Integrand size = 186, antiderivative size = 29 \[ \int \frac {500-200 x-455 x^2+200 x^3-20 x^4+e^4 \left (-190 x^2+80 x^3-8 x^4\right )+\left (-500-275 x+180 x^2-20 x^3+e^4 \left (-380 x+160 x^2-16 x^3\right )\right ) \log (x)+e^4 \left (-190+80 x-8 x^2\right ) \log ^2(x)}{125 x^2-50 x^3+5 x^4+e^4 \left (50 x^2-20 x^3+2 x^4\right )+\left (125 x-50 x^2+5 x^3+e^4 \left (100 x-40 x^2+4 x^3\right )\right ) \log (x)+e^4 \left (50-20 x+2 x^2\right ) \log ^2(x)} \, dx=\frac {x}{5-x}-4 \left (x+\log \left (2+\frac {5 x}{e^4 (x+\log (x))}\right )\right ) \]
Time = 0.11 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.31 \[ \int \frac {500-200 x-455 x^2+200 x^3-20 x^4+e^4 \left (-190 x^2+80 x^3-8 x^4\right )+\left (-500-275 x+180 x^2-20 x^3+e^4 \left (-380 x+160 x^2-16 x^3\right )\right ) \log (x)+e^4 \left (-190+80 x-8 x^2\right ) \log ^2(x)}{125 x^2-50 x^3+5 x^4+e^4 \left (50 x^2-20 x^3+2 x^4\right )+\left (125 x-50 x^2+5 x^3+e^4 \left (100 x-40 x^2+4 x^3\right )\right ) \log (x)+e^4 \left (50-20 x+2 x^2\right ) \log ^2(x)} \, dx=-\frac {5}{-5+x}-4 x+4 \log (x+\log (x))-4 \log \left (5 x+2 e^4 x+2 e^4 \log (x)\right ) \]
Integrate[(500 - 200*x - 455*x^2 + 200*x^3 - 20*x^4 + E^4*(-190*x^2 + 80*x ^3 - 8*x^4) + (-500 - 275*x + 180*x^2 - 20*x^3 + E^4*(-380*x + 160*x^2 - 1 6*x^3))*Log[x] + E^4*(-190 + 80*x - 8*x^2)*Log[x]^2)/(125*x^2 - 50*x^3 + 5 *x^4 + E^4*(50*x^2 - 20*x^3 + 2*x^4) + (125*x - 50*x^2 + 5*x^3 + E^4*(100* x - 40*x^2 + 4*x^3))*Log[x] + E^4*(50 - 20*x + 2*x^2)*Log[x]^2),x]
Time = 1.29 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.38, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.016, Rules used = {7292, 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.
| \(\displaystyle \int \frac {-20 x^4+200 x^3-455 x^2+e^4 \left (-8 x^2+80 x-190\right ) \log ^2(x)+\left (-20 x^3+180 x^2+e^4 \left (-16 x^3+160 x^2-380 x\right )-275 x-500\right ) \log (x)+e^4 \left (-8 x^4+80 x^3-190 x^2\right )-200 x+500}{5 x^4-50 x^3+125 x^2+e^4 \left (2 x^2-20 x+50\right ) \log ^2(x)+\left (5 x^3-50 x^2+e^4 \left (4 x^3-40 x^2+100 x\right )+125 x\right ) \log (x)+e^4 \left (2 x^4-20 x^3+50 x^2\right )} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {-20 x^4+200 x^3-455 x^2+e^4 \left (-8 x^2+80 x-190\right ) \log ^2(x)+\left (-20 x^3+180 x^2+e^4 \left (-16 x^3+160 x^2-380 x\right )-275 x-500\right ) \log (x)+e^4 \left (-8 x^4+80 x^3-190 x^2\right )-200 x+500}{(5-x)^2 (x+\log (x)) \left (5 \left (1+\frac {2 e^4}{5}\right ) x+2 e^4 \log (x)\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {-4 x^2+40 x-95}{(x-5)^2}+\frac {4 (x+1)}{x (x+\log (x))}+\frac {4 \left (-\left (\left (5+2 e^4\right ) x\right )-2 e^4\right )}{x \left (5 \left (1+\frac {2 e^4}{5}\right ) x+2 e^4 \log (x)\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -4 x+\frac {5}{5-x}+4 \log (x+\log (x))-4 \log \left (\left (5+2 e^4\right ) x+2 e^4 \log (x)\right )\) |
Int[(500 - 200*x - 455*x^2 + 200*x^3 - 20*x^4 + E^4*(-190*x^2 + 80*x^3 - 8 *x^4) + (-500 - 275*x + 180*x^2 - 20*x^3 + E^4*(-380*x + 160*x^2 - 16*x^3) )*Log[x] + E^4*(-190 + 80*x - 8*x^2)*Log[x]^2)/(125*x^2 - 50*x^3 + 5*x^4 + E^4*(50*x^2 - 20*x^3 + 2*x^4) + (125*x - 50*x^2 + 5*x^3 + E^4*(100*x - 40 *x^2 + 4*x^3))*Log[x] + E^4*(50 - 20*x + 2*x^2)*Log[x]^2),x]
3.27.99.3.1 Defintions of rubi rules used
Time = 1.44 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.31
| method | result | size |
| risch | \(-\frac {4 x^{2}-20 x +5}{-5+x}+4 \ln \left (x +\ln \left (x \right )\right )-4 \ln \left (\ln \left (x \right )+x +\frac {5 \,{\mathrm e}^{-4} x}{2}\right )\) | \(38\) |
| norman | \(\frac {-4 x^{2}+95}{-5+x}+4 \ln \left (x +\ln \left (x \right )\right )-4 \ln \left (2 \,{\mathrm e}^{4} \ln \left (x \right )+2 x \,{\mathrm e}^{4}+5 x \right )\) | \(40\) |
| default | \(-\frac {4 x^{2}-95}{-5+x}+4 \ln \left (x +\ln \left (x \right )\right )-4 \ln \left (2 \,{\mathrm e}^{4} \ln \left (x \right )+2 x \,{\mathrm e}^{4}+5 x \right )\) | \(41\) |
| parallelrisch | \(\frac {-4 \ln \left (\frac {2 \,{\mathrm e}^{4} \ln \left (x \right )+2 x \,{\mathrm e}^{4}+5 x}{2 \,{\mathrm e}^{4}+5}\right ) x +4 \ln \left (x +\ln \left (x \right )\right ) x -4 x^{2}+95+20 \ln \left (\frac {2 \,{\mathrm e}^{4} \ln \left (x \right )+2 x \,{\mathrm e}^{4}+5 x}{2 \,{\mathrm e}^{4}+5}\right )-20 \ln \left (x +\ln \left (x \right )\right )}{-5+x}\) | \(84\) |
int(((-8*x^2+80*x-190)*exp(4)*ln(x)^2+((-16*x^3+160*x^2-380*x)*exp(4)-20*x ^3+180*x^2-275*x-500)*ln(x)+(-8*x^4+80*x^3-190*x^2)*exp(4)-20*x^4+200*x^3- 455*x^2-200*x+500)/((2*x^2-20*x+50)*exp(4)*ln(x)^2+((4*x^3-40*x^2+100*x)*e xp(4)+5*x^3-50*x^2+125*x)*ln(x)+(2*x^4-20*x^3+50*x^2)*exp(4)+5*x^4-50*x^3+ 125*x^2),x,method=_RETURNVERBOSE)
Time = 0.26 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.66 \[ \int \frac {500-200 x-455 x^2+200 x^3-20 x^4+e^4 \left (-190 x^2+80 x^3-8 x^4\right )+\left (-500-275 x+180 x^2-20 x^3+e^4 \left (-380 x+160 x^2-16 x^3\right )\right ) \log (x)+e^4 \left (-190+80 x-8 x^2\right ) \log ^2(x)}{125 x^2-50 x^3+5 x^4+e^4 \left (50 x^2-20 x^3+2 x^4\right )+\left (125 x-50 x^2+5 x^3+e^4 \left (100 x-40 x^2+4 x^3\right )\right ) \log (x)+e^4 \left (50-20 x+2 x^2\right ) \log ^2(x)} \, dx=-\frac {4 \, x^{2} + 4 \, {\left (x - 5\right )} \log \left (2 \, x e^{4} + 2 \, e^{4} \log \left (x\right ) + 5 \, x\right ) - 4 \, {\left (x - 5\right )} \log \left (x + \log \left (x\right )\right ) - 20 \, x + 5}{x - 5} \]
integrate(((-8*x^2+80*x-190)*exp(4)*log(x)^2+((-16*x^3+160*x^2-380*x)*exp( 4)-20*x^3+180*x^2-275*x-500)*log(x)+(-8*x^4+80*x^3-190*x^2)*exp(4)-20*x^4+ 200*x^3-455*x^2-200*x+500)/((2*x^2-20*x+50)*exp(4)*log(x)^2+((4*x^3-40*x^2 +100*x)*exp(4)+5*x^3-50*x^2+125*x)*log(x)+(2*x^4-20*x^3+50*x^2)*exp(4)+5*x ^4-50*x^3+125*x^2),x, algorithm=\
-(4*x^2 + 4*(x - 5)*log(2*x*e^4 + 2*e^4*log(x) + 5*x) - 4*(x - 5)*log(x + log(x)) - 20*x + 5)/(x - 5)
Time = 0.25 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.28 \[ \int \frac {500-200 x-455 x^2+200 x^3-20 x^4+e^4 \left (-190 x^2+80 x^3-8 x^4\right )+\left (-500-275 x+180 x^2-20 x^3+e^4 \left (-380 x+160 x^2-16 x^3\right )\right ) \log (x)+e^4 \left (-190+80 x-8 x^2\right ) \log ^2(x)}{125 x^2-50 x^3+5 x^4+e^4 \left (50 x^2-20 x^3+2 x^4\right )+\left (125 x-50 x^2+5 x^3+e^4 \left (100 x-40 x^2+4 x^3\right )\right ) \log (x)+e^4 \left (50-20 x+2 x^2\right ) \log ^2(x)} \, dx=- 4 x + 4 \log {\left (x + \log {\left (x \right )} \right )} - 4 \log {\left (\frac {40 x + 16 x e^{4}}{16 e^{4}} + \log {\left (x \right )} \right )} - \frac {5}{x - 5} \]
integrate(((-8*x**2+80*x-190)*exp(4)*ln(x)**2+((-16*x**3+160*x**2-380*x)*e xp(4)-20*x**3+180*x**2-275*x-500)*ln(x)+(-8*x**4+80*x**3-190*x**2)*exp(4)- 20*x**4+200*x**3-455*x**2-200*x+500)/((2*x**2-20*x+50)*exp(4)*ln(x)**2+((4 *x**3-40*x**2+100*x)*exp(4)+5*x**3-50*x**2+125*x)*ln(x)+(2*x**4-20*x**3+50 *x**2)*exp(4)+5*x**4-50*x**3+125*x**2),x)
Time = 0.25 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.62 \[ \int \frac {500-200 x-455 x^2+200 x^3-20 x^4+e^4 \left (-190 x^2+80 x^3-8 x^4\right )+\left (-500-275 x+180 x^2-20 x^3+e^4 \left (-380 x+160 x^2-16 x^3\right )\right ) \log (x)+e^4 \left (-190+80 x-8 x^2\right ) \log ^2(x)}{125 x^2-50 x^3+5 x^4+e^4 \left (50 x^2-20 x^3+2 x^4\right )+\left (125 x-50 x^2+5 x^3+e^4 \left (100 x-40 x^2+4 x^3\right )\right ) \log (x)+e^4 \left (50-20 x+2 x^2\right ) \log ^2(x)} \, dx=-\frac {4 \, x^{2} - 20 \, x + 5}{x - 5} - 4 \, \log \left (\frac {1}{2} \, {\left (x {\left (2 \, e^{4} + 5\right )} + 2 \, e^{4} \log \left (x\right )\right )} e^{\left (-4\right )}\right ) + 4 \, \log \left (x + \log \left (x\right )\right ) \]
integrate(((-8*x^2+80*x-190)*exp(4)*log(x)^2+((-16*x^3+160*x^2-380*x)*exp( 4)-20*x^3+180*x^2-275*x-500)*log(x)+(-8*x^4+80*x^3-190*x^2)*exp(4)-20*x^4+ 200*x^3-455*x^2-200*x+500)/((2*x^2-20*x+50)*exp(4)*log(x)^2+((4*x^3-40*x^2 +100*x)*exp(4)+5*x^3-50*x^2+125*x)*log(x)+(2*x^4-20*x^3+50*x^2)*exp(4)+5*x ^4-50*x^3+125*x^2),x, algorithm=\
-(4*x^2 - 20*x + 5)/(x - 5) - 4*log(1/2*(x*(2*e^4 + 5) + 2*e^4*log(x))*e^( -4)) + 4*log(x + log(x))
Leaf count of result is larger than twice the leaf count of optimal. 77 vs. \(2 (28) = 56\).
Time = 0.31 (sec) , antiderivative size = 77, normalized size of antiderivative = 2.66 \[ \int \frac {500-200 x-455 x^2+200 x^3-20 x^4+e^4 \left (-190 x^2+80 x^3-8 x^4\right )+\left (-500-275 x+180 x^2-20 x^3+e^4 \left (-380 x+160 x^2-16 x^3\right )\right ) \log (x)+e^4 \left (-190+80 x-8 x^2\right ) \log ^2(x)}{125 x^2-50 x^3+5 x^4+e^4 \left (50 x^2-20 x^3+2 x^4\right )+\left (125 x-50 x^2+5 x^3+e^4 \left (100 x-40 x^2+4 x^3\right )\right ) \log (x)+e^4 \left (50-20 x+2 x^2\right ) \log ^2(x)} \, dx=-\frac {4 \, x^{2} + 4 \, x \log \left (2 \, x e^{4} + 2 \, e^{4} \log \left (x\right ) + 5 \, x\right ) - 4 \, x \log \left (-x - \log \left (x\right )\right ) - 20 \, x - 20 \, \log \left (2 \, x e^{4} + 2 \, e^{4} \log \left (x\right ) + 5 \, x\right ) + 20 \, \log \left (-x - \log \left (x\right )\right ) + 5}{x - 5} \]
integrate(((-8*x^2+80*x-190)*exp(4)*log(x)^2+((-16*x^3+160*x^2-380*x)*exp( 4)-20*x^3+180*x^2-275*x-500)*log(x)+(-8*x^4+80*x^3-190*x^2)*exp(4)-20*x^4+ 200*x^3-455*x^2-200*x+500)/((2*x^2-20*x+50)*exp(4)*log(x)^2+((4*x^3-40*x^2 +100*x)*exp(4)+5*x^3-50*x^2+125*x)*log(x)+(2*x^4-20*x^3+50*x^2)*exp(4)+5*x ^4-50*x^3+125*x^2),x, algorithm=\
-(4*x^2 + 4*x*log(2*x*e^4 + 2*e^4*log(x) + 5*x) - 4*x*log(-x - log(x)) - 2 0*x - 20*log(2*x*e^4 + 2*e^4*log(x) + 5*x) + 20*log(-x - log(x)) + 5)/(x - 5)
Timed out. \[ \int \frac {500-200 x-455 x^2+200 x^3-20 x^4+e^4 \left (-190 x^2+80 x^3-8 x^4\right )+\left (-500-275 x+180 x^2-20 x^3+e^4 \left (-380 x+160 x^2-16 x^3\right )\right ) \log (x)+e^4 \left (-190+80 x-8 x^2\right ) \log ^2(x)}{125 x^2-50 x^3+5 x^4+e^4 \left (50 x^2-20 x^3+2 x^4\right )+\left (125 x-50 x^2+5 x^3+e^4 \left (100 x-40 x^2+4 x^3\right )\right ) \log (x)+e^4 \left (50-20 x+2 x^2\right ) \log ^2(x)} \, dx=\int -\frac {200\,x+\ln \left (x\right )\,\left (275\,x+{\mathrm {e}}^4\,\left (16\,x^3-160\,x^2+380\,x\right )-180\,x^2+20\,x^3+500\right )+{\mathrm {e}}^4\,\left (8\,x^4-80\,x^3+190\,x^2\right )+455\,x^2-200\,x^3+20\,x^4+{\mathrm {e}}^4\,{\ln \left (x\right )}^2\,\left (8\,x^2-80\,x+190\right )-500}{\ln \left (x\right )\,\left (125\,x+{\mathrm {e}}^4\,\left (4\,x^3-40\,x^2+100\,x\right )-50\,x^2+5\,x^3\right )+{\mathrm {e}}^4\,\left (2\,x^4-20\,x^3+50\,x^2\right )+125\,x^2-50\,x^3+5\,x^4+{\mathrm {e}}^4\,{\ln \left (x\right )}^2\,\left (2\,x^2-20\,x+50\right )} \,d x \]
int(-(200*x + log(x)*(275*x + exp(4)*(380*x - 160*x^2 + 16*x^3) - 180*x^2 + 20*x^3 + 500) + exp(4)*(190*x^2 - 80*x^3 + 8*x^4) + 455*x^2 - 200*x^3 + 20*x^4 + exp(4)*log(x)^2*(8*x^2 - 80*x + 190) - 500)/(log(x)*(125*x + exp( 4)*(100*x - 40*x^2 + 4*x^3) - 50*x^2 + 5*x^3) + exp(4)*(50*x^2 - 20*x^3 + 2*x^4) + 125*x^2 - 50*x^3 + 5*x^4 + exp(4)*log(x)^2*(2*x^2 - 20*x + 50)),x )
int(-(200*x + log(x)*(275*x + exp(4)*(380*x - 160*x^2 + 16*x^3) - 180*x^2 + 20*x^3 + 500) + exp(4)*(190*x^2 - 80*x^3 + 8*x^4) + 455*x^2 - 200*x^3 + 20*x^4 + exp(4)*log(x)^2*(8*x^2 - 80*x + 190) - 500)/(log(x)*(125*x + exp( 4)*(100*x - 40*x^2 + 4*x^3) - 50*x^2 + 5*x^3) + exp(4)*(50*x^2 - 20*x^3 + 2*x^4) + 125*x^2 - 50*x^3 + 5*x^4 + exp(4)*log(x)^2*(2*x^2 - 20*x + 50)), x)