3.91.58 \(\int \frac {e^x (-4-4 x)-4 e^x x \log (x)}{x} \, dx\)

Optimal. Leaf size=22 \[ \log (4)+4 \left (-9+e^5-e^x-e^x \log (x)\right ) \]

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Rubi [A]  time = 0.04, antiderivative size = 13, normalized size of antiderivative = 0.59, number of steps used = 6, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {14, 2194, 2178, 2554} \begin {gather*} -4 e^x-4 e^x \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^x*(-4 - 4*x) - 4*E^x*x*Log[x])/x,x]

[Out]

-4*E^x - 4*E^x*Log[x]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 2178

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - (c*f)/d))*ExpIntegral
Ei[(f*g*(c + d*x)*Log[F])/d])/d, x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !$UseGamma === True

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 2554

Int[Log[u_]*(v_), x_Symbol] :> With[{w = IntHide[v, x]}, Dist[Log[u], w, x] - Int[SimplifyIntegrand[(w*D[u, x]
)/u, x], x] /; InverseFunctionFreeQ[w, x]] /; InverseFunctionFreeQ[u, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-4 e^x-\frac {4 e^x}{x}-4 e^x \log (x)\right ) \, dx\\ &=-\left (4 \int e^x \, dx\right )-4 \int \frac {e^x}{x} \, dx-4 \int e^x \log (x) \, dx\\ &=-4 e^x-4 \text {Ei}(x)-4 e^x \log (x)+4 \int \frac {e^x}{x} \, dx\\ &=-4 e^x-4 e^x \log (x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 12, normalized size = 0.55 \begin {gather*} -4 \left (e^x+e^x \log (x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^x*(-4 - 4*x) - 4*E^x*x*Log[x])/x,x]

[Out]

-4*(E^x + E^x*Log[x])

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fricas [A]  time = 0.42, size = 11, normalized size = 0.50 \begin {gather*} -4 \, e^{x} \log \relax (x) - 4 \, e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4*e^x*log(x) - 4*e^x

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giac [A]  time = 0.13, size = 11, normalized size = 0.50 \begin {gather*} -4 \, e^{x} \log \relax (x) - 4 \, e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4*e^x*log(x) - 4*e^x

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maple [A]  time = 0.02, size = 12, normalized size = 0.55

method	result	size
norman	\(-4 \,{\mathrm e}^{x} \ln \relax (x )-4 \,{\mathrm e}^{x}\)	\(12\)
risch	\(-4 \,{\mathrm e}^{x} \ln \relax (x )-4 \,{\mathrm e}^{x}\)	\(12\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4*exp(x)*ln(x)-4*exp(x)

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maxima [A]  time = 0.38, size = 11, normalized size = 0.50 \begin {gather*} -4 \, e^{x} \log \relax (x) - 4 \, e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4*e^x*log(x) - 4*e^x

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mupad [B]  time = 7.08, size = 8, normalized size = 0.36 \begin {gather*} -4\,{\mathrm {e}}^x\,\left (\ln \relax (x)+1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4*exp(x)*(log(x) + 1)

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sympy [A]  time = 0.28, size = 10, normalized size = 0.45 \begin {gather*} \left (- 4 \log {\relax (x )} - 4\right ) e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-4*log(x) - 4)*exp(x)

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