Integrand size = 73, antiderivative size = 26 \begin {dmath*} \int \frac {5832-3420 x+668 x^2-58 x^3+2 x^4+\left (-738 x+206 x^2-27 x^3+x^4\right ) \log \left (\frac {1}{x^2}\right )}{\left (2916 x-1710 x^2+334 x^3-29 x^4+x^5\right ) \log \left (\frac {1}{x^2}\right )} \, dx=\log \left (\frac {-4+x+\frac {x (1+2 x)}{(-9+x)^2}}{4 \log \left (\frac {1}{x^2}\right )}\right ) \end {dmath*}
Time = 0.05 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.23 \begin {dmath*} \int \frac {5832-3420 x+668 x^2-58 x^3+2 x^4+\left (-738 x+206 x^2-27 x^3+x^4\right ) \log \left (\frac {1}{x^2}\right )}{\left (2916 x-1710 x^2+334 x^3-29 x^4+x^5\right ) \log \left (\frac {1}{x^2}\right )} \, dx=-2 \log (9-x)+\log \left (324-154 x+20 x^2-x^3\right )-\log \left (\log \left (\frac {1}{x^2}\right )\right ) \end {dmath*}
Integrate[(5832 - 3420*x + 668*x^2 - 58*x^3 + 2*x^4 + (-738*x + 206*x^2 - 27*x^3 + x^4)*Log[x^(-2)])/((2916*x - 1710*x^2 + 334*x^3 - 29*x^4 + x^5)*L og[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.
\(\displaystyle \int \frac {2 x^4-58 x^3+668 x^2+\left (x^4-27 x^3+206 x^2-738 x\right ) \log \left (\frac {1}{x^2}\right )-3420 x+5832}{\left (x^5-29 x^4+334 x^3-1710 x^2+2916 x\right ) \log \left (\frac {1}{x^2}\right )} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {2 x^4-58 x^3+668 x^2+\left (x^4-27 x^3+206 x^2-738 x\right ) \log \left (\frac {1}{x^2}\right )-3420 x+5832}{x \left (x^4-29 x^3+334 x^2-1710 x+2916\right ) \log \left (\frac {1}{x^2}\right )}dx\) |
\(\Big \downarrow \) 2463 |
\(\displaystyle \int \left (\frac {2 x^4-58 x^3+668 x^2+\left (x^4-27 x^3+206 x^2-738 x\right ) \log \left (\frac {1}{x^2}\right )-3420 x+5832}{171 (x-9) x \log \left (\frac {1}{x^2}\right )}+\frac {\left (-x^2+11 x-55\right ) \left (2 x^4-58 x^3+668 x^2+\left (x^4-27 x^3+206 x^2-738 x\right ) \log \left (\frac {1}{x^2}\right )-3420 x+5832\right )}{171 x \left (x^3-20 x^2+154 x-324\right ) \log \left (\frac {1}{x^2}\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {2}{171} \int \frac {x^3-20 x^2+154 x-495}{x \log \left (\frac {1}{x^2}\right )}dx+\frac {2}{171} \int \frac {x^3-20 x^2+154 x-324}{x \log \left (\frac {1}{x^2}\right )}dx+\log \left (-x^3+20 x^2-154 x+324\right )-2 \log (9-x)\) |
Int[(5832 - 3420*x + 668*x^2 - 58*x^3 + 2*x^4 + (-738*x + 206*x^2 - 27*x^3 + x^4)*Log[x^(-2)])/((2916*x - 1710*x^2 + 334*x^3 - 29*x^4 + x^5)*Log[x^( -2)]),x]
3.6.2.3.1 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 = 0.77 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12
method | result | size |
norman | \(-\ln \left (\ln \left (\frac {1}{x^{2}}\right )\right )-2 \ln \left (x -9\right )+\ln \left (x^{3}-20 x^{2}+154 x -324\right )\) | \(29\) |
risch | \(-\ln \left (\ln \left (\frac {1}{x^{2}}\right )\right )-2 \ln \left (x -9\right )+\ln \left (x^{3}-20 x^{2}+154 x -324\right )\) | \(29\) |
parallelrisch | \(-\ln \left (\ln \left (\frac {1}{x^{2}}\right )\right )-2 \ln \left (x -9\right )+\ln \left (x^{3}-20 x^{2}+154 x -324\right )\) | \(29\) |
parts | \(-\ln \left (\ln \left (\frac {1}{x^{2}}\right )\right )-2 \ln \left (x -9\right )+\ln \left (x^{3}-20 x^{2}+154 x -324\right )\) | \(29\) |
derivativedivides | \(-\ln \left (\ln \left (\frac {1}{x^{2}}\right )\right )-\ln \left (\frac {1}{x}\right )+\ln \left (\frac {324}{x^{3}}-\frac {154}{x^{2}}+\frac {20}{x}-1\right )-2 \ln \left (\frac {9}{x}-1\right )\) | \(43\) |
default | \(-\ln \left (\ln \left (\frac {1}{x^{2}}\right )\right )-\ln \left (\frac {1}{x}\right )+\ln \left (\frac {324}{x^{3}}-\frac {154}{x^{2}}+\frac {20}{x}-1\right )-2 \ln \left (\frac {9}{x}-1\right )\) | \(43\) |
int(((x^4-27*x^3+206*x^2-738*x)*ln(1/x^2)+2*x^4-58*x^3+668*x^2-3420*x+5832 )/(x^5-29*x^4+334*x^3-1710*x^2+2916*x)/ln(1/x^2),x,method=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \begin {dmath*} \int \frac {5832-3420 x+668 x^2-58 x^3+2 x^4+\left (-738 x+206 x^2-27 x^3+x^4\right ) \log \left (\frac {1}{x^2}\right )}{\left (2916 x-1710 x^2+334 x^3-29 x^4+x^5\right ) \log \left (\frac {1}{x^2}\right )} \, dx=\log \left (x^{3} - 20 \, x^{2} + 154 \, x - 324\right ) - 2 \, \log \left (x - 9\right ) - \log \left (\log \left (\frac {1}{x^{2}}\right )\right ) \end {dmath*}
integrate(((x^4-27*x^3+206*x^2-738*x)*log(1/x^2)+2*x^4-58*x^3+668*x^2-3420 *x+5832)/(x^5-29*x^4+334*x^3-1710*x^2+2916*x)/log(1/x^2),x, algorithm=\
Time = 0.10 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12 \begin {dmath*} \int \frac {5832-3420 x+668 x^2-58 x^3+2 x^4+\left (-738 x+206 x^2-27 x^3+x^4\right ) \log \left (\frac {1}{x^2}\right )}{\left (2916 x-1710 x^2+334 x^3-29 x^4+x^5\right ) \log \left (\frac {1}{x^2}\right )} \, dx=- 2 \log {\left (x - 9 \right )} + \log {\left (x^{3} - 20 x^{2} + 154 x - 324 \right )} - \log {\left (\log {\left (\frac {1}{x^{2}} \right )} \right )} \end {dmath*}
integrate(((x**4-27*x**3+206*x**2-738*x)*ln(1/x**2)+2*x**4-58*x**3+668*x** 2-3420*x+5832)/(x**5-29*x**4+334*x**3-1710*x**2+2916*x)/ln(1/x**2),x)
Time = 0.23 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \begin {dmath*} \int \frac {5832-3420 x+668 x^2-58 x^3+2 x^4+\left (-738 x+206 x^2-27 x^3+x^4\right ) \log \left (\frac {1}{x^2}\right )}{\left (2916 x-1710 x^2+334 x^3-29 x^4+x^5\right ) \log \left (\frac {1}{x^2}\right )} \, dx=\log \left (x^{3} - 20 \, x^{2} + 154 \, x - 324\right ) - 2 \, \log \left (x - 9\right ) - \log \left (\log \left (x\right )\right ) \end {dmath*}
integrate(((x^4-27*x^3+206*x^2-738*x)*log(1/x^2)+2*x^4-58*x^3+668*x^2-3420 *x+5832)/(x^5-29*x^4+334*x^3-1710*x^2+2916*x)/log(1/x^2),x, algorithm=\
Time = 0.28 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \begin {dmath*} \int \frac {5832-3420 x+668 x^2-58 x^3+2 x^4+\left (-738 x+206 x^2-27 x^3+x^4\right ) \log \left (\frac {1}{x^2}\right )}{\left (2916 x-1710 x^2+334 x^3-29 x^4+x^5\right ) \log \left (\frac {1}{x^2}\right )} \, dx=\log \left (x^{3} - 20 \, x^{2} + 154 \, x - 324\right ) - 2 \, \log \left (x - 9\right ) - \log \left (\log \left (x^{2}\right )\right ) \end {dmath*}
integrate(((x^4-27*x^3+206*x^2-738*x)*log(1/x^2)+2*x^4-58*x^3+668*x^2-3420 *x+5832)/(x^5-29*x^4+334*x^3-1710*x^2+2916*x)/log(1/x^2),x, algorithm=\
Time = 13.65 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \begin {dmath*} \int \frac {5832-3420 x+668 x^2-58 x^3+2 x^4+\left (-738 x+206 x^2-27 x^3+x^4\right ) \log \left (\frac {1}{x^2}\right )}{\left (2916 x-1710 x^2+334 x^3-29 x^4+x^5\right ) \log \left (\frac {1}{x^2}\right )} \, dx=\ln \left (x^3-20\,x^2+154\,x-324\right )-\ln \left (\ln \left (\frac {1}{x^2}\right )\right )-2\,\ln \left (x-9\right ) \end {dmath*}