Integrand size = 110, antiderivative size = 28 \[ \int \frac {-11+409 x+420 x^2+100 x^3+\left (200+520 x+280 x^2+20 x^3\right ) \log (x)+\left (160+180 x+40 x^2\right ) \log ^2(x)+(20+20 x) \log ^3(x)}{-11 x+100 x^2+100 x^3+\left (220 x^2+20 x^3\right ) \log (x)+\left (100 x+40 x^2\right ) \log ^2(x)+20 x \log ^3(x)} \, dx=x+\log \left (\frac {9 x}{20}-x \left (1-(5+\log (x)) \left (x+(x+\log (x))^2\right )\right )\right ) \]
Time = 0.06 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.64 \[ \int \frac {-11+409 x+420 x^2+100 x^3+\left (200+520 x+280 x^2+20 x^3\right ) \log (x)+\left (160+180 x+40 x^2\right ) \log ^2(x)+(20+20 x) \log ^3(x)}{-11 x+100 x^2+100 x^3+\left (220 x^2+20 x^3\right ) \log (x)+\left (100 x+40 x^2\right ) \log ^2(x)+20 x \log ^3(x)} \, dx=x+\log (x)+\log \left (11-100 x-100 x^2-220 x \log (x)-20 x^2 \log (x)-100 \log ^2(x)-40 x \log ^2(x)-20 \log ^3(x)\right ) \]
Integrate[(-11 + 409*x + 420*x^2 + 100*x^3 + (200 + 520*x + 280*x^2 + 20*x ^3)*Log[x] + (160 + 180*x + 40*x^2)*Log[x]^2 + (20 + 20*x)*Log[x]^3)/(-11* x + 100*x^2 + 100*x^3 + (220*x^2 + 20*x^3)*Log[x] + (100*x + 40*x^2)*Log[x ]^2 + 20*x*Log[x]^3),x]
x + Log[x] + Log[11 - 100*x - 100*x^2 - 220*x*Log[x] - 20*x^2*Log[x] - 100 *Log[x]^2 - 40*x*Log[x]^2 - 20*Log[x]^3]
Time = 0.61 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.64, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.018, Rules used = {7293, 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.
\(\displaystyle \int \frac {100 x^3+420 x^2+\left (40 x^2+180 x+160\right ) \log ^2(x)+\left (20 x^3+280 x^2+520 x+200\right ) \log (x)+409 x+(20 x+20) \log ^3(x)-11}{100 x^3+100 x^2+\left (40 x^2+100 x\right ) \log ^2(x)+\left (20 x^3+220 x^2\right ) \log (x)-11 x+20 x \log ^3(x)} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {20 \left (11 x^2+2 x^2 \log (x)+16 x+2 x \log ^2(x)+3 \log ^2(x)+15 x \log (x)+10 \log (x)\right )}{x \left (100 x^2+20 x^2 \log (x)+100 x+20 \log ^3(x)+40 x \log ^2(x)+100 \log ^2(x)+220 x \log (x)-11\right )}+\frac {x+1}{x}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \log \left (-100 x^2-20 x^2 \log (x)-100 x-20 \log ^3(x)-40 x \log ^2(x)-100 \log ^2(x)-220 x \log (x)+11\right )+x+\log (x)\) |
Int[(-11 + 409*x + 420*x^2 + 100*x^3 + (200 + 520*x + 280*x^2 + 20*x^3)*Lo g[x] + (160 + 180*x + 40*x^2)*Log[x]^2 + (20 + 20*x)*Log[x]^3)/(-11*x + 10 0*x^2 + 100*x^3 + (220*x^2 + 20*x^3)*Log[x] + (100*x + 40*x^2)*Log[x]^2 + 20*x*Log[x]^3),x]
x + Log[x] + Log[11 - 100*x - 100*x^2 - 220*x*Log[x] - 20*x^2*Log[x] - 100 *Log[x]^2 - 40*x*Log[x]^2 - 20*Log[x]^3]
3.28.63.3.1 Defintions of rubi rules used
Time = 0.74 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.43
method | result | size |
risch | \(x +\ln \left (x \right )+\ln \left (\ln \left (x \right )^{3}+\left (5+2 x \right ) \ln \left (x \right )^{2}+\left (x^{2}+11 x \right ) \ln \left (x \right )+5 x^{2}+5 x -\frac {11}{20}\right )\) | \(40\) |
parallelrisch | \(\ln \left (x^{2} \ln \left (x \right )+2 x \ln \left (x \right )^{2}+\ln \left (x \right )^{3}+5 x^{2}+11 x \ln \left (x \right )+5 \ln \left (x \right )^{2}+5 x -\frac {11}{20}\right )+x +\ln \left (x \right )\) | \(44\) |
default | \(\ln \left (x \right )+x +\ln \left (20 \ln \left (x \right )^{3}+40 x \ln \left (x \right )^{2}+20 x^{2} \ln \left (x \right )+100 \ln \left (x \right )^{2}+220 x \ln \left (x \right )+100 x^{2}+100 x -11\right )\) | \(47\) |
norman | \(\ln \left (x \right )+x +\ln \left (20 \ln \left (x \right )^{3}+40 x \ln \left (x \right )^{2}+20 x^{2} \ln \left (x \right )+100 \ln \left (x \right )^{2}+220 x \ln \left (x \right )+100 x^{2}+100 x -11\right )\) | \(47\) |
int(((20*x+20)*ln(x)^3+(40*x^2+180*x+160)*ln(x)^2+(20*x^3+280*x^2+520*x+20 0)*ln(x)+100*x^3+420*x^2+409*x-11)/(20*x*ln(x)^3+(40*x^2+100*x)*ln(x)^2+(2 0*x^3+220*x^2)*ln(x)+100*x^3+100*x^2-11*x),x,method=_RETURNVERBOSE)
Time = 0.26 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.54 \[ \int \frac {-11+409 x+420 x^2+100 x^3+\left (200+520 x+280 x^2+20 x^3\right ) \log (x)+\left (160+180 x+40 x^2\right ) \log ^2(x)+(20+20 x) \log ^3(x)}{-11 x+100 x^2+100 x^3+\left (220 x^2+20 x^3\right ) \log (x)+\left (100 x+40 x^2\right ) \log ^2(x)+20 x \log ^3(x)} \, dx=x + \log \left (20 \, {\left (2 \, x + 5\right )} \log \left (x\right )^{2} + 20 \, \log \left (x\right )^{3} + 100 \, x^{2} + 20 \, {\left (x^{2} + 11 \, x\right )} \log \left (x\right ) + 100 \, x - 11\right ) + \log \left (x\right ) \]
integrate(((20*x+20)*log(x)^3+(40*x^2+180*x+160)*log(x)^2+(20*x^3+280*x^2+ 520*x+200)*log(x)+100*x^3+420*x^2+409*x-11)/(20*x*log(x)^3+(40*x^2+100*x)* log(x)^2+(20*x^3+220*x^2)*log(x)+100*x^3+100*x^2-11*x),x, algorithm=\
x + log(20*(2*x + 5)*log(x)^2 + 20*log(x)^3 + 100*x^2 + 20*(x^2 + 11*x)*lo g(x) + 100*x - 11) + log(x)
Time = 0.20 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.50 \[ \int \frac {-11+409 x+420 x^2+100 x^3+\left (200+520 x+280 x^2+20 x^3\right ) \log (x)+\left (160+180 x+40 x^2\right ) \log ^2(x)+(20+20 x) \log ^3(x)}{-11 x+100 x^2+100 x^3+\left (220 x^2+20 x^3\right ) \log (x)+\left (100 x+40 x^2\right ) \log ^2(x)+20 x \log ^3(x)} \, dx=x + \log {\left (x \right )} + \log {\left (5 x^{2} + 5 x + \left (2 x + 5\right ) \log {\left (x \right )}^{2} + \left (x^{2} + 11 x\right ) \log {\left (x \right )} + \log {\left (x \right )}^{3} - \frac {11}{20} \right )} \]
integrate(((20*x+20)*ln(x)**3+(40*x**2+180*x+160)*ln(x)**2+(20*x**3+280*x* *2+520*x+200)*ln(x)+100*x**3+420*x**2+409*x-11)/(20*x*ln(x)**3+(40*x**2+10 0*x)*ln(x)**2+(20*x**3+220*x**2)*ln(x)+100*x**3+100*x**2-11*x),x)
x + log(x) + log(5*x**2 + 5*x + (2*x + 5)*log(x)**2 + (x**2 + 11*x)*log(x) + log(x)**3 - 11/20)
Time = 0.22 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.39 \[ \int \frac {-11+409 x+420 x^2+100 x^3+\left (200+520 x+280 x^2+20 x^3\right ) \log (x)+\left (160+180 x+40 x^2\right ) \log ^2(x)+(20+20 x) \log ^3(x)}{-11 x+100 x^2+100 x^3+\left (220 x^2+20 x^3\right ) \log (x)+\left (100 x+40 x^2\right ) \log ^2(x)+20 x \log ^3(x)} \, dx=x + \log \left ({\left (2 \, x + 5\right )} \log \left (x\right )^{2} + \log \left (x\right )^{3} + 5 \, x^{2} + {\left (x^{2} + 11 \, x\right )} \log \left (x\right ) + 5 \, x - \frac {11}{20}\right ) + \log \left (x\right ) \]
integrate(((20*x+20)*log(x)^3+(40*x^2+180*x+160)*log(x)^2+(20*x^3+280*x^2+ 520*x+200)*log(x)+100*x^3+420*x^2+409*x-11)/(20*x*log(x)^3+(40*x^2+100*x)* log(x)^2+(20*x^3+220*x^2)*log(x)+100*x^3+100*x^2-11*x),x, algorithm=\
Time = 0.27 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.64 \[ \int \frac {-11+409 x+420 x^2+100 x^3+\left (200+520 x+280 x^2+20 x^3\right ) \log (x)+\left (160+180 x+40 x^2\right ) \log ^2(x)+(20+20 x) \log ^3(x)}{-11 x+100 x^2+100 x^3+\left (220 x^2+20 x^3\right ) \log (x)+\left (100 x+40 x^2\right ) \log ^2(x)+20 x \log ^3(x)} \, dx=x + \log \left (20 \, x^{2} \log \left (x\right ) + 40 \, x \log \left (x\right )^{2} + 20 \, \log \left (x\right )^{3} + 100 \, x^{2} + 220 \, x \log \left (x\right ) + 100 \, \log \left (x\right )^{2} + 100 \, x - 11\right ) + \log \left (x\right ) \]
integrate(((20*x+20)*log(x)^3+(40*x^2+180*x+160)*log(x)^2+(20*x^3+280*x^2+ 520*x+200)*log(x)+100*x^3+420*x^2+409*x-11)/(20*x*log(x)^3+(40*x^2+100*x)* log(x)^2+(20*x^3+220*x^2)*log(x)+100*x^3+100*x^2-11*x),x, algorithm=\
x + log(20*x^2*log(x) + 40*x*log(x)^2 + 20*log(x)^3 + 100*x^2 + 220*x*log( x) + 100*log(x)^2 + 100*x - 11) + log(x)
Time = 12.35 (sec) , antiderivative size = 57, normalized size of antiderivative = 2.04 \[ \int \frac {-11+409 x+420 x^2+100 x^3+\left (200+520 x+280 x^2+20 x^3\right ) \log (x)+\left (160+180 x+40 x^2\right ) \log ^2(x)+(20+20 x) \log ^3(x)}{-11 x+100 x^2+100 x^3+\left (220 x^2+20 x^3\right ) \log (x)+\left (100 x+40 x^2\right ) \log ^2(x)+20 x \log ^3(x)} \, dx=\ln \left (20\,x^2\,\ln \left (x\right )+100\,x^2+40\,x\,{\ln \left (x\right )}^2+220\,x\,\ln \left (x\right )+100\,x+20\,{\ln \left (x\right )}^3+100\,{\ln \left (x\right )}^2-11\right )+\frac {x^2\,\ln \left (x\right )+x^3}{x^2} \]
int((409*x + log(x)^2*(180*x + 40*x^2 + 160) + 420*x^2 + 100*x^3 + log(x)^ 3*(20*x + 20) + log(x)*(520*x + 280*x^2 + 20*x^3 + 200) - 11)/(log(x)^2*(1 00*x + 40*x^2) - 11*x + log(x)*(220*x^2 + 20*x^3) + 20*x*log(x)^3 + 100*x^ 2 + 100*x^3),x)