e^{2+2 x} x\right )\right ) \, dx=\int { -{\left (2 \, x e^{\left (2 \, x + 2\right )} + e^{2}\right )} e^{\left (e^{\left (2 \, x\right )}\right )} + e^{2} \,d x } \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.68 \[ \int \left (e^2+e^{e^{2 x}} \left (-e^2-2 e^{2+2 x} x\right )\right ) \, dx=x e^{2} - x e^{\left (e^{\left (2 \, x\right )} + 2\right )} \]

[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)

Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.55 \[ \int \left (e^2+e^{e^{2 x}} \left (-e^2-2 e^{2+2 x} x\right )\right ) \, dx=-x\,{\mathrm {e}}^2\,\left ({\mathrm {e}}^{{\mathrm {e}}^{2\,x}}-1\right ) \]

[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)