Integrand size = 121, antiderivative size = 31 \[ \int \frac {-4+7 x+x^2+x^3+\left (4-8 x+4 x^2\right ) \log (x)+\left (-4-x^2\right ) \log \left (\frac {x^2}{3}\right )}{x^4+\left (2 x^3-2 x^4\right ) \log (x)+\left (x^2-2 x^3+x^4\right ) \log ^2(x)+\left (-2 x^3+\left (-2 x^2+2 x^3\right ) \log (x)\right ) \log \left (\frac {x^2}{3}\right )+x^2 \log ^2\left (\frac {x^2}{3}\right )} \, dx=-\frac {4+x}{x \left (\log (x)+\frac {-x+\log \left (\frac {x^2}{3}\right )}{-1+x}\right )} \]
Timed out. \[ \int \frac {-4+7 x+x^2+x^3+\left (4-8 x+4 x^2\right ) \log (x)+\left (-4-x^2\right ) \log \left (\frac {x^2}{3}\right )}{x^4+\left (2 x^3-2 x^4\right ) \log (x)+\left (x^2-2 x^3+x^4\right ) \log ^2(x)+\left (-2 x^3+\left (-2 x^2+2 x^3\right ) \log (x)\right ) \log \left (\frac {x^2}{3}\right )+x^2 \log ^2\left (\frac {x^2}{3}\right )} \, dx=\text {\$Aborted} \]
Integrate[(-4 + 7*x + x^2 + x^3 + (4 - 8*x + 4*x^2)*Log[x] + (-4 - x^2)*Lo g[x^2/3])/(x^4 + (2*x^3 - 2*x^4)*Log[x] + (x^2 - 2*x^3 + x^4)*Log[x]^2 + ( -2*x^3 + (-2*x^2 + 2*x^3)*Log[x])*Log[x^2/3] + x^2*Log[x^2/3]^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.
\(\displaystyle \int \frac {x^3+x^2+\left (4 x^2-8 x+4\right ) \log (x)+\left (-x^2-4\right ) \log \left (\frac {x^2}{3}\right )+7 x-4}{x^4+x^2 \log ^2\left (\frac {x^2}{3}\right )+\left (2 x^3-2 x^4\right ) \log (x)+\left (\left (2 x^3-2 x^2\right ) \log (x)-2 x^3\right ) \log \left (\frac {x^2}{3}\right )+\left (x^4-2 x^3+x^2\right ) \log ^2(x)} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {x^3+x^2-x^2 \log \left (\frac {x^2}{3}\right )-4 \log \left (x^2\right )+7 x+4 (x-1)^2 \log (x)-4 (1-\log (3))}{x^2 \left (-\log \left (\frac {x^2}{3}\right )+x-(x-1) \log (x)\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {4 (x-1)^2 \log (x)}{x^2 \left (\log \left (\frac {x^2}{3}\right )-x+x \log (x)-\log (x)\right )^2}+\frac {\log (3)-\log \left (x^2\right )}{\left (\log \left (\frac {x^2}{3}\right )-x+x \log (x)-\log (x)\right )^2}-\frac {4 \log \left (x^2\right )}{x^2 \left (\log \left (\frac {x^2}{3}\right )-x+x \log (x)-\log (x)\right )^2}+\frac {x}{\left (\log \left (\frac {x^2}{3}\right )-x+x \log (x)-\log (x)\right )^2}+\frac {7}{x \left (\log \left (\frac {x^2}{3}\right )-x+x \log (x)-\log (x)\right )^2}+\frac {\log (81)-4}{x^2 \left (\log \left (\frac {x^2}{3}\right )-x+x \log (x)-\log (x)\right )^2}+\frac {1}{\left (\log \left (\frac {x^2}{3}\right )-x+x \log (x)-\log (x)\right )^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \log (3) \int \frac {1}{\left (\log (x) x-x-\log (x)+\log \left (\frac {x^2}{3}\right )\right )^2}dx+\int \frac {1}{\left (\log (x) x-x-\log (x)+\log \left (\frac {x^2}{3}\right )\right )^2}dx-(4-\log (81)) \int \frac {1}{x^2 \left (\log (x) x-x-\log (x)+\log \left (\frac {x^2}{3}\right )\right )^2}dx+7 \int \frac {1}{x \left (\log (x) x-x-\log (x)+\log \left (\frac {x^2}{3}\right )\right )^2}dx+\int \frac {x}{\left (\log (x) x-x-\log (x)+\log \left (\frac {x^2}{3}\right )\right )^2}dx+4 \int \frac {\log (x)}{\left (\log (x) x-x-\log (x)+\log \left (\frac {x^2}{3}\right )\right )^2}dx+4 \int \frac {\log (x)}{x^2 \left (\log (x) x-x-\log (x)+\log \left (\frac {x^2}{3}\right )\right )^2}dx-8 \int \frac {\log (x)}{x \left (\log (x) x-x-\log (x)+\log \left (\frac {x^2}{3}\right )\right )^2}dx-\int \frac {\log \left (x^2\right )}{\left (\log (x) x-x-\log (x)+\log \left (\frac {x^2}{3}\right )\right )^2}dx-4 \int \frac {\log \left (x^2\right )}{x^2 \left (\log (x) x-x-\log (x)+\log \left (\frac {x^2}{3}\right )\right )^2}dx\) |
Int[(-4 + 7*x + x^2 + x^3 + (4 - 8*x + 4*x^2)*Log[x] + (-4 - x^2)*Log[x^2/ 3])/(x^4 + (2*x^3 - 2*x^4)*Log[x] + (x^2 - 2*x^3 + x^4)*Log[x]^2 + (-2*x^3 + (-2*x^2 + 2*x^3)*Log[x])*Log[x^2/3] + x^2*Log[x^2/3]^2),x]
3.26.34.3.1 Defintions of rubi rules used
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 1.48 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.03
method | result | size |
default | \(\frac {x^{2}+3 x -4}{x \left (-x \ln \left (x \right )+\ln \left (3\right )+x -\ln \left (x^{2}\right )+\ln \left (x \right )\right )}\) | \(32\) |
parallelrisch | \(\frac {-24 x^{2}-72 x +96}{24 x \left (x \ln \left (x \right )-\ln \left (x \right )+\ln \left (\frac {x^{2}}{3}\right )-x \right )}\) | \(36\) |
risch | \(-\frac {2 \left (x^{2}+3 x -4\right )}{x \left (-i \pi \,\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x \right )^{2}+2 i \pi \operatorname {csgn}\left (i x^{2}\right )^{2} \operatorname {csgn}\left (i x \right )-i \pi \operatorname {csgn}\left (i x^{2}\right )^{3}+2 x \ln \left (x \right )-2 \ln \left (3\right )-2 x +2 \ln \left (x \right )\right )}\) | \(82\) |
int(((-x^2-4)*ln(1/3*x^2)+(4*x^2-8*x+4)*ln(x)+x^3+x^2+7*x-4)/(x^2*ln(1/3*x ^2)^2+((2*x^3-2*x^2)*ln(x)-2*x^3)*ln(1/3*x^2)+(x^4-2*x^3+x^2)*ln(x)^2+(-2* x^4+2*x^3)*ln(x)+x^4),x,method=_RETURNVERBOSE)
Time = 0.27 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.90 \[ \int \frac {-4+7 x+x^2+x^3+\left (4-8 x+4 x^2\right ) \log (x)+\left (-4-x^2\right ) \log \left (\frac {x^2}{3}\right )}{x^4+\left (2 x^3-2 x^4\right ) \log (x)+\left (x^2-2 x^3+x^4\right ) \log ^2(x)+\left (-2 x^3+\left (-2 x^2+2 x^3\right ) \log (x)\right ) \log \left (\frac {x^2}{3}\right )+x^2 \log ^2\left (\frac {x^2}{3}\right )} \, dx=\frac {x^{2} + 3 \, x - 4}{x^{2} + x \log \left (3\right ) - {\left (x^{2} + x\right )} \log \left (x\right )} \]
integrate(((-x^2-4)*log(1/3*x^2)+(4*x^2-8*x+4)*log(x)+x^3+x^2+7*x-4)/(x^2* log(1/3*x^2)^2+((2*x^3-2*x^2)*log(x)-2*x^3)*log(1/3*x^2)+(x^4-2*x^3+x^2)*l og(x)^2+(-2*x^4+2*x^3)*log(x)+x^4),x, algorithm=\
Time = 0.18 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.77 \[ \int \frac {-4+7 x+x^2+x^3+\left (4-8 x+4 x^2\right ) \log (x)+\left (-4-x^2\right ) \log \left (\frac {x^2}{3}\right )}{x^4+\left (2 x^3-2 x^4\right ) \log (x)+\left (x^2-2 x^3+x^4\right ) \log ^2(x)+\left (-2 x^3+\left (-2 x^2+2 x^3\right ) \log (x)\right ) \log \left (\frac {x^2}{3}\right )+x^2 \log ^2\left (\frac {x^2}{3}\right )} \, dx=\frac {- x^{2} - 3 x + 4}{- x^{2} - x \log {\left (3 \right )} + \left (x^{2} + x\right ) \log {\left (x \right )}} \]
integrate(((-x**2-4)*ln(1/3*x**2)+(4*x**2-8*x+4)*ln(x)+x**3+x**2+7*x-4)/(x **2*ln(1/3*x**2)**2+((2*x**3-2*x**2)*ln(x)-2*x**3)*ln(1/3*x**2)+(x**4-2*x* *3+x**2)*ln(x)**2+(-2*x**4+2*x**3)*ln(x)+x**4),x)
Time = 0.31 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.90 \[ \int \frac {-4+7 x+x^2+x^3+\left (4-8 x+4 x^2\right ) \log (x)+\left (-4-x^2\right ) \log \left (\frac {x^2}{3}\right )}{x^4+\left (2 x^3-2 x^4\right ) \log (x)+\left (x^2-2 x^3+x^4\right ) \log ^2(x)+\left (-2 x^3+\left (-2 x^2+2 x^3\right ) \log (x)\right ) \log \left (\frac {x^2}{3}\right )+x^2 \log ^2\left (\frac {x^2}{3}\right )} \, dx=\frac {x^{2} + 3 \, x - 4}{x^{2} + x \log \left (3\right ) - {\left (x^{2} + x\right )} \log \left (x\right )} \]
integrate(((-x^2-4)*log(1/3*x^2)+(4*x^2-8*x+4)*log(x)+x^3+x^2+7*x-4)/(x^2* log(1/3*x^2)^2+((2*x^3-2*x^2)*log(x)-2*x^3)*log(1/3*x^2)+(x^4-2*x^3+x^2)*l og(x)^2+(-2*x^4+2*x^3)*log(x)+x^4),x, algorithm=\
Time = 0.29 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.06 \[ \int \frac {-4+7 x+x^2+x^3+\left (4-8 x+4 x^2\right ) \log (x)+\left (-4-x^2\right ) \log \left (\frac {x^2}{3}\right )}{x^4+\left (2 x^3-2 x^4\right ) \log (x)+\left (x^2-2 x^3+x^4\right ) \log ^2(x)+\left (-2 x^3+\left (-2 x^2+2 x^3\right ) \log (x)\right ) \log \left (\frac {x^2}{3}\right )+x^2 \log ^2\left (\frac {x^2}{3}\right )} \, dx=-\frac {x^{2} + 3 \, x - 4}{x^{2} \log \left (x\right ) - x^{2} - x \log \left (3\right ) + x \log \left (x\right )} \]
integrate(((-x^2-4)*log(1/3*x^2)+(4*x^2-8*x+4)*log(x)+x^3+x^2+7*x-4)/(x^2* log(1/3*x^2)^2+((2*x^3-2*x^2)*log(x)-2*x^3)*log(1/3*x^2)+(x^4-2*x^3+x^2)*l og(x)^2+(-2*x^4+2*x^3)*log(x)+x^4),x, algorithm=\
Time = 8.54 (sec) , antiderivative size = 101, normalized size of antiderivative = 3.26 \[ \int \frac {-4+7 x+x^2+x^3+\left (4-8 x+4 x^2\right ) \log (x)+\left (-4-x^2\right ) \log \left (\frac {x^2}{3}\right )}{x^4+\left (2 x^3-2 x^4\right ) \log (x)+\left (x^2-2 x^3+x^4\right ) \log ^2(x)+\left (-2 x^3+\left (-2 x^2+2 x^3\right ) \log (x)\right ) \log \left (\frac {x^2}{3}\right )+x^2 \log ^2\left (\frac {x^2}{3}\right )} \, dx=-\frac {x+\ln \left (\frac {x^2}{3}\right )\,\left (x^3+3\,x^2-4\,x\right )-4\,x^3-x^4-\ln \left (x\right )\,\left (2\,x^3+6\,x^2-8\,x\right )+4}{\left (x-x^2\,\left (\ln \left (\frac {x^2}{3}\right )-2\,\ln \left (x\right )\right )+x^2+x^3\right )\,\left (x-\ln \left (\frac {x^2}{3}\right )+2\,\ln \left (x\right )-\ln \left (x\right )\,\left (x+1\right )\right )} \]
int((7*x - log(x^2/3)*(x^2 + 4) + log(x)*(4*x^2 - 8*x + 4) + x^2 + x^3 - 4 )/(log(x)*(2*x^3 - 2*x^4) + x^2*log(x^2/3)^2 - log(x^2/3)*(log(x)*(2*x^2 - 2*x^3) + 2*x^3) + x^4 + log(x)^2*(x^2 - 2*x^3 + x^4)),x)