Integrand size = 137, antiderivative size = 23 \[ \int \frac {1530+1002 x-2902 x^2+1394 x^3-186 x^4+2 x^5+\left (-1530 x+1038 x^2-182 x^3+2 x^4\right ) \log (x)+\left (18+12 x-34 x^2+16 x^3-2 x^4+\left (-18 x+12 x^2-2 x^3\right ) \log (x)\right ) \log \left (\frac {-4 x+x^2+(-3+x) \log (x)}{-3+x}\right )}{12 x^2-7 x^3+x^4+\left (9 x-6 x^2+x^3\right ) \log (x)} \, dx=e+\left (85-x+\log \left (x+\frac {x}{3-x}+\log (x)\right )\right )^2 \] Output:
(85-x+ln(1/(3-x)*x+x+ln(x)))^2+exp(1)
Time = 0.04 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {1530+1002 x-2902 x^2+1394 x^3-186 x^4+2 x^5+\left (-1530 x+1038 x^2-182 x^3+2 x^4\right ) \log (x)+\left (18+12 x-34 x^2+16 x^3-2 x^4+\left (-18 x+12 x^2-2 x^3\right ) \log (x)\right ) \log \left (\frac {-4 x+x^2+(-3+x) \log (x)}{-3+x}\right )}{12 x^2-7 x^3+x^4+\left (9 x-6 x^2+x^3\right ) \log (x)} \, dx=\left (-85+x-\log \left (\frac {(-4+x) x}{-3+x}+\log (x)\right )\right )^2 \] Input:
Integrate[(1530 + 1002*x - 2902*x^2 + 1394*x^3 - 186*x^4 + 2*x^5 + (-1530* x + 1038*x^2 - 182*x^3 + 2*x^4)*Log[x] + (18 + 12*x - 34*x^2 + 16*x^3 - 2* x^4 + (-18*x + 12*x^2 - 2*x^3)*Log[x])*Log[(-4*x + x^2 + (-3 + x)*Log[x])/ (-3 + x)])/(12*x^2 - 7*x^3 + x^4 + (9*x - 6*x^2 + x^3)*Log[x]),x]
Output:
(-85 + x - Log[((-4 + x)*x)/(-3 + x) + Log[x]])^2
Time = 0.68 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.022, Rules used = {7239, 27, 7237}
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.
| \(\displaystyle \int \frac {2 x^5-186 x^4+1394 x^3-2902 x^2+\left (2 x^4-182 x^3+1038 x^2-1530 x\right ) \log (x)+\left (-2 x^4+16 x^3-34 x^2+\left (-2 x^3+12 x^2-18 x\right ) \log (x)+12 x+18\right ) \log \left (\frac {x^2-4 x+(x-3) \log (x)}{x-3}\right )+1002 x+1530}{x^4-7 x^3+12 x^2+\left (x^3-6 x^2+9 x\right ) \log (x)} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {2 \left (-x^4+8 x^3-17 x^2+6 x-(x-3)^2 x \log (x)+9\right ) \left (x-\log \left (\frac {(x-4) x}{x-3}+\log (x)\right )-85\right )}{(3-x) x ((x-4) x+(x-3) \log (x))}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \int \frac {\left (-x^4+8 x^3-17 x^2-(3-x)^2 \log (x) x+6 x+9\right ) \left (-x+\log \left (\frac {(4-x) x}{3-x}+\log (x)\right )+85\right )}{(3-x) x ((4-x) x+(3-x) \log (x))}dx\) |
| \(\Big \downarrow \) 7237 |
| \(\displaystyle \left (-x+\log \left (\frac {(4-x) x}{3-x}+\log (x)\right )+85\right )^2\) |
Input:
Int[(1530 + 1002*x - 2902*x^2 + 1394*x^3 - 186*x^4 + 2*x^5 + (-1530*x + 10 38*x^2 - 182*x^3 + 2*x^4)*Log[x] + (18 + 12*x - 34*x^2 + 16*x^3 - 2*x^4 + (-18*x + 12*x^2 - 2*x^3)*Log[x])*Log[(-4*x + x^2 + (-3 + x)*Log[x])/(-3 + x)])/(12*x^2 - 7*x^3 + x^4 + (9*x - 6*x^2 + x^3)*Log[x]),x]
Output:
(85 - x + Log[((4 - x)*x)/(3 - x) + Log[x]])^2
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Si mp[q*(y^(m + 1)/(m + 1)), x] /; !FalseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Leaf count of result is larger than twice the leaf count of optimal. \(74\) vs. \(2(24)=48\).
Time = 2.83 (sec) , antiderivative size = 75, normalized size of antiderivative = 3.26
| method | result | size |
| parallelrisch | \(x^{2}-2 \ln \left (\frac {\ln \left (x \right ) \left (-3+x \right )+x^{2}-4 x}{-3+x}\right ) x +\ln \left (\frac {\ln \left (x \right ) \left (-3+x \right )+x^{2}-4 x}{-3+x}\right )^{2}-170 x +170 \ln \left (\frac {\ln \left (x \right ) \left (-3+x \right )+x^{2}-4 x}{-3+x}\right )\) | \(75\) |
Input:
int((((-2*x^3+12*x^2-18*x)*ln(x)-2*x^4+16*x^3-34*x^2+12*x+18)*ln((ln(x)*(- 3+x)+x^2-4*x)/(-3+x))+(2*x^4-182*x^3+1038*x^2-1530*x)*ln(x)+2*x^5-186*x^4+ 1394*x^3-2902*x^2+1002*x+1530)/((x^3-6*x^2+9*x)*ln(x)+x^4-7*x^3+12*x^2),x, method=_RETURNVERBOSE)
Output:
x^2-2*ln((ln(x)*(-3+x)+x^2-4*x)/(-3+x))*x+ln((ln(x)*(-3+x)+x^2-4*x)/(-3+x) )^2-170*x+170*ln((ln(x)*(-3+x)+x^2-4*x)/(-3+x))
Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (23) = 46\).
Time = 0.09 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.35 \[ \int \frac {1530+1002 x-2902 x^2+1394 x^3-186 x^4+2 x^5+\left (-1530 x+1038 x^2-182 x^3+2 x^4\right ) \log (x)+\left (18+12 x-34 x^2+16 x^3-2 x^4+\left (-18 x+12 x^2-2 x^3\right ) \log (x)\right ) \log \left (\frac {-4 x+x^2+(-3+x) \log (x)}{-3+x}\right )}{12 x^2-7 x^3+x^4+\left (9 x-6 x^2+x^3\right ) \log (x)} \, dx=x^{2} - 2 \, {\left (x - 85\right )} \log \left (\frac {x^{2} + {\left (x - 3\right )} \log \left (x\right ) - 4 \, x}{x - 3}\right ) + \log \left (\frac {x^{2} + {\left (x - 3\right )} \log \left (x\right ) - 4 \, x}{x - 3}\right )^{2} - 170 \, x \] Input:
integrate((((-2*x^3+12*x^2-18*x)*log(x)-2*x^4+16*x^3-34*x^2+12*x+18)*log(( log(x)*(-3+x)+x^2-4*x)/(-3+x))+(2*x^4-182*x^3+1038*x^2-1530*x)*log(x)+2*x^ 5-186*x^4+1394*x^3-2902*x^2+1002*x+1530)/((x^3-6*x^2+9*x)*log(x)+x^4-7*x^3 +12*x^2),x, algorithm="fricas")
Output:
x^2 - 2*(x - 85)*log((x^2 + (x - 3)*log(x) - 4*x)/(x - 3)) + log((x^2 + (x - 3)*log(x) - 4*x)/(x - 3))^2 - 170*x
Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (19) = 38\).
Time = 0.91 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.83 \[ \int \frac {1530+1002 x-2902 x^2+1394 x^3-186 x^4+2 x^5+\left (-1530 x+1038 x^2-182 x^3+2 x^4\right ) \log (x)+\left (18+12 x-34 x^2+16 x^3-2 x^4+\left (-18 x+12 x^2-2 x^3\right ) \log (x)\right ) \log \left (\frac {-4 x+x^2+(-3+x) \log (x)}{-3+x}\right )}{12 x^2-7 x^3+x^4+\left (9 x-6 x^2+x^3\right ) \log (x)} \, dx=x^{2} - 2 x \log {\left (\frac {x^{2} - 4 x + \left (x - 3\right ) \log {\left (x \right )}}{x - 3} \right )} - 170 x + \log {\left (\frac {x^{2} - 4 x + \left (x - 3\right ) \log {\left (x \right )}}{x - 3} \right )}^{2} + 170 \log {\left (\log {\left (x \right )} + \frac {x^{2} - 4 x}{x - 3} \right )} \] Input:
integrate((((-2*x**3+12*x**2-18*x)*ln(x)-2*x**4+16*x**3-34*x**2+12*x+18)*l n((ln(x)*(-3+x)+x**2-4*x)/(-3+x))+(2*x**4-182*x**3+1038*x**2-1530*x)*ln(x) +2*x**5-186*x**4+1394*x**3-2902*x**2+1002*x+1530)/((x**3-6*x**2+9*x)*ln(x) +x**4-7*x**3+12*x**2),x)
Output:
x**2 - 2*x*log((x**2 - 4*x + (x - 3)*log(x))/(x - 3)) - 170*x + log((x**2 - 4*x + (x - 3)*log(x))/(x - 3))**2 + 170*log(log(x) + (x**2 - 4*x)/(x - 3 ))
\[ \int \frac {1530+1002 x-2902 x^2+1394 x^3-186 x^4+2 x^5+\left (-1530 x+1038 x^2-182 x^3+2 x^4\right ) \log (x)+\left (18+12 x-34 x^2+16 x^3-2 x^4+\left (-18 x+12 x^2-2 x^3\right ) \log (x)\right ) \log \left (\frac {-4 x+x^2+(-3+x) \log (x)}{-3+x}\right )}{12 x^2-7 x^3+x^4+\left (9 x-6 x^2+x^3\right ) \log (x)} \, dx=\int { \frac {2 \, {\left (x^{5} - 93 \, x^{4} + 697 \, x^{3} - 1451 \, x^{2} + {\left (x^{4} - 91 \, x^{3} + 519 \, x^{2} - 765 \, x\right )} \log \left (x\right ) - {\left (x^{4} - 8 \, x^{3} + 17 \, x^{2} + {\left (x^{3} - 6 \, x^{2} + 9 \, x\right )} \log \left (x\right ) - 6 \, x - 9\right )} \log \left (\frac {x^{2} + {\left (x - 3\right )} \log \left (x\right ) - 4 \, x}{x - 3}\right ) + 501 \, x + 765\right )}}{x^{4} - 7 \, x^{3} + 12 \, x^{2} + {\left (x^{3} - 6 \, x^{2} + 9 \, x\right )} \log \left (x\right )} \,d x } \] Input:
integrate((((-2*x^3+12*x^2-18*x)*log(x)-2*x^4+16*x^3-34*x^2+12*x+18)*log(( log(x)*(-3+x)+x^2-4*x)/(-3+x))+(2*x^4-182*x^3+1038*x^2-1530*x)*log(x)+2*x^ 5-186*x^4+1394*x^3-2902*x^2+1002*x+1530)/((x^3-6*x^2+9*x)*log(x)+x^4-7*x^3 +12*x^2),x, algorithm="maxima")
Output:
2*integrate((x^5 - 93*x^4 + 697*x^3 - 1451*x^2 + (x^4 - 91*x^3 + 519*x^2 - 765*x)*log(x) - (x^4 - 8*x^3 + 17*x^2 + (x^3 - 6*x^2 + 9*x)*log(x) - 6*x - 9)*log((x^2 + (x - 3)*log(x) - 4*x)/(x - 3)) + 501*x + 765)/(x^4 - 7*x^3 + 12*x^2 + (x^3 - 6*x^2 + 9*x)*log(x)), x)
\[ \int \frac {1530+1002 x-2902 x^2+1394 x^3-186 x^4+2 x^5+\left (-1530 x+1038 x^2-182 x^3+2 x^4\right ) \log (x)+\left (18+12 x-34 x^2+16 x^3-2 x^4+\left (-18 x+12 x^2-2 x^3\right ) \log (x)\right ) \log \left (\frac {-4 x+x^2+(-3+x) \log (x)}{-3+x}\right )}{12 x^2-7 x^3+x^4+\left (9 x-6 x^2+x^3\right ) \log (x)} \, dx=\int { \frac {2 \, {\left (x^{5} - 93 \, x^{4} + 697 \, x^{3} - 1451 \, x^{2} + {\left (x^{4} - 91 \, x^{3} + 519 \, x^{2} - 765 \, x\right )} \log \left (x\right ) - {\left (x^{4} - 8 \, x^{3} + 17 \, x^{2} + {\left (x^{3} - 6 \, x^{2} + 9 \, x\right )} \log \left (x\right ) - 6 \, x - 9\right )} \log \left (\frac {x^{2} + {\left (x - 3\right )} \log \left (x\right ) - 4 \, x}{x - 3}\right ) + 501 \, x + 765\right )}}{x^{4} - 7 \, x^{3} + 12 \, x^{2} + {\left (x^{3} - 6 \, x^{2} + 9 \, x\right )} \log \left (x\right )} \,d x } \] Input:
integrate((((-2*x^3+12*x^2-18*x)*log(x)-2*x^4+16*x^3-34*x^2+12*x+18)*log(( log(x)*(-3+x)+x^2-4*x)/(-3+x))+(2*x^4-182*x^3+1038*x^2-1530*x)*log(x)+2*x^ 5-186*x^4+1394*x^3-2902*x^2+1002*x+1530)/((x^3-6*x^2+9*x)*log(x)+x^4-7*x^3 +12*x^2),x, algorithm="giac")
Output:
integrate(2*(x^5 - 93*x^4 + 697*x^3 - 1451*x^2 + (x^4 - 91*x^3 + 519*x^2 - 765*x)*log(x) - (x^4 - 8*x^3 + 17*x^2 + (x^3 - 6*x^2 + 9*x)*log(x) - 6*x - 9)*log((x^2 + (x - 3)*log(x) - 4*x)/(x - 3)) + 501*x + 765)/(x^4 - 7*x^3 + 12*x^2 + (x^3 - 6*x^2 + 9*x)*log(x)), x)
Timed out. \[ \int \frac {1530+1002 x-2902 x^2+1394 x^3-186 x^4+2 x^5+\left (-1530 x+1038 x^2-182 x^3+2 x^4\right ) \log (x)+\left (18+12 x-34 x^2+16 x^3-2 x^4+\left (-18 x+12 x^2-2 x^3\right ) \log (x)\right ) \log \left (\frac {-4 x+x^2+(-3+x) \log (x)}{-3+x}\right )}{12 x^2-7 x^3+x^4+\left (9 x-6 x^2+x^3\right ) \log (x)} \, dx=\int \frac {1002\,x-\ln \left (x\right )\,\left (-2\,x^4+182\,x^3-1038\,x^2+1530\,x\right )+\ln \left (\frac {\ln \left (x\right )\,\left (x-3\right )-4\,x+x^2}{x-3}\right )\,\left (12\,x-34\,x^2+16\,x^3-2\,x^4-\ln \left (x\right )\,\left (2\,x^3-12\,x^2+18\,x\right )+18\right )-2902\,x^2+1394\,x^3-186\,x^4+2\,x^5+1530}{\ln \left (x\right )\,\left (x^3-6\,x^2+9\,x\right )+12\,x^2-7\,x^3+x^4} \,d x \] Input:
int((1002*x - log(x)*(1530*x - 1038*x^2 + 182*x^3 - 2*x^4) + log((log(x)*( x - 3) - 4*x + x^2)/(x - 3))*(12*x - 34*x^2 + 16*x^3 - 2*x^4 - log(x)*(18* x - 12*x^2 + 2*x^3) + 18) - 2902*x^2 + 1394*x^3 - 186*x^4 + 2*x^5 + 1530)/ (log(x)*(9*x - 6*x^2 + x^3) + 12*x^2 - 7*x^3 + x^4),x)
Output:
int((1002*x - log(x)*(1530*x - 1038*x^2 + 182*x^3 - 2*x^4) + log((log(x)*( x - 3) - 4*x + x^2)/(x - 3))*(12*x - 34*x^2 + 16*x^3 - 2*x^4 - log(x)*(18* x - 12*x^2 + 2*x^3) + 18) - 2902*x^2 + 1394*x^3 - 186*x^4 + 2*x^5 + 1530)/ (log(x)*(9*x - 6*x^2 + x^3) + 12*x^2 - 7*x^3 + x^4), x)
Time = 0.20 (sec) , antiderivative size = 104, normalized size of antiderivative = 4.52 \[ \int \frac {1530+1002 x-2902 x^2+1394 x^3-186 x^4+2 x^5+\left (-1530 x+1038 x^2-182 x^3+2 x^4\right ) \log (x)+\left (18+12 x-34 x^2+16 x^3-2 x^4+\left (-18 x+12 x^2-2 x^3\right ) \log (x)\right ) \log \left (\frac {-4 x+x^2+(-3+x) \log (x)}{-3+x}\right )}{12 x^2-7 x^3+x^4+\left (9 x-6 x^2+x^3\right ) \log (x)} \, dx=180 \,\mathrm {log}\left (\mathrm {log}\left (x \right ) x -3 \,\mathrm {log}\left (x \right )+x^{2}-4 x \right )-180 \,\mathrm {log}\left (x -3\right )+\mathrm {log}\left (\frac {\mathrm {log}\left (x \right ) x -3 \,\mathrm {log}\left (x \right )+x^{2}-4 x}{x -3}\right )^{2}-2 \,\mathrm {log}\left (\frac {\mathrm {log}\left (x \right ) x -3 \,\mathrm {log}\left (x \right )+x^{2}-4 x}{x -3}\right ) x -10 \,\mathrm {log}\left (\frac {\mathrm {log}\left (x \right ) x -3 \,\mathrm {log}\left (x \right )+x^{2}-4 x}{x -3}\right )+x^{2}-170 x \] Input:
int((((-2*x^3+12*x^2-18*x)*log(x)-2*x^4+16*x^3-34*x^2+12*x+18)*log((log(x) *(-3+x)+x^2-4*x)/(-3+x))+(2*x^4-182*x^3+1038*x^2-1530*x)*log(x)+2*x^5-186* x^4+1394*x^3-2902*x^2+1002*x+1530)/((x^3-6*x^2+9*x)*log(x)+x^4-7*x^3+12*x^ 2),x)
Output:
180*log(log(x)*x - 3*log(x) + x**2 - 4*x) - 180*log(x - 3) + log((log(x)*x - 3*log(x) + x**2 - 4*x)/(x - 3))**2 - 2*log((log(x)*x - 3*log(x) + x**2 - 4*x)/(x - 3))*x - 10*log((log(x)*x - 3*log(x) + x**2 - 4*x)/(x - 3)) + x **2 - 170*x