3.47.89 \(\int \frac {-e^{2 x}-2 e^{2 x} x \log (x)}{x} \, dx\)

Optimal. Leaf size=11 \[ 5-e^{2 x} \log (x) \]

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Rubi [A]  time = 0.04, antiderivative size = 9, normalized size of antiderivative = 0.82, number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {14, 2178, 2194, 2554, 12} \begin {gather*} -e^{2 x} \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-E^(2*x) - 2*E^(2*x)*x*Log[x])/x,x]

[Out]

-(E^(2*x)*Log[x])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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 (-\frac {e^{2 x}}{x}-2 e^{2 x} \log (x)\right ) \, dx\\ &=-\left (2 \int e^{2 x} \log (x) \, dx\right )-\int \frac {e^{2 x}}{x} \, dx\\ &=-\text {Ei}(2 x)-e^{2 x} \log (x)+2 \int \frac {e^{2 x}}{2 x} \, dx\\ &=-\text {Ei}(2 x)-e^{2 x} \log (x)+\int \frac {e^{2 x}}{x} \, dx\\ &=-e^{2 x} \log (x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 9, normalized size = 0.82 \begin {gather*} -e^{2 x} \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-E^(2*x) - 2*E^(2*x)*x*Log[x])/x,x]

[Out]

-(E^(2*x)*Log[x])

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fricas [A]  time = 0.90, size = 8, normalized size = 0.73 \begin {gather*} -e^{\left (2 \, x\right )} \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-e^(2*x)*log(x)

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giac [A]  time = 0.22, size = 8, normalized size = 0.73 \begin {gather*} -e^{\left (2 \, x\right )} \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-e^(2*x)*log(x)

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maple [A]  time = 0.04, size = 9, normalized size = 0.82

method	result	size
norman	\(-{\mathrm e}^{2 x} \ln \relax (x )\)	\(9\)
risch	\(-{\mathrm e}^{2 x} \ln \relax (x )\)	\(9\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-exp(x)^2*ln(x)

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maxima [A]  time = 0.39, size = 8, normalized size = 0.73 \begin {gather*} -e^{\left (2 \, x\right )} \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-e^(2*x)*log(x)

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mupad [B]  time = 3.20, size = 8, normalized size = 0.73 \begin {gather*} -{\mathrm {e}}^{2\,x}\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-exp(2*x)*log(x)

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sympy [A]  time = 0.26, size = 8, normalized size = 0.73 \begin {gather*} - e^{2 x} \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x*exp(x)**2*ln(x)-exp(x)**2)/x,x)

[Out]

-exp(2*x)*log(x)

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