\(\int \frac {81+216 x+90 x^2-108 x^3-39 x^4+12 x^5+4 x^6+e^2 (9+18 x-3 x^2-12 x^3+4 x^4)+e (-54-126 x-18 x^2+78 x^3-8 x^5)+(12 x-6 x^2-6 x^3+e (-3 x+4 x^2)) \log (x)}{(81 x+216 x^2+90 x^3-108 x^4-39 x^5+12 x^6+4 x^7+e^2 (9 x+18 x^2-3 x^3-12 x^4+4 x^5)+e (-54 x-126 x^2-18 x^3+78 x^4-8 x^6)) \log (x)} \, dx\) [7585]

Optimal result

Integrand size = 199, antiderivative size = 30 \[ \int \frac {81+216 x+90 x^2-108 x^3-39 x^4+12 x^5+4 x^6+e^2 \left (9+18 x-3 x^2-12 x^3+4 x^4\right )+e \left (-54-126 x-18 x^2+78 x^3-8 x^5\right )+\left (12 x-6 x^2-6 x^3+e \left (-3 x+4 x^2\right )\right ) \log (x)}{\left (81 x+216 x^2+90 x^3-108 x^4-39 x^5+12 x^6+4 x^7+e^2 \left (9 x+18 x^2-3 x^3-12 x^4+4 x^5\right )+e \left (-54 x-126 x^2-18 x^3+78 x^4-8 x^6\right )\right ) \log (x)} \, dx=1-\frac {1}{(-3+e-x) x \left (-3-\frac {3}{x}+2 x\right )}+\log (\log (x)) \]

[Out]

1-1/(exp(1)-3-x)/x/(2*x-3-3/x)+ln(ln(x))

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.87, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.020, Rules used = {6820, 1604, 2339, 29} \[ \int \frac {81+216 x+90 x^2-108 x^3-39 x^4+12 x^5+4 x^6+e^2 \left (9+18 x-3 x^2-12 x^3+4 x^4\right )+e \left (-54-126 x-18 x^2+78 x^3-8 x^5\right )+\left (12 x-6 x^2-6 x^3+e \left (-3 x+4 x^2\right )\right ) \log (x)}{\left (81 x+216 x^2+90 x^3-108 x^4-39 x^5+12 x^6+4 x^7+e^2 \left (9 x+18 x^2-3 x^3-12 x^4+4 x^5\right )+e \left (-54 x-126 x^2-18 x^3+78 x^4-8 x^6\right )\right ) \log (x)} \, dx=\log (\log (x))-\frac {1}{(x-e+3) \left (-2 x^2+3 x+3\right )} \]

[In]

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 -
18*x^3 + 78*x^4 - 8*x^6))*Log[x]),x]

[Out]

-(1/((3 - E + x)*(3 + 3*x - 2*x^2))) + Log[Log[x]]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 1604

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
}, x] && PolyQ[Pp, x] && PolyQ[Qq, x] && PolyQ[Rr, x] && NeQ[m, -1] && NeQ[n, -1]

Rule 2339

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

Rule 6820

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

Rubi steps \begin{align*} \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 )}+\text {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{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.87 \[ \int \frac {81+216 x+90 x^2-108 x^3-39 x^4+12 x^5+4 x^6+e^2 \left (9+18 x-3 x^2-12 x^3+4 x^4\right )+e \left (-54-126 x-18 x^2+78 x^3-8 x^5\right )+\left (12 x-6 x^2-6 x^3+e \left (-3 x+4 x^2\right )\right ) \log (x)}{\left (81 x+216 x^2+90 x^3-108 x^4-39 x^5+12 x^6+4 x^7+e^2 \left (9 x+18 x^2-3 x^3-12 x^4+4 x^5\right )+e \left (-54 x-126 x^2-18 x^3+78 x^4-8 x^6\right )\right ) \log (x)} \, dx=-\frac {1}{(-3+e-x) \left (-3-3 x+2 x^2\right )}+\log (\log (x)) \]

[In]

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*
x^2 - 18*x^3 + 78*x^4 - 8*x^6))*Log[x]),x]

[Out]

-(1/((-3 + E - x)*(-3 - 3*x + 2*x^2))) + Log[Log[x]]

Maple [A] (verified)

Time = 0.83 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.93

method	result	size
default	\(\ln \left (\ln \left (x \right )\right )-\frac {1}{\left (2 x^{2}-3 x -3\right ) \left ({\mathrm e}-3-x \right )}\)	\(28\)
norman	\(\ln \left (\ln \left (x \right )\right )-\frac {1}{\left (2 x^{2}-3 x -3\right ) \left ({\mathrm e}-3-x \right )}\)	\(28\)
parts	\(\ln \left (\ln \left (x \right )\right )-\frac {1}{\left (2 x^{2}-3 x -3\right ) \left ({\mathrm e}-3-x \right )}\)	\(28\)
risch	\(-\frac {1}{2 x^{2} {\mathrm e}-2 x^{3}-3 x \,{\mathrm e}-3 x^{2}-3 \,{\mathrm e}+12 x +9}+\ln \left (\ln \left (x \right )\right )\)	\(40\)
parallelrisch	\(\frac {6-4 \,{\mathrm e}+18 x^{2} \ln \left (\ln \left (x \right )\right )-72 x \ln \left (\ln \left (x \right )\right )-54 \ln \left (\ln \left (x \right )\right )-8 \,{\mathrm e} \ln \left (\ln \left (x \right )\right ) x^{3}-12 \,{\mathrm e}^{2} \ln \left (\ln \left (x \right )\right ) x +8 \,{\mathrm e}^{2} \ln \left (\ln \left (x \right )\right ) x^{2}-12 \,{\mathrm e}^{2} \ln \left (\ln \left (x \right )\right )+12 \ln \left (\ln \left (x \right )\right ) x^{3}+54 \,{\mathrm e} \ln \left (\ln \left (x \right )\right )-24 \,{\mathrm e} \ln \left (\ln \left (x \right )\right ) x^{2}+66 \,{\mathrm e} \ln \left (\ln \left (x \right )\right ) x}{2 \left (2 \,{\mathrm e}-3\right ) \left (2 x^{2} {\mathrm e}-2 x^{3}-3 x \,{\mathrm e}-3 x^{2}-3 \,{\mathrm e}+12 x +9\right )}\)	\(143\)

[In]

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
OSE)

[Out]

ln(ln(x))-1/(2*x^2-3*x-3)/(exp(1)-3-x)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 67 vs. \(2 (30) = 60\).

Time = 0.30 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.23 \[ \int \frac {81+216 x+90 x^2-108 x^3-39 x^4+12 x^5+4 x^6+e^2 \left (9+18 x-3 x^2-12 x^3+4 x^4\right )+e \left (-54-126 x-18 x^2+78 x^3-8 x^5\right )+\left (12 x-6 x^2-6 x^3+e \left (-3 x+4 x^2\right )\right ) \log (x)}{\left (81 x+216 x^2+90 x^3-108 x^4-39 x^5+12 x^6+4 x^7+e^2 \left (9 x+18 x^2-3 x^3-12 x^4+4 x^5\right )+e \left (-54 x-126 x^2-18 x^3+78 x^4-8 x^6\right )\right ) \log (x)} \, dx=\frac {{\left (2 \, x^{3} + 3 \, x^{2} - {\left (2 \, x^{2} - 3 \, x - 3\right )} e - 12 \, x - 9\right )} \log \left (\log \left (x\right )\right ) + 1}{2 \, x^{3} + 3 \, x^{2} - {\left (2 \, x^{2} - 3 \, x - 3\right )} e - 12 \, x - 9} \]

[In]

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
="fricas")

[Out]

((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
- 9)

Sympy [A] (verification not implemented)

Time = 0.91 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.20 \[ \int \frac {81+216 x+90 x^2-108 x^3-39 x^4+12 x^5+4 x^6+e^2 \left (9+18 x-3 x^2-12 x^3+4 x^4\right )+e \left (-54-126 x-18 x^2+78 x^3-8 x^5\right )+\left (12 x-6 x^2-6 x^3+e \left (-3 x+4 x^2\right )\right ) \log (x)}{\left (81 x+216 x^2+90 x^3-108 x^4-39 x^5+12 x^6+4 x^7+e^2 \left (9 x+18 x^2-3 x^3-12 x^4+4 x^5\right )+e \left (-54 x-126 x^2-18 x^3+78 x^4-8 x^6\right )\right ) \log (x)} \, dx=\log {\left (\log {\left (x \right )} \right )} + \frac {1}{2 x^{3} + x^{2} \cdot \left (3 - 2 e\right ) + x \left (-12 + 3 e\right ) - 9 + 3 e} \]

[In]

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
**2+81*x)/ln(x),x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.17 \[ \int \frac {81+216 x+90 x^2-108 x^3-39 x^4+12 x^5+4 x^6+e^2 \left (9+18 x-3 x^2-12 x^3+4 x^4\right )+e \left (-54-126 x-18 x^2+78 x^3-8 x^5\right )+\left (12 x-6 x^2-6 x^3+e \left (-3 x+4 x^2\right )\right ) \log (x)}{\left (81 x+216 x^2+90 x^3-108 x^4-39 x^5+12 x^6+4 x^7+e^2 \left (9 x+18 x^2-3 x^3-12 x^4+4 x^5\right )+e \left (-54 x-126 x^2-18 x^3+78 x^4-8 x^6\right )\right ) \log (x)} \, dx=\frac {1}{2 \, x^{3} - x^{2} {\left (2 \, e - 3\right )} + 3 \, x {\left (e - 4\right )} + 3 \, e - 9} + \log \left (\log \left (x\right )\right ) \]

[In]

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
="maxima")

[Out]

1/(2*x^3 - x^2*(2*e - 3) + 3*x*(e - 4) + 3*e - 9) + log(log(x))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (30) = 60\).

Time = 0.44 (sec) , antiderivative size = 88, normalized size of antiderivative = 2.93 \[ \int \frac {81+216 x+90 x^2-108 x^3-39 x^4+12 x^5+4 x^6+e^2 \left (9+18 x-3 x^2-12 x^3+4 x^4\right )+e \left (-54-126 x-18 x^2+78 x^3-8 x^5\right )+\left (12 x-6 x^2-6 x^3+e \left (-3 x+4 x^2\right )\right ) \log (x)}{\left (81 x+216 x^2+90 x^3-108 x^4-39 x^5+12 x^6+4 x^7+e^2 \left (9 x+18 x^2-3 x^3-12 x^4+4 x^5\right )+e \left (-54 x-126 x^2-18 x^3+78 x^4-8 x^6\right )\right ) \log (x)} \, dx=\frac {2 \, x^{3} \log \left (\log \left (x\right )\right ) - 2 \, x^{2} e \log \left (\log \left (x\right )\right ) + 3 \, x^{2} \log \left (\log \left (x\right )\right ) + 3 \, x e \log \left (\log \left (x\right )\right ) - 12 \, x \log \left (\log \left (x\right )\right ) + 3 \, e \log \left (\log \left (x\right )\right ) - 9 \, \log \left (\log \left (x\right )\right ) + 2}{2 \, x^{3} - 2 \, x^{2} e + 3 \, x^{2} + 3 \, x e - 12 \, x + 3 \, e - 9} \]

[In]

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
="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 14.37 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.20 \[ \int \frac {81+216 x+90 x^2-108 x^3-39 x^4+12 x^5+4 x^6+e^2 \left (9+18 x-3 x^2-12 x^3+4 x^4\right )+e \left (-54-126 x-18 x^2+78 x^3-8 x^5\right )+\left (12 x-6 x^2-6 x^3+e \left (-3 x+4 x^2\right )\right ) \log (x)}{\left (81 x+216 x^2+90 x^3-108 x^4-39 x^5+12 x^6+4 x^7+e^2 \left (9 x+18 x^2-3 x^3-12 x^4+4 x^5\right )+e \left (-54 x-126 x^2-18 x^3+78 x^4-8 x^6\right )\right ) \log (x)} \, dx=\ln \left (\ln \left (x\right )\right )+\frac {1}{2\,x^3+\left (3-2\,\mathrm {e}\right )\,x^2+\left (3\,\mathrm {e}-12\right )\,x+3\,\mathrm {e}-9} \]

[In]

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
*x^2 + 90*x^3 - 108*x^4 - 39*x^5 + 12*x^6 + 4*x^7)),x)

[Out]

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