Integrand size = 151, antiderivative size = 28 \[ \int \frac {320-5184 x+860 x^2+116 x^3+3 x^4+\left (-640 x+4928 x^2-712 x^3-108 x^4-3 x^5\right ) \log \left (-4 x+x^2\right )+\left (-16 x+4 x^2+\left (16 x^2-4 x^3\right ) \log \left (-4 x+x^2\right )\right ) \log \left (x-x^2 \log \left (-4 x+x^2\right )\right )}{4800 x-720 x^2-108 x^3-3 x^4+\left (-4800 x^2+720 x^3+108 x^4+3 x^5\right ) \log \left (-4 x+x^2\right )} \, dx=-8-x+\frac {4 \log \left (x-x^2 \log ((-4+x) x)\right )}{3 (20+x)} \]
Time = 0.11 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {320-5184 x+860 x^2+116 x^3+3 x^4+\left (-640 x+4928 x^2-712 x^3-108 x^4-3 x^5\right ) \log \left (-4 x+x^2\right )+\left (-16 x+4 x^2+\left (16 x^2-4 x^3\right ) \log \left (-4 x+x^2\right )\right ) \log \left (x-x^2 \log \left (-4 x+x^2\right )\right )}{4800 x-720 x^2-108 x^3-3 x^4+\left (-4800 x^2+720 x^3+108 x^4+3 x^5\right ) \log \left (-4 x+x^2\right )} \, dx=\frac {1}{3} \left (-3 x+\frac {4 \log \left (x-x^2 \log ((-4+x) x)\right )}{20+x}\right ) \]
Integrate[(320 - 5184*x + 860*x^2 + 116*x^3 + 3*x^4 + (-640*x + 4928*x^2 - 712*x^3 - 108*x^4 - 3*x^5)*Log[-4*x + x^2] + (-16*x + 4*x^2 + (16*x^2 - 4 *x^3)*Log[-4*x + x^2])*Log[x - x^2*Log[-4*x + x^2]])/(4800*x - 720*x^2 - 1 08*x^3 - 3*x^4 + (-4800*x^2 + 720*x^3 + 108*x^4 + 3*x^5)*Log[-4*x + 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.
\(\displaystyle \int \frac {3 x^4+116 x^3+860 x^2+\left (4 x^2+\left (16 x^2-4 x^3\right ) \log \left (x^2-4 x\right )-16 x\right ) \log \left (x-x^2 \log \left (x^2-4 x\right )\right )+\left (-3 x^5-108 x^4-712 x^3+4928 x^2-640 x\right ) \log \left (x^2-4 x\right )-5184 x+320}{-3 x^4-108 x^3-720 x^2+\left (3 x^5+108 x^4+720 x^3-4800 x^2\right ) \log \left (x^2-4 x\right )+4800 x} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {3 x^4+116 x^3+860 x^2+\left (4 x^2+\left (16 x^2-4 x^3\right ) \log \left (x^2-4 x\right )-16 x\right ) \log \left (x-x^2 \log \left (x^2-4 x\right )\right )+\left (-3 x^5-108 x^4-712 x^3+4928 x^2-640 x\right ) \log \left (x^2-4 x\right )-5184 x+320}{3 (4-x) x (x+20)^2 (1-x \log ((x-4) x))}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{3} \int \frac {3 x^4+116 x^3+860 x^2-5184 x-\left (3 x^5+108 x^4+712 x^3-4928 x^2+640 x\right ) \log \left (x^2-4 x\right )-4 \left (-x^2+4 x-\left (4 x^2-x^3\right ) \log \left (x^2-4 x\right )\right ) \log \left (x-x^2 \log \left (x^2-4 x\right )\right )+320}{(4-x) x (x+20)^2 (1-x \log (-((4-x) x)))}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \frac {1}{3} \int \left (\frac {3 x^3}{(x-4) (x+20)^2 (x \log ((x-4) x)-1)}+\frac {116 x^2}{(x-4) (x+20)^2 (x \log ((x-4) x)-1)}+\frac {860 x}{(x-4) (x+20)^2 (x \log ((x-4) x)-1)}-\frac {4 \log \left (x-x^2 \log ((x-4) x)\right )}{(x+20)^2}-\frac {\left (3 x^2+60 x-8\right ) \log ((x-4) x)}{(x+20) (x \log ((x-4) x)-1)}-\frac {5184}{(x-4) (x+20)^2 (x \log ((x-4) x)-1)}+\frac {320}{(x-4) (x+20)^2 (x \log ((x-4) x)-1) x}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{3} \left (-4 \int \frac {\log \left (x-x^2 \log ((x-4) x)\right )}{(x+20)^2}dx+\frac {2}{3} \int \frac {1}{(x-4) (x \log ((x-4) x)-1)}dx+\frac {1}{5} \int \frac {1}{x (x \log ((x-4) x)-1)}dx+\frac {107}{15} \int \frac {1}{(x+20) (x \log ((x-4) x)-1)}dx-3 x+\frac {2 \log (x)}{5}-\frac {2}{5} \log (x+20)\right )\) |
Int[(320 - 5184*x + 860*x^2 + 116*x^3 + 3*x^4 + (-640*x + 4928*x^2 - 712*x ^3 - 108*x^4 - 3*x^5)*Log[-4*x + x^2] + (-16*x + 4*x^2 + (16*x^2 - 4*x^3)* Log[-4*x + x^2])*Log[x - x^2*Log[-4*x + x^2]])/(4800*x - 720*x^2 - 108*x^3 - 3*x^4 + (-4800*x^2 + 720*x^3 + 108*x^4 + 3*x^5)*Log[-4*x + x^2]),x]
3.2.74.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Time = 5.90 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.29
method | result | size |
parallelrisch | \(\frac {-19200-432 x^{2}-9600 x +576 \ln \left (-x^{2} \ln \left (x^{2}-4 x \right )+x \right )}{8640+432 x}\) | \(36\) |
int((((-4*x^3+16*x^2)*ln(x^2-4*x)+4*x^2-16*x)*ln(-x^2*ln(x^2-4*x)+x)+(-3*x ^5-108*x^4-712*x^3+4928*x^2-640*x)*ln(x^2-4*x)+3*x^4+116*x^3+860*x^2-5184* x+320)/((3*x^5+108*x^4+720*x^3-4800*x^2)*ln(x^2-4*x)-3*x^4-108*x^3-720*x^2 +4800*x),x,method=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.21 \[ \int \frac {320-5184 x+860 x^2+116 x^3+3 x^4+\left (-640 x+4928 x^2-712 x^3-108 x^4-3 x^5\right ) \log \left (-4 x+x^2\right )+\left (-16 x+4 x^2+\left (16 x^2-4 x^3\right ) \log \left (-4 x+x^2\right )\right ) \log \left (x-x^2 \log \left (-4 x+x^2\right )\right )}{4800 x-720 x^2-108 x^3-3 x^4+\left (-4800 x^2+720 x^3+108 x^4+3 x^5\right ) \log \left (-4 x+x^2\right )} \, dx=-\frac {3 \, x^{2} + 60 \, x - 4 \, \log \left (-x^{2} \log \left (x^{2} - 4 \, x\right ) + x\right )}{3 \, {\left (x + 20\right )}} \]
integrate((((-4*x^3+16*x^2)*log(x^2-4*x)+4*x^2-16*x)*log(-x^2*log(x^2-4*x) +x)+(-3*x^5-108*x^4-712*x^3+4928*x^2-640*x)*log(x^2-4*x)+3*x^4+116*x^3+860 *x^2-5184*x+320)/((3*x^5+108*x^4+720*x^3-4800*x^2)*log(x^2-4*x)-3*x^4-108* x^3-720*x^2+4800*x),x, algorithm=\
Time = 0.33 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.79 \[ \int \frac {320-5184 x+860 x^2+116 x^3+3 x^4+\left (-640 x+4928 x^2-712 x^3-108 x^4-3 x^5\right ) \log \left (-4 x+x^2\right )+\left (-16 x+4 x^2+\left (16 x^2-4 x^3\right ) \log \left (-4 x+x^2\right )\right ) \log \left (x-x^2 \log \left (-4 x+x^2\right )\right )}{4800 x-720 x^2-108 x^3-3 x^4+\left (-4800 x^2+720 x^3+108 x^4+3 x^5\right ) \log \left (-4 x+x^2\right )} \, dx=- x + \frac {4 \log {\left (- x^{2} \log {\left (x^{2} - 4 x \right )} + x \right )}}{3 x + 60} \]
integrate((((-4*x**3+16*x**2)*ln(x**2-4*x)+4*x**2-16*x)*ln(-x**2*ln(x**2-4 *x)+x)+(-3*x**5-108*x**4-712*x**3+4928*x**2-640*x)*ln(x**2-4*x)+3*x**4+116 *x**3+860*x**2-5184*x+320)/((3*x**5+108*x**4+720*x**3-4800*x**2)*ln(x**2-4 *x)-3*x**4-108*x**3-720*x**2+4800*x),x)
Time = 0.24 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.32 \[ \int \frac {320-5184 x+860 x^2+116 x^3+3 x^4+\left (-640 x+4928 x^2-712 x^3-108 x^4-3 x^5\right ) \log \left (-4 x+x^2\right )+\left (-16 x+4 x^2+\left (16 x^2-4 x^3\right ) \log \left (-4 x+x^2\right )\right ) \log \left (x-x^2 \log \left (-4 x+x^2\right )\right )}{4800 x-720 x^2-108 x^3-3 x^4+\left (-4800 x^2+720 x^3+108 x^4+3 x^5\right ) \log \left (-4 x+x^2\right )} \, dx=-\frac {3 \, x^{2} + 60 \, x - 4 \, \log \left (-x \log \left (x - 4\right ) - x \log \left (x\right ) + 1\right ) - 4 \, \log \left (x\right )}{3 \, {\left (x + 20\right )}} \]
integrate((((-4*x^3+16*x^2)*log(x^2-4*x)+4*x^2-16*x)*log(-x^2*log(x^2-4*x) +x)+(-3*x^5-108*x^4-712*x^3+4928*x^2-640*x)*log(x^2-4*x)+3*x^4+116*x^3+860 *x^2-5184*x+320)/((3*x^5+108*x^4+720*x^3-4800*x^2)*log(x^2-4*x)-3*x^4-108* x^3-720*x^2+4800*x),x, algorithm=\
Time = 0.36 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.21 \[ \int \frac {320-5184 x+860 x^2+116 x^3+3 x^4+\left (-640 x+4928 x^2-712 x^3-108 x^4-3 x^5\right ) \log \left (-4 x+x^2\right )+\left (-16 x+4 x^2+\left (16 x^2-4 x^3\right ) \log \left (-4 x+x^2\right )\right ) \log \left (x-x^2 \log \left (-4 x+x^2\right )\right )}{4800 x-720 x^2-108 x^3-3 x^4+\left (-4800 x^2+720 x^3+108 x^4+3 x^5\right ) \log \left (-4 x+x^2\right )} \, dx=-x + \frac {4 \, \log \left (-x \log \left (x^{2} - 4 \, x\right ) + 1\right )}{3 \, {\left (x + 20\right )}} + \frac {4 \, \log \left (x\right )}{3 \, {\left (x + 20\right )}} \]
integrate((((-4*x^3+16*x^2)*log(x^2-4*x)+4*x^2-16*x)*log(-x^2*log(x^2-4*x) +x)+(-3*x^5-108*x^4-712*x^3+4928*x^2-640*x)*log(x^2-4*x)+3*x^4+116*x^3+860 *x^2-5184*x+320)/((3*x^5+108*x^4+720*x^3-4800*x^2)*log(x^2-4*x)-3*x^4-108* x^3-720*x^2+4800*x),x, algorithm=\
Time = 10.17 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {320-5184 x+860 x^2+116 x^3+3 x^4+\left (-640 x+4928 x^2-712 x^3-108 x^4-3 x^5\right ) \log \left (-4 x+x^2\right )+\left (-16 x+4 x^2+\left (16 x^2-4 x^3\right ) \log \left (-4 x+x^2\right )\right ) \log \left (x-x^2 \log \left (-4 x+x^2\right )\right )}{4800 x-720 x^2-108 x^3-3 x^4+\left (-4800 x^2+720 x^3+108 x^4+3 x^5\right ) \log \left (-4 x+x^2\right )} \, dx=\frac {4\,\ln \left (x-x^2\,\ln \left (x^2-4\,x\right )\right )}{3\,\left (x+20\right )}-x \]