Integrand size = 209, antiderivative size = 28 \[ \int \frac {-175-120 x-1254 x^3-1001 x^4-300 x^5-40 x^6-2 x^7+\left (-60-40 x+8 x^2-1000 x^3-800 x^4-200 x^5-16 x^6\right ) \log (x)+\left (-5+x^2-300 x^3-200 x^4-36 x^5\right ) \log ^2(x)+\left (-40 x^3-16 x^4\right ) \log ^3(x)-2 x^3 \log ^4(x)}{625 x^2+500 x^3+150 x^4+20 x^5+x^6+\left (500 x^2+400 x^3+100 x^4+8 x^5\right ) \log (x)+\left (150 x^2+100 x^3+18 x^4\right ) \log ^2(x)+\left (20 x^2+8 x^3\right ) \log ^3(x)+x^2 \log ^4(x)} \, dx=\frac {\left (5+x^2\right ) \left (-x+\frac {1}{2 x \log (x)+(5+x+\log (x))^2}\right )}{x} \]
Time = 0.10 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.29 \[ \int \frac {-175-120 x-1254 x^3-1001 x^4-300 x^5-40 x^6-2 x^7+\left (-60-40 x+8 x^2-1000 x^3-800 x^4-200 x^5-16 x^6\right ) \log (x)+\left (-5+x^2-300 x^3-200 x^4-36 x^5\right ) \log ^2(x)+\left (-40 x^3-16 x^4\right ) \log ^3(x)-2 x^3 \log ^4(x)}{625 x^2+500 x^3+150 x^4+20 x^5+x^6+\left (500 x^2+400 x^3+100 x^4+8 x^5\right ) \log (x)+\left (150 x^2+100 x^3+18 x^4\right ) \log ^2(x)+\left (20 x^2+8 x^3\right ) \log ^3(x)+x^2 \log ^4(x)} \, dx=-x^2+\frac {5+x^2}{x \left ((5+x)^2+2 (5+2 x) \log (x)+\log ^2(x)\right )} \]
Integrate[(-175 - 120*x - 1254*x^3 - 1001*x^4 - 300*x^5 - 40*x^6 - 2*x^7 + (-60 - 40*x + 8*x^2 - 1000*x^3 - 800*x^4 - 200*x^5 - 16*x^6)*Log[x] + (-5 + x^2 - 300*x^3 - 200*x^4 - 36*x^5)*Log[x]^2 + (-40*x^3 - 16*x^4)*Log[x]^ 3 - 2*x^3*Log[x]^4)/(625*x^2 + 500*x^3 + 150*x^4 + 20*x^5 + x^6 + (500*x^2 + 400*x^3 + 100*x^4 + 8*x^5)*Log[x] + (150*x^2 + 100*x^3 + 18*x^4)*Log[x] ^2 + (20*x^2 + 8*x^3)*Log[x]^3 + x^2*Log[x]^4),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.
| \(\displaystyle \int \frac {-2 x^7-40 x^6-300 x^5-1001 x^4-1254 x^3-2 x^3 \log ^4(x)+\left (-16 x^4-40 x^3\right ) \log ^3(x)+\left (-36 x^5-200 x^4-300 x^3+x^2-5\right ) \log ^2(x)+\left (-16 x^6-200 x^5-800 x^4-1000 x^3+8 x^2-40 x-60\right ) \log (x)-120 x-175}{x^6+20 x^5+150 x^4+500 x^3+625 x^2+x^2 \log ^4(x)+\left (8 x^3+20 x^2\right ) \log ^3(x)+\left (18 x^4+100 x^3+150 x^2\right ) \log ^2(x)+\left (8 x^5+100 x^4+400 x^3+500 x^2\right ) \log (x)} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {-2 x^7-40 x^6-300 x^5-1001 x^4-1254 x^3-2 x^3 \log ^4(x)+\left (-16 x^4-40 x^3\right ) \log ^3(x)+\left (-36 x^5-200 x^4-300 x^3+x^2-5\right ) \log ^2(x)+\left (-16 x^6-200 x^5-800 x^4-1000 x^3+8 x^2-40 x-60\right ) \log (x)-120 x-175}{x^2 \left (x^2+10 x+\log ^2(x)+4 x \log (x)+10 \log (x)+25\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {x^2-5}{x^2 \left (x^2+10 x+\log ^2(x)+4 x \log (x)+10 \log (x)+25\right )}-\frac {2 \left (x^2+5\right ) \left (x^2+7 x+2 x \log (x)+\log (x)+5\right )}{x^2 \left (x^2+10 x+\log ^2(x)+4 x \log (x)+10 \log (x)+25\right )^2}-2 x\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -20 \int \frac {1}{\left (x^2+4 \log (x) x+10 x+\log ^2(x)+10 \log (x)+25\right )^2}dx-50 \int \frac {1}{x^2 \left (x^2+4 \log (x) x+10 x+\log ^2(x)+10 \log (x)+25\right )^2}dx-70 \int \frac {1}{x \left (x^2+4 \log (x) x+10 x+\log ^2(x)+10 \log (x)+25\right )^2}dx-14 \int \frac {x}{\left (x^2+4 \log (x) x+10 x+\log ^2(x)+10 \log (x)+25\right )^2}dx-2 \int \frac {x^2}{\left (x^2+4 \log (x) x+10 x+\log ^2(x)+10 \log (x)+25\right )^2}dx-2 \int \frac {\log (x)}{\left (x^2+4 \log (x) x+10 x+\log ^2(x)+10 \log (x)+25\right )^2}dx-10 \int \frac {\log (x)}{x^2 \left (x^2+4 \log (x) x+10 x+\log ^2(x)+10 \log (x)+25\right )^2}dx-20 \int \frac {\log (x)}{x \left (x^2+4 \log (x) x+10 x+\log ^2(x)+10 \log (x)+25\right )^2}dx-4 \int \frac {x \log (x)}{\left (x^2+4 \log (x) x+10 x+\log ^2(x)+10 \log (x)+25\right )^2}dx+\int \frac {1}{x^2+4 \log (x) x+10 x+\log ^2(x)+10 \log (x)+25}dx-5 \int \frac {1}{x^2 \left (x^2+4 \log (x) x+10 x+\log ^2(x)+10 \log (x)+25\right )}dx-x^2\) |
Int[(-175 - 120*x - 1254*x^3 - 1001*x^4 - 300*x^5 - 40*x^6 - 2*x^7 + (-60 - 40*x + 8*x^2 - 1000*x^3 - 800*x^4 - 200*x^5 - 16*x^6)*Log[x] + (-5 + x^2 - 300*x^3 - 200*x^4 - 36*x^5)*Log[x]^2 + (-40*x^3 - 16*x^4)*Log[x]^3 - 2* x^3*Log[x]^4)/(625*x^2 + 500*x^3 + 150*x^4 + 20*x^5 + x^6 + (500*x^2 + 400 *x^3 + 100*x^4 + 8*x^5)*Log[x] + (150*x^2 + 100*x^3 + 18*x^4)*Log[x]^2 + ( 20*x^2 + 8*x^3)*Log[x]^3 + x^2*Log[x]^4),x]
3.9.42.3.1 Defintions of rubi rules used
Time = 0.54 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.39
| method | result | size |
| risch | \(-x^{2}+\frac {x^{2}+5}{x \left (\ln \left (x \right )^{2}+4 x \ln \left (x \right )+x^{2}+10 \ln \left (x \right )+10 x +25\right )}\) | \(39\) |
| default | \(-x^{2}+\frac {5}{\left (5+\ln \left (x \right )\right )^{2} x}+\frac {x \ln \left (x \right )^{2}+10 x \ln \left (x \right )-20 \ln \left (x \right )+20 x -50}{\left (5+\ln \left (x \right )\right )^{2} \left (\ln \left (x \right )^{2}+4 x \ln \left (x \right )+x^{2}+10 \ln \left (x \right )+10 x +25\right )}\) | \(68\) |
| parallelrisch | \(\frac {5-4 x^{4} \ln \left (x \right )-x^{3} \ln \left (x \right )^{2}-10 x^{4}-25 x^{3}+x^{2}-x^{5}-10 x^{3} \ln \left (x \right )}{x \left (\ln \left (x \right )^{2}+4 x \ln \left (x \right )+x^{2}+10 \ln \left (x \right )+10 x +25\right )}\) | \(71\) |
int((-2*x^3*ln(x)^4+(-16*x^4-40*x^3)*ln(x)^3+(-36*x^5-200*x^4-300*x^3+x^2- 5)*ln(x)^2+(-16*x^6-200*x^5-800*x^4-1000*x^3+8*x^2-40*x-60)*ln(x)-2*x^7-40 *x^6-300*x^5-1001*x^4-1254*x^3-120*x-175)/(x^2*ln(x)^4+(8*x^3+20*x^2)*ln(x )^3+(18*x^4+100*x^3+150*x^2)*ln(x)^2+(8*x^5+100*x^4+400*x^3+500*x^2)*ln(x) +x^6+20*x^5+150*x^4+500*x^3+625*x^2),x,method=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 78 vs. \(2 (29) = 58\).
Time = 0.26 (sec) , antiderivative size = 78, normalized size of antiderivative = 2.79 \[ \int \frac {-175-120 x-1254 x^3-1001 x^4-300 x^5-40 x^6-2 x^7+\left (-60-40 x+8 x^2-1000 x^3-800 x^4-200 x^5-16 x^6\right ) \log (x)+\left (-5+x^2-300 x^3-200 x^4-36 x^5\right ) \log ^2(x)+\left (-40 x^3-16 x^4\right ) \log ^3(x)-2 x^3 \log ^4(x)}{625 x^2+500 x^3+150 x^4+20 x^5+x^6+\left (500 x^2+400 x^3+100 x^4+8 x^5\right ) \log (x)+\left (150 x^2+100 x^3+18 x^4\right ) \log ^2(x)+\left (20 x^2+8 x^3\right ) \log ^3(x)+x^2 \log ^4(x)} \, dx=-\frac {x^{5} + x^{3} \log \left (x\right )^{2} + 10 \, x^{4} + 25 \, x^{3} - x^{2} + 2 \, {\left (2 \, x^{4} + 5 \, x^{3}\right )} \log \left (x\right ) - 5}{x^{3} + x \log \left (x\right )^{2} + 10 \, x^{2} + 2 \, {\left (2 \, x^{2} + 5 \, x\right )} \log \left (x\right ) + 25 \, x} \]
integrate((-2*x^3*log(x)^4+(-16*x^4-40*x^3)*log(x)^3+(-36*x^5-200*x^4-300* x^3+x^2-5)*log(x)^2+(-16*x^6-200*x^5-800*x^4-1000*x^3+8*x^2-40*x-60)*log(x )-2*x^7-40*x^6-300*x^5-1001*x^4-1254*x^3-120*x-175)/(x^2*log(x)^4+(8*x^3+2 0*x^2)*log(x)^3+(18*x^4+100*x^3+150*x^2)*log(x)^2+(8*x^5+100*x^4+400*x^3+5 00*x^2)*log(x)+x^6+20*x^5+150*x^4+500*x^3+625*x^2),x, algorithm=\
-(x^5 + x^3*log(x)^2 + 10*x^4 + 25*x^3 - x^2 + 2*(2*x^4 + 5*x^3)*log(x) - 5)/(x^3 + x*log(x)^2 + 10*x^2 + 2*(2*x^2 + 5*x)*log(x) + 25*x)
Time = 0.10 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.32 \[ \int \frac {-175-120 x-1254 x^3-1001 x^4-300 x^5-40 x^6-2 x^7+\left (-60-40 x+8 x^2-1000 x^3-800 x^4-200 x^5-16 x^6\right ) \log (x)+\left (-5+x^2-300 x^3-200 x^4-36 x^5\right ) \log ^2(x)+\left (-40 x^3-16 x^4\right ) \log ^3(x)-2 x^3 \log ^4(x)}{625 x^2+500 x^3+150 x^4+20 x^5+x^6+\left (500 x^2+400 x^3+100 x^4+8 x^5\right ) \log (x)+\left (150 x^2+100 x^3+18 x^4\right ) \log ^2(x)+\left (20 x^2+8 x^3\right ) \log ^3(x)+x^2 \log ^4(x)} \, dx=- x^{2} + \frac {x^{2} + 5}{x^{3} + 10 x^{2} + x \log {\left (x \right )}^{2} + 25 x + \left (4 x^{2} + 10 x\right ) \log {\left (x \right )}} \]
integrate((-2*x**3*ln(x)**4+(-16*x**4-40*x**3)*ln(x)**3+(-36*x**5-200*x**4 -300*x**3+x**2-5)*ln(x)**2+(-16*x**6-200*x**5-800*x**4-1000*x**3+8*x**2-40 *x-60)*ln(x)-2*x**7-40*x**6-300*x**5-1001*x**4-1254*x**3-120*x-175)/(x**2* ln(x)**4+(8*x**3+20*x**2)*ln(x)**3+(18*x**4+100*x**3+150*x**2)*ln(x)**2+(8 *x**5+100*x**4+400*x**3+500*x**2)*ln(x)+x**6+20*x**5+150*x**4+500*x**3+625 *x**2),x)
Leaf count of result is larger than twice the leaf count of optimal. 78 vs. \(2 (29) = 58\).
Time = 0.26 (sec) , antiderivative size = 78, normalized size of antiderivative = 2.79 \[ \int \frac {-175-120 x-1254 x^3-1001 x^4-300 x^5-40 x^6-2 x^7+\left (-60-40 x+8 x^2-1000 x^3-800 x^4-200 x^5-16 x^6\right ) \log (x)+\left (-5+x^2-300 x^3-200 x^4-36 x^5\right ) \log ^2(x)+\left (-40 x^3-16 x^4\right ) \log ^3(x)-2 x^3 \log ^4(x)}{625 x^2+500 x^3+150 x^4+20 x^5+x^6+\left (500 x^2+400 x^3+100 x^4+8 x^5\right ) \log (x)+\left (150 x^2+100 x^3+18 x^4\right ) \log ^2(x)+\left (20 x^2+8 x^3\right ) \log ^3(x)+x^2 \log ^4(x)} \, dx=-\frac {x^{5} + x^{3} \log \left (x\right )^{2} + 10 \, x^{4} + 25 \, x^{3} - x^{2} + 2 \, {\left (2 \, x^{4} + 5 \, x^{3}\right )} \log \left (x\right ) - 5}{x^{3} + x \log \left (x\right )^{2} + 10 \, x^{2} + 2 \, {\left (2 \, x^{2} + 5 \, x\right )} \log \left (x\right ) + 25 \, x} \]
integrate((-2*x^3*log(x)^4+(-16*x^4-40*x^3)*log(x)^3+(-36*x^5-200*x^4-300* x^3+x^2-5)*log(x)^2+(-16*x^6-200*x^5-800*x^4-1000*x^3+8*x^2-40*x-60)*log(x )-2*x^7-40*x^6-300*x^5-1001*x^4-1254*x^3-120*x-175)/(x^2*log(x)^4+(8*x^3+2 0*x^2)*log(x)^3+(18*x^4+100*x^3+150*x^2)*log(x)^2+(8*x^5+100*x^4+400*x^3+5 00*x^2)*log(x)+x^6+20*x^5+150*x^4+500*x^3+625*x^2),x, algorithm=\
-(x^5 + x^3*log(x)^2 + 10*x^4 + 25*x^3 - x^2 + 2*(2*x^4 + 5*x^3)*log(x) - 5)/(x^3 + x*log(x)^2 + 10*x^2 + 2*(2*x^2 + 5*x)*log(x) + 25*x)
Time = 0.34 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.57 \[ \int \frac {-175-120 x-1254 x^3-1001 x^4-300 x^5-40 x^6-2 x^7+\left (-60-40 x+8 x^2-1000 x^3-800 x^4-200 x^5-16 x^6\right ) \log (x)+\left (-5+x^2-300 x^3-200 x^4-36 x^5\right ) \log ^2(x)+\left (-40 x^3-16 x^4\right ) \log ^3(x)-2 x^3 \log ^4(x)}{625 x^2+500 x^3+150 x^4+20 x^5+x^6+\left (500 x^2+400 x^3+100 x^4+8 x^5\right ) \log (x)+\left (150 x^2+100 x^3+18 x^4\right ) \log ^2(x)+\left (20 x^2+8 x^3\right ) \log ^3(x)+x^2 \log ^4(x)} \, dx=-x^{2} + \frac {x^{2} + 5}{x^{3} + 4 \, x^{2} \log \left (x\right ) + x \log \left (x\right )^{2} + 10 \, x^{2} + 10 \, x \log \left (x\right ) + 25 \, x} \]
integrate((-2*x^3*log(x)^4+(-16*x^4-40*x^3)*log(x)^3+(-36*x^5-200*x^4-300* x^3+x^2-5)*log(x)^2+(-16*x^6-200*x^5-800*x^4-1000*x^3+8*x^2-40*x-60)*log(x )-2*x^7-40*x^6-300*x^5-1001*x^4-1254*x^3-120*x-175)/(x^2*log(x)^4+(8*x^3+2 0*x^2)*log(x)^3+(18*x^4+100*x^3+150*x^2)*log(x)^2+(8*x^5+100*x^4+400*x^3+5 00*x^2)*log(x)+x^6+20*x^5+150*x^4+500*x^3+625*x^2),x, algorithm=\
Timed out. \[ \int \frac {-175-120 x-1254 x^3-1001 x^4-300 x^5-40 x^6-2 x^7+\left (-60-40 x+8 x^2-1000 x^3-800 x^4-200 x^5-16 x^6\right ) \log (x)+\left (-5+x^2-300 x^3-200 x^4-36 x^5\right ) \log ^2(x)+\left (-40 x^3-16 x^4\right ) \log ^3(x)-2 x^3 \log ^4(x)}{625 x^2+500 x^3+150 x^4+20 x^5+x^6+\left (500 x^2+400 x^3+100 x^4+8 x^5\right ) \log (x)+\left (150 x^2+100 x^3+18 x^4\right ) \log ^2(x)+\left (20 x^2+8 x^3\right ) \log ^3(x)+x^2 \log ^4(x)} \, dx=\int -\frac {120\,x+{\ln \left (x\right )}^3\,\left (16\,x^4+40\,x^3\right )+2\,x^3\,{\ln \left (x\right )}^4+\ln \left (x\right )\,\left (16\,x^6+200\,x^5+800\,x^4+1000\,x^3-8\,x^2+40\,x+60\right )+1254\,x^3+1001\,x^4+300\,x^5+40\,x^6+2\,x^7+{\ln \left (x\right )}^2\,\left (36\,x^5+200\,x^4+300\,x^3-x^2+5\right )+175}{{\ln \left (x\right )}^3\,\left (8\,x^3+20\,x^2\right )+x^2\,{\ln \left (x\right )}^4+\ln \left (x\right )\,\left (8\,x^5+100\,x^4+400\,x^3+500\,x^2\right )+{\ln \left (x\right )}^2\,\left (18\,x^4+100\,x^3+150\,x^2\right )+625\,x^2+500\,x^3+150\,x^4+20\,x^5+x^6} \,d x \]
int(-(120*x + log(x)^3*(40*x^3 + 16*x^4) + 2*x^3*log(x)^4 + log(x)*(40*x - 8*x^2 + 1000*x^3 + 800*x^4 + 200*x^5 + 16*x^6 + 60) + 1254*x^3 + 1001*x^4 + 300*x^5 + 40*x^6 + 2*x^7 + log(x)^2*(300*x^3 - x^2 + 200*x^4 + 36*x^5 + 5) + 175)/(log(x)^3*(20*x^2 + 8*x^3) + x^2*log(x)^4 + log(x)*(500*x^2 + 4 00*x^3 + 100*x^4 + 8*x^5) + log(x)^2*(150*x^2 + 100*x^3 + 18*x^4) + 625*x^ 2 + 500*x^3 + 150*x^4 + 20*x^5 + x^6),x)
int(-(120*x + log(x)^3*(40*x^3 + 16*x^4) + 2*x^3*log(x)^4 + log(x)*(40*x - 8*x^2 + 1000*x^3 + 800*x^4 + 200*x^5 + 16*x^6 + 60) + 1254*x^3 + 1001*x^4 + 300*x^5 + 40*x^6 + 2*x^7 + log(x)^2*(300*x^3 - x^2 + 200*x^4 + 36*x^5 + 5) + 175)/(log(x)^3*(20*x^2 + 8*x^3) + x^2*log(x)^4 + log(x)*(500*x^2 + 4 00*x^3 + 100*x^4 + 8*x^5) + log(x)^2*(150*x^2 + 100*x^3 + 18*x^4) + 625*x^ 2 + 500*x^3 + 150*x^4 + 20*x^5 + x^6), x)