\(\int \frac {1}{4} e^{e^x} (-9 e+e^x (54+e (-18-9 x))) \, dx\) [292]

Optimal result

Integrand size = 26, antiderivative size = 17 \[ \int \frac {1}{4} e^{e^x} \left (-9 e+e^x (54+e (-18-9 x))\right ) \, dx=\frac {9}{4} e^{e^x} (6-e (2+x)) \]

[Out]

9/4*exp(exp(x))*(6-(2+x)*exp(1))

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {12, 2326} \[ \int \frac {1}{4} e^{e^x} \left (-9 e+e^x (54+e (-18-9 x))\right ) \, dx=\frac {9}{4} e^{e^x} (6-e (x+2)) \]

[In]

Int[(E^E^x*(-9*E + E^x*(54 + E*(-18 - 9*x))))/4,x]

[Out]

(9*E^E^x*(6 - E*(2 + x)))/4

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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}& = \frac {1}{4} \int e^{e^x} \left (-9 e+e^x (54+e (-18-9 x))\right ) \, dx \\ & = \frac {9}{4} e^{e^x} (6-e (2+x)) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.94 \[ \int \frac {1}{4} e^{e^x} \left (-9 e+e^x (54+e (-18-9 x))\right ) \, dx=-\frac {9}{4} e^{e^x} (-6+e (2+x)) \]

[In]

Integrate[(E^E^x*(-9*E + E^x*(54 + E*(-18 - 9*x))))/4,x]

[Out]

(-9*E^E^x*(-6 + E*(2 + x)))/4

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00

method	result	size
risch	\(\frac {\left (-9 x \,{\mathrm e}-18 \,{\mathrm e}+54\right ) {\mathrm e}^{{\mathrm e}^{x}}}{4}\)	\(17\)
norman	\(\left (-\frac {9 \,{\mathrm e}}{2}+\frac {27}{2}\right ) {\mathrm e}^{{\mathrm e}^{x}}-\frac {9 x \,{\mathrm e} \,{\mathrm e}^{{\mathrm e}^{x}}}{4}\)	\(20\)
parallelrisch	\(-\frac {9 x \,{\mathrm e} \,{\mathrm e}^{{\mathrm e}^{x}}}{4}-\frac {9 \,{\mathrm e} \,{\mathrm e}^{{\mathrm e}^{x}}}{2}+\frac {27 \,{\mathrm e}^{{\mathrm e}^{x}}}{2}\)	\(22\)

[In]

int(1/4*(((-9*x-18)*exp(1)+54)*exp(x)-9*exp(1))*exp(exp(x)),x,method=_RETURNVERBOSE)

[Out]

1/4*(-9*x*exp(1)-18*exp(1)+54)*exp(exp(x))

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int \frac {1}{4} e^{e^x} \left (-9 e+e^x (54+e (-18-9 x))\right ) \, dx=-\frac {9}{4} \, {\left ({\left (x + 2\right )} e - 6\right )} e^{\left (e^{x}\right )} \]

[In]

integrate(1/4*(((-9*x-18)*exp(1)+54)*exp(x)-9*exp(1))*exp(exp(x)),x, algorithm="fricas")

[Out]

-9/4*((x + 2)*e - 6)*e^(e^x)

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.12 \[ \int \frac {1}{4} e^{e^x} \left (-9 e+e^x (54+e (-18-9 x))\right ) \, dx=\frac {\left (- 9 e x - 18 e + 54\right ) e^{e^{x}}}{4} \]

[In]

integrate(1/4*(((-9*x-18)*exp(1)+54)*exp(x)-9*exp(1))*exp(exp(x)),x)

[Out]

(-9*E*x - 18*E + 54)*exp(exp(x))/4

Maxima [F]

\[ \int \frac {1}{4} e^{e^x} \left (-9 e+e^x (54+e (-18-9 x))\right ) \, dx=\int { -\frac {9}{4} \, {\left ({\left ({\left (x + 2\right )} e - 6\right )} e^{x} + e\right )} e^{\left (e^{x}\right )} \,d x } \]

[In]

integrate(1/4*(((-9*x-18)*exp(1)+54)*exp(x)-9*exp(1))*exp(exp(x)),x, algorithm="maxima")

[Out]

-9/4*Ei(e^x)*e - 9/4*x*e^(e^x + 1) - 9/2*e^(e^x + 1) + 27/2*e^(e^x) + 9/4*integrate(e^(e^x + 1), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 30 vs. \(2 (13) = 26\).

Time = 0.26 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.76 \[ \int \frac {1}{4} e^{e^x} \left (-9 e+e^x (54+e (-18-9 x))\right ) \, dx=-\frac {9}{4} \, {\left (x e^{\left (x + e^{x} + 1\right )} + 2 \, e^{\left (x + e^{x} + 1\right )} - 6 \, e^{\left (x + e^{x}\right )}\right )} e^{\left (-x\right )} \]

[In]

integrate(1/4*(((-9*x-18)*exp(1)+54)*exp(x)-9*exp(1))*exp(exp(x)),x, algorithm="giac")

[Out]

-9/4*(x*e^(x + e^x + 1) + 2*e^(x + e^x + 1) - 6*e^(x + e^x))*e^(-x)

Mupad [B] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int \frac {1}{4} e^{e^x} \left (-9 e+e^x (54+e (-18-9 x))\right ) \, dx=-\frac {9\,{\mathrm {e}}^{{\mathrm {e}}^x}\,\left (2\,\mathrm {e}+x\,\mathrm {e}-6\right )}{4} \]

[In]

int(-(exp(exp(x))*(9*exp(1) + exp(x)*(exp(1)*(9*x + 18) - 54)))/4,x)

[Out]

-(9*exp(exp(x))*(2*exp(1) + x*exp(1) - 6))/4