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

Optimal. Leaf size=17 \[ \frac {9}{4} e^{e^x} (6-e (2+x)) \]

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Rubi [A]  time = 0.02, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {12, 2288} \begin {gather*} \frac {9}{4} e^{e^x} (6-e (x+2)) \end {gather*}

Antiderivative was successfully verified.

[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 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 {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 {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 16, normalized size = 0.94 \begin {gather*} -\frac {9}{4} e^{e^x} (-6+e (2+x)) \end {gather*}

Antiderivative was successfully verified.

[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

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fricas [A]  time = 0.69, size = 13, normalized size = 0.76 \begin {gather*} -\frac {9}{4} \, {\left ({\left (x + 2\right )} e - 6\right )} e^{\left (e^{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [B]  time = 0.64, size = 30, normalized size = 1.76 \begin {gather*} -\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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [A]  time = 0.07, size = 17, normalized size = 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	\(\left (-\frac {9 \,{\mathrm e}}{2}+\frac {27}{2}\right ) {\mathrm e}^{{\mathrm e}^{x}}-\frac {9 x \,{\mathrm e} \,{\mathrm e}^{{\mathrm e}^{x}}}{4}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[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))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\frac {9}{4} \, {\rm Ei}\left (e^{x}\right ) e - \frac {9}{4} \, x e^{\left (e^{x} + 1\right )} - \frac {9}{2} \, e^{\left (e^{x} + 1\right )} + \frac {27}{2} \, e^{\left (e^{x}\right )} + \frac {9}{4} \, \int e^{\left (e^{x} + 1\right )}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 0.08, size = 15, normalized size = 0.88 \begin {gather*} -\frac {9\,{\mathrm {e}}^{{\mathrm {e}}^x}\,\left (2\,\mathrm {e}+x\,\mathrm {e}-6\right )}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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sympy [A]  time = 0.18, size = 19, normalized size = 1.12 \begin {gather*} \frac {\left (- 9 e x - 18 e + 54\right ) e^{e^{x}}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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