3.70.43 \(\int \frac {e^{4+2 x} (-4+8 x)}{x^2 \log (3)} \, dx\)

Optimal. Leaf size=26 \[ -e^4+\frac {4 e^4 \left (1+\frac {e^{2 x}}{x}\right )}{\log (3)} \]

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Rubi [A]  time = 0.03, antiderivative size = 16, normalized size of antiderivative = 0.62, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {12, 2197} \begin {gather*} \frac {4 e^{2 x+4}}{x \log (3)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(4 + 2*x)*(-4 + 8*x))/(x^2*Log[3]),x]

[Out]

(4*E^(4 + 2*x))/(x*Log[3])

Rule 12

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

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 {\int \frac {e^{4+2 x} (-4+8 x)}{x^2} \, dx}{\log (3)}\\ &=\frac {4 e^{4+2 x}}{x \log (3)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 16, normalized size = 0.62 \begin {gather*} \frac {4 e^{4+2 x}}{x \log (3)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(4 + 2*x)*(-4 + 8*x))/(x^2*Log[3]),x]

[Out]

(4*E^(4 + 2*x))/(x*Log[3])

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fricas [A]  time = 0.92, size = 15, normalized size = 0.58 \begin {gather*} \frac {4 \, e^{\left (2 \, x + 4\right )}}{x \log \relax (3)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*x-4)*exp(4)*exp(x)^2/x^2/log(3),x, algorithm="fricas")

[Out]

4*e^(2*x + 4)/(x*log(3))

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giac [A]  time = 0.19, size = 15, normalized size = 0.58 \begin {gather*} \frac {4 \, e^{\left (2 \, x + 4\right )}}{x \log \relax (3)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*x-4)*exp(4)*exp(x)^2/x^2/log(3),x, algorithm="giac")

[Out]

4*e^(2*x + 4)/(x*log(3))

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maple [A]  time = 0.05, size = 16, normalized size = 0.62

method	result	size
gosper	\(\frac {4 \,{\mathrm e}^{4} {\mathrm e}^{2 x}}{\ln \relax (3) x}\)	\(16\)
default	\(\frac {4 \,{\mathrm e}^{4} {\mathrm e}^{2 x}}{\ln \relax (3) x}\)	\(16\)
norman	\(\frac {4 \,{\mathrm e}^{4} {\mathrm e}^{2 x}}{\ln \relax (3) x}\)	\(16\)
risch	\(\frac {4 \,{\mathrm e}^{2 x +4}}{x \ln \relax (3)}\)	\(16\)
meijerg	\(\frac {8 \,{\mathrm e}^{4} \left (\ln \relax (x )+\ln \relax (2)+i \pi -\ln \left (-2 x \right )-\expIntegralEi \left (1, -2 x \right )\right )}{\ln \relax (3)}+\frac {8 \,{\mathrm e}^{4} \left (\frac {1}{2 x}+1-\ln \relax (x )-\ln \relax (2)-i \pi -\frac {4 x +2}{4 x}+\frac {{\mathrm e}^{2 x}}{2 x}+\ln \left (-2 x \right )+\expIntegralEi \left (1, -2 x \right )\right )}{\ln \relax (3)}\)	\(87\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((8*x-4)*exp(4)*exp(x)^2/x^2/ln(3),x,method=_RETURNVERBOSE)

[Out]

4*exp(4)*exp(x)^2/ln(3)/x

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maxima [C]  time = 0.42, size = 23, normalized size = 0.88 \begin {gather*} \frac {8 \, {\left ({\rm Ei}\left (2 \, x\right ) e^{4} - e^{4} \Gamma \left (-1, -2 \, x\right )\right )}}{\log \relax (3)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*x-4)*exp(4)*exp(x)^2/x^2/log(3),x, algorithm="maxima")

[Out]

8*(Ei(2*x)*e^4 - e^4*gamma(-1, -2*x))/log(3)

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mupad [B]  time = 4.13, size = 15, normalized size = 0.58 \begin {gather*} \frac {4\,{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^4}{x\,\ln \relax (3)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(2*x)*exp(4)*(8*x - 4))/(x^2*log(3)),x)

[Out]

(4*exp(2*x)*exp(4))/(x*log(3))

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sympy [A]  time = 0.10, size = 14, normalized size = 0.54 \begin {gather*} \frac {4 e^{4} e^{2 x}}{x \log {\relax (3 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*x-4)*exp(4)*exp(x)**2/x**2/ln(3),x)

[Out]

4*exp(4)*exp(2*x)/(x*log(3))

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