3.29.14 \(\int \frac {e^{-8+4 x} (2-24 x+8 x \log (x))}{x} \, dx\)

Optimal. Leaf size=19 \[ 1-2 e^{-4 (2-x)} (3-\log (x)) \]

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Rubi [A]  time = 0.05, antiderivative size = 21, normalized size of antiderivative = 1.11, number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {2288} \begin {gather*} -\frac {2 e^{4 x-8} (3 x-x \log (x))}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(-8 + 4*x)*(2 - 24*x + 8*x*Log[x]))/x,x]

[Out]

(-2*E^(-8 + 4*x)*(3*x - x*Log[x]))/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 {2 e^{-8+4 x} (3 x-x \log (x))}{x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 13, normalized size = 0.68 \begin {gather*} 2 e^{-8+4 x} (-3+\log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(-8 + 4*x)*(2 - 24*x + 8*x*Log[x]))/x,x]

[Out]

2*E^(-8 + 4*x)*(-3 + Log[x])

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fricas [A]  time = 1.67, size = 19, normalized size = 1.00 \begin {gather*} 2 \, e^{\left (4 \, x - 8\right )} \log \relax (x) - 6 \, e^{\left (4 \, x - 8\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*x*log(x)-24*x+2)/x/exp(-4*x+8),x, algorithm="fricas")

[Out]

2*e^(4*x - 8)*log(x) - 6*e^(4*x - 8)

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giac [A]  time = 0.30, size = 18, normalized size = 0.95 \begin {gather*} 2 \, {\left (e^{\left (4 \, x\right )} \log \relax (x) - 3 \, e^{\left (4 \, x\right )}\right )} e^{\left (-8\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*x*log(x)-24*x+2)/x/exp(-4*x+8),x, algorithm="giac")

[Out]

2*(e^(4*x)*log(x) - 3*e^(4*x))*e^(-8)

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maple [A]  time = 0.03, size = 16, normalized size = 0.84

method	result	size
norman	\(\left (2 \ln \relax (x )-6\right ) {\mathrm e}^{4 x -8}\)	\(16\)
risch	\(2 \ln \relax (x ) {\mathrm e}^{4 x -8}-6 \,{\mathrm e}^{4 x -8}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((8*x*ln(x)-24*x+2)/x/exp(-4*x+8),x,method=_RETURNVERBOSE)

[Out]

(2*ln(x)-6)/exp(-4*x+8)

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maxima [A]  time = 0.78, size = 19, normalized size = 1.00 \begin {gather*} 2 \, e^{\left (4 \, x - 8\right )} \log \relax (x) - 6 \, e^{\left (4 \, x - 8\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*x*log(x)-24*x+2)/x/exp(-4*x+8),x, algorithm="maxima")

[Out]

2*e^(4*x - 8)*log(x) - 6*e^(4*x - 8)

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mupad [B]  time = 1.86, size = 13, normalized size = 0.68 \begin {gather*} {\mathrm {e}}^{4\,x-8}\,\left (2\,\ln \relax (x)-6\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(4*x - 8)*(8*x*log(x) - 24*x + 2))/x,x)

[Out]

exp(4*x - 8)*(2*log(x) - 6)

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sympy [A]  time = 0.29, size = 12, normalized size = 0.63 \begin {gather*} \left (2 \log {\relax (x )} - 6\right ) e^{4 x - 8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*x*ln(x)-24*x+2)/x/exp(-4*x+8),x)

[Out]

(2*log(x) - 6)*exp(4*x - 8)

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