∫ 81+216x+90x2−108x3−39x4+12x5+4x6+e2(9+18x−3x2−12x3+4x4)+e(−54−126x−18x2+78x3−8x5)+(12x−6x2−6x3+e(−3x+4x2)) log(x)_____________________________________________________________________________________________________ (81x+216x2+90x3−108x4−39x5+12x6+4x7+e2(9x+18x2−3x3−12x4+4x5)+e(−54x−126x2−18x3+78x4−8x6)) log(x) dx

Optimal. Leaf size=30 \[ 1-\frac {1}{(-3+e-x) x \left (-3-\frac {3}{x}+2 x\right )}+\log (\log (x)) \]

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Rubi [A] time = 0.45, antiderivative size = 26, normalized size of antiderivative = 0.87, number of steps used = 5, number of rules used = 4, integrand size = 199, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.020, Rules used = {6688, 1590, 2302, 29} \begin {gather*} \log (\log (x))-\frac {1}{(x-e+3) \left (-2 x^2+3 x+3\right )} \end {gather*}

Int[(81 + 216*x + 90*x^2 - 108*x^3 - 39*x^4 + 12*x^5 + 4*x^6 + E^2*(9 + 18*x - 3*x^2 - 12*x^3 + 4*x^4) + E*(-5

4 - 126*x - 18*x^2 + 78*x^3 - 8*x^5) + (12*x - 6*x^2 - 6*x^3 + E*(-3*x + 4*x^2))*Log[x])/((81*x + 216*x^2 + 90

*x^3 - 108*x^4 - 39*x^5 + 12*x^6 + 4*x^7 + E^2*(9*x + 18*x^2 - 3*x^3 - 12*x^4 + 4*x^5) + E*(-54*x - 126*x^2 -

Int[(Pp_)*(Qq_)^(m_.)*(Rr_)^(n_.), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x], r = Expon[Rr, x]}, S

imp[(Coeff[Pp, x, p]*x^(p - q - r + 1)*Qq^(m + 1)*Rr^(n + 1))/((p + m*q + n*r + 1)*Coeff[Qq, x, q]*Coeff[Rr, x

, r]), x] /; NeQ[p + m*q + n*r + 1, 0] && EqQ[(p + m*q + n*r + 1)*Coeff[Qq, x, q]*Coeff[Rr, x, r]*Pp, Coeff[Pp

, x, p]*x^(p - q - r)*((p - q - r + 1)*Qq*Rr + (m + 1)*x*Rr*D[Qq, x] + (n + 1)*x*Qq*D[Rr, x])]] /; FreeQ[{m, n

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L

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

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {3 (4-e)-2 (3-2 e) x-6 x^2}{(3-e+x)^2 \left (3+3 x-2 x^2\right )^2}+\frac {1}{x \log (x)}\right ) \, dx\\ &=\int \frac {3 (4-e)-2 (3-2 e) x-6 x^2}{(3-e+x)^2 \left (3+3 x-2 x^2\right )^2} \, dx+\int \frac {1}{x \log (x)} \, dx\\ &=-\frac {1}{(3-e+x) \left (3+3 x-2 x^2\right )}+\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\log (x)\right )\\ &=-\frac {1}{(3-e+x) \left (3+3 x-2 x^2\right )}+\log (\log (x))\\ \end {aligned} \end {gather*}

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Mathematica [A] time = 0.08, size = 26, normalized size = 0.87 \begin {gather*} -\frac {1}{(-3+e-x) \left (-3-3 x+2 x^2\right )}+\log (\log (x)) \end {gather*}

Integrate[(81 + 216*x + 90*x^2 - 108*x^3 - 39*x^4 + 12*x^5 + 4*x^6 + E^2*(9 + 18*x - 3*x^2 - 12*x^3 + 4*x^4) +

E*(-54 - 126*x - 18*x^2 + 78*x^3 - 8*x^5) + (12*x - 6*x^2 - 6*x^3 + E*(-3*x + 4*x^2))*Log[x])/((81*x + 216*x^

2 + 90*x^3 - 108*x^4 - 39*x^5 + 12*x^6 + 4*x^7 + E^2*(9*x + 18*x^2 - 3*x^3 - 12*x^4 + 4*x^5) + E*(-54*x - 126*

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fricas [B] time = 1.27, size = 67, normalized size = 2.23 \begin {gather*} \frac {{\left (2 \, x^{3} + 3 \, x^{2} - {\left (2 \, x^{2} - 3 \, x - 3\right )} e - 12 \, x - 9\right )} \log \left (\log \relax (x)\right ) + 1}{2 \, x^{3} + 3 \, x^{2} - {\left (2 \, x^{2} - 3 \, x - 3\right )} e - 12 \, x - 9} \end {gather*}

integrate((((4*x^2-3*x)*exp(1)-6*x^3-6*x^2+12*x)*log(x)+(4*x^4-12*x^3-3*x^2+18*x+9)*exp(1)^2+(-8*x^5+78*x^3-18

*x^2-126*x-54)*exp(1)+4*x^6+12*x^5-39*x^4-108*x^3+90*x^2+216*x+81)/((4*x^5-12*x^4-3*x^3+18*x^2+9*x)*exp(1)^2+(

-8*x^6+78*x^4-18*x^3-126*x^2-54*x)*exp(1)+4*x^7+12*x^6-39*x^5-108*x^4+90*x^3+216*x^2+81*x)/log(x),x, algorithm

((2*x^3 + 3*x^2 - (2*x^2 - 3*x - 3)*e - 12*x - 9)*log(log(x)) + 1)/(2*x^3 + 3*x^2 - (2*x^2 - 3*x - 3)*e - 12*x

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giac [B] time = 0.42, size = 88, normalized size = 2.93 \begin {gather*} \frac {2 \, x^{3} \log \left (\log \relax (x)\right ) - 2 \, x^{2} e \log \left (\log \relax (x)\right ) + 3 \, x^{2} \log \left (\log \relax (x)\right ) + 3 \, x e \log \left (\log \relax (x)\right ) - 12 \, x \log \left (\log \relax (x)\right ) + 3 \, e \log \left (\log \relax (x)\right ) - 9 \, \log \left (\log \relax (x)\right ) + 2}{2 \, x^{3} - 2 \, x^{2} e + 3 \, x^{2} + 3 \, x e - 12 \, x + 3 \, e - 9} \end {gather*}

integrate((((4*x^2-3*x)*exp(1)-6*x^3-6*x^2+12*x)*log(x)+(4*x^4-12*x^3-3*x^2+18*x+9)*exp(1)^2+(-8*x^5+78*x^3-18

*x^2-126*x-54)*exp(1)+4*x^6+12*x^5-39*x^4-108*x^3+90*x^2+216*x+81)/((4*x^5-12*x^4-3*x^3+18*x^2+9*x)*exp(1)^2+(

-8*x^6+78*x^4-18*x^3-126*x^2-54*x)*exp(1)+4*x^7+12*x^6-39*x^5-108*x^4+90*x^3+216*x^2+81*x)/log(x),x, algorithm

(2*x^3*log(log(x)) - 2*x^2*e*log(log(x)) + 3*x^2*log(log(x)) + 3*x*e*log(log(x)) - 12*x*log(log(x)) + 3*e*log(

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method	result	size

default	\(\ln \left (\ln \relax (x )\right )-\frac {1}{\left (2 x^{2}-3 x -3\right ) \left ({\mathrm e}-3-x \right )}\)	\(28\)
norman	\(\ln \left (\ln \relax (x )\right )-\frac {1}{\left (2 x^{2}-3 x -3\right ) \left ({\mathrm e}-3-x \right )}\)	\(28\)
risch	\(-\frac {1}{2 x^{2} {\mathrm e}-2 x^{3}-3 x \,{\mathrm e}-3 x^{2}-3 \,{\mathrm e}+12 x +9}+\ln \left (\ln \relax (x )\right )\)	\(40\)

int((((4*x^2-3*x)*exp(1)-6*x^3-6*x^2+12*x)*ln(x)+(4*x^4-12*x^3-3*x^2+18*x+9)*exp(1)^2+(-8*x^5+78*x^3-18*x^2-12

6*x-54)*exp(1)+4*x^6+12*x^5-39*x^4-108*x^3+90*x^2+216*x+81)/((4*x^5-12*x^4-3*x^3+18*x^2+9*x)*exp(1)^2+(-8*x^6+

78*x^4-18*x^3-126*x^2-54*x)*exp(1)+4*x^7+12*x^6-39*x^5-108*x^4+90*x^3+216*x^2+81*x)/ln(x),x,method=_RETURNVERB

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maxima [A] time = 0.60, size = 35, normalized size = 1.17 \begin {gather*} \frac {1}{2 \, x^{3} - x^{2} {\left (2 \, e - 3\right )} + 3 \, x {\left (e - 4\right )} + 3 \, e - 9} + \log \left (\log \relax (x)\right ) \end {gather*}

integrate((((4*x^2-3*x)*exp(1)-6*x^3-6*x^2+12*x)*log(x)+(4*x^4-12*x^3-3*x^2+18*x+9)*exp(1)^2+(-8*x^5+78*x^3-18

*x^2-126*x-54)*exp(1)+4*x^6+12*x^5-39*x^4-108*x^3+90*x^2+216*x+81)/((4*x^5-12*x^4-3*x^3+18*x^2+9*x)*exp(1)^2+(

-8*x^6+78*x^4-18*x^3-126*x^2-54*x)*exp(1)+4*x^7+12*x^6-39*x^5-108*x^4+90*x^3+216*x^2+81*x)/log(x),x, algorithm

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mupad [B] time = 5.03, size = 36, normalized size = 1.20 \begin {gather*} \ln \left (\ln \relax (x)\right )+\frac {1}{2\,x^3+\left (3-2\,\mathrm {e}\right )\,x^2+\left (3\,\mathrm {e}-12\right )\,x+3\,\mathrm {e}-9} \end {gather*}

int((216*x - log(x)*(exp(1)*(3*x - 4*x^2) - 12*x + 6*x^2 + 6*x^3) + exp(2)*(18*x - 3*x^2 - 12*x^3 + 4*x^4 + 9)

- exp(1)*(126*x + 18*x^2 - 78*x^3 + 8*x^5 + 54) + 90*x^2 - 108*x^3 - 39*x^4 + 12*x^5 + 4*x^6 + 81)/(log(x)*(8

1*x + exp(2)*(9*x + 18*x^2 - 3*x^3 - 12*x^4 + 4*x^5) - exp(1)*(54*x + 126*x^2 + 18*x^3 - 78*x^4 + 8*x^6) + 216

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sympy [A] time = 1.42, size = 36, normalized size = 1.20 \begin {gather*} \log {\left (\log {\relax (x )} \right )} + \frac {1}{2 x^{3} + x^{2} \left (3 - 2 e\right ) + x \left (-12 + 3 e\right ) - 9 + 3 e} \end {gather*}

integrate((((4*x**2-3*x)*exp(1)-6*x**3-6*x**2+12*x)*ln(x)+(4*x**4-12*x**3-3*x**2+18*x+9)*exp(1)**2+(-8*x**5+78

*x**3-18*x**2-126*x-54)*exp(1)+4*x**6+12*x**5-39*x**4-108*x**3+90*x**2+216*x+81)/((4*x**5-12*x**4-3*x**3+18*x*

*2+9*x)*exp(1)**2+(-8*x**6+78*x**4-18*x**3-126*x**2-54*x)*exp(1)+4*x**7+12*x**6-39*x**5-108*x**4+90*x**3+216*x

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