Integrand size = 199, antiderivative size = 30 \[ \int \frac {81+216 x+90 x^2-108 x^3-39 x^4+12 x^5+4 x^6+e^2 \left (9+18 x-3 x^2-12 x^3+4 x^4\right )+e \left (-54-126 x-18 x^2+78 x^3-8 x^5\right )+\left (12 x-6 x^2-6 x^3+e \left (-3 x+4 x^2\right )\right ) \log (x)}{\left (81 x+216 x^2+90 x^3-108 x^4-39 x^5+12 x^6+4 x^7+e^2 \left (9 x+18 x^2-3 x^3-12 x^4+4 x^5\right )+e \left (-54 x-126 x^2-18 x^3+78 x^4-8 x^6\right )\right ) \log (x)} \, dx=1-\frac {1}{(-3+e-x) x \left (-3-\frac {3}{x}+2 x\right )}+\log (\log (x)) \]
Time = 0.08 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.87 \[ \int \frac {81+216 x+90 x^2-108 x^3-39 x^4+12 x^5+4 x^6+e^2 \left (9+18 x-3 x^2-12 x^3+4 x^4\right )+e \left (-54-126 x-18 x^2+78 x^3-8 x^5\right )+\left (12 x-6 x^2-6 x^3+e \left (-3 x+4 x^2\right )\right ) \log (x)}{\left (81 x+216 x^2+90 x^3-108 x^4-39 x^5+12 x^6+4 x^7+e^2 \left (9 x+18 x^2-3 x^3-12 x^4+4 x^5\right )+e \left (-54 x-126 x^2-18 x^3+78 x^4-8 x^6\right )\right ) \log (x)} \, dx=-\frac {1}{(-3+e-x) \left (-3-3 x+2 x^2\right )}+\log (\log (x)) \]
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]
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {4 x^6+12 x^5-39 x^4-108 x^3+90 x^2+\left (-6 x^3-6 x^2+e \left (4 x^2-3 x\right )+12 x\right ) \log (x)+e \left (-8 x^5+78 x^3-18 x^2-126 x-54\right )+e^2 \left (4 x^4-12 x^3-3 x^2+18 x+9\right )+216 x+81}{\left (4 x^7+12 x^6-39 x^5-108 x^4+90 x^3+216 x^2+e \left (-8 x^6+78 x^4-18 x^3-126 x^2-54 x\right )+e^2 \left (4 x^5-12 x^4-3 x^3+18 x^2+9 x\right )+81 x\right ) \log (x)} \, dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {4 x^6+12 x^5-39 x^4-108 x^3+90 x^2+\left (-6 x^3-6 x^2+e \left (4 x^2-3 x\right )+12 x\right ) \log (x)+e \left (-8 x^5+78 x^3-18 x^2-126 x-54\right )+e^2 \left (4 x^4-12 x^3-3 x^2+18 x+9\right )+216 x+81}{x \left (4 x^6+4 (3-2 e) x^5-\left (39-4 e^2\right ) x^4-6 (9-2 e) (2-e) x^3+3 \left (30-6 e-e^2\right ) x^2+18 (3-e) (4-e) x+9 (3-e)^2\right ) \log (x)}dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (\frac {2 (4 e-15) \left (4 x^6+12 x^5-39 x^4-108 x^3+90 x^2+\left (-6 x^3-6 x^2+e \left (4 x^2-3 x\right )+12 x\right ) \log (x)+e \left (-8 x^5+78 x^3-18 x^2-126 x-54\right )+e^2 \left (4 x^4-12 x^3-3 x^2+18 x+9\right )+216 x+81\right )}{\left (24-15 e+2 e^2\right )^3 (-x+e-3) x \log (x)}+\frac {4 x^6+12 x^5-39 x^4-108 x^3+90 x^2+\left (-6 x^3-6 x^2+e \left (4 x^2-3 x\right )+12 x\right ) \log (x)+e \left (-8 x^5+78 x^3-18 x^2-126 x-54\right )+e^2 \left (4 x^4-12 x^3-3 x^2+18 x+9\right )+216 x+81}{\left (24-15 e+2 e^2\right )^2 (-x+e-3)^2 x \log (x)}+\frac {2 \left (2 (15-4 e) x-3 \left (37-17 e+2 e^2\right )\right ) \left (4 x^6+12 x^5-39 x^4-108 x^3+90 x^2+\left (-6 x^3-6 x^2+e \left (4 x^2-3 x\right )+12 x\right ) \log (x)+e \left (-8 x^5+78 x^3-18 x^2-126 x-54\right )+e^2 \left (4 x^4-12 x^3-3 x^2+18 x+9\right )+216 x+81\right )}{\left (24-15 e+2 e^2\right )^3 x \left (-2 x^2+3 x+3\right ) \log (x)}+\frac {\left (-2 (15-4 e) x+4 e^2-36 e+87\right ) \left (4 x^6+12 x^5-39 x^4-108 x^3+90 x^2+\left (-6 x^3-6 x^2+e \left (4 x^2-3 x\right )+12 x\right ) \log (x)+e \left (-8 x^5+78 x^3-18 x^2-126 x-54\right )+e^2 \left (4 x^4-12 x^3-3 x^2+18 x+9\right )+216 x+81\right )}{\left (24-15 e+2 e^2\right )^2 x \left (-2 x^2+3 x+3\right )^2 \log (x)}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\int \frac {\left (-2 x^2+3 x+3\right )^2}{x \log (x)}dx}{\left (24-15 e+2 e^2\right )^2}+\frac {2 \int \frac {(x-e+3)^2 \left (3 \left (37-17 e+2 e^2\right )-2 (15-4 e) x\right ) \left (2 x^2-3 x-3\right )}{x \log (x)}dx}{\left (24-15 e+2 e^2\right )^3}-\frac {2 (15-4 e) \int \frac {(-x+e-3) \left (2 x^2-3 x-3\right )^2}{x \log (x)}dx}{\left (24-15 e+2 e^2\right )^3}+\frac {\int \frac {(x-e+3)^2 \left (-2 (15-4 e) x+4 e^2-36 e+87\right )}{x \log (x)}dx}{\left (24-15 e+2 e^2\right )^2}-\frac {(x-e+3) \left (-2 (15-4 e) x+4 e^2-36 e+87\right )}{\left (24-15 e+2 e^2\right )^2 \left (-2 x^2+3 x+3\right )}-\frac {6 x}{\left (24-15 e+2 e^2\right )^2}-\frac {2 (9-2 e) (12-e) x}{\left (24-15 e+2 e^2\right )^3}+\frac {4 (15-4 e) (6-e) x}{\left (24-15 e+2 e^2\right )^3}+\frac {1}{\left (24-15 e+2 e^2\right ) (x-e+3)}\) |
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*(-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]
3.16.85.3.1 Defintions of rubi rules used
Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p *r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ erQ[p] && !MonomialQ[Px, x] && (ILtQ[p, 0] || !PolyQ[u, x])
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u, Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && Gt Q[Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0]
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 | \(\frac {6-4 \,{\mathrm e}+18 x^{2} \ln \left (\ln \left (x \right )\right )-72 x \ln \left (\ln \left (x \right )\right )-54 \ln \left (\ln \left (x \right )\right )-8 \,{\mathrm e} \ln \left (\ln \left (x \right )\right ) x^{3}-12 \,{\mathrm e}^{2} \ln \left (\ln \left (x \right )\right ) x +8 \,{\mathrm e}^{2} \ln \left (\ln \left (x \right )\right ) x^{2}-12 \,{\mathrm e}^{2} \ln \left (\ln \left (x \right )\right )+12 \ln \left (\ln \left (x \right )\right ) x^{3}+54 \,{\mathrm e} \ln \left (\ln \left (x \right )\right )-24 \,{\mathrm e} \ln \left (\ln \left (x \right )\right ) x^{2}+66 \,{\mathrm e} \ln \left (\ln \left (x \right )\right ) x}{2 \left (2 \,{\mathrm e}-3\right ) \left (2 x^{2} {\mathrm e}-2 x^{3}-3 x \,{\mathrm e}-3 x^{2}-3 \,{\mathrm e}+12 x +9\right )}\) | \(143\) |
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^2-126*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,method=_RETURNVERBOSE)
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 \[ \int \frac {81+216 x+90 x^2-108 x^3-39 x^4+12 x^5+4 x^6+e^2 \left (9+18 x-3 x^2-12 x^3+4 x^4\right )+e \left (-54-126 x-18 x^2+78 x^3-8 x^5\right )+\left (12 x-6 x^2-6 x^3+e \left (-3 x+4 x^2\right )\right ) \log (x)}{\left (81 x+216 x^2+90 x^3-108 x^4-39 x^5+12 x^6+4 x^7+e^2 \left (9 x+18 x^2-3 x^3-12 x^4+4 x^5\right )+e \left (-54 x-126 x^2-18 x^3+78 x^4-8 x^6\right )\right ) \log (x)} \, dx=\frac {{\left (2 \, x^{3} + 3 \, x^{2} - {\left (2 \, x^{2} - 3 \, x - 3\right )} e - 12 \, x - 9\right )} \log \left (\log \left (x\right )\right ) + 1}{2 \, x^{3} + 3 \, x^{2} - {\left (2 \, x^{2} - 3 \, x - 3\right )} e - 12 \, x - 9} \]
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 - 9)
Time = 0.91 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.20 \[ \int \frac {81+216 x+90 x^2-108 x^3-39 x^4+12 x^5+4 x^6+e^2 \left (9+18 x-3 x^2-12 x^3+4 x^4\right )+e \left (-54-126 x-18 x^2+78 x^3-8 x^5\right )+\left (12 x-6 x^2-6 x^3+e \left (-3 x+4 x^2\right )\right ) \log (x)}{\left (81 x+216 x^2+90 x^3-108 x^4-39 x^5+12 x^6+4 x^7+e^2 \left (9 x+18 x^2-3 x^3-12 x^4+4 x^5\right )+e \left (-54 x-126 x^2-18 x^3+78 x^4-8 x^6\right )\right ) \log (x)} \, dx=\log {\left (\log {\left (x \right )} \right )} + \frac {1}{2 x^{3} + x^{2} \cdot \left (3 - 2 e\right ) + x \left (-12 + 3 e\right ) - 9 + 3 e} \]
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)
Time = 0.25 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.17 \[ \int \frac {81+216 x+90 x^2-108 x^3-39 x^4+12 x^5+4 x^6+e^2 \left (9+18 x-3 x^2-12 x^3+4 x^4\right )+e \left (-54-126 x-18 x^2+78 x^3-8 x^5\right )+\left (12 x-6 x^2-6 x^3+e \left (-3 x+4 x^2\right )\right ) \log (x)}{\left (81 x+216 x^2+90 x^3-108 x^4-39 x^5+12 x^6+4 x^7+e^2 \left (9 x+18 x^2-3 x^3-12 x^4+4 x^5\right )+e \left (-54 x-126 x^2-18 x^3+78 x^4-8 x^6\right )\right ) \log (x)} \, dx=\frac {1}{2 \, x^{3} - x^{2} {\left (2 \, e - 3\right )} + 3 \, x {\left (e - 4\right )} + 3 \, e - 9} + \log \left (\log \left (x\right )\right ) \]
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=\
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 \[ \int \frac {81+216 x+90 x^2-108 x^3-39 x^4+12 x^5+4 x^6+e^2 \left (9+18 x-3 x^2-12 x^3+4 x^4\right )+e \left (-54-126 x-18 x^2+78 x^3-8 x^5\right )+\left (12 x-6 x^2-6 x^3+e \left (-3 x+4 x^2\right )\right ) \log (x)}{\left (81 x+216 x^2+90 x^3-108 x^4-39 x^5+12 x^6+4 x^7+e^2 \left (9 x+18 x^2-3 x^3-12 x^4+4 x^5\right )+e \left (-54 x-126 x^2-18 x^3+78 x^4-8 x^6\right )\right ) \log (x)} \, dx=\frac {2 \, x^{3} \log \left (\log \left (x\right )\right ) - 2 \, x^{2} e \log \left (\log \left (x\right )\right ) + 3 \, x^{2} \log \left (\log \left (x\right )\right ) + 3 \, x e \log \left (\log \left (x\right )\right ) - 12 \, x \log \left (\log \left (x\right )\right ) + 3 \, e \log \left (\log \left (x\right )\right ) - 9 \, \log \left (\log \left (x\right )\right ) + 2}{2 \, x^{3} - 2 \, x^{2} e + 3 \, x^{2} + 3 \, x e - 12 \, x + 3 \, e - 9} \]
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(l og(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)
Time = 14.37 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.20 \[ \int \frac {81+216 x+90 x^2-108 x^3-39 x^4+12 x^5+4 x^6+e^2 \left (9+18 x-3 x^2-12 x^3+4 x^4\right )+e \left (-54-126 x-18 x^2+78 x^3-8 x^5\right )+\left (12 x-6 x^2-6 x^3+e \left (-3 x+4 x^2\right )\right ) \log (x)}{\left (81 x+216 x^2+90 x^3-108 x^4-39 x^5+12 x^6+4 x^7+e^2 \left (9 x+18 x^2-3 x^3-12 x^4+4 x^5\right )+e \left (-54 x-126 x^2-18 x^3+78 x^4-8 x^6\right )\right ) \log (x)} \, dx=\ln \left (\ln \left (x\right )\right )+\frac {1}{2\,x^3+\left (3-2\,\mathrm {e}\right )\,x^2+\left (3\,\mathrm {e}-12\right )\,x+3\,\mathrm {e}-9} \]
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)*(81* 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)