3.74.28 \(\int (2 x+4 x \log (x)+e^{e^{-4-e+x}} (-x+(-2 x-e^{-4-e+x} x^2) \log (x))) \, dx\)

Optimal. Leaf size=20 \[ \left (2-e^{e^{-4-e+x}}\right ) x^2 \log (x) \]

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Rubi [A]  time = 0.43, antiderivative size = 25, normalized size of antiderivative = 1.25, number of steps used = 5, number of rules used = 4, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2304, 6742, 2288, 2554} \begin {gather*} 2 x^2 \log (x)-e^{e^{x-e-4}} x^2 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[2*x + 4*x*Log[x] + E^E^(-4 - E + x)*(-x + (-2*x - E^(-4 - E + x)*x^2)*Log[x]),x]

[Out]

2*x^2*Log[x] - E^E^(-4 - E + x)*x^2*Log[x]

Rule 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rule 2304

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log[c*x^
n]))/(d*(m + 1)), x] - Simp[(b*n*(d*x)^(m + 1))/(d*(m + 1)^2), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

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 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=x^2+4 \int x \log (x) \, dx+\int e^{e^{-4-e+x}} \left (-x+\left (-2 x-e^{-4-e+x} x^2\right ) \log (x)\right ) \, dx\\ &=2 x^2 \log (x)+\int \left (-e^{e^{-4-e+x}} x-e^{-4-e+e^{-4-e+x}} x \left (2 e^{4+e}+e^x x\right ) \log (x)\right ) \, dx\\ &=2 x^2 \log (x)-\int e^{e^{-4-e+x}} x \, dx-\int e^{-4-e+e^{-4-e+x}} x \left (2 e^{4+e}+e^x x\right ) \log (x) \, dx\\ &=2 x^2 \log (x)-e^{e^{-4-e+x}} x^2 \log (x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.22, size = 19, normalized size = 0.95 \begin {gather*} -\left (\left (-2+e^{e^{-4-e+x}}\right ) x^2 \log (x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[2*x + 4*x*Log[x] + E^E^(-4 - E + x)*(-x + (-2*x - E^(-4 - E + x)*x^2)*Log[x]),x]

[Out]

-((-2 + E^E^(-4 - E + x))*x^2*Log[x])

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fricas [A]  time = 0.49, size = 24, normalized size = 1.20 \begin {gather*} -x^{2} e^{\left (e^{\left (x - e - 4\right )}\right )} \log \relax (x) + 2 \, x^{2} \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^2*exp(x-exp(1)-4)-2*x)*log(x)-x)*exp(exp(x-exp(1)-4))+4*x*log(x)+2*x,x, algorithm="fricas")

[Out]

-x^2*e^(e^(x - e - 4))*log(x) + 2*x^2*log(x)

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giac [A]  time = 0.14, size = 24, normalized size = 1.20 \begin {gather*} -x^{2} e^{\left (e^{\left (x - e - 4\right )}\right )} \log \relax (x) + 2 \, x^{2} \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^2*exp(x-exp(1)-4)-2*x)*log(x)-x)*exp(exp(x-exp(1)-4))+4*x*log(x)+2*x,x, algorithm="giac")

[Out]

-x^2*e^(e^(x - e - 4))*log(x) + 2*x^2*log(x)

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maple [A]  time = 0.06, size = 25, normalized size = 1.25

method	result	size
risch	\(-\ln \relax (x ) x^{2} {\mathrm e}^{{\mathrm e}^{x -{\mathrm e}-4}}+2 x^{2} \ln \relax (x )\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-x^2*exp(x-exp(1)-4)-2*x)*ln(x)-x)*exp(exp(x-exp(1)-4))+4*x*ln(x)+2*x,x,method=_RETURNVERBOSE)

[Out]

-ln(x)*x^2*exp(exp(x-exp(1)-4))+2*x^2*ln(x)

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maxima [A]  time = 0.36, size = 24, normalized size = 1.20 \begin {gather*} -x^{2} e^{\left (e^{\left (x - e - 4\right )}\right )} \log \relax (x) + 2 \, x^{2} \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^2*exp(x-exp(1)-4)-2*x)*log(x)-x)*exp(exp(x-exp(1)-4))+4*x*log(x)+2*x,x, algorithm="maxima")

[Out]

-x^2*e^(e^(x - e - 4))*log(x) + 2*x^2*log(x)

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mupad [B]  time = 4.56, size = 20, normalized size = 1.00 \begin {gather*} -x^2\,\ln \relax (x)\,\left ({\mathrm {e}}^{{\mathrm {e}}^{-\mathrm {e}}\,{\mathrm {e}}^{-4}\,{\mathrm {e}}^x}-2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2*x - exp(exp(x - exp(1) - 4))*(x + log(x)*(2*x + x^2*exp(x - exp(1) - 4))) + 4*x*log(x),x)

[Out]

-x^2*log(x)*(exp(exp(-exp(1))*exp(-4)*exp(x)) - 2)

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sympy [A]  time = 12.55, size = 24, normalized size = 1.20 \begin {gather*} - x^{2} e^{e^{x - 4 - e}} \log {\relax (x )} + 2 x^{2} \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x**2*exp(x-exp(1)-4)-2*x)*ln(x)-x)*exp(exp(x-exp(1)-4))+4*x*ln(x)+2*x,x)

[Out]

-x**2*exp(exp(x - 4 - E))*log(x) + 2*x**2*log(x)

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