3.171 \(\int e^x \log (x) \, dx\)

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

[Out]

-Ei(x)+exp(x)*ln(x)

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Rubi [A]  time = 0.02, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 3, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2194, 2554, 2178} \[ e^x \log (x)-\text {ExpIntegralEi}(x) \]

Antiderivative was successfully verified.

[In]

Int[E^x*Log[x],x]

[Out]

-ExpIntegralEi[x] + E^x*Log[x]

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 {align*} \int e^x \log (x) \, dx &=e^x \log (x)-\int \frac {e^x}{x} \, dx\\ &=-\text {Ei}(x)+e^x \log (x)\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 11, normalized size = 1.00 \[ e^x \log (x)-\text {Ei}(x) \]

Antiderivative was successfully verified.

[In]

Integrate[E^x*Log[x],x]

[Out]

-ExpIntegralEi[x] + E^x*Log[x]

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fricas [A]  time = 0.43, size = 10, normalized size = 0.91 \[ e^{x} \log \relax (x) - {\rm Ei}\relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*log(x),x, algorithm="fricas")

[Out]

e^x*log(x) - Ei(x)

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giac [A]  time = 1.19, size = 10, normalized size = 0.91 \[ e^{x} \log \relax (x) - {\rm Ei}\relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*log(x),x, algorithm="giac")

[Out]

e^x*log(x) - Ei(x)

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maple [A]  time = 0.01, size = 12, normalized size = 1.09 \[ {\mathrm e}^{x} \ln \relax (x )+\Ei \left (1, -x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x)*ln(x),x)

[Out]

exp(x)*ln(x)+Ei(1,-x)

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maxima [A]  time = 0.53, size = 10, normalized size = 0.91 \[ e^{x} \log \relax (x) - {\rm Ei}\relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*log(x),x, algorithm="maxima")

[Out]

e^x*log(x) - Ei(x)

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mupad [B]  time = 0.03, size = 10, normalized size = 0.91 \[ {\mathrm {e}}^x\,\ln \relax (x)-\mathrm {ei}\relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x)*log(x),x)

[Out]

exp(x)*log(x) - ei(x)

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sympy [A]  time = 1.95, size = 8, normalized size = 0.73 \[ e^{x} \log {\relax (x )} - \operatorname {Ei}{\relax (x )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*ln(x),x)

[Out]

exp(x)*log(x) - Ei(x)

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