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

Optimal. Leaf size=22 \[ \text{ExpIntegralEi}(x)-e^x-e^x \log (x)+e^x x \log (x) \]

[Out]

-E^x + ExpIntegralEi[x] - E^x*Log[x] + E^x*x*Log[x]

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Rubi [A]  time = 0.0521265, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.714, Rules used = {2176, 2194, 2554, 2199, 2178} \[ \text{ExpIntegralEi}(x)-e^x-e^x \log (x)+e^x x \log (x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-E^x + ExpIntegralEi[x] - E^x*Log[x] + E^x*x*Log[x]

Rule 2176

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m
*(b*F^(g*(e + f*x)))^n)/(f*g*n*Log[F]), x] - Dist[(d*m)/(f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !$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]

Rule 2199

Int[(F_)^((c_.)*(v_))*(u_)^(m_.)*(w_), x_Symbol] :> Int[ExpandIntegrand[F^(c*ExpandToSum[v, x]), w*NormalizePo
werOfLinear[u, x]^m, x], x] /; FreeQ[{F, c}, x] && PolynomialQ[w, x] && LinearQ[v, x] && PowerOfLinearQ[u, x]
&& IntegerQ[m] &&  !$UseGamma === True

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 x \log (x) \, dx &=-e^x \log (x)+e^x x \log (x)-\int \frac{e^x (-1+x)}{x} \, dx\\ &=-e^x \log (x)+e^x x \log (x)-\int \left (e^x-\frac{e^x}{x}\right ) \, dx\\ &=-e^x \log (x)+e^x x \log (x)-\int e^x \, dx+\int \frac{e^x}{x} \, dx\\ &=-e^x+\text{Ei}(x)-e^x \log (x)+e^x x \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0110205, size = 17, normalized size = 0.77 \[ \text{ExpIntegralEi}(x)-e^x+e^x (x-1) \log (x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-E^x + ExpIntegralEi[x] + E^x*(-1 + x)*Log[x]

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Maple [A]  time = 0.013, size = 21, normalized size = 1. \begin{align*} \left ( -1+x \right ){{\rm e}^{x}}\ln \left ( x \right ) -{\it Ei} \left ( 1,-x \right ) -{{\rm e}^{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.02932, size = 20, normalized size = 0.91 \begin{align*}{\left (x - 1\right )} e^{x} \log \left (x\right ) +{\rm Ei}\left (x\right ) - e^{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x - 1)*e^x*log(x) + Ei(x) - e^x

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Fricas [A]  time = 1.85356, size = 46, normalized size = 2.09 \begin{align*}{\left (x - 1\right )} e^{x} \log \left (x\right ) +{\rm Ei}\left (x\right ) - e^{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x - 1)*e^x*log(x) + Ei(x) - e^x

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Sympy [A]  time = 3.19639, size = 17, normalized size = 0.77 \begin{align*} \left (x e^{x} - e^{x}\right ) \log{\left (x \right )} - e^{x} + \operatorname{Ei}{\left (x \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.07831, size = 20, normalized size = 0.91 \begin{align*}{\left (x - 1\right )} e^{x} \log \left (x\right ) +{\rm Ei}\left (x\right ) - e^{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x - 1)*e^x*log(x) + Ei(x) - e^x