Integrand size = 90, antiderivative size = 25 \begin {dmath*} \int \frac {(90-45 x) \log (2-x)+\left (12800 x^5+1600 x^6+50 x^7\right ) \log (\log (2-x))+\left (-64000 x^4+22400 x^5+4450 x^6+175 x^7\right ) \log (2-x) \log ^2(\log (2-x))}{(-18+9 x) \log (2-x)} \, dx=x \left (-5+\frac {25}{9} x^4 (16+x)^2 \log ^2(\log (2-x))\right ) \end {dmath*}
\begin {dmath*} \int \frac {(90-45 x) \log (2-x)+\left (12800 x^5+1600 x^6+50 x^7\right ) \log (\log (2-x))+\left (-64000 x^4+22400 x^5+4450 x^6+175 x^7\right ) \log (2-x) \log ^2(\log (2-x))}{(-18+9 x) \log (2-x)} \, dx=\int \frac {(90-45 x) \log (2-x)+\left (12800 x^5+1600 x^6+50 x^7\right ) \log (\log (2-x))+\left (-64000 x^4+22400 x^5+4450 x^6+175 x^7\right ) \log (2-x) \log ^2(\log (2-x))}{(-18+9 x) \log (2-x)} \, dx \end {dmath*}
Integrate[((90 - 45*x)*Log[2 - x] + (12800*x^5 + 1600*x^6 + 50*x^7)*Log[Lo g[2 - x]] + (-64000*x^4 + 22400*x^5 + 4450*x^6 + 175*x^7)*Log[2 - x]*Log[L og[2 - x]]^2)/((-18 + 9*x)*Log[2 - x]),x]
Integrate[((90 - 45*x)*Log[2 - x] + (12800*x^5 + 1600*x^6 + 50*x^7)*Log[Lo g[2 - x]] + (-64000*x^4 + 22400*x^5 + 4450*x^6 + 175*x^7)*Log[2 - x]*Log[L og[2 - x]]^2)/((-18 + 9*x)*Log[2 - 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.
\(\displaystyle \int \frac {\left (50 x^7+1600 x^6+12800 x^5\right ) \log (\log (2-x))+\left (175 x^7+4450 x^6+22400 x^5-64000 x^4\right ) \log (2-x) \log ^2(\log (2-x))+(90-45 x) \log (2-x)}{(9 x-18) \log (2-x)} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {50 (x+16)^2 x^5 \log (\log (2-x))}{9 (x-2) \log (2-x)}+\frac {25}{9} (x+16) (7 x+80) x^4 \log ^2(\log (2-x))-5\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -28800 \text {Subst}\left (\int \frac {\log (\log (x))}{\log (x)}dx,x,2-x\right )+\frac {175}{9} \int x^6 \log ^2(\log (2-x))dx+\frac {50}{9} \int \frac {x^6 \log (\log (2-x))}{\log (2-x)}dx+\frac {1600}{3} \int x^5 \log ^2(\log (2-x))dx+\frac {1700}{9} \int \frac {x^5 \log (\log (2-x))}{\log (2-x)}dx+\frac {32000}{9} \int x^4 \log ^2(\log (2-x))dx+1800 \int \frac {x^4 \log (\log (2-x))}{\log (2-x)}dx+3600 \int \frac {x^3 \log (\log (2-x))}{\log (2-x)}dx+7200 \int \frac {x^2 \log (\log (2-x))}{\log (2-x)}dx+14400 \int \frac {x \log (\log (2-x))}{\log (2-x)}dx-5 x+28800 \log ^2(\log (2-x))\) |
Int[((90 - 45*x)*Log[2 - x] + (12800*x^5 + 1600*x^6 + 50*x^7)*Log[Log[2 - x]] + (-64000*x^4 + 22400*x^5 + 4450*x^6 + 175*x^7)*Log[2 - x]*Log[Log[2 - x]]^2)/((-18 + 9*x)*Log[2 - x]),x]
3.11.5.3.1 Defintions of rubi rules used
Time = 1.51 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.24
method | result | size |
risch | \(\left (\frac {25}{9} x^{7}+\frac {800}{9} x^{6}+\frac {6400}{9} x^{5}\right ) \ln \left (\ln \left (2-x \right )\right )^{2}-5 x\) | \(31\) |
parallelrisch | \(\frac {25 \ln \left (\ln \left (2-x \right )\right )^{2} x^{7}}{9}+\frac {800 \ln \left (\ln \left (2-x \right )\right )^{2} x^{6}}{9}+\frac {6400 \ln \left (\ln \left (2-x \right )\right )^{2} x^{5}}{9}-5-5 x\) | \(48\) |
int(((175*x^7+4450*x^6+22400*x^5-64000*x^4)*ln(2-x)*ln(ln(2-x))^2+(50*x^7+ 1600*x^6+12800*x^5)*ln(ln(2-x))+(-45*x+90)*ln(2-x))/(9*x-18)/ln(2-x),x,met hod=_RETURNVERBOSE)
Time = 0.27 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.16 \begin {dmath*} \int \frac {(90-45 x) \log (2-x)+\left (12800 x^5+1600 x^6+50 x^7\right ) \log (\log (2-x))+\left (-64000 x^4+22400 x^5+4450 x^6+175 x^7\right ) \log (2-x) \log ^2(\log (2-x))}{(-18+9 x) \log (2-x)} \, dx=\frac {25}{9} \, {\left (x^{7} + 32 \, x^{6} + 256 \, x^{5}\right )} \log \left (\log \left (-x + 2\right )\right )^{2} - 5 \, x \end {dmath*}
integrate(((175*x^7+4450*x^6+22400*x^5-64000*x^4)*log(2-x)*log(log(2-x))^2 +(50*x^7+1600*x^6+12800*x^5)*log(log(2-x))+(-45*x+90)*log(2-x))/(9*x-18)/l og(2-x),x, algorithm=\
Time = 0.32 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.24 \begin {dmath*} \int \frac {(90-45 x) \log (2-x)+\left (12800 x^5+1600 x^6+50 x^7\right ) \log (\log (2-x))+\left (-64000 x^4+22400 x^5+4450 x^6+175 x^7\right ) \log (2-x) \log ^2(\log (2-x))}{(-18+9 x) \log (2-x)} \, dx=- 5 x + \left (\frac {25 x^{7}}{9} + \frac {800 x^{6}}{9} + \frac {6400 x^{5}}{9}\right ) \log {\left (\log {\left (2 - x \right )} \right )}^{2} \end {dmath*}
integrate(((175*x**7+4450*x**6+22400*x**5-64000*x**4)*ln(2-x)*ln(ln(2-x))* *2+(50*x**7+1600*x**6+12800*x**5)*ln(ln(2-x))+(-45*x+90)*ln(2-x))/(9*x-18) /ln(2-x),x)
Time = 0.29 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.72 \begin {dmath*} \int \frac {(90-45 x) \log (2-x)+\left (12800 x^5+1600 x^6+50 x^7\right ) \log (\log (2-x))+\left (-64000 x^4+22400 x^5+4450 x^6+175 x^7\right ) \log (2-x) \log ^2(\log (2-x))}{(-18+9 x) \log (2-x)} \, dx=\frac {25}{9} \, {\left (x^{7} + 32 \, x^{6} + 256 \, x^{5}\right )} \log \left (\log \left (-x + 2\right )\right )^{2} - 5 \, x - 10 \, \log \left (x - 2\right ) + 10 \, \log \left (-x + 2\right ) \end {dmath*}
integrate(((175*x^7+4450*x^6+22400*x^5-64000*x^4)*log(2-x)*log(log(2-x))^2 +(50*x^7+1600*x^6+12800*x^5)*log(log(2-x))+(-45*x+90)*log(2-x))/(9*x-18)/l og(2-x),x, algorithm=\
Time = 0.33 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.16 \begin {dmath*} \int \frac {(90-45 x) \log (2-x)+\left (12800 x^5+1600 x^6+50 x^7\right ) \log (\log (2-x))+\left (-64000 x^4+22400 x^5+4450 x^6+175 x^7\right ) \log (2-x) \log ^2(\log (2-x))}{(-18+9 x) \log (2-x)} \, dx=\frac {25}{9} \, {\left (x^{7} + 32 \, x^{6} + 256 \, x^{5}\right )} \log \left (\log \left (-x + 2\right )\right )^{2} - 5 \, x \end {dmath*}
integrate(((175*x^7+4450*x^6+22400*x^5-64000*x^4)*log(2-x)*log(log(2-x))^2 +(50*x^7+1600*x^6+12800*x^5)*log(log(2-x))+(-45*x+90)*log(2-x))/(9*x-18)/l og(2-x),x, algorithm=\
Time = 16.98 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.20 \begin {dmath*} \int \frac {(90-45 x) \log (2-x)+\left (12800 x^5+1600 x^6+50 x^7\right ) \log (\log (2-x))+\left (-64000 x^4+22400 x^5+4450 x^6+175 x^7\right ) \log (2-x) \log ^2(\log (2-x))}{(-18+9 x) \log (2-x)} \, dx={\ln \left (\ln \left (2-x\right )\right )}^2\,\left (\frac {25\,x^7}{9}+\frac {800\,x^6}{9}+\frac {6400\,x^5}{9}\right )-5\,x \end {dmath*}