3.96.16 \(\int \frac {1}{4} e^{-13-6 e^{\frac {3 x}{e^3}}} (-e^3+e^{\frac {3 x}{e^3}} (-18+18 x)) \, dx\)

Optimal. Leaf size=23 \[ \frac {1}{4} e^{-10-6 e^{\frac {3 x}{e^3}}} (1-x) \]

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Rubi [A]  time = 0.03, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {12, 2288} \begin {gather*} \frac {1}{4} e^{-6 e^{\frac {3 x}{e^3}}-10} (1-x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(-13 - 6*E^((3*x)/E^3))*(-E^3 + E^((3*x)/E^3)*(-18 + 18*x)))/4,x]

[Out]

(E^(-10 - 6*E^((3*x)/E^3))*(1 - 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^{-13-6 e^{\frac {3 x}{e^3}}} \left (-e^3+e^{\frac {3 x}{e^3}} (-18+18 x)\right ) \, dx\\ &=\frac {1}{4} e^{-10-6 e^{\frac {3 x}{e^3}}} (1-x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 21, normalized size = 0.91 \begin {gather*} -\frac {1}{4} e^{-10-6 e^{\frac {3 x}{e^3}}} (-1+x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(-13 - 6*E^((3*x)/E^3))*(-E^3 + E^((3*x)/E^3)*(-18 + 18*x)))/4,x]

[Out]

-1/4*(E^(-10 - 6*E^((3*x)/E^3))*(-1 + x))

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fricas [A]  time = 0.54, size = 16, normalized size = 0.70 \begin {gather*} -\frac {1}{4} \, {\left (x - 1\right )} e^{\left (-6 \, e^{\left (3 \, x e^{\left (-3\right )}\right )} - 10\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*((18*x-18)*exp(3*x/exp(3))-exp(3))/exp(3)/exp(3*exp(3*x/exp(3))+5)^2,x, algorithm="fricas")

[Out]

-1/4*(x - 1)*e^(-6*e^(3*x*e^(-3)) - 10)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{4} \, {\left (18 \, {\left (x - 1\right )} e^{\left (3 \, x e^{\left (-3\right )}\right )} - e^{3}\right )} e^{\left (-6 \, e^{\left (3 \, x e^{\left (-3\right )}\right )} - 13\right )}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*((18*x-18)*exp(3*x/exp(3))-exp(3))/exp(3)/exp(3*exp(3*x/exp(3))+5)^2,x, algorithm="giac")

[Out]

integrate(1/4*(18*(x - 1)*e^(3*x*e^(-3)) - e^3)*e^(-6*e^(3*x*e^(-3)) - 13), x)

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maple [A]  time = 0.08, size = 17, normalized size = 0.74

method	result	size
risch	\(-\frac {\left (x -1\right ) {\mathrm e}^{-6 \,{\mathrm e}^{3 x \,{\mathrm e}^{-3}}-10}}{4}\)	\(17\)
norman	\(\left (\frac {1}{4}-\frac {x}{4}\right ) {\mathrm e}^{-6 \,{\mathrm e}^{3 x \,{\mathrm e}^{-3}}-10}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/4*((18*x-18)*exp(3*x/exp(3))-exp(3))/exp(3)/exp(3*exp(3*x/exp(3))+5)^2,x,method=_RETURNVERBOSE)

[Out]

-1/4*(x-1)*exp(-6*exp(3*x*exp(-3))-10)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\frac {1}{12} \, {\rm Ei}\left (-6 \, e^{\left (3 \, x e^{\left (-3\right )}\right )}\right ) e^{\left (-7\right )} - \frac {1}{4} \, x e^{\left (-6 \, e^{\left (3 \, x e^{\left (-3\right )}\right )} - 10\right )} + \frac {1}{4} \, e^{\left (-6 \, e^{\left (3 \, x e^{\left (-3\right )}\right )} - 10\right )} + \frac {1}{4} \, \int e^{\left (-6 \, e^{\left (3 \, x e^{\left (-3\right )}\right )} - 10\right )}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*((18*x-18)*exp(3*x/exp(3))-exp(3))/exp(3)/exp(3*exp(3*x/exp(3))+5)^2,x, algorithm="maxima")

[Out]

-1/12*Ei(-6*e^(3*x*e^(-3)))*e^(-7) - 1/4*x*e^(-6*e^(3*x*e^(-3)) - 10) + 1/4*e^(-6*e^(3*x*e^(-3)) - 10) + 1/4*i
ntegrate(e^(-6*e^(3*x*e^(-3)) - 10), x)

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mupad [B]  time = 0.15, size = 16, normalized size = 0.70 \begin {gather*} -\frac {{\mathrm {e}}^{-10}\,{\mathrm {e}}^{-6\,{\mathrm {e}}^{3\,x\,{\mathrm {e}}^{-3}}}\,\left (x-1\right )}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(-3)*exp(- 6*exp(3*x*exp(-3)) - 10)*(exp(3)/4 - (exp(3*x*exp(-3))*(18*x - 18))/4),x)

[Out]

-(exp(-10)*exp(-6*exp(3*x*exp(-3)))*(x - 1))/4

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sympy [A]  time = 0.19, size = 19, normalized size = 0.83 \begin {gather*} \frac {\left (1 - x\right ) e^{- 6 e^{\frac {3 x}{e^{3}}} - 10}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*((18*x-18)*exp(3*x/exp(3))-exp(3))/exp(3)/exp(3*exp(3*x/exp(3))+5)**2,x)

[Out]

(1 - x)*exp(-6*exp(3*x*exp(-3)) - 10)/4

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