Integrand size = 93, antiderivative size = 26 \[ \int \frac {-800-800 x+\left (1600-400 x-800 x^2\right ) \log (x)+\left (-5 x+2 x^3\right ) \log ^3(x)+\left ((400+400 x) \log (x)+\left (-x-x^2\right ) \log ^3(x)\right ) \log \left (\frac {400-x \log ^2(x)}{\log ^2(x)}\right )}{(-400-400 x) \log (x)+\left (x+x^2\right ) \log ^3(x)} \, dx=5-3 \log (1+x)+x \left (-1+x-\log \left (-x+\frac {400}{\log ^2(x)}\right )\right ) \]
Time = 0.08 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.04 \[ \int \frac {-800-800 x+\left (1600-400 x-800 x^2\right ) \log (x)+\left (-5 x+2 x^3\right ) \log ^3(x)+\left ((400+400 x) \log (x)+\left (-x-x^2\right ) \log ^3(x)\right ) \log \left (\frac {400-x \log ^2(x)}{\log ^2(x)}\right )}{(-400-400 x) \log (x)+\left (x+x^2\right ) \log ^3(x)} \, dx=-x+x^2-3 \log (1+x)-x \log \left (-x+\frac {400}{\log ^2(x)}\right ) \]
Integrate[(-800 - 800*x + (1600 - 400*x - 800*x^2)*Log[x] + (-5*x + 2*x^3) *Log[x]^3 + ((400 + 400*x)*Log[x] + (-x - x^2)*Log[x]^3)*Log[(400 - x*Log[ x]^2)/Log[x]^2])/((-400 - 400*x)*Log[x] + (x + x^2)*Log[x]^3),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.
\(\displaystyle \int \frac {\left (2 x^3-5 x\right ) \log ^3(x)+\left (\left (-x^2-x\right ) \log ^3(x)+(400 x+400) \log (x)\right ) \log \left (\frac {400-x \log ^2(x)}{\log ^2(x)}\right )+\left (-800 x^2-400 x+1600\right ) \log (x)-800 x-800}{\left (x^2+x\right ) \log ^3(x)+(-400 x-400) \log (x)} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {-\left (\left (2 x^3-5 x\right ) \log ^3(x)\right )-\left (\left (-x^2-x\right ) \log ^3(x)+(400 x+400) \log (x)\right ) \log \left (\frac {400-x \log ^2(x)}{\log ^2(x)}\right )-\left (-800 x^2-400 x+1600\right ) \log (x)+800 x+800}{(x+1) \log (x) \left (400-x \log ^2(x)\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {x \left (2 x^2-5\right ) \log ^2(x)}{(x+1) \left (x \log ^2(x)-400\right )}-\frac {400 \left (2 x^2+x-4\right )}{(x+1) \left (x \log ^2(x)-400\right )}-\log \left (\frac {400}{\log ^2(x)}-x\right )-\frac {800 x}{(x+1) \left (x \log ^2(x)-400\right ) \log (x)}-\frac {800}{(x+1) \left (x \log ^2(x)-400\right ) \log (x)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -400 \int \frac {1}{400-x \log ^2(x)}dx-800 \int \frac {1}{x \log ^2(x)-400}dx-2 \int \frac {x \log (x)}{x \log ^2(x)-400}dx-\int \log \left (\frac {400}{\log ^2(x)}-x\right )dx+2 \int \frac {1}{(x+1) \log (x)}dx+2 \int \frac {x}{(x+1) \log (x)}dx+x^2-2 x-3 \log (x+1)\) |
Int[(-800 - 800*x + (1600 - 400*x - 800*x^2)*Log[x] + (-5*x + 2*x^3)*Log[x ]^3 + ((400 + 400*x)*Log[x] + (-x - x^2)*Log[x]^3)*Log[(400 - x*Log[x]^2)/ Log[x]^2])/((-400 - 400*x)*Log[x] + (x + x^2)*Log[x]^3),x]
3.10.45.3.1 Defintions of rubi rules used
Time = 9.55 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.23
method | result | size |
parallelrisch | \(x^{2}-x \ln \left (-\frac {x \ln \left (x \right )^{2}-400}{\ln \left (x \right )^{2}}\right )-3 \ln \left (1+x \right )-x\) | \(32\) |
risch | \(-x \ln \left (x \ln \left (x \right )^{2}-400\right )+2 x \ln \left (\ln \left (x \right )\right )+i \pi x \operatorname {csgn}\left (i \ln \left (x \right )^{2}\right )^{2} \operatorname {csgn}\left (i \ln \left (x \right )\right )+\frac {i \pi x \,\operatorname {csgn}\left (\frac {i}{\ln \left (x \right )^{2}}\right ) \operatorname {csgn}\left (i \left (x \ln \left (x \right )^{2}-400\right )\right ) \operatorname {csgn}\left (\frac {i \left (x \ln \left (x \right )^{2}-400\right )}{\ln \left (x \right )^{2}}\right )}{2}-\frac {i \pi x \operatorname {csgn}\left (i \ln \left (x \right )^{2}\right )^{3}}{2}-\frac {i \pi x \,\operatorname {csgn}\left (\frac {i}{\ln \left (x \right )^{2}}\right ) {\operatorname {csgn}\left (\frac {i \left (x \ln \left (x \right )^{2}-400\right )}{\ln \left (x \right )^{2}}\right )}^{2}}{2}-\frac {i \pi x \,\operatorname {csgn}\left (i \left (x \ln \left (x \right )^{2}-400\right )\right ) {\operatorname {csgn}\left (\frac {i \left (x \ln \left (x \right )^{2}-400\right )}{\ln \left (x \right )^{2}}\right )}^{2}}{2}-\frac {i \pi x {\operatorname {csgn}\left (\frac {i \left (x \ln \left (x \right )^{2}-400\right )}{\ln \left (x \right )^{2}}\right )}^{3}}{2}+i \pi x {\operatorname {csgn}\left (\frac {i \left (x \ln \left (x \right )^{2}-400\right )}{\ln \left (x \right )^{2}}\right )}^{2}-i \pi x -\frac {i \pi x \,\operatorname {csgn}\left (i \ln \left (x \right )^{2}\right ) \operatorname {csgn}\left (i \ln \left (x \right )\right )^{2}}{2}+x^{2}-x -3 \ln \left (1+x \right )\) | \(247\) |
int((((-x^2-x)*ln(x)^3+(400*x+400)*ln(x))*ln((-x*ln(x)^2+400)/ln(x)^2)+(2* x^3-5*x)*ln(x)^3+(-800*x^2-400*x+1600)*ln(x)-800*x-800)/((x^2+x)*ln(x)^3+( -400*x-400)*ln(x)),x,method=_RETURNVERBOSE)
Time = 0.26 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.19 \[ \int \frac {-800-800 x+\left (1600-400 x-800 x^2\right ) \log (x)+\left (-5 x+2 x^3\right ) \log ^3(x)+\left ((400+400 x) \log (x)+\left (-x-x^2\right ) \log ^3(x)\right ) \log \left (\frac {400-x \log ^2(x)}{\log ^2(x)}\right )}{(-400-400 x) \log (x)+\left (x+x^2\right ) \log ^3(x)} \, dx=x^{2} - x \log \left (-\frac {x \log \left (x\right )^{2} - 400}{\log \left (x\right )^{2}}\right ) - x - 3 \, \log \left (x + 1\right ) \]
integrate((((-x^2-x)*log(x)^3+(400*x+400)*log(x))*log((-x*log(x)^2+400)/lo g(x)^2)+(2*x^3-5*x)*log(x)^3+(-800*x^2-400*x+1600)*log(x)-800*x-800)/((x^2 +x)*log(x)^3+(-400*x-400)*log(x)),x, algorithm=\
Time = 0.21 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.04 \[ \int \frac {-800-800 x+\left (1600-400 x-800 x^2\right ) \log (x)+\left (-5 x+2 x^3\right ) \log ^3(x)+\left ((400+400 x) \log (x)+\left (-x-x^2\right ) \log ^3(x)\right ) \log \left (\frac {400-x \log ^2(x)}{\log ^2(x)}\right )}{(-400-400 x) \log (x)+\left (x+x^2\right ) \log ^3(x)} \, dx=x^{2} - x \log {\left (\frac {- x \log {\left (x \right )}^{2} + 400}{\log {\left (x \right )}^{2}} \right )} - x - 3 \log {\left (x + 1 \right )} \]
integrate((((-x**2-x)*ln(x)**3+(400*x+400)*ln(x))*ln((-x*ln(x)**2+400)/ln( x)**2)+(2*x**3-5*x)*ln(x)**3+(-800*x**2-400*x+1600)*ln(x)-800*x-800)/((x** 2+x)*ln(x)**3+(-400*x-400)*ln(x)),x)
Time = 0.21 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.23 \[ \int \frac {-800-800 x+\left (1600-400 x-800 x^2\right ) \log (x)+\left (-5 x+2 x^3\right ) \log ^3(x)+\left ((400+400 x) \log (x)+\left (-x-x^2\right ) \log ^3(x)\right ) \log \left (\frac {400-x \log ^2(x)}{\log ^2(x)}\right )}{(-400-400 x) \log (x)+\left (x+x^2\right ) \log ^3(x)} \, dx=x^{2} - x \log \left (-x \log \left (x\right )^{2} + 400\right ) + 2 \, x \log \left (\log \left (x\right )\right ) - x - 3 \, \log \left (x + 1\right ) \]
integrate((((-x^2-x)*log(x)^3+(400*x+400)*log(x))*log((-x*log(x)^2+400)/lo g(x)^2)+(2*x^3-5*x)*log(x)^3+(-800*x^2-400*x+1600)*log(x)-800*x-800)/((x^2 +x)*log(x)^3+(-400*x-400)*log(x)),x, algorithm=\
Time = 0.33 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.27 \[ \int \frac {-800-800 x+\left (1600-400 x-800 x^2\right ) \log (x)+\left (-5 x+2 x^3\right ) \log ^3(x)+\left ((400+400 x) \log (x)+\left (-x-x^2\right ) \log ^3(x)\right ) \log \left (\frac {400-x \log ^2(x)}{\log ^2(x)}\right )}{(-400-400 x) \log (x)+\left (x+x^2\right ) \log ^3(x)} \, dx=x^{2} - x \log \left (-x \log \left (x\right )^{2} + 400\right ) + x \log \left (\log \left (x\right )^{2}\right ) - x - 3 \, \log \left (x + 1\right ) \]
integrate((((-x^2-x)*log(x)^3+(400*x+400)*log(x))*log((-x*log(x)^2+400)/lo g(x)^2)+(2*x^3-5*x)*log(x)^3+(-800*x^2-400*x+1600)*log(x)-800*x-800)/((x^2 +x)*log(x)^3+(-400*x-400)*log(x)),x, algorithm=\
Time = 9.91 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.19 \[ \int \frac {-800-800 x+\left (1600-400 x-800 x^2\right ) \log (x)+\left (-5 x+2 x^3\right ) \log ^3(x)+\left ((400+400 x) \log (x)+\left (-x-x^2\right ) \log ^3(x)\right ) \log \left (\frac {400-x \log ^2(x)}{\log ^2(x)}\right )}{(-400-400 x) \log (x)+\left (x+x^2\right ) \log ^3(x)} \, dx=x^2-3\,\ln \left (x+1\right )-x\,\ln \left (-\frac {x\,{\ln \left (x\right )}^2-400}{{\ln \left (x\right )}^2}\right )-x \]