3.2.72 \(\int e^x x \log (x) \, dx\) [172]

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

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.04, antiderivative size = 22, normalized size of antiderivative = 1.00, 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 = {2207, 2225, 2634, 2230, 2209} \begin {gather*} \text {ExpIntegralEi}(x)-e^x-e^x \log (x)+e^x x \log (x) \end {gather*}

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 2207

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] &&  !TrueQ[$UseGamma]

Rule 2209

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

Rule 2225

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 2230

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] &&  !TrueQ[$UseGamma]

Rule 2634

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 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 = 0.01, size = 17, normalized size = 0.77 \begin {gather*} -e^x+\text {Ei}(x)+e^x (-1+x) \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 0.01, size = 21, normalized size = 0.95

method	result	size
risch	\(\left (-1+x \right ) {\mathrm e}^{x} \ln \left (x \right )-\expIntegral \left (1, -x \right )-{\mathrm e}^{x}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x)*x*ln(x),x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 5.68, size = 15, normalized size = 0.68 \begin {gather*} {\left (x - 1\right )} e^{x} \log \left (x\right ) + {\rm Ei}\left (x\right ) - e^{x} \end {gather*}

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

________________________________________________________________________________________

Fricas [A]
time = 0.46, size = 15, normalized size = 0.68 \begin {gather*} {\left (x - 1\right )} e^{x} \log \left (x\right ) + {\rm Ei}\left (x\right ) - e^{x} \end {gather*}

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

________________________________________________________________________________________

Sympy [A]
time = 1.67, size = 17, normalized size = 0.77 \begin {gather*} \left (x e^{x} - e^{x}\right ) \log {\left (x \right )} - e^{x} + \operatorname {Ei}{\left (x \right )} \end {gather*}

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)

________________________________________________________________________________________

Giac [A]
time = 0.75, size = 15, normalized size = 0.68 \begin {gather*} {\left (x - 1\right )} e^{x} \log \left (x\right ) + {\rm Ei}\left (x\right ) - e^{x} \end {gather*}

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

________________________________________________________________________________________

Mupad [B]
time = 0.22, size = 28, normalized size = 1.27 \begin {gather*} \mathrm {ei}\left (x\right )-\frac {x\,{\mathrm {e}}^x+x\,{\mathrm {e}}^x\,\ln \left (x\right )-x^2\,{\mathrm {e}}^x\,\ln \left (x\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ei(x) - (x*exp(x) + x*exp(x)*log(x) - x^2*exp(x)*log(x))/x

________________________________________________________________________________________