Integrand size = 109, antiderivative size = 26 \[ \int \frac {660+775 x-241 x^2-31 x^3+8 x^4+e^2 \left (24-10 x+x^2\right )+\left (160+120 x-72 x^2+8 x^3\right ) \log (4-x)}{660 x-137 x^2-39 x^3+8 x^4+e^2 \left (20-9 x+x^2\right )+\left (160 x-72 x^2+8 x^3\right ) \log (4-x)} \, dx=x+\log \left (\frac {e^2+x+8 x (4+x+\log (4-x))}{-5+x}\right ) \]
Time = 0.06 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.23 \[ \int \frac {660+775 x-241 x^2-31 x^3+8 x^4+e^2 \left (24-10 x+x^2\right )+\left (160+120 x-72 x^2+8 x^3\right ) \log (4-x)}{660 x-137 x^2-39 x^3+8 x^4+e^2 \left (20-9 x+x^2\right )+\left (160 x-72 x^2+8 x^3\right ) \log (4-x)} \, dx=x-\log (5-x)+\log \left (e^2+33 x+8 x^2+8 x \log (4-x)\right ) \]
Integrate[(660 + 775*x - 241*x^2 - 31*x^3 + 8*x^4 + E^2*(24 - 10*x + x^2) + (160 + 120*x - 72*x^2 + 8*x^3)*Log[4 - x])/(660*x - 137*x^2 - 39*x^3 + 8 *x^4 + E^2*(20 - 9*x + x^2) + (160*x - 72*x^2 + 8*x^3)*Log[4 - x]),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.
\(\displaystyle \int \frac {8 x^4-31 x^3-241 x^2+e^2 \left (x^2-10 x+24\right )+\left (8 x^3-72 x^2+120 x+160\right ) \log (4-x)+775 x+660}{8 x^4-39 x^3-137 x^2+e^2 \left (x^2-9 x+20\right )+\left (8 x^3-72 x^2+160 x\right ) \log (4-x)+660 x} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {8 x^4-31 x^3-241 x^2+e^2 \left (x^2-10 x+24\right )+\left (8 x^3-72 x^2+120 x+160\right ) \log (4-x)+775 x+660}{\left (x^2-9 x+20\right ) \left (8 x^2+33 x+8 x \log (4-x)+e^2\right )}dx\) |
\(\Big \downarrow \) 7279 |
\(\displaystyle \int \left (\frac {x^2-5 x-5}{(x-5) x}+\frac {8 x^3-24 x^2-e^2 x+4 e^2}{(x-4) x \left (8 x^2+33 x+8 x \log (4-x)+e^2\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 8 \int \frac {1}{8 x^2+8 \log (4-x) x+33 x+e^2}dx+32 \int \frac {1}{(x-4) \left (8 x^2+8 \log (4-x) x+33 x+e^2\right )}dx-e^2 \int \frac {1}{x \left (8 x^2+8 \log (4-x) x+33 x+e^2\right )}dx+8 \int \frac {x}{8 x^2+8 \log (4-x) x+33 x+e^2}dx+x-\log (5-x)+\log (x)\) |
Int[(660 + 775*x - 241*x^2 - 31*x^3 + 8*x^4 + E^2*(24 - 10*x + x^2) + (160 + 120*x - 72*x^2 + 8*x^3)*Log[4 - x])/(660*x - 137*x^2 - 39*x^3 + 8*x^4 + E^2*(20 - 9*x + x^2) + (160*x - 72*x^2 + 8*x^3)*Log[4 - x]),x]
3.4.36.3.1 Defintions of rubi rules used
Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ {v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
Time = 4.32 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.15
method | result | size |
norman | \(x -\ln \left (-5+x \right )+\ln \left (8 x^{2}+8 \ln \left (-x +4\right ) x +{\mathrm e}^{2}+33 x \right )\) | \(30\) |
parallelrisch | \(8-\ln \left (-5+x \right )+\ln \left (x^{2}+\ln \left (-x +4\right ) x +\frac {{\mathrm e}^{2}}{8}+\frac {33 x}{8}\right )+x\) | \(30\) |
risch | \(x +\ln \left (x \right )-\ln \left (-5+x \right )+\ln \left (\ln \left (-x +4\right )+\frac {8 x^{2}+{\mathrm e}^{2}+33 x}{8 x}\right )\) | \(35\) |
derivativedivides | \(x -4+\ln \left (-8 \left (-x +4\right ) \ln \left (-x +4\right )+8 \left (-x +4\right )^{2}+{\mathrm e}^{2}+32 \ln \left (-x +4\right )+97 x -128\right )-\ln \left (5-x \right )\) | \(50\) |
default | \(x -4+\ln \left (-8 \left (-x +4\right ) \ln \left (-x +4\right )+8 \left (-x +4\right )^{2}+{\mathrm e}^{2}+32 \ln \left (-x +4\right )+97 x -128\right )-\ln \left (5-x \right )\) | \(50\) |
int(((8*x^3-72*x^2+120*x+160)*ln(-x+4)+(x^2-10*x+24)*exp(2)+8*x^4-31*x^3-2 41*x^2+775*x+660)/((8*x^3-72*x^2+160*x)*ln(-x+4)+(x^2-9*x+20)*exp(2)+8*x^4 -39*x^3-137*x^2+660*x),x,method=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.35 \[ \int \frac {660+775 x-241 x^2-31 x^3+8 x^4+e^2 \left (24-10 x+x^2\right )+\left (160+120 x-72 x^2+8 x^3\right ) \log (4-x)}{660 x-137 x^2-39 x^3+8 x^4+e^2 \left (20-9 x+x^2\right )+\left (160 x-72 x^2+8 x^3\right ) \log (4-x)} \, dx=x - \log \left (x - 5\right ) + \log \left (x\right ) + \log \left (\frac {8 \, x^{2} + 8 \, x \log \left (-x + 4\right ) + 33 \, x + e^{2}}{x}\right ) \]
integrate(((8*x^3-72*x^2+120*x+160)*log(-x+4)+(x^2-10*x+24)*exp(2)+8*x^4-3 1*x^3-241*x^2+775*x+660)/((8*x^3-72*x^2+160*x)*log(-x+4)+(x^2-9*x+20)*exp( 2)+8*x^4-39*x^3-137*x^2+660*x),x, algorithm=\
Time = 0.20 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.19 \[ \int \frac {660+775 x-241 x^2-31 x^3+8 x^4+e^2 \left (24-10 x+x^2\right )+\left (160+120 x-72 x^2+8 x^3\right ) \log (4-x)}{660 x-137 x^2-39 x^3+8 x^4+e^2 \left (20-9 x+x^2\right )+\left (160 x-72 x^2+8 x^3\right ) \log (4-x)} \, dx=x + \log {\left (x \right )} - \log {\left (x - 5 \right )} + \log {\left (\log {\left (4 - x \right )} + \frac {8 x^{2} + 33 x + e^{2}}{8 x} \right )} \]
integrate(((8*x**3-72*x**2+120*x+160)*ln(-x+4)+(x**2-10*x+24)*exp(2)+8*x** 4-31*x**3-241*x**2+775*x+660)/((8*x**3-72*x**2+160*x)*ln(-x+4)+(x**2-9*x+2 0)*exp(2)+8*x**4-39*x**3-137*x**2+660*x),x)
Time = 0.24 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.38 \[ \int \frac {660+775 x-241 x^2-31 x^3+8 x^4+e^2 \left (24-10 x+x^2\right )+\left (160+120 x-72 x^2+8 x^3\right ) \log (4-x)}{660 x-137 x^2-39 x^3+8 x^4+e^2 \left (20-9 x+x^2\right )+\left (160 x-72 x^2+8 x^3\right ) \log (4-x)} \, dx=x - \log \left (x - 5\right ) + \log \left (x\right ) + \log \left (\frac {8 \, x^{2} + 8 \, x \log \left (-x + 4\right ) + 33 \, x + e^{2}}{8 \, x}\right ) \]
integrate(((8*x^3-72*x^2+120*x+160)*log(-x+4)+(x^2-10*x+24)*exp(2)+8*x^4-3 1*x^3-241*x^2+775*x+660)/((8*x^3-72*x^2+160*x)*log(-x+4)+(x^2-9*x+20)*exp( 2)+8*x^4-39*x^3-137*x^2+660*x),x, algorithm=\
Time = 0.27 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.73 \[ \int \frac {660+775 x-241 x^2-31 x^3+8 x^4+e^2 \left (24-10 x+x^2\right )+\left (160+120 x-72 x^2+8 x^3\right ) \log (4-x)}{660 x-137 x^2-39 x^3+8 x^4+e^2 \left (20-9 x+x^2\right )+\left (160 x-72 x^2+8 x^3\right ) \log (4-x)} \, dx=x + \log \left (8 \, {\left (x - 4\right )}^{2} + 8 \, {\left (x - 4\right )} \log \left (-x + 4\right ) + 97 \, x + e^{2} + 32 \, \log \left (-x + 4\right ) - 128\right ) - \log \left (-x + 5\right ) - 4 \]
integrate(((8*x^3-72*x^2+120*x+160)*log(-x+4)+(x^2-10*x+24)*exp(2)+8*x^4-3 1*x^3-241*x^2+775*x+660)/((8*x^3-72*x^2+160*x)*log(-x+4)+(x^2-9*x+20)*exp( 2)+8*x^4-39*x^3-137*x^2+660*x),x, algorithm=\
x + log(8*(x - 4)^2 + 8*(x - 4)*log(-x + 4) + 97*x + e^2 + 32*log(-x + 4) - 128) - log(-x + 5) - 4
Timed out. \[ \int \frac {660+775 x-241 x^2-31 x^3+8 x^4+e^2 \left (24-10 x+x^2\right )+\left (160+120 x-72 x^2+8 x^3\right ) \log (4-x)}{660 x-137 x^2-39 x^3+8 x^4+e^2 \left (20-9 x+x^2\right )+\left (160 x-72 x^2+8 x^3\right ) \log (4-x)} \, dx=\int \frac {775\,x+\ln \left (4-x\right )\,\left (8\,x^3-72\,x^2+120\,x+160\right )+{\mathrm {e}}^2\,\left (x^2-10\,x+24\right )-241\,x^2-31\,x^3+8\,x^4+660}{660\,x+\ln \left (4-x\right )\,\left (8\,x^3-72\,x^2+160\,x\right )+{\mathrm {e}}^2\,\left (x^2-9\,x+20\right )-137\,x^2-39\,x^3+8\,x^4} \,d x \]
int((775*x + log(4 - x)*(120*x - 72*x^2 + 8*x^3 + 160) + exp(2)*(x^2 - 10* x + 24) - 241*x^2 - 31*x^3 + 8*x^4 + 660)/(660*x + log(4 - x)*(160*x - 72* x^2 + 8*x^3) + exp(2)*(x^2 - 9*x + 20) - 137*x^2 - 39*x^3 + 8*x^4),x)