Integrand size = 47, antiderivative size = 24 \[ \int \frac {80-16 x-4 x^2+\left (80+4 x^2\right ) \log \left (-\frac {x}{\log (5)}\right )}{400-160 x-24 x^2+8 x^3+x^4} \, dx=\frac {x \log \left (-\frac {x}{\log (5)}\right )}{5-\left (1+\frac {x}{4}\right ) x} \]
Time = 0.41 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.88 \[ \int \frac {80-16 x-4 x^2+\left (80+4 x^2\right ) \log \left (-\frac {x}{\log (5)}\right )}{400-160 x-24 x^2+8 x^3+x^4} \, dx=-\frac {4 x \log \left (-\frac {x}{\log (5)}\right )}{-20+4 x+x^2} \]
Integrate[(80 - 16*x - 4*x^2 + (80 + 4*x^2)*Log[-(x/Log[5])])/(400 - 160*x - 24*x^2 + 8*x^3 + x^4),x]
Leaf count is larger than twice the leaf count of optimal. \(192\) vs. \(2(24)=48\).
Time = 1.15 (sec) , antiderivative size = 192, normalized size of antiderivative = 8.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {2463, 2009}
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.
\(\displaystyle \int \frac {-4 x^2+\left (4 x^2+80\right ) \log \left (-\frac {x}{\log (5)}\right )-16 x+80}{x^4+8 x^3-24 x^2-160 x+400} \, dx\) |
\(\Big \downarrow \) 2463 |
\(\displaystyle \int \left (\frac {-4 x^2+\left (4 x^2+80\right ) \log \left (-\frac {x}{\log (5)}\right )-16 x+80}{96 \sqrt {6} \left (-2 x+4 \sqrt {6}-4\right )}+\frac {-4 x^2+\left (4 x^2+80\right ) \log \left (-\frac {x}{\log (5)}\right )-16 x+80}{96 \sqrt {6} \left (2 x+4 \sqrt {6}+4\right )}+\frac {-4 x^2+\left (4 x^2+80\right ) \log \left (-\frac {x}{\log (5)}\right )-16 x+80}{24 \left (-2 x+4 \sqrt {6}-4\right )^2}+\frac {-4 x^2+\left (4 x^2+80\right ) \log \left (-\frac {x}{\log (5)}\right )-16 x+80}{24 \left (2 x+4 \sqrt {6}+4\right )^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {\left (x+2 \left (1-\sqrt {6}\right )\right )^2}{96 \sqrt {6}}+\frac {\left (x+2 \left (1+\sqrt {6}\right )\right )^2}{96 \sqrt {6}}+\frac {1}{144} \left (6+\sqrt {6}\right ) x+\frac {1}{144} \left (6-\sqrt {6}\right ) x-\frac {x}{6}-\frac {1}{144} \left (6+\sqrt {6}\right ) x \log \left (-\frac {x}{\log (5)}\right )-\frac {1}{144} \left (6-\sqrt {6}\right ) x \log \left (-\frac {x}{\log (5)}\right )+\frac {1}{12} x \log \left (-\frac {x}{\log (5)}\right )+\frac {x \log \left (-\frac {x}{\log (5)}\right )}{\sqrt {6} \left (x+2 \left (1+\sqrt {6}\right )\right )}-\frac {x \log \left (-\frac {x}{\log (5)}\right )}{\sqrt {6} \left (x+2 \left (1-\sqrt {6}\right )\right )}\) |
Int[(80 - 16*x - 4*x^2 + (80 + 4*x^2)*Log[-(x/Log[5])])/(400 - 160*x - 24* x^2 + 8*x^3 + x^4),x]
-1/6*x + ((6 - Sqrt[6])*x)/144 + ((6 + Sqrt[6])*x)/144 - (2*(1 - Sqrt[6]) + x)^2/(96*Sqrt[6]) + (2*(1 + Sqrt[6]) + x)^2/(96*Sqrt[6]) + (x*Log[-(x/Lo g[5])])/12 - ((6 - Sqrt[6])*x*Log[-(x/Log[5])])/144 - ((6 + Sqrt[6])*x*Log [-(x/Log[5])])/144 - (x*Log[-(x/Log[5])])/(Sqrt[6]*(2*(1 - Sqrt[6]) + x)) + (x*Log[-(x/Log[5])])/(Sqrt[6]*(2*(1 + Sqrt[6]) + x))
3.16.78.3.1 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]
Time = 0.10 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92
method | result | size |
norman | \(-\frac {4 x \ln \left (-\frac {x}{\ln \left (5\right )}\right )}{x^{2}+4 x -20}\) | \(22\) |
risch | \(-\frac {4 x \ln \left (-\frac {x}{\ln \left (5\right )}\right )}{x^{2}+4 x -20}\) | \(22\) |
parallelrisch | \(-\frac {4 x \ln \left (-\frac {x}{\ln \left (5\right )}\right )}{x^{2}+4 x -20}\) | \(22\) |
derivativedivides | \(-\ln \left (5\right ) \left (\frac {\sqrt {6}\, \ln \left (-\frac {x}{\ln \left (5\right )}\right ) \left (\ln \left (\frac {x +2 \sqrt {6}+2}{2 \sqrt {6}+2}\right )-\ln \left (\frac {-x +2 \sqrt {6}-2}{2 \sqrt {6}-2}\right )\right )}{6 \ln \left (5\right )}-\frac {\ln \left (-\frac {x}{\ln \left (5\right )}\right ) \left (\ln \left (\frac {x +2 \sqrt {6}+2}{2 \sqrt {6}+2}\right ) \sqrt {6}\, x^{2}-\ln \left (\frac {-x +2 \sqrt {6}-2}{2 \sqrt {6}-2}\right ) \sqrt {6}\, x^{2}+4 \ln \left (\frac {x +2 \sqrt {6}+2}{2 \sqrt {6}+2}\right ) \sqrt {6}\, x -4 \ln \left (\frac {-x +2 \sqrt {6}-2}{2 \sqrt {6}-2}\right ) \sqrt {6}\, x -20 \sqrt {6}\, \ln \left (\frac {x +2 \sqrt {6}+2}{2 \sqrt {6}+2}\right )+20 \sqrt {6}\, \ln \left (\frac {-x +2 \sqrt {6}-2}{2 \sqrt {6}-2}\right )-24 x \right )}{6 \left (x^{2}+4 x -20\right ) \ln \left (5\right )}\right )\) | \(243\) |
default | \(-\ln \left (5\right ) \left (\frac {\sqrt {6}\, \ln \left (-\frac {x}{\ln \left (5\right )}\right ) \left (\ln \left (\frac {x +2 \sqrt {6}+2}{2 \sqrt {6}+2}\right )-\ln \left (\frac {-x +2 \sqrt {6}-2}{2 \sqrt {6}-2}\right )\right )}{6 \ln \left (5\right )}-\frac {\ln \left (-\frac {x}{\ln \left (5\right )}\right ) \left (\ln \left (\frac {x +2 \sqrt {6}+2}{2 \sqrt {6}+2}\right ) \sqrt {6}\, x^{2}-\ln \left (\frac {-x +2 \sqrt {6}-2}{2 \sqrt {6}-2}\right ) \sqrt {6}\, x^{2}+4 \ln \left (\frac {x +2 \sqrt {6}+2}{2 \sqrt {6}+2}\right ) \sqrt {6}\, x -4 \ln \left (\frac {-x +2 \sqrt {6}-2}{2 \sqrt {6}-2}\right ) \sqrt {6}\, x -20 \sqrt {6}\, \ln \left (\frac {x +2 \sqrt {6}+2}{2 \sqrt {6}+2}\right )+20 \sqrt {6}\, \ln \left (\frac {-x +2 \sqrt {6}-2}{2 \sqrt {6}-2}\right )-24 x \right )}{6 \left (x^{2}+4 x -20\right ) \ln \left (5\right )}\right )\) | \(243\) |
parts | \(\frac {\sqrt {6}\, \operatorname {arctanh}\left (\frac {\left (4+2 x \right ) \sqrt {6}}{24}\right )}{3}-4 \ln \left (5\right ) \left (-\frac {\sqrt {6}\, \operatorname {arctanh}\left (\frac {\left (-2 x \ln \left (5\right )-4 \ln \left (5\right )\right ) \sqrt {6}}{24 \ln \left (5\right )}\right )}{12 \ln \left (5\right )}-\frac {\ln \left (-\frac {x}{\ln \left (5\right )}\right ) \left (\ln \left (\frac {x +2 \sqrt {6}+2}{2 \sqrt {6}+2}\right ) \sqrt {6}\, x^{2}-\ln \left (\frac {-x +2 \sqrt {6}-2}{2 \sqrt {6}-2}\right ) \sqrt {6}\, x^{2}+4 \ln \left (\frac {x +2 \sqrt {6}+2}{2 \sqrt {6}+2}\right ) \sqrt {6}\, x -4 \ln \left (\frac {-x +2 \sqrt {6}-2}{2 \sqrt {6}-2}\right ) \sqrt {6}\, x -20 \sqrt {6}\, \ln \left (\frac {x +2 \sqrt {6}+2}{2 \sqrt {6}+2}\right )+20 \sqrt {6}\, \ln \left (\frac {-x +2 \sqrt {6}-2}{2 \sqrt {6}-2}\right )-24 x \right )}{24 \left (x^{2}+4 x -20\right ) \ln \left (5\right )}+\frac {\sqrt {6}\, \ln \left (-\frac {x}{\ln \left (5\right )}\right ) \left (\ln \left (\frac {x +2 \sqrt {6}+2}{2 \sqrt {6}+2}\right )-\ln \left (\frac {-x +2 \sqrt {6}-2}{2 \sqrt {6}-2}\right )\right )}{24 \ln \left (5\right )}\right )\) | \(289\) |
int(((4*x^2+80)*ln(-x/ln(5))-4*x^2-16*x+80)/(x^4+8*x^3-24*x^2-160*x+400),x ,method=_RETURNVERBOSE)
Time = 0.26 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.88 \[ \int \frac {80-16 x-4 x^2+\left (80+4 x^2\right ) \log \left (-\frac {x}{\log (5)}\right )}{400-160 x-24 x^2+8 x^3+x^4} \, dx=-\frac {4 \, x \log \left (-\frac {x}{\log \left (5\right )}\right )}{x^{2} + 4 \, x - 20} \]
integrate(((4*x^2+80)*log(-x/log(5))-4*x^2-16*x+80)/(x^4+8*x^3-24*x^2-160* x+400),x, algorithm=\
Time = 0.07 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83 \[ \int \frac {80-16 x-4 x^2+\left (80+4 x^2\right ) \log \left (-\frac {x}{\log (5)}\right )}{400-160 x-24 x^2+8 x^3+x^4} \, dx=- \frac {4 x \log {\left (- \frac {x}{\log {\left (5 \right )}} \right )}}{x^{2} + 4 x - 20} \]
Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (20) = 40\).
Time = 0.33 (sec) , antiderivative size = 73, normalized size of antiderivative = 3.04 \[ \int \frac {80-16 x-4 x^2+\left (80+4 x^2\right ) \log \left (-\frac {x}{\log (5)}\right )}{400-160 x-24 x^2+8 x^3+x^4} \, dx=-\frac {4 \, {\left (x \log \left (-x\right ) - x \log \left (\log \left (5\right )\right )\right )}}{x^{2} + 4 \, x - 20} + \frac {7 \, x - 10}{3 \, {\left (x^{2} + 4 \, x - 20\right )}} - \frac {5 \, {\left (x + 2\right )}}{3 \, {\left (x^{2} + 4 \, x - 20\right )}} - \frac {2 \, {\left (x - 10\right )}}{3 \, {\left (x^{2} + 4 \, x - 20\right )}} \]
integrate(((4*x^2+80)*log(-x/log(5))-4*x^2-16*x+80)/(x^4+8*x^3-24*x^2-160* x+400),x, algorithm=\
-4*(x*log(-x) - x*log(log(5)))/(x^2 + 4*x - 20) + 1/3*(7*x - 10)/(x^2 + 4* x - 20) - 5/3*(x + 2)/(x^2 + 4*x - 20) - 2/3*(x - 10)/(x^2 + 4*x - 20)
Time = 0.27 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.42 \[ \int \frac {80-16 x-4 x^2+\left (80+4 x^2\right ) \log \left (-\frac {x}{\log (5)}\right )}{400-160 x-24 x^2+8 x^3+x^4} \, dx=-\frac {4 \, x \log \left (-x\right )}{x^{2} + 4 \, x - 20} + \frac {4 \, x \log \left (\log \left (5\right )\right )}{x^{2} + 4 \, x - 20} \]
integrate(((4*x^2+80)*log(-x/log(5))-4*x^2-16*x+80)/(x^4+8*x^3-24*x^2-160* x+400),x, algorithm=\
Time = 12.88 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.96 \[ \int \frac {80-16 x-4 x^2+\left (80+4 x^2\right ) \log \left (-\frac {x}{\log (5)}\right )}{400-160 x-24 x^2+8 x^3+x^4} \, dx=-\frac {4\,x\,\left (\ln \left (-x\right )-\ln \left (\ln \left (5\right )\right )\right )}{x^2+4\,x-20} \]