Integrand size = 77, antiderivative size = 27 \[ \int \frac {32-72 x+72 x^2-30 x^3+4 x^4+\left (8 x-20 x^2+2 x^3+2 x^4\right ) \log \left (x^2\right )}{\left (16 x-36 x^2+36 x^3-15 x^4+2 x^5\right ) \log \left (x^2\right )} \, dx=\log \left (\frac {(-4+x)^2 \log \left (x^2\right )}{1-\frac {4}{2-x}+2 x}\right ) \]
Time = 0.40 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.22 \[ \int \frac {32-72 x+72 x^2-30 x^3+4 x^4+\left (8 x-20 x^2+2 x^3+2 x^4\right ) \log \left (x^2\right )}{\left (16 x-36 x^2+36 x^3-15 x^4+2 x^5\right ) \log \left (x^2\right )} \, dx=\log (2-x)+2 \log (4-x)-\log \left (2-3 x+2 x^2\right )+\log \left (\log \left (x^2\right )\right ) \]
Integrate[(32 - 72*x + 72*x^2 - 30*x^3 + 4*x^4 + (8*x - 20*x^2 + 2*x^3 + 2 *x^4)*Log[x^2])/((16*x - 36*x^2 + 36*x^3 - 15*x^4 + 2*x^5)*Log[x^2]),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.
\(\displaystyle \int \frac {4 x^4-30 x^3+72 x^2+\left (2 x^4+2 x^3-20 x^2+8 x\right ) \log \left (x^2\right )-72 x+32}{\left (2 x^5-15 x^4+36 x^3-36 x^2+16 x\right ) \log \left (x^2\right )} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {4 x^4-30 x^3+72 x^2+\left (2 x^4+2 x^3-20 x^2+8 x\right ) \log \left (x^2\right )-72 x+32}{x \left (2 x^4-15 x^3+36 x^2-36 x+16\right ) \log \left (x^2\right )}dx\) |
\(\Big \downarrow \) 2463 |
\(\displaystyle \int \left (\frac {4 x^4-30 x^3+72 x^2+\left (2 x^4+2 x^3-20 x^2+8 x\right ) \log \left (x^2\right )-72 x+32}{44 (x-4) x \log \left (x^2\right )}-\frac {4 x^4-30 x^3+72 x^2+\left (2 x^4+2 x^3-20 x^2+8 x\right ) \log \left (x^2\right )-72 x+32}{8 (x-2) x \log \left (x^2\right )}+\frac {(18 x+1) \left (4 x^4-30 x^3+72 x^2+\left (2 x^4+2 x^3-20 x^2+8 x\right ) \log \left (x^2\right )-72 x+32\right )}{88 x \left (2 x^2-3 x+2\right ) \log \left (x^2\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {1}{4} \int \frac {2 x^3-11 x^2+14 x-8}{x \log \left (x^2\right )}dx+\frac {1}{22} \int \frac {2 x^3-7 x^2+8 x-4}{x \log \left (x^2\right )}dx+\frac {1}{44} \int \frac {18 x^3-107 x^2+138 x+8}{x \log \left (x^2\right )}dx-\log \left (2 x^2-3 x+2\right )+\log (2-x)+2 \log (4-x)\) |
Int[(32 - 72*x + 72*x^2 - 30*x^3 + 4*x^4 + (8*x - 20*x^2 + 2*x^3 + 2*x^4)* Log[x^2])/((16*x - 36*x^2 + 36*x^3 - 15*x^4 + 2*x^5)*Log[x^2]),x]
3.19.65.3.1 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 = 0.09 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.04
method | result | size |
parallelrisch | \(\ln \left (\ln \left (x^{2}\right )\right )+\ln \left (-2+x \right )+2 \ln \left (x -4\right )-\ln \left (x^{2}-\frac {3}{2} x +1\right )\) | \(28\) |
default | \(\ln \left (\ln \left (x^{2}\right )\right )-\ln \left (2 x^{2}-3 x +2\right )+\ln \left (-2+x \right )+2 \ln \left (x -4\right )\) | \(30\) |
norman | \(\ln \left (\ln \left (x^{2}\right )\right )-\ln \left (2 x^{2}-3 x +2\right )+\ln \left (-2+x \right )+2 \ln \left (x -4\right )\) | \(30\) |
risch | \(\ln \left (\ln \left (x^{2}\right )\right )-\ln \left (2 x^{2}-3 x +2\right )+\ln \left (-2+x \right )+2 \ln \left (x -4\right )\) | \(30\) |
parts | \(\ln \left (\ln \left (x^{2}\right )\right )-\ln \left (2 x^{2}-3 x +2\right )+\ln \left (-2+x \right )+2 \ln \left (x -4\right )\) | \(30\) |
int(((2*x^4+2*x^3-20*x^2+8*x)*ln(x^2)+4*x^4-30*x^3+72*x^2-72*x+32)/(2*x^5- 15*x^4+36*x^3-36*x^2+16*x)/ln(x^2),x,method=_RETURNVERBOSE)
Time = 0.24 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {32-72 x+72 x^2-30 x^3+4 x^4+\left (8 x-20 x^2+2 x^3+2 x^4\right ) \log \left (x^2\right )}{\left (16 x-36 x^2+36 x^3-15 x^4+2 x^5\right ) \log \left (x^2\right )} \, dx=-\log \left (2 \, x^{2} - 3 \, x + 2\right ) + \log \left (x - 2\right ) + 2 \, \log \left (x - 4\right ) + \log \left (\log \left (x^{2}\right )\right ) \]
integrate(((2*x^4+2*x^3-20*x^2+8*x)*log(x^2)+4*x^4-30*x^3+72*x^2-72*x+32)/ (2*x^5-15*x^4+36*x^3-36*x^2+16*x)/log(x^2),x, algorithm=\
Time = 0.09 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {32-72 x+72 x^2-30 x^3+4 x^4+\left (8 x-20 x^2+2 x^3+2 x^4\right ) \log \left (x^2\right )}{\left (16 x-36 x^2+36 x^3-15 x^4+2 x^5\right ) \log \left (x^2\right )} \, dx=2 \log {\left (x - 4 \right )} + \log {\left (x - 2 \right )} - \log {\left (2 x^{2} - 3 x + 2 \right )} + \log {\left (\log {\left (x^{2} \right )} \right )} \]
integrate(((2*x**4+2*x**3-20*x**2+8*x)*ln(x**2)+4*x**4-30*x**3+72*x**2-72* x+32)/(2*x**5-15*x**4+36*x**3-36*x**2+16*x)/ln(x**2),x)
Time = 0.24 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {32-72 x+72 x^2-30 x^3+4 x^4+\left (8 x-20 x^2+2 x^3+2 x^4\right ) \log \left (x^2\right )}{\left (16 x-36 x^2+36 x^3-15 x^4+2 x^5\right ) \log \left (x^2\right )} \, dx=-\log \left (2 \, x^{2} - 3 \, x + 2\right ) + \log \left (x - 2\right ) + 2 \, \log \left (x - 4\right ) + \log \left (\log \left (x\right )\right ) \]
integrate(((2*x^4+2*x^3-20*x^2+8*x)*log(x^2)+4*x^4-30*x^3+72*x^2-72*x+32)/ (2*x^5-15*x^4+36*x^3-36*x^2+16*x)/log(x^2),x, algorithm=\
Time = 0.28 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {32-72 x+72 x^2-30 x^3+4 x^4+\left (8 x-20 x^2+2 x^3+2 x^4\right ) \log \left (x^2\right )}{\left (16 x-36 x^2+36 x^3-15 x^4+2 x^5\right ) \log \left (x^2\right )} \, dx=-\log \left (2 \, x^{2} - 3 \, x + 2\right ) + \log \left (x - 2\right ) + 2 \, \log \left (x - 4\right ) + \log \left (\log \left (x^{2}\right )\right ) \]
integrate(((2*x^4+2*x^3-20*x^2+8*x)*log(x^2)+4*x^4-30*x^3+72*x^2-72*x+32)/ (2*x^5-15*x^4+36*x^3-36*x^2+16*x)/log(x^2),x, algorithm=\
Time = 11.34 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {32-72 x+72 x^2-30 x^3+4 x^4+\left (8 x-20 x^2+2 x^3+2 x^4\right ) \log \left (x^2\right )}{\left (16 x-36 x^2+36 x^3-15 x^4+2 x^5\right ) \log \left (x^2\right )} \, dx=\ln \left (x-2\right )+2\,\ln \left (x-4\right )-\ln \left (x^2-\frac {3\,x}{2}+1\right )+\ln \left (\ln \left (x^2\right )\right ) \]