Integrand size = 177, antiderivative size = 29 \[ \int \frac {-16+8 x-13 x^2+2 x^3+\left (4080-2024 x+247 x^2+5 x^3-x^4\right ) \log ^2\left (\frac {-3060+753 x+3 x^2-3 x^3}{-4+x}\right )}{-4080+2024 x-247 x^2-5 x^3+x^4+\left (-8160 x+4048 x^2-494 x^3-10 x^4+2 x^5\right ) \log \left (\frac {-3060+753 x+3 x^2-3 x^3}{-4+x}\right )+\left (-4080 x^2+2024 x^3-247 x^4-5 x^5+x^6\right ) \log ^2\left (\frac {-3060+753 x+3 x^2-3 x^3}{-4+x}\right )} \, dx=\frac {x}{x^2+\frac {x}{\log \left (3 \left (255+x+\frac {x^3}{4-x}\right )\right )}} \]
Time = 0.08 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.24 \[ \int \frac {-16+8 x-13 x^2+2 x^3+\left (4080-2024 x+247 x^2+5 x^3-x^4\right ) \log ^2\left (\frac {-3060+753 x+3 x^2-3 x^3}{-4+x}\right )}{-4080+2024 x-247 x^2-5 x^3+x^4+\left (-8160 x+4048 x^2-494 x^3-10 x^4+2 x^5\right ) \log \left (\frac {-3060+753 x+3 x^2-3 x^3}{-4+x}\right )+\left (-4080 x^2+2024 x^3-247 x^4-5 x^5+x^6\right ) \log ^2\left (\frac {-3060+753 x+3 x^2-3 x^3}{-4+x}\right )} \, dx=\frac {1}{x}-\frac {1}{x \left (1+x \log \left (-\frac {3 \left (1020-251 x-x^2+x^3\right )}{-4+x}\right )\right )} \]
Integrate[(-16 + 8*x - 13*x^2 + 2*x^3 + (4080 - 2024*x + 247*x^2 + 5*x^3 - x^4)*Log[(-3060 + 753*x + 3*x^2 - 3*x^3)/(-4 + x)]^2)/(-4080 + 2024*x - 2 47*x^2 - 5*x^3 + x^4 + (-8160*x + 4048*x^2 - 494*x^3 - 10*x^4 + 2*x^5)*Log [(-3060 + 753*x + 3*x^2 - 3*x^3)/(-4 + x)] + (-4080*x^2 + 2024*x^3 - 247*x ^4 - 5*x^5 + x^6)*Log[(-3060 + 753*x + 3*x^2 - 3*x^3)/(-4 + 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.
| \(\displaystyle \int \frac {2 x^3-13 x^2+\left (-x^4+5 x^3+247 x^2-2024 x+4080\right ) \log ^2\left (\frac {-3 x^3+3 x^2+753 x-3060}{x-4}\right )+8 x-16}{x^4-5 x^3-247 x^2+\left (2 x^5-10 x^4-494 x^3+4048 x^2-8160 x\right ) \log \left (\frac {-3 x^3+3 x^2+753 x-3060}{x-4}\right )+\left (x^6-5 x^5-247 x^4+2024 x^3-4080 x^2\right ) \log ^2\left (\frac {-3 x^3+3 x^2+753 x-3060}{x-4}\right )+2024 x-4080} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {-2 x^3+13 x^2-\left (-x^4+5 x^3+247 x^2-2024 x+4080\right ) \log ^2\left (\frac {-3 x^3+3 x^2+753 x-3060}{x-4}\right )-8 x+16}{\left (-x^4+5 x^3+247 x^2-2024 x+4080\right ) \left (x \log \left (-\frac {3 \left (x^3-x^2-251 x+1020\right )}{x-4}\right )+1\right )^2}dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (\frac {\left (x^2+3 x-239\right ) \left (-2 x^3+13 x^2-\left (-x^4+5 x^3+247 x^2-2024 x+4080\right ) \log ^2\left (\frac {-3 x^3+3 x^2+753 x-3060}{x-4}\right )-8 x+16\right )}{64 \left (x^3-x^2-251 x+1020\right ) \left (x \log \left (-\frac {3 \left (x^3-x^2-251 x+1020\right )}{x-4}\right )+1\right )^2}-\frac {-2 x^3+13 x^2-\left (-x^4+5 x^3+247 x^2-2024 x+4080\right ) \log ^2\left (\frac {-3 x^3+3 x^2+753 x-3060}{x-4}\right )-8 x+16}{64 (x-4) \left (x \log \left (-\frac {3 \left (x^3-x^2-251 x+1020\right )}{x-4}\right )+1\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {-2 x^3+13 x^2-\left (-x^4+5 x^3+247 x^2-2024 x+4080\right ) \log ^2\left (-\frac {3 \left (x^3-x^2-251 x+1020\right )}{x-4}\right )-8 x+16}{(4-x) \left (x^3-x^2-251 x+1020\right ) \left (x \log \left (-\frac {3 \left (x^3-x^2-251 x+1020\right )}{x-4}\right )+1\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {1}{x^2}+\frac {2}{x^2 \left (x \log \left (-\frac {3 \left (x^3-x^2-251 x+1020\right )}{x-4}\right )+1\right )}+\frac {2 x^5-14 x^4+13 x^3+231 x^2-2024 x+4080}{(x-4) x^2 \left (x^3-x^2-251 x+1020\right ) \left (x \log \left (-\frac {3 \left (x^3-x^2-251 x+1020\right )}{x-4}\right )+1\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\int \frac {1}{(x-4) \left (x \log \left (-\frac {3 \left (x^3-x^2-251 x+1020\right )}{x-4}\right )+1\right )^2}dx-\int \frac {1}{x^2 \left (x \log \left (-\frac {3 \left (x^3-x^2-251 x+1020\right )}{x-4}\right )+1\right )^2}dx-251 \int \frac {1}{\left (x^3-x^2-251 x+1020\right ) \left (x \log \left (-\frac {3 \left (x^3-x^2-251 x+1020\right )}{x-4}\right )+1\right )^2}dx-2 \int \frac {x}{\left (x^3-x^2-251 x+1020\right ) \left (x \log \left (-\frac {3 \left (x^3-x^2-251 x+1020\right )}{x-4}\right )+1\right )^2}dx+3 \int \frac {x^2}{\left (x^3-x^2-251 x+1020\right ) \left (x \log \left (-\frac {3 \left (x^3-x^2-251 x+1020\right )}{x-4}\right )+1\right )^2}dx+2 \int \frac {1}{x^2 \left (x \log \left (-\frac {3 \left (x^3-x^2-251 x+1020\right )}{x-4}\right )+1\right )}dx+\frac {1}{x}\) |
Int[(-16 + 8*x - 13*x^2 + 2*x^3 + (4080 - 2024*x + 247*x^2 + 5*x^3 - x^4)* Log[(-3060 + 753*x + 3*x^2 - 3*x^3)/(-4 + x)]^2)/(-4080 + 2024*x - 247*x^2 - 5*x^3 + x^4 + (-8160*x + 4048*x^2 - 494*x^3 - 10*x^4 + 2*x^5)*Log[(-306 0 + 753*x + 3*x^2 - 3*x^3)/(-4 + x)] + (-4080*x^2 + 2024*x^3 - 247*x^4 - 5 *x^5 + x^6)*Log[(-3060 + 753*x + 3*x^2 - 3*x^3)/(-4 + x)]^2),x]
3.28.71.3.1 Defintions of rubi rules used
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]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 1.04 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.31
| method | result | size |
| risch | \(\frac {1}{x}-\frac {1}{x \left (x \ln \left (\frac {-3 x^{3}+3 x^{2}+753 x -3060}{x -4}\right )+1\right )}\) | \(38\) |
| parallelrisch | \(\frac {\ln \left (-\frac {3 \left (x^{3}-x^{2}-251 x +1020\right )}{x -4}\right )}{x \ln \left (-\frac {3 \left (x^{3}-x^{2}-251 x +1020\right )}{x -4}\right )+1}\) | \(50\) |
| norman | \(\frac {\ln \left (\frac {-3 x^{3}+3 x^{2}+753 x -3060}{x -4}\right )}{x \ln \left (\frac {-3 x^{3}+3 x^{2}+753 x -3060}{x -4}\right )+1}\) | \(52\) |
int(((-x^4+5*x^3+247*x^2-2024*x+4080)*ln((-3*x^3+3*x^2+753*x-3060)/(x-4))^ 2+2*x^3-13*x^2+8*x-16)/((x^6-5*x^5-247*x^4+2024*x^3-4080*x^2)*ln((-3*x^3+3 *x^2+753*x-3060)/(x-4))^2+(2*x^5-10*x^4-494*x^3+4048*x^2-8160*x)*ln((-3*x^ 3+3*x^2+753*x-3060)/(x-4))+x^4-5*x^3-247*x^2+2024*x-4080),x,method=_RETURN VERBOSE)
Time = 0.29 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.69 \[ \int \frac {-16+8 x-13 x^2+2 x^3+\left (4080-2024 x+247 x^2+5 x^3-x^4\right ) \log ^2\left (\frac {-3060+753 x+3 x^2-3 x^3}{-4+x}\right )}{-4080+2024 x-247 x^2-5 x^3+x^4+\left (-8160 x+4048 x^2-494 x^3-10 x^4+2 x^5\right ) \log \left (\frac {-3060+753 x+3 x^2-3 x^3}{-4+x}\right )+\left (-4080 x^2+2024 x^3-247 x^4-5 x^5+x^6\right ) \log ^2\left (\frac {-3060+753 x+3 x^2-3 x^3}{-4+x}\right )} \, dx=\frac {\log \left (-\frac {3 \, {\left (x^{3} - x^{2} - 251 \, x + 1020\right )}}{x - 4}\right )}{x \log \left (-\frac {3 \, {\left (x^{3} - x^{2} - 251 \, x + 1020\right )}}{x - 4}\right ) + 1} \]
integrate(((-x^4+5*x^3+247*x^2-2024*x+4080)*log((-3*x^3+3*x^2+753*x-3060)/ (x-4))^2+2*x^3-13*x^2+8*x-16)/((x^6-5*x^5-247*x^4+2024*x^3-4080*x^2)*log(( -3*x^3+3*x^2+753*x-3060)/(x-4))^2+(2*x^5-10*x^4-494*x^3+4048*x^2-8160*x)*l og((-3*x^3+3*x^2+753*x-3060)/(x-4))+x^4-5*x^3-247*x^2+2024*x-4080),x, algo rithm=\
log(-3*(x^3 - x^2 - 251*x + 1020)/(x - 4))/(x*log(-3*(x^3 - x^2 - 251*x + 1020)/(x - 4)) + 1)
Time = 0.14 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {-16+8 x-13 x^2+2 x^3+\left (4080-2024 x+247 x^2+5 x^3-x^4\right ) \log ^2\left (\frac {-3060+753 x+3 x^2-3 x^3}{-4+x}\right )}{-4080+2024 x-247 x^2-5 x^3+x^4+\left (-8160 x+4048 x^2-494 x^3-10 x^4+2 x^5\right ) \log \left (\frac {-3060+753 x+3 x^2-3 x^3}{-4+x}\right )+\left (-4080 x^2+2024 x^3-247 x^4-5 x^5+x^6\right ) \log ^2\left (\frac {-3060+753 x+3 x^2-3 x^3}{-4+x}\right )} \, dx=- \frac {1}{x^{2} \log {\left (\frac {- 3 x^{3} + 3 x^{2} + 753 x - 3060}{x - 4} \right )} + x} + \frac {1}{x} \]
integrate(((-x**4+5*x**3+247*x**2-2024*x+4080)*ln((-3*x**3+3*x**2+753*x-30 60)/(x-4))**2+2*x**3-13*x**2+8*x-16)/((x**6-5*x**5-247*x**4+2024*x**3-4080 *x**2)*ln((-3*x**3+3*x**2+753*x-3060)/(x-4))**2+(2*x**5-10*x**4-494*x**3+4 048*x**2-8160*x)*ln((-3*x**3+3*x**2+753*x-3060)/(x-4))+x**4-5*x**3-247*x** 2+2024*x-4080),x)
Result contains complex when optimal does not.
Time = 0.33 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.14 \[ \int \frac {-16+8 x-13 x^2+2 x^3+\left (4080-2024 x+247 x^2+5 x^3-x^4\right ) \log ^2\left (\frac {-3060+753 x+3 x^2-3 x^3}{-4+x}\right )}{-4080+2024 x-247 x^2-5 x^3+x^4+\left (-8160 x+4048 x^2-494 x^3-10 x^4+2 x^5\right ) \log \left (\frac {-3060+753 x+3 x^2-3 x^3}{-4+x}\right )+\left (-4080 x^2+2024 x^3-247 x^4-5 x^5+x^6\right ) \log ^2\left (\frac {-3060+753 x+3 x^2-3 x^3}{-4+x}\right )} \, dx=\frac {i \, \pi + \log \left (3\right ) + \log \left (x^{3} - x^{2} - 251 \, x + 1020\right ) - \log \left (x - 4\right )}{{\left (i \, \pi + \log \left (3\right )\right )} x + x \log \left (x^{3} - x^{2} - 251 \, x + 1020\right ) - x \log \left (x - 4\right ) + 1} \]
integrate(((-x^4+5*x^3+247*x^2-2024*x+4080)*log((-3*x^3+3*x^2+753*x-3060)/ (x-4))^2+2*x^3-13*x^2+8*x-16)/((x^6-5*x^5-247*x^4+2024*x^3-4080*x^2)*log(( -3*x^3+3*x^2+753*x-3060)/(x-4))^2+(2*x^5-10*x^4-494*x^3+4048*x^2-8160*x)*l og((-3*x^3+3*x^2+753*x-3060)/(x-4))+x^4-5*x^3-247*x^2+2024*x-4080),x, algo rithm=\
(I*pi + log(3) + log(x^3 - x^2 - 251*x + 1020) - log(x - 4))/((I*pi + log( 3))*x + x*log(x^3 - x^2 - 251*x + 1020) - x*log(x - 4) + 1)
Time = 0.73 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.21 \[ \int \frac {-16+8 x-13 x^2+2 x^3+\left (4080-2024 x+247 x^2+5 x^3-x^4\right ) \log ^2\left (\frac {-3060+753 x+3 x^2-3 x^3}{-4+x}\right )}{-4080+2024 x-247 x^2-5 x^3+x^4+\left (-8160 x+4048 x^2-494 x^3-10 x^4+2 x^5\right ) \log \left (\frac {-3060+753 x+3 x^2-3 x^3}{-4+x}\right )+\left (-4080 x^2+2024 x^3-247 x^4-5 x^5+x^6\right ) \log ^2\left (\frac {-3060+753 x+3 x^2-3 x^3}{-4+x}\right )} \, dx=-\frac {1}{x^{2} \log \left (-\frac {3 \, {\left (x^{3} - x^{2} - 251 \, x + 1020\right )}}{x - 4}\right ) + x} + \frac {1}{x} \]
integrate(((-x^4+5*x^3+247*x^2-2024*x+4080)*log((-3*x^3+3*x^2+753*x-3060)/ (x-4))^2+2*x^3-13*x^2+8*x-16)/((x^6-5*x^5-247*x^4+2024*x^3-4080*x^2)*log(( -3*x^3+3*x^2+753*x-3060)/(x-4))^2+(2*x^5-10*x^4-494*x^3+4048*x^2-8160*x)*l og((-3*x^3+3*x^2+753*x-3060)/(x-4))+x^4-5*x^3-247*x^2+2024*x-4080),x, algo rithm=\
Time = 15.25 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.24 \[ \int \frac {-16+8 x-13 x^2+2 x^3+\left (4080-2024 x+247 x^2+5 x^3-x^4\right ) \log ^2\left (\frac {-3060+753 x+3 x^2-3 x^3}{-4+x}\right )}{-4080+2024 x-247 x^2-5 x^3+x^4+\left (-8160 x+4048 x^2-494 x^3-10 x^4+2 x^5\right ) \log \left (\frac {-3060+753 x+3 x^2-3 x^3}{-4+x}\right )+\left (-4080 x^2+2024 x^3-247 x^4-5 x^5+x^6\right ) \log ^2\left (\frac {-3060+753 x+3 x^2-3 x^3}{-4+x}\right )} \, dx=\frac {1}{x}-\frac {1}{x+x^2\,\ln \left (\frac {-3\,x^3+3\,x^2+753\,x-3060}{x-4}\right )} \]
int(-(8*x + log((753*x + 3*x^2 - 3*x^3 - 3060)/(x - 4))^2*(247*x^2 - 2024* x + 5*x^3 - x^4 + 4080) - 13*x^2 + 2*x^3 - 16)/(log((753*x + 3*x^2 - 3*x^3 - 3060)/(x - 4))*(8160*x - 4048*x^2 + 494*x^3 + 10*x^4 - 2*x^5) - 2024*x + 247*x^2 + 5*x^3 - x^4 + log((753*x + 3*x^2 - 3*x^3 - 3060)/(x - 4))^2*(4 080*x^2 - 2024*x^3 + 247*x^4 + 5*x^5 - x^6) + 4080),x)