3.63.95 \(\int e^{\frac {1+e^{67} (-20+3 x-e^2 x)}{e^{67}}} (3-e^2) \, dx\)

Optimal. Leaf size=15 \[ e^{-20+\frac {1}{e^{67}}-\left (-3+e^2\right ) x} \]

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Rubi [A]  time = 0.02, antiderivative size = 25, normalized size of antiderivative = 1.67, number of steps used = 3, number of rules used = 3, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {12, 2227, 2194} \begin {gather*} e^{\frac {e^{67} \left (3-e^2\right ) x-20 e^{67}+1}{e^{67}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^((1 + E^67*(-20 + 3*x - E^2*x))/E^67)*(3 - E^2),x]

[Out]

E^((1 - 20*E^67 + E^67*(3 - E^2)*x)/E^67)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 2227

Int[(u_.)*(F_)^((a_.) + (b_.)*(v_)), x_Symbol] :> Int[u*F^(a + b*NormalizePowerOfLinear[v, x]), x] /; FreeQ[{F
, a, b}, x] && PolynomialQ[u, x] && PowerOfLinearQ[v, x] &&  !PowerOfLinearMatchQ[v, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\left (3-e^2\right ) \int e^{\frac {1+e^{67} \left (-20+3 x-e^2 x\right )}{e^{67}}} \, dx\\ &=\left (3-e^2\right ) \int e^{\frac {1-20 e^{67}+e^{67} \left (3-e^2\right ) x}{e^{67}}} \, dx\\ &=e^{\frac {1-20 e^{67}+e^{67} \left (3-e^2\right ) x}{e^{67}}}\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.05, size = 32, normalized size = 2.13 \begin {gather*} -\frac {e^{-20+\frac {1}{e^{67}}+3 x-e^2 x} \left (-3+e^2\right )}{3-e^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^((1 + E^67*(-20 + 3*x - E^2*x))/E^67)*(3 - E^2),x]

[Out]

-((E^(-20 + E^(-67) + 3*x - E^2*x)*(-3 + E^2))/(3 - E^2))

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fricas [A]  time = 0.94, size = 20, normalized size = 1.33 \begin {gather*} e^{\left (-{\left (x e^{69} - {\left (3 \, x - 20\right )} e^{67} - 1\right )} e^{\left (-67\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-exp(2)+3)*exp(((-exp(2)*x+3*x-20)*exp(67)+1)/exp(67)),x, algorithm="fricas")

[Out]

e^(-(x*e^69 - (3*x - 20)*e^67 - 1)*e^(-67))

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giac [A]  time = 0.17, size = 19, normalized size = 1.27 \begin {gather*} e^{\left (-{\left ({\left (x e^{2} - 3 \, x + 20\right )} e^{67} - 1\right )} e^{\left (-67\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-exp(2)+3)*exp(((-exp(2)*x+3*x-20)*exp(67)+1)/exp(67)),x, algorithm="giac")

[Out]

e^(-((x*e^2 - 3*x + 20)*e^67 - 1)*e^(-67))

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maple [A]  time = 0.05, size = 22, normalized size = 1.47

method	result	size
derivativedivides	\({\mathrm e}^{\left (\left (-{\mathrm e}^{2} x +3 x -20\right ) {\mathrm e}^{67}+1\right ) {\mathrm e}^{-67}}\)	\(22\)
default	\({\mathrm e}^{\left (\left (-{\mathrm e}^{2} x +3 x -20\right ) {\mathrm e}^{67}+1\right ) {\mathrm e}^{-67}}\)	\(22\)
norman	\({\mathrm e}^{\left (\left (-{\mathrm e}^{2} x +3 x -20\right ) {\mathrm e}^{67}+1\right ) {\mathrm e}^{-67}}\)	\(22\)
gosper	\({\mathrm e}^{-\left ({\mathrm e}^{2} {\mathrm e}^{67} x -3 \,{\mathrm e}^{67} x +20 \,{\mathrm e}^{67}-1\right ) {\mathrm e}^{-67}}\)	\(25\)
risch	\(\frac {{\mathrm e}^{\left (-x \,{\mathrm e}^{69}+3 \,{\mathrm e}^{67} x -20 \,{\mathrm e}^{67}+1\right ) {\mathrm e}^{-67}} {\mathrm e}^{2}}{{\mathrm e}^{2}-3}-\frac {3 \,{\mathrm e}^{\left (-x \,{\mathrm e}^{69}+3 \,{\mathrm e}^{67} x -20 \,{\mathrm e}^{67}+1\right ) {\mathrm e}^{-67}}}{{\mathrm e}^{2}-3}\)	\(59\)
meijerg	\(-\frac {{\mathrm e}^{2+{\mathrm e}^{-67} \left (-20 \,{\mathrm e}^{67}+1\right )} \left (1-{\mathrm e}^{-x \left ({\mathrm e}^{2}-3\right )}\right )}{{\mathrm e}^{2}-3}+\frac {3 \,{\mathrm e}^{{\mathrm e}^{-67} \left (-20 \,{\mathrm e}^{67}+1\right )} \left (1-{\mathrm e}^{-x \left ({\mathrm e}^{2}-3\right )}\right )}{{\mathrm e}^{2}-3}\)	\(64\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-exp(2)+3)*exp(((-exp(2)*x+3*x-20)*exp(67)+1)/exp(67)),x,method=_RETURNVERBOSE)

[Out]

exp(((-exp(2)*x+3*x-20)*exp(67)+1)/exp(67))

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maxima [A]  time = 0.36, size = 19, normalized size = 1.27 \begin {gather*} e^{\left (-{\left ({\left (x e^{2} - 3 \, x + 20\right )} e^{67} - 1\right )} e^{\left (-67\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-exp(2)+3)*exp(((-exp(2)*x+3*x-20)*exp(67)+1)/exp(67)),x, algorithm="maxima")

[Out]

e^(-((x*e^2 - 3*x + 20)*e^67 - 1)*e^(-67))

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mupad [B]  time = 0.09, size = 16, normalized size = 1.07 \begin {gather*} {\mathrm {e}}^{3\,x}\,{\mathrm {e}}^{-20}\,{\mathrm {e}}^{-x\,{\mathrm {e}}^2}\,{\mathrm {e}}^{{\mathrm {e}}^{-67}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(-exp(-67)*(exp(67)*(x*exp(2) - 3*x + 20) - 1))*(exp(2) - 3),x)

[Out]

exp(3*x)*exp(-20)*exp(-x*exp(2))*exp(exp(-67))

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sympy [A]  time = 0.12, size = 19, normalized size = 1.27 \begin {gather*} e^{\frac {\left (- x e^{2} + 3 x - 20\right ) e^{67} + 1}{e^{67}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-exp(2)+3)*exp(((-exp(2)*x+3*x-20)*exp(67)+1)/exp(67)),x)

[Out]

exp(((-x*exp(2) + 3*x - 20)*exp(67) + 1)*exp(-67))

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