Integrand size = 53, antiderivative size = 27 \[ \int \frac {-12+4 x+\left (-4 x-24 x^3+8 x^4\right ) \log (x)+\left (-36 x^3+8 x^4\right ) \log ^2(x)}{9 x-6 x^2+x^3} \, dx=-2-\frac {4 x \left (\frac {\log (x)}{x}+x^2 \log ^2(x)\right )}{3-x} \] Output:
-2-x/(3/4-1/4*x)*(ln(x)/x+x^2*ln(x)^2)
Time = 0.07 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.26 \[ \int \frac {-12+4 x+\left (-4 x-24 x^3+8 x^4\right ) \log (x)+\left (-36 x^3+8 x^4\right ) \log ^2(x)}{9 x-6 x^2+x^3} \, dx=\frac {4}{3} \left (-\log (3-x)+\log (-3+x)+\frac {3 \log (x) \left (1+x^3 \log (x)\right )}{-3+x}\right ) \] Input:
Integrate[(-12 + 4*x + (-4*x - 24*x^3 + 8*x^4)*Log[x] + (-36*x^3 + 8*x^4)* Log[x]^2)/(9*x - 6*x^2 + x^3),x]
Output:
(4*(-Log[3 - x] + Log[-3 + x] + (3*Log[x]*(1 + x^3*Log[x]))/(-3 + x)))/3
Time = 1.20 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.96, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {2026, 7277, 25, 7293, 2009}
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.
\(\displaystyle \int \frac {\left (8 x^4-36 x^3\right ) \log ^2(x)+\left (8 x^4-24 x^3-4 x\right ) \log (x)+4 x-12}{x^3-6 x^2+9 x} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {\left (8 x^4-36 x^3\right ) \log ^2(x)+\left (8 x^4-24 x^3-4 x\right ) \log (x)+4 x-12}{x \left (x^2-6 x+9\right )}dx\) |
\(\Big \downarrow \) 7277 |
\(\displaystyle 4 \int -\frac {\left (9 x^3-2 x^4\right ) \log ^2(x)+\left (-2 x^4+6 x^3+x\right ) \log (x)-x+3}{(3-x)^2 x}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -4 \int \frac {\left (9 x^3-2 x^4\right ) \log ^2(x)+\left (-2 x^4+6 x^3+x\right ) \log (x)-x+3}{(3-x)^2 x}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -4 \int \left (-\frac {x^2 (2 x-9) \log ^2(x)}{(x-3)^2}-\frac {\left (2 x^3-6 x^2-1\right ) \log (x)}{(x-3)^2}+\frac {1}{(3-x) x}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -4 \left (-x^2 \log ^2(x)+\frac {9 x \log ^2(x)}{3-x}-3 x \log ^2(x)+\frac {x \log (x)}{3 (3-x)}+\frac {\log (x)}{3}\right )\) |
Input:
Int[(-12 + 4*x + (-4*x - 24*x^3 + 8*x^4)*Log[x] + (-36*x^3 + 8*x^4)*Log[x] ^2)/(9*x - 6*x^2 + x^3),x]
Output:
-4*(Log[x]/3 + (x*Log[x])/(3*(3 - x)) - 3*x*Log[x]^2 + (9*x*Log[x]^2)/(3 - x) - x^2*Log[x]^2)
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_)*((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.))^(p_.), x_Symbol] :> Simp[1/(4^p*c^p) Int[u*(b + 2*c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n} , x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p] && !AlgebraicFu nctionQ[u, x]
Time = 0.63 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.78
method | result | size |
norman | \(\frac {4 \ln \left (x \right )+4 x^{3} \ln \left (x \right )^{2}}{-3+x}\) | \(21\) |
parallelrisch | \(\frac {4 \ln \left (x \right )+4 x^{3} \ln \left (x \right )^{2}}{-3+x}\) | \(21\) |
risch | \(\frac {4 x^{3} \ln \left (x \right )^{2}}{-3+x}+\frac {4 \ln \left (x \right )}{-3+x}\) | \(25\) |
orering | \(\frac {\left (14 x^{7}-114 x^{6}+45 x^{4}+27 x^{3}-9 x +27\right ) \left (\left (8 x^{4}-36 x^{3}\right ) \ln \left (x \right )^{2}+\left (8 x^{4}-24 x^{3}-4 x \right ) \ln \left (x \right )+4 x -12\right )}{\left (16 x^{6}-162 x^{5}+90 x^{3}-243 x^{2}+9\right ) \left (x^{3}-6 x^{2}+9 x \right )}-\frac {3 x^{2} \left (4 x^{6}-24 x^{5}+54 x^{2}-9\right ) \left (\frac {\left (32 x^{3}-108 x^{2}\right ) \ln \left (x \right )^{2}+\frac {2 \left (8 x^{4}-36 x^{3}\right ) \ln \left (x \right )}{x}+\left (32 x^{3}-72 x^{2}-4\right ) \ln \left (x \right )+\frac {8 x^{4}-24 x^{3}-4 x}{x}+4}{x^{3}-6 x^{2}+9 x}-\frac {\left (\left (8 x^{4}-36 x^{3}\right ) \ln \left (x \right )^{2}+\left (8 x^{4}-24 x^{3}-4 x \right ) \ln \left (x \right )+4 x -12\right ) \left (3 x^{2}-12 x +9\right )}{\left (x^{3}-6 x^{2}+9 x \right )^{2}}\right )}{2 \left (16 x^{6}-162 x^{5}+90 x^{3}-243 x^{2}+9\right )}+\frac {x^{2} \left (4 x^{6}-18 x^{3}+9\right ) \left (-3+x \right ) \left (\frac {\left (96 x^{2}-216 x \right ) \ln \left (x \right )^{2}+\frac {4 \left (32 x^{3}-108 x^{2}\right ) \ln \left (x \right )}{x}+\frac {16 x^{4}-72 x^{3}}{x^{2}}-\frac {2 \left (8 x^{4}-36 x^{3}\right ) \ln \left (x \right )}{x^{2}}+\left (96 x^{2}-144 x \right ) \ln \left (x \right )+\frac {64 x^{3}-144 x^{2}-8}{x}-\frac {8 x^{4}-24 x^{3}-4 x}{x^{2}}}{x^{3}-6 x^{2}+9 x}-\frac {2 \left (\left (32 x^{3}-108 x^{2}\right ) \ln \left (x \right )^{2}+\frac {2 \left (8 x^{4}-36 x^{3}\right ) \ln \left (x \right )}{x}+\left (32 x^{3}-72 x^{2}-4\right ) \ln \left (x \right )+\frac {8 x^{4}-24 x^{3}-4 x}{x}+4\right ) \left (3 x^{2}-12 x +9\right )}{\left (x^{3}-6 x^{2}+9 x \right )^{2}}+\frac {2 \left (\left (8 x^{4}-36 x^{3}\right ) \ln \left (x \right )^{2}+\left (8 x^{4}-24 x^{3}-4 x \right ) \ln \left (x \right )+4 x -12\right ) \left (3 x^{2}-12 x +9\right )^{2}}{\left (x^{3}-6 x^{2}+9 x \right )^{3}}-\frac {\left (\left (8 x^{4}-36 x^{3}\right ) \ln \left (x \right )^{2}+\left (8 x^{4}-24 x^{3}-4 x \right ) \ln \left (x \right )+4 x -12\right ) \left (6 x -12\right )}{\left (x^{3}-6 x^{2}+9 x \right )^{2}}\right )}{32 x^{6}-324 x^{5}+180 x^{3}-486 x^{2}+18}\) | \(694\) |
Input:
int(((8*x^4-36*x^3)*ln(x)^2+(8*x^4-24*x^3-4*x)*ln(x)+4*x-12)/(x^3-6*x^2+9* x),x,method=_RETURNVERBOSE)
Output:
(4*ln(x)+4*x^3*ln(x)^2)/(-3+x)
Time = 0.09 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.67 \[ \int \frac {-12+4 x+\left (-4 x-24 x^3+8 x^4\right ) \log (x)+\left (-36 x^3+8 x^4\right ) \log ^2(x)}{9 x-6 x^2+x^3} \, dx=\frac {4 \, {\left (x^{3} \log \left (x\right )^{2} + \log \left (x\right )\right )}}{x - 3} \] Input:
integrate(((8*x^4-36*x^3)*log(x)^2+(8*x^4-24*x^3-4*x)*log(x)+4*x-12)/(x^3- 6*x^2+9*x),x, algorithm="fricas")
Output:
4*(x^3*log(x)^2 + log(x))/(x - 3)
Time = 0.11 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.74 \[ \int \frac {-12+4 x+\left (-4 x-24 x^3+8 x^4\right ) \log (x)+\left (-36 x^3+8 x^4\right ) \log ^2(x)}{9 x-6 x^2+x^3} \, dx=\frac {4 x^{3} \log {\left (x \right )}^{2}}{x - 3} + \frac {4 \log {\left (x \right )}}{x - 3} \] Input:
integrate(((8*x**4-36*x**3)*ln(x)**2+(8*x**4-24*x**3-4*x)*ln(x)+4*x-12)/(x **3-6*x**2+9*x),x)
Output:
4*x**3*log(x)**2/(x - 3) + 4*log(x)/(x - 3)
Time = 0.07 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89 \[ \int \frac {-12+4 x+\left (-4 x-24 x^3+8 x^4\right ) \log (x)+\left (-36 x^3+8 x^4\right ) \log ^2(x)}{9 x-6 x^2+x^3} \, dx=\frac {4 \, x^{3} \log \left (x\right )^{2}}{x - 3} + \frac {4 \, \log \left (x\right )}{x - 3} \] Input:
integrate(((8*x^4-36*x^3)*log(x)^2+(8*x^4-24*x^3-4*x)*log(x)+4*x-12)/(x^3- 6*x^2+9*x),x, algorithm="maxima")
Output:
4*x^3*log(x)^2/(x - 3) + 4*log(x)/(x - 3)
\[ \int \frac {-12+4 x+\left (-4 x-24 x^3+8 x^4\right ) \log (x)+\left (-36 x^3+8 x^4\right ) \log ^2(x)}{9 x-6 x^2+x^3} \, dx=\int { \frac {4 \, {\left ({\left (2 \, x^{4} - 9 \, x^{3}\right )} \log \left (x\right )^{2} + {\left (2 \, x^{4} - 6 \, x^{3} - x\right )} \log \left (x\right ) + x - 3\right )}}{x^{3} - 6 \, x^{2} + 9 \, x} \,d x } \] Input:
integrate(((8*x^4-36*x^3)*log(x)^2+(8*x^4-24*x^3-4*x)*log(x)+4*x-12)/(x^3- 6*x^2+9*x),x, algorithm="giac")
Output:
integrate(4*((2*x^4 - 9*x^3)*log(x)^2 + (2*x^4 - 6*x^3 - x)*log(x) + x - 3 )/(x^3 - 6*x^2 + 9*x), x)
Time = 3.48 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.63 \[ \int \frac {-12+4 x+\left (-4 x-24 x^3+8 x^4\right ) \log (x)+\left (-36 x^3+8 x^4\right ) \log ^2(x)}{9 x-6 x^2+x^3} \, dx=\frac {4\,\ln \left (x\right )\,\left (x^3\,\ln \left (x\right )+1\right )}{x-3} \] Input:
int(-(log(x)^2*(36*x^3 - 8*x^4) - 4*x + log(x)*(4*x + 24*x^3 - 8*x^4) + 12 )/(9*x - 6*x^2 + x^3),x)
Output:
(4*log(x)*(x^3*log(x) + 1))/(x - 3)
Time = 0.16 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.63 \[ \int \frac {-12+4 x+\left (-4 x-24 x^3+8 x^4\right ) \log (x)+\left (-36 x^3+8 x^4\right ) \log ^2(x)}{9 x-6 x^2+x^3} \, dx=\frac {4 \,\mathrm {log}\left (x \right ) \left (\mathrm {log}\left (x \right ) x^{3}+1\right )}{x -3} \] Input:
int(((8*x^4-36*x^3)*log(x)^2+(8*x^4-24*x^3-4*x)*log(x)+4*x-12)/(x^3-6*x^2+ 9*x),x)
Output:
(4*log(x)*(log(x)*x**3 + 1))/(x - 3)