Integrand size = 129, antiderivative size = 33 \[ \int \frac {-800 x-40 x^5+\left (800 x+800 x^3-280 x^5-40 x^7\right ) \log (x)+\left (-100-200 x^2+75 x^4+10 x^6\right ) \log ^3(x)+\left (-80 x^5 \log (x)+20 x^4 \log ^3(x)\right ) \log \left (\frac {-4 x^2+x \log ^2(x)}{\log ^2(x)}\right )}{\left (-1600 x^2-160 x^6-4 x^{10}\right ) \log (x)+\left (400 x+40 x^5+x^9\right ) \log ^3(x)} \, dx=\frac {-4-x^2-\log \left (x-\frac {4 x^2}{\log ^2(x)}\right )}{4+\frac {x^4}{5}} \] Output:
(-4-x^2-ln(x-4*x^2/ln(x)^2))/(4+1/5*x^4)
Time = 0.08 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.79 \[ \int \frac {-800 x-40 x^5+\left (800 x+800 x^3-280 x^5-40 x^7\right ) \log (x)+\left (-100-200 x^2+75 x^4+10 x^6\right ) \log ^3(x)+\left (-80 x^5 \log (x)+20 x^4 \log ^3(x)\right ) \log \left (\frac {-4 x^2+x \log ^2(x)}{\log ^2(x)}\right )}{\left (-1600 x^2-160 x^6-4 x^{10}\right ) \log (x)+\left (400 x+40 x^5+x^9\right ) \log ^3(x)} \, dx=-\frac {5 \left (4+x^2+\log \left (x-\frac {4 x^2}{\log ^2(x)}\right )\right )}{20+x^4} \] Input:
Integrate[(-800*x - 40*x^5 + (800*x + 800*x^3 - 280*x^5 - 40*x^7)*Log[x] + (-100 - 200*x^2 + 75*x^4 + 10*x^6)*Log[x]^3 + (-80*x^5*Log[x] + 20*x^4*Lo g[x]^3)*Log[(-4*x^2 + x*Log[x]^2)/Log[x]^2])/((-1600*x^2 - 160*x^6 - 4*x^1 0)*Log[x] + (400*x + 40*x^5 + x^9)*Log[x]^3),x]
Output:
(-5*(4 + x^2 + Log[x - (4*x^2)/Log[x]^2]))/(20 + x^4)
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.
| \(\displaystyle \int \frac {-40 x^5+\left (-40 x^7-280 x^5+800 x^3+800 x\right ) \log (x)+\left (10 x^6+75 x^4-200 x^2-100\right ) \log ^3(x)+\left (20 x^4 \log ^3(x)-80 x^5 \log (x)\right ) \log \left (\frac {x \log ^2(x)-4 x^2}{\log ^2(x)}\right )-800 x}{\left (x^9+40 x^5+400 x\right ) \log ^3(x)+\left (-4 x^{10}-160 x^6-1600 x^2\right ) \log (x)} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {40 x^5-\left (-40 x^7-280 x^5+800 x^3+800 x\right ) \log (x)-\left (10 x^6+75 x^4-200 x^2-100\right ) \log ^3(x)-\left (20 x^4 \log ^3(x)-80 x^5 \log (x)\right ) \log \left (\frac {x \log ^2(x)-4 x^2}{\log ^2(x)}\right )+800 x}{x \left (x^4+20\right )^2 \log (x) \left (4 x-\log ^2(x)\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {40 x^4}{\left (x^4+20\right )^2 \log (x) \left (4 x-\log ^2(x)\right )}+\frac {800}{\left (x^4+20\right )^2 \log (x) \left (4 x-\log ^2(x)\right )}+\frac {40 \left (x^6+7 x^4-20 x^2-20\right )}{\left (x^4+20\right )^2 \left (4 x-\log ^2(x)\right )}-\frac {5 \left (2 x^6+15 x^4-40 x^2-20\right ) \log ^2(x)}{\left (x^4+20\right )^2 x \left (4 x-\log ^2(x)\right )}+\frac {20 x^3 \log \left (x-\frac {4 x^2}{\log ^2(x)}\right )}{\left (x^4+20\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 200 \int \frac {1}{x \left (x^4+20\right )^2 \log (x)}dx+10 \int \frac {x^3}{\left (x^4+20\right )^2 \log (x)}dx+20 \int \frac {x^3 \log \left (x-\frac {4 x^2}{\log ^2(x)}\right )}{\left (x^4+20\right )^2}dx-\left (\frac {1}{4}-\frac {i}{4}\right ) \sqrt [4]{5} \int \frac {1}{\left ((1-i) \sqrt [4]{5}-x\right ) \left (4 x-\log ^2(x)\right )}dx-\left (\frac {1}{4}+\frac {i}{4}\right ) \sqrt [4]{5} \int \frac {1}{\left ((1+i) \sqrt [4]{5}-x\right ) \left (4 x-\log ^2(x)\right )}dx-\left (\frac {1}{4}-\frac {i}{4}\right ) \sqrt [4]{5} \int \frac {1}{\left (x+(1-i) \sqrt [4]{5}\right ) \left (4 x-\log ^2(x)\right )}dx-\left (\frac {1}{4}+\frac {i}{4}\right ) \sqrt [4]{5} \int \frac {1}{\left (x+(1+i) \sqrt [4]{5}\right ) \left (4 x-\log ^2(x)\right )}dx+\frac {1}{8} \int \frac {\log (x)}{\left ((1-i) \sqrt [4]{5}-x\right ) \left (4 x-\log ^2(x)\right )}dx+\frac {1}{8} \int \frac {\log (x)}{\left ((1+i) \sqrt [4]{5}-x\right ) \left (4 x-\log ^2(x)\right )}dx-\frac {1}{8} \int \frac {\log (x)}{\left (x+(1-i) \sqrt [4]{5}\right ) \left (4 x-\log ^2(x)\right )}dx-\frac {1}{8} \int \frac {\log (x)}{\left (x+(1+i) \sqrt [4]{5}\right ) \left (4 x-\log ^2(x)\right )}dx-\int \frac {1}{\log ^2(x)-4 x}dx+\frac {1}{16} \log \left (x^4+20\right )-\frac {x \left (5 x-x^3\right )}{x^4+20}-\frac {1}{4} \log \left (\log ^2(x)-4 x\right )-\frac {\log (x)}{4}\) |
Input:
Int[(-800*x - 40*x^5 + (800*x + 800*x^3 - 280*x^5 - 40*x^7)*Log[x] + (-100 - 200*x^2 + 75*x^4 + 10*x^6)*Log[x]^3 + (-80*x^5*Log[x] + 20*x^4*Log[x]^3 )*Log[(-4*x^2 + x*Log[x]^2)/Log[x]^2])/((-1600*x^2 - 160*x^6 - 4*x^10)*Log [x] + (400*x + 40*x^5 + x^9)*Log[x]^3),x]
Output:
$Aborted
Time = 11.31 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.03
| method | result | size |
| parallelrisch | \(\frac {-160-40 x^{2}-40 \ln \left (\frac {x \left (\ln \left (x \right )^{2}-4 x \right )}{\ln \left (x \right )^{2}}\right )}{8 x^{4}+160}\) | \(34\) |
| risch | \(-\frac {5 \ln \left (-\frac {\ln \left (x \right )^{2}}{4}+x \right )}{x^{4}+20}-\frac {5 \left (-i \pi \,\operatorname {csgn}\left (i \left (\frac {\ln \left (x \right )^{2}}{4}-x \right )\right ) \operatorname {csgn}\left (\frac {i \left (\frac {\ln \left (x \right )^{2}}{4}-x \right )}{\ln \left (x \right )^{2}}\right ) \operatorname {csgn}\left (\frac {i}{\ln \left (x \right )^{2}}\right )+i \pi {\operatorname {csgn}\left (\frac {i \left (\frac {\ln \left (x \right )^{2}}{4}-x \right ) x}{\ln \left (x \right )^{2}}\right )}^{2} \operatorname {csgn}\left (i x \right )+i \pi \operatorname {csgn}\left (i \ln \left (x \right )\right )^{2} \operatorname {csgn}\left (i \ln \left (x \right )^{2}\right )-2 i \pi {\operatorname {csgn}\left (\frac {i \left (\frac {\ln \left (x \right )^{2}}{4}-x \right ) x}{\ln \left (x \right )^{2}}\right )}^{2}+2 i \pi -i \pi \,\operatorname {csgn}\left (\frac {i \left (\frac {\ln \left (x \right )^{2}}{4}-x \right )}{\ln \left (x \right )^{2}}\right ) \operatorname {csgn}\left (\frac {i \left (\frac {\ln \left (x \right )^{2}}{4}-x \right ) x}{\ln \left (x \right )^{2}}\right ) \operatorname {csgn}\left (i x \right )+i \pi {\operatorname {csgn}\left (\frac {i \left (\frac {\ln \left (x \right )^{2}}{4}-x \right )}{\ln \left (x \right )^{2}}\right )}^{3}+i \pi {\operatorname {csgn}\left (\frac {i \left (\frac {\ln \left (x \right )^{2}}{4}-x \right )}{\ln \left (x \right )^{2}}\right )}^{2} \operatorname {csgn}\left (\frac {i}{\ln \left (x \right )^{2}}\right )-i \pi {\operatorname {csgn}\left (\frac {i \left (\frac {\ln \left (x \right )^{2}}{4}-x \right ) x}{\ln \left (x \right )^{2}}\right )}^{3}-2 i \pi \,\operatorname {csgn}\left (i \ln \left (x \right )\right ) \operatorname {csgn}\left (i \ln \left (x \right )^{2}\right )^{2}-i \pi \,\operatorname {csgn}\left (i \left (\frac {\ln \left (x \right )^{2}}{4}-x \right )\right ) {\operatorname {csgn}\left (\frac {i \left (\frac {\ln \left (x \right )^{2}}{4}-x \right )}{\ln \left (x \right )^{2}}\right )}^{2}+i \pi \operatorname {csgn}\left (i \ln \left (x \right )^{2}\right )^{3}-i \pi \,\operatorname {csgn}\left (\frac {i \left (\frac {\ln \left (x \right )^{2}}{4}-x \right )}{\ln \left (x \right )^{2}}\right ) {\operatorname {csgn}\left (\frac {i \left (\frac {\ln \left (x \right )^{2}}{4}-x \right ) x}{\ln \left (x \right )^{2}}\right )}^{2}+8+2 x^{2}+4 \ln \left (2\right )+2 \ln \left (x \right )-4 \ln \left (\ln \left (x \right )\right )\right )}{2 \left (x^{4}+20\right )}\) | \(414\) |
Input:
int(((20*x^4*ln(x)^3-80*x^5*ln(x))*ln((x*ln(x)^2-4*x^2)/ln(x)^2)+(10*x^6+7 5*x^4-200*x^2-100)*ln(x)^3+(-40*x^7-280*x^5+800*x^3+800*x)*ln(x)-40*x^5-80 0*x)/((x^9+40*x^5+400*x)*ln(x)^3+(-4*x^10-160*x^6-1600*x^2)*ln(x)),x,metho d=_RETURNVERBOSE)
Output:
1/8*(-160-40*x^2-40*ln(x*(ln(x)^2-4*x)/ln(x)^2))/(x^4+20)
Time = 0.09 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.97 \[ \int \frac {-800 x-40 x^5+\left (800 x+800 x^3-280 x^5-40 x^7\right ) \log (x)+\left (-100-200 x^2+75 x^4+10 x^6\right ) \log ^3(x)+\left (-80 x^5 \log (x)+20 x^4 \log ^3(x)\right ) \log \left (\frac {-4 x^2+x \log ^2(x)}{\log ^2(x)}\right )}{\left (-1600 x^2-160 x^6-4 x^{10}\right ) \log (x)+\left (400 x+40 x^5+x^9\right ) \log ^3(x)} \, dx=-\frac {5 \, {\left (x^{2} + \log \left (\frac {x \log \left (x\right )^{2} - 4 \, x^{2}}{\log \left (x\right )^{2}}\right ) + 4\right )}}{x^{4} + 20} \] Input:
integrate(((20*x^4*log(x)^3-80*x^5*log(x))*log((x*log(x)^2-4*x^2)/log(x)^2 )+(10*x^6+75*x^4-200*x^2-100)*log(x)^3+(-40*x^7-280*x^5+800*x^3+800*x)*log (x)-40*x^5-800*x)/((x^9+40*x^5+400*x)*log(x)^3+(-4*x^10-160*x^6-1600*x^2)* log(x)),x, algorithm="fricas")
Output:
-5*(x^2 + log((x*log(x)^2 - 4*x^2)/log(x)^2) + 4)/(x^4 + 20)
Time = 0.25 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.12 \[ \int \frac {-800 x-40 x^5+\left (800 x+800 x^3-280 x^5-40 x^7\right ) \log (x)+\left (-100-200 x^2+75 x^4+10 x^6\right ) \log ^3(x)+\left (-80 x^5 \log (x)+20 x^4 \log ^3(x)\right ) \log \left (\frac {-4 x^2+x \log ^2(x)}{\log ^2(x)}\right )}{\left (-1600 x^2-160 x^6-4 x^{10}\right ) \log (x)+\left (400 x+40 x^5+x^9\right ) \log ^3(x)} \, dx=\frac {- 5 x^{2} - 20}{x^{4} + 20} - \frac {5 \log {\left (\frac {- 4 x^{2} + x \log {\left (x \right )}^{2}}{\log {\left (x \right )}^{2}} \right )}}{x^{4} + 20} \] Input:
integrate(((20*x**4*ln(x)**3-80*x**5*ln(x))*ln((x*ln(x)**2-4*x**2)/ln(x)** 2)+(10*x**6+75*x**4-200*x**2-100)*ln(x)**3+(-40*x**7-280*x**5+800*x**3+800 *x)*ln(x)-40*x**5-800*x)/((x**9+40*x**5+400*x)*ln(x)**3+(-4*x**10-160*x**6 -1600*x**2)*ln(x)),x)
Output:
(-5*x**2 - 20)/(x**4 + 20) - 5*log((-4*x**2 + x*log(x)**2)/log(x)**2)/(x** 4 + 20)
Time = 0.14 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.91 \[ \int \frac {-800 x-40 x^5+\left (800 x+800 x^3-280 x^5-40 x^7\right ) \log (x)+\left (-100-200 x^2+75 x^4+10 x^6\right ) \log ^3(x)+\left (-80 x^5 \log (x)+20 x^4 \log ^3(x)\right ) \log \left (\frac {-4 x^2+x \log ^2(x)}{\log ^2(x)}\right )}{\left (-1600 x^2-160 x^6-4 x^{10}\right ) \log (x)+\left (400 x+40 x^5+x^9\right ) \log ^3(x)} \, dx=-\frac {5 \, {\left (x^{2} + \log \left (\log \left (x\right )^{2} - 4 \, x\right ) + \log \left (x\right ) - 2 \, \log \left (\log \left (x\right )\right ) + 4\right )}}{x^{4} + 20} \] Input:
integrate(((20*x^4*log(x)^3-80*x^5*log(x))*log((x*log(x)^2-4*x^2)/log(x)^2 )+(10*x^6+75*x^4-200*x^2-100)*log(x)^3+(-40*x^7-280*x^5+800*x^3+800*x)*log (x)-40*x^5-800*x)/((x^9+40*x^5+400*x)*log(x)^3+(-4*x^10-160*x^6-1600*x^2)* log(x)),x, algorithm="maxima")
Output:
-5*(x^2 + log(log(x)^2 - 4*x) + log(x) - 2*log(log(x)) + 4)/(x^4 + 20)
Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (26) = 52\).
Time = 0.18 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.76 \[ \int \frac {-800 x-40 x^5+\left (800 x+800 x^3-280 x^5-40 x^7\right ) \log (x)+\left (-100-200 x^2+75 x^4+10 x^6\right ) \log ^3(x)+\left (-80 x^5 \log (x)+20 x^4 \log ^3(x)\right ) \log \left (\frac {-4 x^2+x \log ^2(x)}{\log ^2(x)}\right )}{\left (-1600 x^2-160 x^6-4 x^{10}\right ) \log (x)+\left (400 x+40 x^5+x^9\right ) \log ^3(x)} \, dx=-\frac {5 \, {\left (x^{2} + 4\right )}}{x^{4} + 20} - \frac {5 \, \log \left (\log \left (x\right )^{2} - 4 \, x\right )}{x^{4} + 20} + \frac {5 \, \log \left (\log \left (x\right )^{2}\right )}{x^{4} + 20} - \frac {5 \, \log \left (x\right )}{x^{4} + 20} \] Input:
integrate(((20*x^4*log(x)^3-80*x^5*log(x))*log((x*log(x)^2-4*x^2)/log(x)^2 )+(10*x^6+75*x^4-200*x^2-100)*log(x)^3+(-40*x^7-280*x^5+800*x^3+800*x)*log (x)-40*x^5-800*x)/((x^9+40*x^5+400*x)*log(x)^3+(-4*x^10-160*x^6-1600*x^2)* log(x)),x, algorithm="giac")
Output:
-5*(x^2 + 4)/(x^4 + 20) - 5*log(log(x)^2 - 4*x)/(x^4 + 20) + 5*log(log(x)^ 2)/(x^4 + 20) - 5*log(x)/(x^4 + 20)
Time = 2.71 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.33 \[ \int \frac {-800 x-40 x^5+\left (800 x+800 x^3-280 x^5-40 x^7\right ) \log (x)+\left (-100-200 x^2+75 x^4+10 x^6\right ) \log ^3(x)+\left (-80 x^5 \log (x)+20 x^4 \log ^3(x)\right ) \log \left (\frac {-4 x^2+x \log ^2(x)}{\log ^2(x)}\right )}{\left (-1600 x^2-160 x^6-4 x^{10}\right ) \log (x)+\left (400 x+40 x^5+x^9\right ) \log ^3(x)} \, dx=-\frac {5\,x^2+20}{x^4+20}-\frac {5\,\ln \left (\frac {x\,{\ln \left (x\right )}^2-4\,x^2}{{\ln \left (x\right )}^2}\right )}{x^4+20} \] Input:
int((800*x + log((x*log(x)^2 - 4*x^2)/log(x)^2)*(80*x^5*log(x) - 20*x^4*lo g(x)^3) - log(x)*(800*x + 800*x^3 - 280*x^5 - 40*x^7) + log(x)^3*(200*x^2 - 75*x^4 - 10*x^6 + 100) + 40*x^5)/(log(x)*(1600*x^2 + 160*x^6 + 4*x^10) - log(x)^3*(400*x + 40*x^5 + x^9)),x)
Output:
- (5*x^2 + 20)/(x^4 + 20) - (5*log((x*log(x)^2 - 4*x^2)/log(x)^2))/(x^4 + 20)
Time = 0.20 (sec) , antiderivative size = 92, normalized size of antiderivative = 2.79 \[ \int \frac {-800 x-40 x^5+\left (800 x+800 x^3-280 x^5-40 x^7\right ) \log (x)+\left (-100-200 x^2+75 x^4+10 x^6\right ) \log ^3(x)+\left (-80 x^5 \log (x)+20 x^4 \log ^3(x)\right ) \log \left (\frac {-4 x^2+x \log ^2(x)}{\log ^2(x)}\right )}{\left (-1600 x^2-160 x^6-4 x^{10}\right ) \log (x)+\left (400 x+40 x^5+x^9\right ) \log ^3(x)} \, dx=\frac {2 \,\mathrm {log}\left (\mathrm {log}\left (x \right )\right ) x^{4}+40 \,\mathrm {log}\left (\mathrm {log}\left (x \right )\right )-\mathrm {log}\left (\mathrm {log}\left (x \right )^{2}-4 x \right ) x^{4}-20 \,\mathrm {log}\left (\mathrm {log}\left (x \right )^{2}-4 x \right )+\mathrm {log}\left (\frac {\mathrm {log}\left (x \right )^{2} x -4 x^{2}}{\mathrm {log}\left (x \right )^{2}}\right ) x^{4}-\mathrm {log}\left (x \right ) x^{4}-20 \,\mathrm {log}\left (x \right )+4 x^{4}-20 x^{2}}{4 x^{4}+80} \] Input:
int(((20*x^4*log(x)^3-80*x^5*log(x))*log((x*log(x)^2-4*x^2)/log(x)^2)+(10* x^6+75*x^4-200*x^2-100)*log(x)^3+(-40*x^7-280*x^5+800*x^3+800*x)*log(x)-40 *x^5-800*x)/((x^9+40*x^5+400*x)*log(x)^3+(-4*x^10-160*x^6-1600*x^2)*log(x) ),x)
Output:
(2*log(log(x))*x**4 + 40*log(log(x)) - log(log(x)**2 - 4*x)*x**4 - 20*log( log(x)**2 - 4*x) + log((log(x)**2*x - 4*x**2)/log(x)**2)*x**4 - log(x)*x** 4 - 20*log(x) + 4*x**4 - 20*x**2)/(4*(x**4 + 20))