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

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

[Out]

-ExpIntegralEi[x] + E^x*Log[x]

________________________________________________________________________________________

Rubi [A]  time = 0.0159023, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 3, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, 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 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]

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

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*}

Mathematica [A]  time = 0.0055287, size = 11, normalized size = 1. \[ e^x \log (x)-\text{ExpIntegralEi}(x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-ExpIntegralEi[x] + E^x*Log[x]

________________________________________________________________________________________

Maple [A]  time = 0.01, size = 12, normalized size = 1.1 \begin{align*}{{\rm e}^{x}}\ln \left ( x \right ) +{\it Ei} \left ( 1,-x \right ) \end{align*}

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)

________________________________________________________________________________________

Maxima [A]  time = 1.0295, size = 14, normalized size = 1.27 \begin{align*} e^{x} \log \left (x\right ) -{\rm Ei}\left (x\right ) \end{align*}

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)

________________________________________________________________________________________

Fricas [A]  time = 1.7651, size = 27, normalized size = 2.45 \begin{align*} e^{x} \log \left (x\right ) -{\rm Ei}\left (x\right ) \end{align*}

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)

________________________________________________________________________________________

Sympy [A]  time = 1.98923, size = 8, normalized size = 0.73 \begin{align*} e^{x} \log{\left (x \right )} - \operatorname{Ei}{\left (x \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(x)*log(x) - Ei(x)

________________________________________________________________________________________

Giac [A]  time = 1.07051, size = 14, normalized size = 1.27 \begin{align*} e^{x} \log \left (x\right ) -{\rm Ei}\left (x\right ) \end{align*}

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)