Integrand size = 88, antiderivative size = 27 \[ \int \frac {\left (-125-55 x-6 x^2+50 x^4+20 x^5+2 x^6\right ) \log (4)+\left (-5 x+200 x^4+80 x^5+8 x^6\right ) \log (4) \log (x) \log \left (\frac {2}{\log (x)}\right )}{\left (25 x+10 x^2+x^3\right ) \log (x) \log ^2\left (\frac {2}{\log (x)}\right )} \, dx=\frac {\left (-5+2 x^4-\frac {x}{5+x}\right ) \log (4)}{\log \left (\frac {2}{\log (x)}\right )} \] Output:
2*ln(2)/ln(2/ln(x))*(2*x^4-x/(5+x)-5)
Time = 0.19 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.19 \[ \int \frac {\left (-125-55 x-6 x^2+50 x^4+20 x^5+2 x^6\right ) \log (4)+\left (-5 x+200 x^4+80 x^5+8 x^6\right ) \log (4) \log (x) \log \left (\frac {2}{\log (x)}\right )}{\left (25 x+10 x^2+x^3\right ) \log (x) \log ^2\left (\frac {2}{\log (x)}\right )} \, dx=\frac {\left (-25-6 x+10 x^4+2 x^5\right ) \log (4)}{(5+x) \log \left (\frac {2}{\log (x)}\right )} \] Input:
Integrate[((-125 - 55*x - 6*x^2 + 50*x^4 + 20*x^5 + 2*x^6)*Log[4] + (-5*x + 200*x^4 + 80*x^5 + 8*x^6)*Log[4]*Log[x]*Log[2/Log[x]])/((25*x + 10*x^2 + x^3)*Log[x]*Log[2/Log[x]]^2),x]
Output:
((-25 - 6*x + 10*x^4 + 2*x^5)*Log[4])/((5 + x)*Log[2/Log[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.
\(\displaystyle \int \frac {\left (8 x^6+80 x^5+200 x^4-5 x\right ) \log (4) \log (x) \log \left (\frac {2}{\log (x)}\right )+\left (2 x^6+20 x^5+50 x^4-6 x^2-55 x-125\right ) \log (4)}{\left (x^3+10 x^2+25 x\right ) \log (x) \log ^2\left (\frac {2}{\log (x)}\right )} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {\left (8 x^6+80 x^5+200 x^4-5 x\right ) \log (4) \log (x) \log \left (\frac {2}{\log (x)}\right )+\left (2 x^6+20 x^5+50 x^4-6 x^2-55 x-125\right ) \log (4)}{x \left (x^2+10 x+25\right ) \log (x) \log ^2\left (\frac {2}{\log (x)}\right )}dx\) |
\(\Big \downarrow \) 2007 |
\(\displaystyle \int \frac {\left (8 x^6+80 x^5+200 x^4-5 x\right ) \log (4) \log (x) \log \left (\frac {2}{\log (x)}\right )+\left (2 x^6+20 x^5+50 x^4-6 x^2-55 x-125\right ) \log (4)}{x (x+5)^2 \log (x) \log ^2\left (\frac {2}{\log (x)}\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {\left (2 x^5+10 x^4-6 x-25\right ) \log (4)}{x (x+5) \log (x) \log ^2\left (\frac {2}{\log (x)}\right )}+\frac {\left (8 x^5+80 x^4+200 x^3-5\right ) \log (4)}{(x+5)^2 \log \left (\frac {2}{\log (x)}\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \log (4) \int \frac {x^3}{\log (x) \log ^2\left (\frac {2}{\log (x)}\right )}dx+8 \log (4) \int \frac {x^3}{\log \left (\frac {2}{\log (x)}\right )}dx-\log (4) \int \frac {1}{(x+5) \log (x) \log ^2\left (\frac {2}{\log (x)}\right )}dx-5 \log (4) \int \frac {1}{(x+5)^2 \log \left (\frac {2}{\log (x)}\right )}dx-\frac {5 \log (4)}{\log \left (\frac {2}{\log (x)}\right )}\) |
Input:
Int[((-125 - 55*x - 6*x^2 + 50*x^4 + 20*x^5 + 2*x^6)*Log[4] + (-5*x + 200* x^4 + 80*x^5 + 8*x^6)*Log[4]*Log[x]*Log[2/Log[x]])/((25*x + 10*x^2 + x^3)* Log[x]*Log[2/Log[x]]^2),x]
Output:
$Aborted
Time = 2.41 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.37
method | result | size |
risch | \(\frac {4 \ln \left (2\right ) \left (2 x^{5}+10 x^{4}-6 x -25\right )}{\left (5+x \right ) \left (2 \ln \left (2\right )-2 \ln \left (\ln \left (x \right )\right )\right )}\) | \(37\) |
parallelrisch | \(\frac {4 x^{5} \ln \left (2\right )+20 x^{4} \ln \left (2\right )-12 x \ln \left (2\right )-50 \ln \left (2\right )}{\ln \left (\frac {2}{\ln \left (x \right )}\right ) \left (5+x \right )}\) | \(40\) |
Input:
int((2*(8*x^6+80*x^5+200*x^4-5*x)*ln(2)*ln(x)*ln(2/ln(x))+2*(2*x^6+20*x^5+ 50*x^4-6*x^2-55*x-125)*ln(2))/(x^3+10*x^2+25*x)/ln(x)/ln(2/ln(x))^2,x,meth od=_RETURNVERBOSE)
Output:
4*ln(2)*(2*x^5+10*x^4-6*x-25)/(5+x)/(2*ln(2)-2*ln(ln(x)))
Time = 0.09 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.22 \[ \int \frac {\left (-125-55 x-6 x^2+50 x^4+20 x^5+2 x^6\right ) \log (4)+\left (-5 x+200 x^4+80 x^5+8 x^6\right ) \log (4) \log (x) \log \left (\frac {2}{\log (x)}\right )}{\left (25 x+10 x^2+x^3\right ) \log (x) \log ^2\left (\frac {2}{\log (x)}\right )} \, dx=\frac {2 \, {\left (2 \, x^{5} + 10 \, x^{4} - 6 \, x - 25\right )} \log \left (2\right )}{{\left (x + 5\right )} \log \left (\frac {2}{\log \left (x\right )}\right )} \] Input:
integrate((2*(8*x^6+80*x^5+200*x^4-5*x)*log(2)*log(x)*log(2/log(x))+2*(2*x ^6+20*x^5+50*x^4-6*x^2-55*x-125)*log(2))/(x^3+10*x^2+25*x)/log(x)/log(2/lo g(x))^2,x, algorithm="fricas")
Output:
2*(2*x^5 + 10*x^4 - 6*x - 25)*log(2)/((x + 5)*log(2/log(x)))
Time = 0.10 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.37 \[ \int \frac {\left (-125-55 x-6 x^2+50 x^4+20 x^5+2 x^6\right ) \log (4)+\left (-5 x+200 x^4+80 x^5+8 x^6\right ) \log (4) \log (x) \log \left (\frac {2}{\log (x)}\right )}{\left (25 x+10 x^2+x^3\right ) \log (x) \log ^2\left (\frac {2}{\log (x)}\right )} \, dx=\frac {4 x^{5} \log {\left (2 \right )} + 20 x^{4} \log {\left (2 \right )} - 12 x \log {\left (2 \right )} - 50 \log {\left (2 \right )}}{\left (x + 5\right ) \log {\left (\frac {2}{\log {\left (x \right )}} \right )}} \] Input:
integrate((2*(8*x**6+80*x**5+200*x**4-5*x)*ln(2)*ln(x)*ln(2/ln(x))+2*(2*x* *6+20*x**5+50*x**4-6*x**2-55*x-125)*ln(2))/(x**3+10*x**2+25*x)/ln(x)/ln(2/ ln(x))**2,x)
Output:
(4*x**5*log(2) + 20*x**4*log(2) - 12*x*log(2) - 50*log(2))/((x + 5)*log(2/ log(x)))
Time = 0.17 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.67 \[ \int \frac {\left (-125-55 x-6 x^2+50 x^4+20 x^5+2 x^6\right ) \log (4)+\left (-5 x+200 x^4+80 x^5+8 x^6\right ) \log (4) \log (x) \log \left (\frac {2}{\log (x)}\right )}{\left (25 x+10 x^2+x^3\right ) \log (x) \log ^2\left (\frac {2}{\log (x)}\right )} \, dx=\frac {2 \, {\left (2 \, x^{5} \log \left (2\right ) + 10 \, x^{4} \log \left (2\right ) - 6 \, x \log \left (2\right ) - 25 \, \log \left (2\right )\right )}}{x \log \left (2\right ) - {\left (x + 5\right )} \log \left (\log \left (x\right )\right ) + 5 \, \log \left (2\right )} \] Input:
integrate((2*(8*x^6+80*x^5+200*x^4-5*x)*log(2)*log(x)*log(2/log(x))+2*(2*x ^6+20*x^5+50*x^4-6*x^2-55*x-125)*log(2))/(x^3+10*x^2+25*x)/log(x)/log(2/lo g(x))^2,x, algorithm="maxima")
Output:
2*(2*x^5*log(2) + 10*x^4*log(2) - 6*x*log(2) - 25*log(2))/(x*log(2) - (x + 5)*log(log(x)) + 5*log(2))
Time = 0.14 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.78 \[ \int \frac {\left (-125-55 x-6 x^2+50 x^4+20 x^5+2 x^6\right ) \log (4)+\left (-5 x+200 x^4+80 x^5+8 x^6\right ) \log (4) \log (x) \log \left (\frac {2}{\log (x)}\right )}{\left (25 x+10 x^2+x^3\right ) \log (x) \log ^2\left (\frac {2}{\log (x)}\right )} \, dx=\frac {2 \, {\left (2 \, x^{5} \log \left (2\right ) + 10 \, x^{4} \log \left (2\right ) - 6 \, x \log \left (2\right ) - 25 \, \log \left (2\right )\right )}}{x \log \left (2\right ) - x \log \left (\log \left (x\right )\right ) + 5 \, \log \left (2\right ) - 5 \, \log \left (\log \left (x\right )\right )} \] Input:
integrate((2*(8*x^6+80*x^5+200*x^4-5*x)*log(2)*log(x)*log(2/log(x))+2*(2*x ^6+20*x^5+50*x^4-6*x^2-55*x-125)*log(2))/(x^3+10*x^2+25*x)/log(x)/log(2/lo g(x))^2,x, algorithm="giac")
Output:
2*(2*x^5*log(2) + 10*x^4*log(2) - 6*x*log(2) - 25*log(2))/(x*log(2) - x*lo g(log(x)) + 5*log(2) - 5*log(log(x)))
Time = 7.89 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.22 \[ \int \frac {\left (-125-55 x-6 x^2+50 x^4+20 x^5+2 x^6\right ) \log (4)+\left (-5 x+200 x^4+80 x^5+8 x^6\right ) \log (4) \log (x) \log \left (\frac {2}{\log (x)}\right )}{\left (25 x+10 x^2+x^3\right ) \log (x) \log ^2\left (\frac {2}{\log (x)}\right )} \, dx=-\frac {2\,\ln \left (2\right )\,\left (-2\,x^5-10\,x^4+6\,x+25\right )}{\ln \left (\frac {2}{\ln \left (x\right )}\right )\,\left (x+5\right )} \] Input:
int(-(2*log(2)*(55*x + 6*x^2 - 50*x^4 - 20*x^5 - 2*x^6 + 125) - 2*log(2/lo g(x))*log(2)*log(x)*(200*x^4 - 5*x + 80*x^5 + 8*x^6))/(log(2/log(x))^2*log (x)*(25*x + 10*x^2 + x^3)),x)
Output:
-(2*log(2)*(6*x - 10*x^4 - 2*x^5 + 25))/(log(2/log(x))*(x + 5))
Time = 0.17 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.22 \[ \int \frac {\left (-125-55 x-6 x^2+50 x^4+20 x^5+2 x^6\right ) \log (4)+\left (-5 x+200 x^4+80 x^5+8 x^6\right ) \log (4) \log (x) \log \left (\frac {2}{\log (x)}\right )}{\left (25 x+10 x^2+x^3\right ) \log (x) \log ^2\left (\frac {2}{\log (x)}\right )} \, dx=\frac {2 \,\mathrm {log}\left (2\right ) \left (2 x^{5}+10 x^{4}-6 x -25\right )}{\mathrm {log}\left (\frac {2}{\mathrm {log}\left (x \right )}\right ) \left (x +5\right )} \] Input:
int((2*(8*x^6+80*x^5+200*x^4-5*x)*log(2)*log(x)*log(2/log(x))+2*(2*x^6+20* x^5+50*x^4-6*x^2-55*x-125)*log(2))/(x^3+10*x^2+25*x)/log(x)/log(2/log(x))^ 2,x)
Output:
(2*log(2)*(2*x**5 + 10*x**4 - 6*x - 25))/(log(2/log(x))*(x + 5))