Integrand size = 74, antiderivative size = 23 \[ \int \frac {-10 x-10 x \log \left (\frac {100}{2401 x^2}\right ) \log \left (5 \log \left (\frac {100}{2401 x^2}\right )\right )+30 x \log \left (\frac {100}{2401 x^2}\right ) \log ^2\left (5 \log \left (\frac {100}{2401 x^2}\right )\right )}{\log \left (\frac {100}{2401 x^2}\right ) \log ^2\left (5 \log \left (\frac {100}{2401 x^2}\right )\right )} \, dx=-5+x^2 \left (15-\frac {5}{\log \left (5 \log \left (\frac {100}{2401 x^2}\right )\right )}\right ) \] Output:
(15-5/ln(5*ln(100/2401/x^2)))*x^2-5
Time = 0.18 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04 \[ \int \frac {-10 x-10 x \log \left (\frac {100}{2401 x^2}\right ) \log \left (5 \log \left (\frac {100}{2401 x^2}\right )\right )+30 x \log \left (\frac {100}{2401 x^2}\right ) \log ^2\left (5 \log \left (\frac {100}{2401 x^2}\right )\right )}{\log \left (\frac {100}{2401 x^2}\right ) \log ^2\left (5 \log \left (\frac {100}{2401 x^2}\right )\right )} \, dx=15 x^2-\frac {5 x^2}{\log \left (5 \log \left (\frac {100}{2401 x^2}\right )\right )} \] Input:
Integrate[(-10*x - 10*x*Log[100/(2401*x^2)]*Log[5*Log[100/(2401*x^2)]] + 3 0*x*Log[100/(2401*x^2)]*Log[5*Log[100/(2401*x^2)]]^2)/(Log[100/(2401*x^2)] *Log[5*Log[100/(2401*x^2)]]^2),x]
Output:
15*x^2 - (5*x^2)/Log[5*Log[100/(2401*x^2)]]
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.
\(\displaystyle \int \frac {30 x \log \left (\frac {100}{2401 x^2}\right ) \log ^2\left (5 \log \left (\frac {100}{2401 x^2}\right )\right )-10 x \log \left (\frac {100}{2401 x^2}\right ) \log \left (5 \log \left (\frac {100}{2401 x^2}\right )\right )-10 x}{\log \left (\frac {100}{2401 x^2}\right ) \log ^2\left (5 \log \left (\frac {100}{2401 x^2}\right )\right )} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {10 x \left (3 \log \left (\frac {100}{2401 x^2}\right ) \log ^2\left (5 \log \left (\frac {100}{2401 x^2}\right )\right )-\log \left (\frac {100}{2401 x^2}\right ) \log \left (5 \log \left (\frac {100}{2401 x^2}\right )\right )-1\right )}{\log \left (\frac {100}{2401 x^2}\right ) \log ^2\left (5 \log \left (\frac {100}{2401 x^2}\right )\right )}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 10 \int -\frac {x \left (-3 \log \left (\frac {100}{2401 x^2}\right ) \log ^2\left (5 \log \left (\frac {100}{2401 x^2}\right )\right )+\log \left (\frac {100}{2401 x^2}\right ) \log \left (5 \log \left (\frac {100}{2401 x^2}\right )\right )+1\right )}{\log \left (\frac {100}{2401 x^2}\right ) \log ^2\left (5 \log \left (\frac {100}{2401 x^2}\right )\right )}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -10 \int \frac {x \left (-3 \log \left (\frac {100}{2401 x^2}\right ) \log ^2\left (5 \log \left (\frac {100}{2401 x^2}\right )\right )+\log \left (\frac {100}{2401 x^2}\right ) \log \left (5 \log \left (\frac {100}{2401 x^2}\right )\right )+1\right )}{\log \left (\frac {100}{2401 x^2}\right ) \log ^2\left (5 \log \left (\frac {100}{2401 x^2}\right )\right )}dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle -10 \int x \left (\frac {1}{\log \left (5 \log \left (\frac {100}{2401 x^2}\right )\right )}-3+\frac {1}{\log ^2\left (5 \log \left (\frac {100}{2401 x^2}\right )\right ) \log \left (\frac {100}{2401 x^2}\right )}\right )dx\) |
\(\Big \downarrow \) 2010 |
\(\displaystyle -10 \int \left (\frac {x}{\log \left (5 \log \left (\frac {100}{2401 x^2}\right )\right )}+\frac {x}{\log \left (\frac {100}{2401 x^2}\right ) \log ^2\left (5 \log \left (\frac {100}{2401 x^2}\right )\right )}-3 x\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -10 \left (\frac {1}{2} \text {Subst}\left (\int \frac {1}{\log \left (\frac {100}{2401 x}\right ) \log ^2\left (5 \log \left (\frac {100}{2401 x}\right )\right )}dx,x,x^2\right )+\frac {1}{2} \text {Subst}\left (\int \frac {1}{\log \left (5 \log \left (\frac {100}{2401 x}\right )\right )}dx,x,x^2\right )-\frac {3 x^2}{2}\right )\) |
Input:
Int[(-10*x - 10*x*Log[100/(2401*x^2)]*Log[5*Log[100/(2401*x^2)]] + 30*x*Lo g[100/(2401*x^2)]*Log[5*Log[100/(2401*x^2)]]^2)/(Log[100/(2401*x^2)]*Log[5 *Log[100/(2401*x^2)]]^2),x]
Output:
$Aborted
Time = 1.30 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.43
method | result | size |
norman | \(\frac {-5 x^{2}+15 x^{2} \ln \left (5 \ln \left (\frac {100}{2401 x^{2}}\right )\right )}{\ln \left (5 \ln \left (\frac {100}{2401 x^{2}}\right )\right )}\) | \(33\) |
parallelrisch | \(\frac {-5 x^{2}+15 x^{2} \ln \left (5 \ln \left (\frac {100}{2401 x^{2}}\right )\right )}{\ln \left (5 \ln \left (\frac {100}{2401 x^{2}}\right )\right )}\) | \(33\) |
Input:
int((30*x*ln(100/2401/x^2)*ln(5*ln(100/2401/x^2))^2-10*x*ln(100/2401/x^2)* ln(5*ln(100/2401/x^2))-10*x)/ln(100/2401/x^2)/ln(5*ln(100/2401/x^2))^2,x,m ethod=_RETURNVERBOSE)
Output:
(-5*x^2+15*x^2*ln(5*ln(100/2401/x^2)))/ln(5*ln(100/2401/x^2))
Time = 0.10 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.43 \[ \int \frac {-10 x-10 x \log \left (\frac {100}{2401 x^2}\right ) \log \left (5 \log \left (\frac {100}{2401 x^2}\right )\right )+30 x \log \left (\frac {100}{2401 x^2}\right ) \log ^2\left (5 \log \left (\frac {100}{2401 x^2}\right )\right )}{\log \left (\frac {100}{2401 x^2}\right ) \log ^2\left (5 \log \left (\frac {100}{2401 x^2}\right )\right )} \, dx=\frac {5 \, {\left (3 \, x^{2} \log \left (5 \, \log \left (\frac {100}{2401 \, x^{2}}\right )\right ) - x^{2}\right )}}{\log \left (5 \, \log \left (\frac {100}{2401 \, x^{2}}\right )\right )} \] Input:
integrate((30*x*log(100/2401/x^2)*log(5*log(100/2401/x^2))^2-10*x*log(100/ 2401/x^2)*log(5*log(100/2401/x^2))-10*x)/log(100/2401/x^2)/log(5*log(100/2 401/x^2))^2,x, algorithm="fricas")
Output:
5*(3*x^2*log(5*log(100/2401/x^2)) - x^2)/log(5*log(100/2401/x^2))
Time = 0.08 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int \frac {-10 x-10 x \log \left (\frac {100}{2401 x^2}\right ) \log \left (5 \log \left (\frac {100}{2401 x^2}\right )\right )+30 x \log \left (\frac {100}{2401 x^2}\right ) \log ^2\left (5 \log \left (\frac {100}{2401 x^2}\right )\right )}{\log \left (\frac {100}{2401 x^2}\right ) \log ^2\left (5 \log \left (\frac {100}{2401 x^2}\right )\right )} \, dx=15 x^{2} - \frac {5 x^{2}}{\log {\left (5 \log {\left (\frac {100}{2401 x^{2}} \right )} \right )}} \] Input:
integrate((30*x*ln(100/2401/x**2)*ln(5*ln(100/2401/x**2))**2-10*x*ln(100/2 401/x**2)*ln(5*ln(100/2401/x**2))-10*x)/ln(100/2401/x**2)/ln(5*ln(100/2401 /x**2))**2,x)
Output:
15*x**2 - 5*x**2/log(5*log(100/(2401*x**2)))
Result contains complex when optimal does not.
Time = 0.16 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.87 \[ \int \frac {-10 x-10 x \log \left (\frac {100}{2401 x^2}\right ) \log \left (5 \log \left (\frac {100}{2401 x^2}\right )\right )+30 x \log \left (\frac {100}{2401 x^2}\right ) \log ^2\left (5 \log \left (\frac {100}{2401 x^2}\right )\right )}{\log \left (\frac {100}{2401 x^2}\right ) \log ^2\left (5 \log \left (\frac {100}{2401 x^2}\right )\right )} \, dx=15 \, x^{2} - \frac {10 \, x^{2}}{2 i \, \pi + 2 \, \log \left (5\right ) + 2 \, \log \left (2\right ) + 2 \, \log \left (2 \, \log \left (7\right ) - \log \left (5\right ) - \log \left (2\right ) + \log \left (x\right )\right )} \] Input:
integrate((30*x*log(100/2401/x^2)*log(5*log(100/2401/x^2))^2-10*x*log(100/ 2401/x^2)*log(5*log(100/2401/x^2))-10*x)/log(100/2401/x^2)/log(5*log(100/2 401/x^2))^2,x, algorithm="maxima")
Output:
15*x^2 - 10*x^2/(2*I*pi + 2*log(5) + 2*log(2) + 2*log(2*log(7) - log(5) - log(2) + log(x)))
Leaf count of result is larger than twice the leaf count of optimal. 79 vs. \(2 (20) = 40\).
Time = 0.29 (sec) , antiderivative size = 79, normalized size of antiderivative = 3.43 \[ \int \frac {-10 x-10 x \log \left (\frac {100}{2401 x^2}\right ) \log \left (5 \log \left (\frac {100}{2401 x^2}\right )\right )+30 x \log \left (\frac {100}{2401 x^2}\right ) \log ^2\left (5 \log \left (\frac {100}{2401 x^2}\right )\right )}{\log \left (\frac {100}{2401 x^2}\right ) \log ^2\left (5 \log \left (\frac {100}{2401 x^2}\right )\right )} \, dx=15 \, x^{2} - \frac {5 \, x^{2} \log \left (\frac {100}{2401 \, x^{2}}\right )}{2 \, \log \left (5\right )^{2} + 2 \, \log \left (5\right ) \log \left (2\right ) - \log \left (5\right ) \log \left (2401 \, x^{2}\right ) + 2 \, \log \left (5\right ) \log \left (\log \left (\frac {100}{2401 \, x^{2}}\right )\right ) + 2 \, \log \left (2\right ) \log \left (\log \left (\frac {100}{2401 \, x^{2}}\right )\right ) - \log \left (2401 \, x^{2}\right ) \log \left (\log \left (\frac {100}{2401 \, x^{2}}\right )\right )} \] Input:
integrate((30*x*log(100/2401/x^2)*log(5*log(100/2401/x^2))^2-10*x*log(100/ 2401/x^2)*log(5*log(100/2401/x^2))-10*x)/log(100/2401/x^2)/log(5*log(100/2 401/x^2))^2,x, algorithm="giac")
Output:
15*x^2 - 5*x^2*log(100/2401/x^2)/(2*log(5)^2 + 2*log(5)*log(2) - log(5)*lo g(2401*x^2) + 2*log(5)*log(log(100/2401/x^2)) + 2*log(2)*log(log(100/2401/ x^2)) - log(2401*x^2)*log(log(100/2401/x^2)))
Time = 7.90 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {-10 x-10 x \log \left (\frac {100}{2401 x^2}\right ) \log \left (5 \log \left (\frac {100}{2401 x^2}\right )\right )+30 x \log \left (\frac {100}{2401 x^2}\right ) \log ^2\left (5 \log \left (\frac {100}{2401 x^2}\right )\right )}{\log \left (\frac {100}{2401 x^2}\right ) \log ^2\left (5 \log \left (\frac {100}{2401 x^2}\right )\right )} \, dx=15\,x^2-\frac {5\,x^2}{\ln \left (5\,\ln \left (\frac {100}{2401\,x^2}\right )\right )} \] Input:
int(-(10*x - 30*x*log(5*log(100/(2401*x^2)))^2*log(100/(2401*x^2)) + 10*x* log(5*log(100/(2401*x^2)))*log(100/(2401*x^2)))/(log(5*log(100/(2401*x^2)) )^2*log(100/(2401*x^2))),x)
Output:
15*x^2 - (5*x^2)/log(5*log(100/(2401*x^2)))
Time = 0.18 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.26 \[ \int \frac {-10 x-10 x \log \left (\frac {100}{2401 x^2}\right ) \log \left (5 \log \left (\frac {100}{2401 x^2}\right )\right )+30 x \log \left (\frac {100}{2401 x^2}\right ) \log ^2\left (5 \log \left (\frac {100}{2401 x^2}\right )\right )}{\log \left (\frac {100}{2401 x^2}\right ) \log ^2\left (5 \log \left (\frac {100}{2401 x^2}\right )\right )} \, dx=\frac {5 x^{2} \left (3 \,\mathrm {log}\left (5 \,\mathrm {log}\left (\frac {100}{2401 x^{2}}\right )\right )-1\right )}{\mathrm {log}\left (5 \,\mathrm {log}\left (\frac {100}{2401 x^{2}}\right )\right )} \] Input:
int((30*x*log(100/2401/x^2)*log(5*log(100/2401/x^2))^2-10*x*log(100/2401/x ^2)*log(5*log(100/2401/x^2))-10*x)/log(100/2401/x^2)/log(5*log(100/2401/x^ 2))^2,x)
Output:
(5*x**2*(3*log(5*log(100/(2401*x**2))) - 1))/log(5*log(100/(2401*x**2)))