Integrand size = 237, antiderivative size = 31 \[ \int \frac {-4+6 x-2 x^2+\left (2 x-x^2+x \log (x)\right ) \log (24-12 x+12 \log (x))+\left (-4+4 x-x^2+(-2+x) \log (x)\right ) \log (24-12 x+12 \log (x)) \log \left (\frac {1}{(-8+4 x) \log ^2(24-12 x+12 \log (x))}\right )}{\left (-20 x^2+20 x^3-5 x^4+\left (-10 x^2+5 x^3\right ) \log (x)\right ) \log (24-12 x+12 \log (x))+\left (-40 x+40 x^2-10 x^3+\left (-20 x+10 x^2\right ) \log (x)\right ) \log (24-12 x+12 \log (x)) \log \left (\frac {1}{(-8+4 x) \log ^2(24-12 x+12 \log (x))}\right )+\left (-20+20 x-5 x^2+(-10+5 x) \log (x)\right ) \log (24-12 x+12 \log (x)) \log ^2\left (\frac {1}{(-8+4 x) \log ^2(24-12 x+12 \log (x))}\right )} \, dx=\frac {x}{5 \left (x+\log \left (\frac {1}{4 (-2+x) \log ^2(12 (2-x+\log (x)))}\right )\right )} \] Output:
x/(5*x+5*ln(1/4/ln(12*ln(x)-12*x+24)^2/(-2+x)))
Time = 0.10 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int \frac {-4+6 x-2 x^2+\left (2 x-x^2+x \log (x)\right ) \log (24-12 x+12 \log (x))+\left (-4+4 x-x^2+(-2+x) \log (x)\right ) \log (24-12 x+12 \log (x)) \log \left (\frac {1}{(-8+4 x) \log ^2(24-12 x+12 \log (x))}\right )}{\left (-20 x^2+20 x^3-5 x^4+\left (-10 x^2+5 x^3\right ) \log (x)\right ) \log (24-12 x+12 \log (x))+\left (-40 x+40 x^2-10 x^3+\left (-20 x+10 x^2\right ) \log (x)\right ) \log (24-12 x+12 \log (x)) \log \left (\frac {1}{(-8+4 x) \log ^2(24-12 x+12 \log (x))}\right )+\left (-20+20 x-5 x^2+(-10+5 x) \log (x)\right ) \log (24-12 x+12 \log (x)) \log ^2\left (\frac {1}{(-8+4 x) \log ^2(24-12 x+12 \log (x))}\right )} \, dx=\frac {x}{5 \left (x+\log \left (\frac {1}{4 (-2+x) \log ^2(12 (2-x+\log (x)))}\right )\right )} \] Input:
Integrate[(-4 + 6*x - 2*x^2 + (2*x - x^2 + x*Log[x])*Log[24 - 12*x + 12*Lo g[x]] + (-4 + 4*x - x^2 + (-2 + x)*Log[x])*Log[24 - 12*x + 12*Log[x]]*Log[ 1/((-8 + 4*x)*Log[24 - 12*x + 12*Log[x]]^2)])/((-20*x^2 + 20*x^3 - 5*x^4 + (-10*x^2 + 5*x^3)*Log[x])*Log[24 - 12*x + 12*Log[x]] + (-40*x + 40*x^2 - 10*x^3 + (-20*x + 10*x^2)*Log[x])*Log[24 - 12*x + 12*Log[x]]*Log[1/((-8 + 4*x)*Log[24 - 12*x + 12*Log[x]]^2)] + (-20 + 20*x - 5*x^2 + (-10 + 5*x)*Lo g[x])*Log[24 - 12*x + 12*Log[x]]*Log[1/((-8 + 4*x)*Log[24 - 12*x + 12*Log[ x]]^2)]^2),x]
Output:
x/(5*(x + Log[1/(4*(-2 + x)*Log[12*(2 - x + Log[x])]^2)]))
Time = 1.80 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.16, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.017, Rules used = {7239, 27, 7262, 17}
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.
| \(\displaystyle \int \frac {-2 x^2+\left (-x^2+4 x+(x-2) \log (x)-4\right ) \log (-12 x+12 \log (x)+24) \log \left (\frac {1}{(4 x-8) \log ^2(-12 x+12 \log (x)+24)}\right )+\left (-x^2+2 x+x \log (x)\right ) \log (-12 x+12 \log (x)+24)+6 x-4}{\left (-5 x^2+20 x+(5 x-10) \log (x)-20\right ) \log (-12 x+12 \log (x)+24) \log ^2\left (\frac {1}{(4 x-8) \log ^2(-12 x+12 \log (x)+24)}\right )+\left (-10 x^3+40 x^2+\left (10 x^2-20 x\right ) \log (x)-40 x\right ) \log (-12 x+12 \log (x)+24) \log \left (\frac {1}{(4 x-8) \log ^2(-12 x+12 \log (x)+24)}\right )+\left (-5 x^4+20 x^3-20 x^2+\left (5 x^3-10 x^2\right ) \log (x)\right ) \log (-12 x+12 \log (x)+24)} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {2 \left (x^2-3 x+2\right )+(x-\log (x)-2) \log (12 (-x+\log (x)+2)) \left (x+(x-2) \log \left (\frac {1}{4 (x-2) \log ^2(12 (-x+\log (x)+2))}\right )\right )}{5 (2-x) (-x+\log (x)+2) \log (12 (-x+\log (x)+2)) \left (x+\log \left (\frac {1}{4 (x-2) \log ^2(12 (-x+\log (x)+2))}\right )\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} \int \frac {2 \left (x^2-3 x+2\right )-(-x+\log (x)+2) \log (12 (-x+\log (x)+2)) \left (x-(2-x) \log \left (-\frac {1}{4 (2-x) \log ^2(12 (-x+\log (x)+2))}\right )\right )}{(2-x) (-x+\log (x)+2) \log (12 (-x+\log (x)+2)) \left (x+\log \left (-\frac {1}{4 (2-x) \log ^2(12 (-x+\log (x)+2))}\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7262 |
| \(\displaystyle \frac {1}{5} \int \frac {1}{\left (\frac {x}{\log \left (-\frac {1}{4 (2-x) \log ^2(12 (-x+\log (x)+2))}\right )}+1\right )^2}d\frac {x}{\log \left (-\frac {1}{4 (2-x) \log ^2(12 (-x+\log (x)+2))}\right )}\) |
| \(\Big \downarrow \) 17 |
| \(\displaystyle -\frac {1}{5 \left (\frac {x}{\log \left (-\frac {1}{4 (2-x) \log ^2(12 (-x+\log (x)+2))}\right )}+1\right )}\) |
Input:
Int[(-4 + 6*x - 2*x^2 + (2*x - x^2 + x*Log[x])*Log[24 - 12*x + 12*Log[x]] + (-4 + 4*x - x^2 + (-2 + x)*Log[x])*Log[24 - 12*x + 12*Log[x]]*Log[1/((-8 + 4*x)*Log[24 - 12*x + 12*Log[x]]^2)])/((-20*x^2 + 20*x^3 - 5*x^4 + (-10* x^2 + 5*x^3)*Log[x])*Log[24 - 12*x + 12*Log[x]] + (-40*x + 40*x^2 - 10*x^3 + (-20*x + 10*x^2)*Log[x])*Log[24 - 12*x + 12*Log[x]]*Log[1/((-8 + 4*x)*L og[24 - 12*x + 12*Log[x]]^2)] + (-20 + 20*x - 5*x^2 + (-10 + 5*x)*Log[x])* Log[24 - 12*x + 12*Log[x]]*Log[1/((-8 + 4*x)*Log[24 - 12*x + 12*Log[x]]^2) ]^2),x]
Output:
-1/5*1/(1 + x/Log[-1/4*1/((2 - x)*Log[12*(2 - x + Log[x])]^2)])
Int[(c_.)*((a_.) + (b_.)*(x_))^(m_.), x_Symbol] :> Simp[c*((a + b*x)^(m + 1 )/(b*(m + 1))), x] /; FreeQ[{a, b, c, m}, x] && NeQ[m, -1]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Int[(u_)*((a_.)*(v_)^(p_.) + (b_.)*(w_)^(q_.))^(m_.), x_Symbol] :> With[{c = Simplify[u/(p*w*D[v, x] - q*v*D[w, x])]}, Simp[c*p Subst[Int[(b + a*x^p )^m, x], x, v*w^(m*q + 1)], x] /; FreeQ[c, x]] /; FreeQ[{a, b, m, p, q}, x] && EqQ[p + q*(m*p + 1), 0] && IntegerQ[p] && IntegerQ[m]
Time = 39.47 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.94
| method | result | size |
| parallelrisch | \(\frac {x}{5 x +5 \ln \left (\frac {1}{\left (4 x -8\right ) \ln \left (12 \ln \left (x \right )-12 x +24\right )^{2}}\right )}\) | \(29\) |
| risch | \(\frac {2 x}{5 \left (-i \pi \,\operatorname {csgn}\left (\frac {i}{-2+x}\right ) \operatorname {csgn}\left (\frac {i}{\ln \left (12 \ln \left (x \right )-12 x +24\right )^{2}}\right ) \operatorname {csgn}\left (\frac {i}{\ln \left (12 \ln \left (x \right )-12 x +24\right )^{2} \left (-2+x \right )}\right )+i \pi \,\operatorname {csgn}\left (\frac {i}{-2+x}\right ) \operatorname {csgn}\left (\frac {i}{\ln \left (12 \ln \left (x \right )-12 x +24\right )^{2} \left (-2+x \right )}\right )^{2}+i \pi \,\operatorname {csgn}\left (\frac {i}{\ln \left (12 \ln \left (x \right )-12 x +24\right )^{2}}\right ) \operatorname {csgn}\left (\frac {i}{\ln \left (12 \ln \left (x \right )-12 x +24\right )^{2} \left (-2+x \right )}\right )^{2}+i \pi \operatorname {csgn}\left (i \ln \left (12 \ln \left (x \right )-12 x +24\right )\right )^{2} \operatorname {csgn}\left (i \ln \left (12 \ln \left (x \right )-12 x +24\right )^{2}\right )-2 i \pi \,\operatorname {csgn}\left (i \ln \left (12 \ln \left (x \right )-12 x +24\right )\right ) \operatorname {csgn}\left (i \ln \left (12 \ln \left (x \right )-12 x +24\right )^{2}\right )^{2}+i \pi \operatorname {csgn}\left (i \ln \left (12 \ln \left (x \right )-12 x +24\right )^{2}\right )^{3}-i \pi \operatorname {csgn}\left (\frac {i}{\ln \left (12 \ln \left (x \right )-12 x +24\right )^{2} \left (-2+x \right )}\right )^{3}-4 \ln \left (2\right )+2 x -2 \ln \left (-2+x \right )-4 \ln \left (\ln \left (12 \ln \left (x \right )-12 x +24\right )\right )\right )}\) | \(283\) |
| default | \(\frac {2 x}{5 \left (-i \pi \,\operatorname {csgn}\left (\frac {i}{-2+x}\right ) \operatorname {csgn}\left (\frac {i}{\left (\ln \left (12\right )+\ln \left (2+\ln \left (x \right )-x \right )\right )^{2}}\right ) \operatorname {csgn}\left (\frac {i}{\left (-2+x \right ) \left (\ln \left (12\right )+\ln \left (2+\ln \left (x \right )-x \right )\right )^{2}}\right )+i \pi \,\operatorname {csgn}\left (\frac {i}{-2+x}\right ) \operatorname {csgn}\left (\frac {i}{\left (-2+x \right ) \left (\ln \left (12\right )+\ln \left (2+\ln \left (x \right )-x \right )\right )^{2}}\right )^{2}+i \pi \,\operatorname {csgn}\left (\frac {i}{\left (\ln \left (12\right )+\ln \left (2+\ln \left (x \right )-x \right )\right )^{2}}\right ) \operatorname {csgn}\left (\frac {i}{\left (-2+x \right ) \left (\ln \left (12\right )+\ln \left (2+\ln \left (x \right )-x \right )\right )^{2}}\right )^{2}+i \pi \operatorname {csgn}\left (i \left (\ln \left (12\right )+\ln \left (2+\ln \left (x \right )-x \right )\right )\right )^{2} \operatorname {csgn}\left (i \left (\ln \left (12\right )+\ln \left (2+\ln \left (x \right )-x \right )\right )^{2}\right )-2 i \pi \,\operatorname {csgn}\left (i \left (\ln \left (12\right )+\ln \left (2+\ln \left (x \right )-x \right )\right )\right ) \operatorname {csgn}\left (i \left (\ln \left (12\right )+\ln \left (2+\ln \left (x \right )-x \right )\right )^{2}\right )^{2}+i \pi \operatorname {csgn}\left (i \left (\ln \left (12\right )+\ln \left (2+\ln \left (x \right )-x \right )\right )^{2}\right )^{3}-i \pi \operatorname {csgn}\left (\frac {i}{\left (-2+x \right ) \left (\ln \left (12\right )+\ln \left (2+\ln \left (x \right )-x \right )\right )^{2}}\right )^{3}-4 \ln \left (2\right )+2 x -2 \ln \left (-2+x \right )-4 \ln \left (2 \ln \left (2\right )+\ln \left (3\right )+\ln \left (2+\ln \left (x \right )-x \right )\right )\right )}\) | \(299\) |
Input:
int((((-2+x)*ln(x)-x^2+4*x-4)*ln(12*ln(x)-12*x+24)*ln(1/(4*x-8)/ln(12*ln(x )-12*x+24)^2)+(x*ln(x)-x^2+2*x)*ln(12*ln(x)-12*x+24)-2*x^2+6*x-4)/(((5*x-1 0)*ln(x)-5*x^2+20*x-20)*ln(12*ln(x)-12*x+24)*ln(1/(4*x-8)/ln(12*ln(x)-12*x +24)^2)^2+((10*x^2-20*x)*ln(x)-10*x^3+40*x^2-40*x)*ln(12*ln(x)-12*x+24)*ln (1/(4*x-8)/ln(12*ln(x)-12*x+24)^2)+((5*x^3-10*x^2)*ln(x)-5*x^4+20*x^3-20*x ^2)*ln(12*ln(x)-12*x+24)),x,method=_RETURNVERBOSE)
Output:
1/5*x/(x+ln(1/(4*x-8)/ln(12*ln(x)-12*x+24)^2))
Time = 0.22 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.87 \[ \int \frac {-4+6 x-2 x^2+\left (2 x-x^2+x \log (x)\right ) \log (24-12 x+12 \log (x))+\left (-4+4 x-x^2+(-2+x) \log (x)\right ) \log (24-12 x+12 \log (x)) \log \left (\frac {1}{(-8+4 x) \log ^2(24-12 x+12 \log (x))}\right )}{\left (-20 x^2+20 x^3-5 x^4+\left (-10 x^2+5 x^3\right ) \log (x)\right ) \log (24-12 x+12 \log (x))+\left (-40 x+40 x^2-10 x^3+\left (-20 x+10 x^2\right ) \log (x)\right ) \log (24-12 x+12 \log (x)) \log \left (\frac {1}{(-8+4 x) \log ^2(24-12 x+12 \log (x))}\right )+\left (-20+20 x-5 x^2+(-10+5 x) \log (x)\right ) \log (24-12 x+12 \log (x)) \log ^2\left (\frac {1}{(-8+4 x) \log ^2(24-12 x+12 \log (x))}\right )} \, dx=\frac {x}{5 \, {\left (x + \log \left (\frac {1}{4 \, {\left (x - 2\right )} \log \left (-12 \, x + 12 \, \log \left (x\right ) + 24\right )^{2}}\right )\right )}} \] Input:
integrate((((-2+x)*log(x)-x^2+4*x-4)*log(12*log(x)-12*x+24)*log(1/(4*x-8)/ log(12*log(x)-12*x+24)^2)+(x*log(x)-x^2+2*x)*log(12*log(x)-12*x+24)-2*x^2+ 6*x-4)/(((5*x-10)*log(x)-5*x^2+20*x-20)*log(12*log(x)-12*x+24)*log(1/(4*x- 8)/log(12*log(x)-12*x+24)^2)^2+((10*x^2-20*x)*log(x)-10*x^3+40*x^2-40*x)*l og(12*log(x)-12*x+24)*log(1/(4*x-8)/log(12*log(x)-12*x+24)^2)+((5*x^3-10*x ^2)*log(x)-5*x^4+20*x^3-20*x^2)*log(12*log(x)-12*x+24)),x, algorithm="fric as")
Output:
1/5*x/(x + log(1/4/((x - 2)*log(-12*x + 12*log(x) + 24)^2)))
Time = 0.97 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.87 \[ \int \frac {-4+6 x-2 x^2+\left (2 x-x^2+x \log (x)\right ) \log (24-12 x+12 \log (x))+\left (-4+4 x-x^2+(-2+x) \log (x)\right ) \log (24-12 x+12 \log (x)) \log \left (\frac {1}{(-8+4 x) \log ^2(24-12 x+12 \log (x))}\right )}{\left (-20 x^2+20 x^3-5 x^4+\left (-10 x^2+5 x^3\right ) \log (x)\right ) \log (24-12 x+12 \log (x))+\left (-40 x+40 x^2-10 x^3+\left (-20 x+10 x^2\right ) \log (x)\right ) \log (24-12 x+12 \log (x)) \log \left (\frac {1}{(-8+4 x) \log ^2(24-12 x+12 \log (x))}\right )+\left (-20+20 x-5 x^2+(-10+5 x) \log (x)\right ) \log (24-12 x+12 \log (x)) \log ^2\left (\frac {1}{(-8+4 x) \log ^2(24-12 x+12 \log (x))}\right )} \, dx=\frac {x}{5 x + 5 \log {\left (\frac {1}{\left (4 x - 8\right ) \log {\left (- 12 x + 12 \log {\left (x \right )} + 24 \right )}^{2}} \right )}} \] Input:
integrate((((-2+x)*ln(x)-x**2+4*x-4)*ln(12*ln(x)-12*x+24)*ln(1/(4*x-8)/ln( 12*ln(x)-12*x+24)**2)+(x*ln(x)-x**2+2*x)*ln(12*ln(x)-12*x+24)-2*x**2+6*x-4 )/(((5*x-10)*ln(x)-5*x**2+20*x-20)*ln(12*ln(x)-12*x+24)*ln(1/(4*x-8)/ln(12 *ln(x)-12*x+24)**2)**2+((10*x**2-20*x)*ln(x)-10*x**3+40*x**2-40*x)*ln(12*l n(x)-12*x+24)*ln(1/(4*x-8)/ln(12*ln(x)-12*x+24)**2)+((5*x**3-10*x**2)*ln(x )-5*x**4+20*x**3-20*x**2)*ln(12*ln(x)-12*x+24)),x)
Output:
x/(5*x + 5*log(1/((4*x - 8)*log(-12*x + 12*log(x) + 24)**2)))
\[ \int \frac {-4+6 x-2 x^2+\left (2 x-x^2+x \log (x)\right ) \log (24-12 x+12 \log (x))+\left (-4+4 x-x^2+(-2+x) \log (x)\right ) \log (24-12 x+12 \log (x)) \log \left (\frac {1}{(-8+4 x) \log ^2(24-12 x+12 \log (x))}\right )}{\left (-20 x^2+20 x^3-5 x^4+\left (-10 x^2+5 x^3\right ) \log (x)\right ) \log (24-12 x+12 \log (x))+\left (-40 x+40 x^2-10 x^3+\left (-20 x+10 x^2\right ) \log (x)\right ) \log (24-12 x+12 \log (x)) \log \left (\frac {1}{(-8+4 x) \log ^2(24-12 x+12 \log (x))}\right )+\left (-20+20 x-5 x^2+(-10+5 x) \log (x)\right ) \log (24-12 x+12 \log (x)) \log ^2\left (\frac {1}{(-8+4 x) \log ^2(24-12 x+12 \log (x))}\right )} \, dx=\int { \frac {{\left (x^{2} - {\left (x - 2\right )} \log \left (x\right ) - 4 \, x + 4\right )} \log \left (-12 \, x + 12 \, \log \left (x\right ) + 24\right ) \log \left (\frac {1}{4 \, {\left (x - 2\right )} \log \left (-12 \, x + 12 \, \log \left (x\right ) + 24\right )^{2}}\right ) + 2 \, x^{2} + {\left (x^{2} - x \log \left (x\right ) - 2 \, x\right )} \log \left (-12 \, x + 12 \, \log \left (x\right ) + 24\right ) - 6 \, x + 4}{5 \, {\left ({\left (x^{2} - {\left (x - 2\right )} \log \left (x\right ) - 4 \, x + 4\right )} \log \left (-12 \, x + 12 \, \log \left (x\right ) + 24\right ) \log \left (\frac {1}{4 \, {\left (x - 2\right )} \log \left (-12 \, x + 12 \, \log \left (x\right ) + 24\right )^{2}}\right )^{2} + 2 \, {\left (x^{3} - 4 \, x^{2} - {\left (x^{2} - 2 \, x\right )} \log \left (x\right ) + 4 \, x\right )} \log \left (-12 \, x + 12 \, \log \left (x\right ) + 24\right ) \log \left (\frac {1}{4 \, {\left (x - 2\right )} \log \left (-12 \, x + 12 \, \log \left (x\right ) + 24\right )^{2}}\right ) + {\left (x^{4} - 4 \, x^{3} + 4 \, x^{2} - {\left (x^{3} - 2 \, x^{2}\right )} \log \left (x\right )\right )} \log \left (-12 \, x + 12 \, \log \left (x\right ) + 24\right )\right )}} \,d x } \] Input:
integrate((((-2+x)*log(x)-x^2+4*x-4)*log(12*log(x)-12*x+24)*log(1/(4*x-8)/ log(12*log(x)-12*x+24)^2)+(x*log(x)-x^2+2*x)*log(12*log(x)-12*x+24)-2*x^2+ 6*x-4)/(((5*x-10)*log(x)-5*x^2+20*x-20)*log(12*log(x)-12*x+24)*log(1/(4*x- 8)/log(12*log(x)-12*x+24)^2)^2+((10*x^2-20*x)*log(x)-10*x^3+40*x^2-40*x)*l og(12*log(x)-12*x+24)*log(1/(4*x-8)/log(12*log(x)-12*x+24)^2)+((5*x^3-10*x ^2)*log(x)-5*x^4+20*x^3-20*x^2)*log(12*log(x)-12*x+24)),x, algorithm="maxi ma")
Output:
-1/5*((-I*pi*(log(3) + 2*log(2)) - log(3)^2 - 4*log(3)*log(2) - 4*log(2)^2 )*x^4 + (I*pi*(5*log(3) + 10*log(2) + 2) + 5*log(3)^2 + 4*(5*log(3) + 1)*l og(2) + 20*log(2)^2 + 2*log(3))*x^3 - 6*(I*pi*(log(3) + 2*log(2) + 1) + lo g(3)^2 + 2*(2*log(3) + 1)*log(2) + 4*log(2)^2 + log(3))*x^2 - 4*(-I*pi - l og(3) - 2*log(2))*x - (x^4*(log(3) + 2*log(2)) - x^3*(5*log(3) + 10*log(2) + 2) + 6*x^2*(log(3) + 2*log(2) + 1) - (x^3*(log(3) + 2*log(2)) - 3*x^2*( log(3) + 2*log(2)))*log(x) - 4*x)*log(x - log(x) - 2) + ((I*pi*(log(3) + 2 *log(2)) + log(3)^2 + 4*log(3)*log(2) + 4*log(2)^2)*x^3 - 3*(I*pi*(log(3) + 2*log(2)) + log(3)^2 + 4*log(3)*log(2) + 4*log(2)^2)*x^2)*log(x) + ((-I* pi - log(3) - 2*log(2))*x^4 - 5*(-I*pi - log(3) - 2*log(2))*x^3 - 6*(I*pi + log(3) + 2*log(2))*x^2 - (x^4 - 5*x^3 + 6*x^2 - (x^3 - 3*x^2)*log(x))*lo g(x - log(x) - 2) + ((I*pi + log(3) + 2*log(2))*x^3 - 3*(I*pi + log(3) + 2 *log(2))*x^2)*log(x))*log(-x + log(x) + 2))/((I*pi*(log(3) + 2*log(2)) + l og(3)^2 + 4*log(3)*log(2) + 4*log(2)^2)*x^4 - (4*(2*log(3) + 5)*log(2)^2 + 8*log(2)^3 + I*pi*(2*(log(3) + 5)*log(2) + 4*log(2)^2 + 5*log(3)) + 5*log (3)^2 + 2*(log(3)^2 + 10*log(3) + 2)*log(2) + 2*log(3))*x^3 + 2*(4*(5*log( 3) + 4)*log(2)^2 + 20*log(2)^3 + I*pi*((5*log(3) + 6)*log(2) + 10*log(2)^2 + 3*log(3)) + 3*log(3)^2 + (5*log(3)^2 + 14*log(3) + 6)*log(2) + 3*log(3) )*x^2 - 4*(6*(2*log(3) + 1)*log(2)^2 + 12*log(2)^3 + 3*I*pi*(log(3)*log(2) + 2*log(2)^2) + (3*log(3)^2 + 3*log(3) + 2)*log(2) + log(3))*x + 8*log...
Leaf count of result is larger than twice the leaf count of optimal. 1588 vs. \(2 (27) = 54\).
Time = 96.25 (sec) , antiderivative size = 1588, normalized size of antiderivative = 51.23 \[ \int \frac {-4+6 x-2 x^2+\left (2 x-x^2+x \log (x)\right ) \log (24-12 x+12 \log (x))+\left (-4+4 x-x^2+(-2+x) \log (x)\right ) \log (24-12 x+12 \log (x)) \log \left (\frac {1}{(-8+4 x) \log ^2(24-12 x+12 \log (x))}\right )}{\left (-20 x^2+20 x^3-5 x^4+\left (-10 x^2+5 x^3\right ) \log (x)\right ) \log (24-12 x+12 \log (x))+\left (-40 x+40 x^2-10 x^3+\left (-20 x+10 x^2\right ) \log (x)\right ) \log (24-12 x+12 \log (x)) \log \left (\frac {1}{(-8+4 x) \log ^2(24-12 x+12 \log (x))}\right )+\left (-20+20 x-5 x^2+(-10+5 x) \log (x)\right ) \log (24-12 x+12 \log (x)) \log ^2\left (\frac {1}{(-8+4 x) \log ^2(24-12 x+12 \log (x))}\right )} \, dx=\text {Too large to display} \] Input:
integrate((((-2+x)*log(x)-x^2+4*x-4)*log(12*log(x)-12*x+24)*log(1/(4*x-8)/ log(12*log(x)-12*x+24)^2)+(x*log(x)-x^2+2*x)*log(12*log(x)-12*x+24)-2*x^2+ 6*x-4)/(((5*x-10)*log(x)-5*x^2+20*x-20)*log(12*log(x)-12*x+24)*log(1/(4*x- 8)/log(12*log(x)-12*x+24)^2)^2+((10*x^2-20*x)*log(x)-10*x^3+40*x^2-40*x)*l og(12*log(x)-12*x+24)*log(1/(4*x-8)/log(12*log(x)-12*x+24)^2)+((5*x^3-10*x ^2)*log(x)-5*x^4+20*x^3-20*x^2)*log(12*log(x)-12*x+24)),x, algorithm="giac ")
Output:
1/5*(2*x^4*log(2)*log(-12*x + 12*log(x) + 24) - 2*x^3*log(2)*log(x)*log(-1 2*x + 12*log(x) + 24) + x^4*log(-3*x + 3*log(x) + 6)*log(-12*x + 12*log(x) + 24) - x^3*log(x)*log(-3*x + 3*log(x) + 6)*log(-12*x + 12*log(x) + 24) - 10*x^3*log(2)*log(-12*x + 12*log(x) + 24) + 6*x^2*log(2)*log(x)*log(-12*x + 12*log(x) + 24) - 5*x^3*log(-3*x + 3*log(x) + 6)*log(-12*x + 12*log(x) + 24) + 3*x^2*log(x)*log(-3*x + 3*log(x) + 6)*log(-12*x + 12*log(x) + 24) - 2*x^3*log(-12*x + 12*log(x) + 24) + 12*x^2*log(2)*log(-12*x + 12*log(x) + 24) + 6*x^2*log(-3*x + 3*log(x) + 6)*log(-12*x + 12*log(x) + 24) + 6*x^2 *log(-12*x + 12*log(x) + 24) - 4*x*log(-12*x + 12*log(x) + 24))/(2*x^4*log (2)*log(-12*x + 12*log(x) + 24) - 4*x^3*log(2)^2*log(-12*x + 12*log(x) + 2 4) - 2*x^3*log(2)*log(x*log(-12*x + 12*log(x) + 24)^2 - 2*log(-12*x + 12*l og(x) + 24)^2)*log(-12*x + 12*log(x) + 24) - 2*x^3*log(2)*log(x)*log(-12*x + 12*log(x) + 24) + 4*x^2*log(2)^2*log(x)*log(-12*x + 12*log(x) + 24) + 2 *x^2*log(2)*log(x*log(-12*x + 12*log(x) + 24)^2 - 2*log(-12*x + 12*log(x) + 24)^2)*log(x)*log(-12*x + 12*log(x) + 24) + x^4*log(-3*x + 3*log(x) + 6) *log(-12*x + 12*log(x) + 24) - 2*x^3*log(2)*log(-3*x + 3*log(x) + 6)*log(- 12*x + 12*log(x) + 24) - x^3*log(x*log(-12*x + 12*log(x) + 24)^2 - 2*log(- 12*x + 12*log(x) + 24)^2)*log(-3*x + 3*log(x) + 6)*log(-12*x + 12*log(x) + 24) - x^3*log(x)*log(-3*x + 3*log(x) + 6)*log(-12*x + 12*log(x) + 24) + 2 *x^2*log(2)*log(x)*log(-3*x + 3*log(x) + 6)*log(-12*x + 12*log(x) + 24)...
Timed out. \[ \int \frac {-4+6 x-2 x^2+\left (2 x-x^2+x \log (x)\right ) \log (24-12 x+12 \log (x))+\left (-4+4 x-x^2+(-2+x) \log (x)\right ) \log (24-12 x+12 \log (x)) \log \left (\frac {1}{(-8+4 x) \log ^2(24-12 x+12 \log (x))}\right )}{\left (-20 x^2+20 x^3-5 x^4+\left (-10 x^2+5 x^3\right ) \log (x)\right ) \log (24-12 x+12 \log (x))+\left (-40 x+40 x^2-10 x^3+\left (-20 x+10 x^2\right ) \log (x)\right ) \log (24-12 x+12 \log (x)) \log \left (\frac {1}{(-8+4 x) \log ^2(24-12 x+12 \log (x))}\right )+\left (-20+20 x-5 x^2+(-10+5 x) \log (x)\right ) \log (24-12 x+12 \log (x)) \log ^2\left (\frac {1}{(-8+4 x) \log ^2(24-12 x+12 \log (x))}\right )} \, dx=-\int \frac {6\,x+\ln \left (12\,\ln \left (x\right )-12\,x+24\right )\,\left (2\,x+x\,\ln \left (x\right )-x^2\right )-2\,x^2+\ln \left (\frac {1}{{\ln \left (12\,\ln \left (x\right )-12\,x+24\right )}^2\,\left (4\,x-8\right )}\right )\,\ln \left (12\,\ln \left (x\right )-12\,x+24\right )\,\left (4\,x+\ln \left (x\right )\,\left (x-2\right )-x^2-4\right )-4}{-\ln \left (12\,\ln \left (x\right )-12\,x+24\right )\,\left (20\,x+\ln \left (x\right )\,\left (5\,x-10\right )-5\,x^2-20\right )\,{\ln \left (\frac {1}{{\ln \left (12\,\ln \left (x\right )-12\,x+24\right )}^2\,\left (4\,x-8\right )}\right )}^2+\ln \left (12\,\ln \left (x\right )-12\,x+24\right )\,\left (40\,x+\ln \left (x\right )\,\left (20\,x-10\,x^2\right )-40\,x^2+10\,x^3\right )\,\ln \left (\frac {1}{{\ln \left (12\,\ln \left (x\right )-12\,x+24\right )}^2\,\left (4\,x-8\right )}\right )+\ln \left (12\,\ln \left (x\right )-12\,x+24\right )\,\left (\ln \left (x\right )\,\left (10\,x^2-5\,x^3\right )+20\,x^2-20\,x^3+5\,x^4\right )} \,d x \] Input:
int(-(6*x + log(12*log(x) - 12*x + 24)*(2*x + x*log(x) - x^2) - 2*x^2 + lo g(1/(log(12*log(x) - 12*x + 24)^2*(4*x - 8)))*log(12*log(x) - 12*x + 24)*( 4*x + log(x)*(x - 2) - x^2 - 4) - 4)/(log(12*log(x) - 12*x + 24)*(log(x)*( 10*x^2 - 5*x^3) + 20*x^2 - 20*x^3 + 5*x^4) - log(1/(log(12*log(x) - 12*x + 24)^2*(4*x - 8)))^2*log(12*log(x) - 12*x + 24)*(20*x + log(x)*(5*x - 10) - 5*x^2 - 20) + log(1/(log(12*log(x) - 12*x + 24)^2*(4*x - 8)))*log(12*log (x) - 12*x + 24)*(40*x + log(x)*(20*x - 10*x^2) - 40*x^2 + 10*x^3)),x)
Output:
-int((6*x + log(12*log(x) - 12*x + 24)*(2*x + x*log(x) - x^2) - 2*x^2 + lo g(1/(log(12*log(x) - 12*x + 24)^2*(4*x - 8)))*log(12*log(x) - 12*x + 24)*( 4*x + log(x)*(x - 2) - x^2 - 4) - 4)/(log(12*log(x) - 12*x + 24)*(log(x)*( 10*x^2 - 5*x^3) + 20*x^2 - 20*x^3 + 5*x^4) - log(1/(log(12*log(x) - 12*x + 24)^2*(4*x - 8)))^2*log(12*log(x) - 12*x + 24)*(20*x + log(x)*(5*x - 10) - 5*x^2 - 20) + log(1/(log(12*log(x) - 12*x + 24)^2*(4*x - 8)))*log(12*log (x) - 12*x + 24)*(40*x + log(x)*(20*x - 10*x^2) - 40*x^2 + 10*x^3)), x)
Time = 0.28 (sec) , antiderivative size = 72, normalized size of antiderivative = 2.32 \[ \int \frac {-4+6 x-2 x^2+\left (2 x-x^2+x \log (x)\right ) \log (24-12 x+12 \log (x))+\left (-4+4 x-x^2+(-2+x) \log (x)\right ) \log (24-12 x+12 \log (x)) \log \left (\frac {1}{(-8+4 x) \log ^2(24-12 x+12 \log (x))}\right )}{\left (-20 x^2+20 x^3-5 x^4+\left (-10 x^2+5 x^3\right ) \log (x)\right ) \log (24-12 x+12 \log (x))+\left (-40 x+40 x^2-10 x^3+\left (-20 x+10 x^2\right ) \log (x)\right ) \log (24-12 x+12 \log (x)) \log \left (\frac {1}{(-8+4 x) \log ^2(24-12 x+12 \log (x))}\right )+\left (-20+20 x-5 x^2+(-10+5 x) \log (x)\right ) \log (24-12 x+12 \log (x)) \log ^2\left (\frac {1}{(-8+4 x) \log ^2(24-12 x+12 \log (x))}\right )} \, dx=-\frac {\mathrm {log}\left (4 \mathrm {log}\left (12 \,\mathrm {log}\left (x \right )-12 x +24\right )^{2} x -8 \mathrm {log}\left (12 \,\mathrm {log}\left (x \right )-12 x +24\right )^{2}\right )}{5 \,\mathrm {log}\left (4 \mathrm {log}\left (12 \,\mathrm {log}\left (x \right )-12 x +24\right )^{2} x -8 \mathrm {log}\left (12 \,\mathrm {log}\left (x \right )-12 x +24\right )^{2}\right )-5 x} \] Input:
int((((-2+x)*log(x)-x^2+4*x-4)*log(12*log(x)-12*x+24)*log(1/(4*x-8)/log(12 *log(x)-12*x+24)^2)+(x*log(x)-x^2+2*x)*log(12*log(x)-12*x+24)-2*x^2+6*x-4) /(((5*x-10)*log(x)-5*x^2+20*x-20)*log(12*log(x)-12*x+24)*log(1/(4*x-8)/log (12*log(x)-12*x+24)^2)^2+((10*x^2-20*x)*log(x)-10*x^3+40*x^2-40*x)*log(12* log(x)-12*x+24)*log(1/(4*x-8)/log(12*log(x)-12*x+24)^2)+((5*x^3-10*x^2)*lo g(x)-5*x^4+20*x^3-20*x^2)*log(12*log(x)-12*x+24)),x)
Output:
( - log(4*log(12*log(x) - 12*x + 24)**2*x - 8*log(12*log(x) - 12*x + 24)** 2))/(5*(log(4*log(12*log(x) - 12*x + 24)**2*x - 8*log(12*log(x) - 12*x + 2 4)**2) - x))