Integrand size = 78, antiderivative size = 22 \[ \int \frac {-24+12 x+\left (36 x-36 \log \left (x^2\right )\right ) \log \left (-x+\log \left (x^2\right )\right )+\left (x^4-x^3 \log \left (x^2\right )\right ) \log ^2\left (-x+\log \left (x^2\right )\right )}{\left (-x^5+x^4 \log \left (x^2\right )\right ) \log ^2\left (-x+\log \left (x^2\right )\right )} \, dx=1-\log (x)+\frac {12}{x^3 \log \left (-x+\log \left (x^2\right )\right )} \]
Time = 0.27 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95 \[ \int \frac {-24+12 x+\left (36 x-36 \log \left (x^2\right )\right ) \log \left (-x+\log \left (x^2\right )\right )+\left (x^4-x^3 \log \left (x^2\right )\right ) \log ^2\left (-x+\log \left (x^2\right )\right )}{\left (-x^5+x^4 \log \left (x^2\right )\right ) \log ^2\left (-x+\log \left (x^2\right )\right )} \, dx=-\log (x)+\frac {12}{x^3 \log \left (-x+\log \left (x^2\right )\right )} \]
Integrate[(-24 + 12*x + (36*x - 36*Log[x^2])*Log[-x + Log[x^2]] + (x^4 - x ^3*Log[x^2])*Log[-x + Log[x^2]]^2)/((-x^5 + x^4*Log[x^2])*Log[-x + Log[x^2 ]]^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.
\(\displaystyle \int \frac {\left (36 x-36 \log \left (x^2\right )\right ) \log \left (\log \left (x^2\right )-x\right )+\left (x^4-x^3 \log \left (x^2\right )\right ) \log ^2\left (\log \left (x^2\right )-x\right )+12 x-24}{\left (x^4 \log \left (x^2\right )-x^5\right ) \log ^2\left (\log \left (x^2\right )-x\right )} \, dx\) |
\(\Big \downarrow \) 3041 |
\(\displaystyle \int \frac {\left (36 x-36 \log \left (x^2\right )\right ) \log \left (\log \left (x^2\right )-x\right )+\left (x^4-x^3 \log \left (x^2\right )\right ) \log ^2\left (\log \left (x^2\right )-x\right )+12 x-24}{x^4 \left (\log \left (x^2\right )-x\right ) \log ^2\left (\log \left (x^2\right )-x\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (-\frac {12 (x-2)}{x^4 \left (x-\log \left (x^2\right )\right ) \log ^2\left (\log \left (x^2\right )-x\right )}-\frac {36}{x^4 \log \left (\log \left (x^2\right )-x\right )}-\frac {1}{x}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 24 \int \frac {1}{x^4 \left (x-\log \left (x^2\right )\right ) \log ^2\left (\log \left (x^2\right )-x\right )}dx-36 \int \frac {1}{x^4 \log \left (\log \left (x^2\right )-x\right )}dx-12 \int \frac {1}{x^3 \left (x-\log \left (x^2\right )\right ) \log ^2\left (\log \left (x^2\right )-x\right )}dx-\log (x)\) |
Int[(-24 + 12*x + (36*x - 36*Log[x^2])*Log[-x + Log[x^2]] + (x^4 - x^3*Log [x^2])*Log[-x + Log[x^2]]^2)/((-x^5 + x^4*Log[x^2])*Log[-x + Log[x^2]]^2), x]
3.28.61.3.1 Defintions of rubi rules used
Int[(u_.)*((a_.)*(x_)^(m_.) + Log[(c_.)*(x_)^(n_.)]^(q_.)*(b_.)*(x_)^(r_.)) ^(p_.), x_Symbol] :> Int[u*x^(p*r)*(a*x^(m - r) + b*Log[c*x^n]^q)^p, x] /; FreeQ[{a, b, c, m, n, p, q, r}, x] && IntegerQ[p]
Time = 0.74 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.68
method | result | size |
parallelrisch | \(\frac {24-\ln \left (\ln \left (x^{2}\right )-x \right ) \ln \left (x^{2}\right ) x^{3}}{2 x^{3} \ln \left (\ln \left (x^{2}\right )-x \right )}\) | \(37\) |
int(((-x^3*ln(x^2)+x^4)*ln(ln(x^2)-x)^2+(-36*ln(x^2)+36*x)*ln(ln(x^2)-x)+1 2*x-24)/(x^4*ln(x^2)-x^5)/ln(ln(x^2)-x)^2,x,method=_RETURNVERBOSE)
Time = 0.27 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.59 \[ \int \frac {-24+12 x+\left (36 x-36 \log \left (x^2\right )\right ) \log \left (-x+\log \left (x^2\right )\right )+\left (x^4-x^3 \log \left (x^2\right )\right ) \log ^2\left (-x+\log \left (x^2\right )\right )}{\left (-x^5+x^4 \log \left (x^2\right )\right ) \log ^2\left (-x+\log \left (x^2\right )\right )} \, dx=-\frac {x^{3} \log \left (x^{2}\right ) \log \left (-x + \log \left (x^{2}\right )\right ) - 24}{2 \, x^{3} \log \left (-x + \log \left (x^{2}\right )\right )} \]
integrate(((-x^3*log(x^2)+x^4)*log(log(x^2)-x)^2+(-36*log(x^2)+36*x)*log(l og(x^2)-x)+12*x-24)/(x^4*log(x^2)-x^5)/log(log(x^2)-x)^2,x, algorithm=\
Time = 0.09 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.68 \[ \int \frac {-24+12 x+\left (36 x-36 \log \left (x^2\right )\right ) \log \left (-x+\log \left (x^2\right )\right )+\left (x^4-x^3 \log \left (x^2\right )\right ) \log ^2\left (-x+\log \left (x^2\right )\right )}{\left (-x^5+x^4 \log \left (x^2\right )\right ) \log ^2\left (-x+\log \left (x^2\right )\right )} \, dx=- \log {\left (x \right )} + \frac {12}{x^{3} \log {\left (- x + \log {\left (x^{2} \right )} \right )}} \]
integrate(((-x**3*ln(x**2)+x**4)*ln(ln(x**2)-x)**2+(-36*ln(x**2)+36*x)*ln( ln(x**2)-x)+12*x-24)/(x**4*ln(x**2)-x**5)/ln(ln(x**2)-x)**2,x)
Time = 0.22 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95 \[ \int \frac {-24+12 x+\left (36 x-36 \log \left (x^2\right )\right ) \log \left (-x+\log \left (x^2\right )\right )+\left (x^4-x^3 \log \left (x^2\right )\right ) \log ^2\left (-x+\log \left (x^2\right )\right )}{\left (-x^5+x^4 \log \left (x^2\right )\right ) \log ^2\left (-x+\log \left (x^2\right )\right )} \, dx=\frac {12}{x^{3} \log \left (-x + 2 \, \log \left (x\right )\right )} - \log \left (x\right ) \]
integrate(((-x^3*log(x^2)+x^4)*log(log(x^2)-x)^2+(-36*log(x^2)+36*x)*log(l og(x^2)-x)+12*x-24)/(x^4*log(x^2)-x^5)/log(log(x^2)-x)^2,x, algorithm=\
Time = 0.34 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95 \[ \int \frac {-24+12 x+\left (36 x-36 \log \left (x^2\right )\right ) \log \left (-x+\log \left (x^2\right )\right )+\left (x^4-x^3 \log \left (x^2\right )\right ) \log ^2\left (-x+\log \left (x^2\right )\right )}{\left (-x^5+x^4 \log \left (x^2\right )\right ) \log ^2\left (-x+\log \left (x^2\right )\right )} \, dx=\frac {12}{x^{3} \log \left (-x + \log \left (x^{2}\right )\right )} - \log \left (x\right ) \]
integrate(((-x^3*log(x^2)+x^4)*log(log(x^2)-x)^2+(-36*log(x^2)+36*x)*log(l og(x^2)-x)+12*x-24)/(x^4*log(x^2)-x^5)/log(log(x^2)-x)^2,x, algorithm=\
Time = 13.94 (sec) , antiderivative size = 52, normalized size of antiderivative = 2.36 \[ \int \frac {-24+12 x+\left (36 x-36 \log \left (x^2\right )\right ) \log \left (-x+\log \left (x^2\right )\right )+\left (x^4-x^3 \log \left (x^2\right )\right ) \log ^2\left (-x+\log \left (x^2\right )\right )}{\left (-x^5+x^4 \log \left (x^2\right )\right ) \log ^2\left (-x+\log \left (x^2\right )\right )} \, dx=\frac {36}{2\,x^2-x^3}-\frac {36\,x}{2\,x^3-x^4}-\ln \left (x\right )+\frac {12}{x^3\,\ln \left (\ln \left (x^2\right )-x\right )} \]