3.29.43 \(\int \frac {e^{e^3 (-9+3 x)} (-2+3 e^3 x)}{x^3} \, dx\)

Optimal. Leaf size=14 \[ \frac {e^{3 e^3 (-3+x)}}{x^2} \]

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Rubi [A]  time = 0.04, antiderivative size = 16, normalized size of antiderivative = 1.14, number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {2197} \begin {gather*} \frac {e^{-3 e^3 (3-x)}}{x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(E^3*(-9 + 3*x))*(-2 + 3*E^3*x))/x^3,x]

[Out]

1/(E^(3*E^3*(3 - x))*x^2)

Rule 2197

Int[(F_)^((c_.)*(v_))*(u_)^(m_.)*(w_), x_Symbol] :> With[{b = Coefficient[v, x, 1], d = Coefficient[u, x, 0],
e = Coefficient[u, x, 1], f = Coefficient[w, x, 0], g = Coefficient[w, x, 1]}, Simp[(g*u^(m + 1)*F^(c*v))/(b*c
*e*Log[F]), x] /; EqQ[e*g*(m + 1) - b*c*(e*f - d*g)*Log[F], 0]] /; FreeQ[{F, c, m}, x] && LinearQ[{u, v, w}, x
]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {e^{-3 e^3 (3-x)}}{x^2}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 14, normalized size = 1.00 \begin {gather*} \frac {e^{3 e^3 (-3+x)}}{x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(E^3*(-9 + 3*x))*(-2 + 3*E^3*x))/x^3,x]

[Out]

E^(3*E^3*(-3 + x))/x^2

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fricas [A]  time = 0.42, size = 12, normalized size = 0.86 \begin {gather*} \frac {e^{\left (3 \, {\left (x - 3\right )} e^{3}\right )}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x*exp(3)-2)*exp((3*x-9)*exp(3))/x^3,x, algorithm="fricas")

[Out]

e^(3*(x - 3)*e^3)/x^2

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giac [A]  time = 0.33, size = 15, normalized size = 1.07 \begin {gather*} \frac {e^{\left (3 \, x e^{3} - 9 \, e^{3}\right )}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x*exp(3)-2)*exp((3*x-9)*exp(3))/x^3,x, algorithm="giac")

[Out]

e^(3*x*e^3 - 9*e^3)/x^2

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maple [A]  time = 0.16, size = 13, normalized size = 0.93

method	result	size
gosper	\(\frac {{\mathrm e}^{3 \,{\mathrm e}^{3} \left (x -3\right )}}{x^{2}}\)	\(13\)
risch	\(\frac {{\mathrm e}^{3 \,{\mathrm e}^{3} \left (x -3\right )}}{x^{2}}\)	\(13\)
norman	\(\frac {{\mathrm e}^{\left (3 x -9\right ) {\mathrm e}^{3}}}{x^{2}}\)	\(14\)
derivativedivides	\(\frac {{\mathrm e}^{3 x \,{\mathrm e}^{3}-9 \,{\mathrm e}^{3}}}{x^{2}}\)	\(16\)
default	\(\frac {{\mathrm e}^{3 x \,{\mathrm e}^{3}-9 \,{\mathrm e}^{3}}}{x^{2}}\)	\(16\)
meijerg	\(-9 \,{\mathrm e}^{6-9 \,{\mathrm e}^{3}} \left (\frac {{\mathrm e}^{-3}}{3 x}-2-\ln \relax (x )-\ln \relax (3)-i \pi -\frac {{\mathrm e}^{-3} \left (6 x \,{\mathrm e}^{3}+2\right )}{6 x}+\frac {{\mathrm e}^{3 x \,{\mathrm e}^{3}-3}}{3 x}+\ln \left (-3 x \,{\mathrm e}^{3}\right )+\expIntegralEi \left (1, -3 x \,{\mathrm e}^{3}\right )\right )-18 \,{\mathrm e}^{6-9 \,{\mathrm e}^{3}} \left (-\frac {{\mathrm e}^{-6}}{18 x^{2}}-\frac {{\mathrm e}^{-3}}{3 x}+\frac {3}{4}+\frac {\ln \relax (x )}{2}+\frac {\ln \relax (3)}{2}+\frac {i \pi }{2}+\frac {{\mathrm e}^{-6} \left (81 x^{2} {\mathrm e}^{6}+36 x \,{\mathrm e}^{3}+6\right )}{108 x^{2}}-\frac {{\mathrm e}^{-6+3 x \,{\mathrm e}^{3}} \left (9 x \,{\mathrm e}^{3}+3\right )}{54 x^{2}}-\frac {\ln \left (-3 x \,{\mathrm e}^{3}\right )}{2}-\frac {\expIntegralEi \left (1, -3 x \,{\mathrm e}^{3}\right )}{2}\right )\)	\(167\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x*exp(3)-2)*exp((3*x-9)*exp(3))/x^3,x,method=_RETURNVERBOSE)

[Out]

1/x^2*exp(3*exp(3)*(x-3))

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maxima [C]  time = 0.71, size = 33, normalized size = 2.36 \begin {gather*} 9 \, e^{\left (-9 \, e^{3} + 6\right )} \Gamma \left (-1, -3 \, x e^{3}\right ) + 18 \, e^{\left (-9 \, e^{3} + 6\right )} \Gamma \left (-2, -3 \, x e^{3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x*exp(3)-2)*exp((3*x-9)*exp(3))/x^3,x, algorithm="maxima")

[Out]

9*e^(-9*e^3 + 6)*gamma(-1, -3*x*e^3) + 18*e^(-9*e^3 + 6)*gamma(-2, -3*x*e^3)

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mupad [B]  time = 0.09, size = 15, normalized size = 1.07 \begin {gather*} \frac {{\mathrm {e}}^{-9\,{\mathrm {e}}^3}\,{\mathrm {e}}^{3\,x\,{\mathrm {e}}^3}}{x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(exp(3)*(3*x - 9))*(3*x*exp(3) - 2))/x^3,x)

[Out]

(exp(-9*exp(3))*exp(3*x*exp(3)))/x^2

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sympy [A]  time = 0.11, size = 12, normalized size = 0.86 \begin {gather*} \frac {e^{\left (3 x - 9\right ) e^{3}}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x*exp(3)-2)*exp((3*x-9)*exp(3))/x**3,x)

[Out]

exp((3*x - 9)*exp(3))/x**2

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