Integrand size = 50, antiderivative size = 18 \[ \int \frac {16 \log (x)+(-2+2 \log (x)) \log \left (4 x^2\right ) \log \left (\log ^2\left (4 x^2\right )\right )}{x^2 \log \left (4 x^2\right ) \log ^3\left (\log ^2\left (4 x^2\right )\right )} \, dx=-\frac {2 \log (x)}{x \log ^2\left (\log ^2\left (4 x^2\right )\right )} \] Output:
-2/x/ln(ln(4*x^2)^2)^2*ln(x)
Time = 0.15 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {16 \log (x)+(-2+2 \log (x)) \log \left (4 x^2\right ) \log \left (\log ^2\left (4 x^2\right )\right )}{x^2 \log \left (4 x^2\right ) \log ^3\left (\log ^2\left (4 x^2\right )\right )} \, dx=-\frac {2 \log (x)}{x \log ^2\left (\log ^2\left (4 x^2\right )\right )} \] Input:
Integrate[(16*Log[x] + (-2 + 2*Log[x])*Log[4*x^2]*Log[Log[4*x^2]^2])/(x^2* Log[4*x^2]*Log[Log[4*x^2]^2]^3),x]
Output:
(-2*Log[x])/(x*Log[Log[4*x^2]^2]^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.
\(\displaystyle \int \frac {(2 \log (x)-2) \log \left (4 x^2\right ) \log \left (\log ^2\left (4 x^2\right )\right )+16 \log (x)}{x^2 \log \left (4 x^2\right ) \log ^3\left (\log ^2\left (4 x^2\right )\right )} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {2 (\log (x)-1)}{x^2 \log ^2\left (\log ^2\left (4 x^2\right )\right )}+\frac {16 \log (x)}{x^2 \log \left (4 x^2\right ) \log ^3\left (\log ^2\left (4 x^2\right )\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -2 \int \frac {1}{x^2 \log ^2\left (\log ^2\left (4 x^2\right )\right )}dx+2 \int \frac {\log (x)}{x^2 \log ^2\left (\log ^2\left (4 x^2\right )\right )}dx+16 \int \frac {\log (x)}{x^2 \log \left (4 x^2\right ) \log ^3\left (\log ^2\left (4 x^2\right )\right )}dx\) |
Input:
Int[(16*Log[x] + (-2 + 2*Log[x])*Log[4*x^2]*Log[Log[4*x^2]^2])/(x^2*Log[4* x^2]*Log[Log[4*x^2]^2]^3),x]
Output:
$Aborted
Time = 8.48 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06
method | result | size |
parallelrisch | \(-\frac {2 \ln \left (x \right )}{x {\ln \left (\ln \left (4 x^{2}\right )^{2}\right )}^{2}}\) | \(19\) |
risch | \(\frac {8 \ln \left (x \right )}{x {\left (2 \pi {\operatorname {csgn}\left (i \left (\pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )-2 \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}+\pi \operatorname {csgn}\left (i x^{2}\right )^{3}+4 i \ln \left (2\right )+4 i \ln \left (x \right )\right )^{2}\right )}^{2}+\pi {\operatorname {csgn}\left (-i \pi \,\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x \right )^{2}+2 i \pi \operatorname {csgn}\left (i x^{2}\right )^{2} \operatorname {csgn}\left (i x \right )-i \pi \operatorname {csgn}\left (i x^{2}\right )^{3}+4 \ln \left (x \right )+4 \ln \left (2\right )\right )}^{2} \operatorname {csgn}\left (i \left (\pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )-2 \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}+\pi \operatorname {csgn}\left (i x^{2}\right )^{3}+4 i \ln \left (2\right )+4 i \ln \left (x \right )\right )^{2}\right )+2 \pi \,\operatorname {csgn}\left (-i \pi \,\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x \right )^{2}+2 i \pi \operatorname {csgn}\left (i x^{2}\right )^{2} \operatorname {csgn}\left (i x \right )-i \pi \operatorname {csgn}\left (i x^{2}\right )^{3}+4 \ln \left (x \right )+4 \ln \left (2\right )\right ) {\operatorname {csgn}\left (i \left (\pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )-2 \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}+\pi \operatorname {csgn}\left (i x^{2}\right )^{3}+4 i \ln \left (2\right )+4 i \ln \left (x \right )\right )^{2}\right )}^{2}-\pi {\operatorname {csgn}\left (i \left (\pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )-2 \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}+\pi \operatorname {csgn}\left (i x^{2}\right )^{3}+4 i \ln \left (2\right )+4 i \ln \left (x \right )\right )^{2}\right )}^{3}-2 \pi -4 i \ln \left (2\right )+4 i \ln \left (\pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )-2 \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}+\pi \operatorname {csgn}\left (i x^{2}\right )^{3}+4 i \ln \left (2\right )+4 i \ln \left (x \right )\right )\right )}^{2}}\) | \(459\) |
Input:
int(((2*ln(x)-2)*ln(4*x^2)*ln(ln(4*x^2)^2)+16*ln(x))/x^2/ln(4*x^2)/ln(ln(4 *x^2)^2)^3,x,method=_RETURNVERBOSE)
Output:
-2/x/ln(ln(4*x^2)^2)^2*ln(x)
Time = 0.09 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.61 \[ \int \frac {16 \log (x)+(-2+2 \log (x)) \log \left (4 x^2\right ) \log \left (\log ^2\left (4 x^2\right )\right )}{x^2 \log \left (4 x^2\right ) \log ^3\left (\log ^2\left (4 x^2\right )\right )} \, dx=-\frac {2 \, \log \left (x\right )}{x \log \left (4 \, \log \left (2\right )^{2} + 8 \, \log \left (2\right ) \log \left (x\right ) + 4 \, \log \left (x\right )^{2}\right )^{2}} \] Input:
integrate(((2*log(x)-2)*log(4*x^2)*log(log(4*x^2)^2)+16*log(x))/x^2/log(4* x^2)/log(log(4*x^2)^2)^3,x, algorithm="fricas")
Output:
-2*log(x)/(x*log(4*log(2)^2 + 8*log(2)*log(x) + 4*log(x)^2)^2)
Time = 0.08 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {16 \log (x)+(-2+2 \log (x)) \log \left (4 x^2\right ) \log \left (\log ^2\left (4 x^2\right )\right )}{x^2 \log \left (4 x^2\right ) \log ^3\left (\log ^2\left (4 x^2\right )\right )} \, dx=- \frac {2 \log {\left (x \right )}}{x \log {\left (\left (2 \log {\left (x \right )} + \log {\left (4 \right )}\right )^{2} \right )}^{2}} \] Input:
integrate(((2*ln(x)-2)*ln(4*x**2)*ln(ln(4*x**2)**2)+16*ln(x))/x**2/ln(4*x* *2)/ln(ln(4*x**2)**2)**3,x)
Output:
-2*log(x)/(x*log((2*log(x) + log(4))**2)**2)
Time = 0.18 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.89 \[ \int \frac {16 \log (x)+(-2+2 \log (x)) \log \left (4 x^2\right ) \log \left (\log ^2\left (4 x^2\right )\right )}{x^2 \log \left (4 x^2\right ) \log ^3\left (\log ^2\left (4 x^2\right )\right )} \, dx=-\frac {\log \left (x\right )}{2 \, {\left (x \log \left (2\right )^{2} + 2 \, x \log \left (2\right ) \log \left (\log \left (2\right ) + \log \left (x\right )\right ) + x \log \left (\log \left (2\right ) + \log \left (x\right )\right )^{2}\right )}} \] Input:
integrate(((2*log(x)-2)*log(4*x^2)*log(log(4*x^2)^2)+16*log(x))/x^2/log(4* x^2)/log(log(4*x^2)^2)^3,x, algorithm="maxima")
Output:
-1/2*log(x)/(x*log(2)^2 + 2*x*log(2)*log(log(2) + log(x)) + x*log(log(2) + log(x))^2)
Leaf count of result is larger than twice the leaf count of optimal. 257 vs. \(2 (18) = 36\).
Time = 0.47 (sec) , antiderivative size = 257, normalized size of antiderivative = 14.28 \[ \int \frac {16 \log (x)+(-2+2 \log (x)) \log \left (4 x^2\right ) \log \left (\log ^2\left (4 x^2\right )\right )}{x^2 \log \left (4 x^2\right ) \log ^3\left (\log ^2\left (4 x^2\right )\right )} \, dx=-\frac {2 \, \log \left (2\right )^{2} \log \left (4 \, x^{2}\right ) \log \left (x\right ) \log \left (\log \left (4 \, x^{2}\right )^{2}\right ) - \log \left (2\right ) \log \left (4 \, x^{2}\right )^{2} \log \left (x\right ) \log \left (\log \left (4 \, x^{2}\right )^{2}\right ) + 4 \, \log \left (2\right ) \log \left (4 \, x^{2}\right ) \log \left (x\right )^{2} \log \left (\log \left (4 \, x^{2}\right )^{2}\right ) - \log \left (4 \, x^{2}\right )^{2} \log \left (x\right )^{2} \log \left (\log \left (4 \, x^{2}\right )^{2}\right ) + 2 \, \log \left (4 \, x^{2}\right ) \log \left (x\right )^{3} \log \left (\log \left (4 \, x^{2}\right )^{2}\right ) - 2 \, \log \left (2\right )^{2} \log \left (4 \, x^{2}\right ) \log \left (\log \left (4 \, x^{2}\right )^{2}\right ) + \log \left (2\right ) \log \left (4 \, x^{2}\right )^{2} \log \left (\log \left (4 \, x^{2}\right )^{2}\right ) - 2 \, \log \left (2\right ) \log \left (4 \, x^{2}\right ) \log \left (x\right ) \log \left (\log \left (4 \, x^{2}\right )^{2}\right ) + 4 \, \log \left (2\right ) \log \left (4 \, x^{2}\right ) \log \left (x\right ) + 4 \, \log \left (4 \, x^{2}\right ) \log \left (x\right )^{2}}{4 \, {\left (x \log \left (2\right )^{2} \log \left (\log \left (4 \, x^{2}\right )^{2}\right )^{2} + 2 \, x \log \left (2\right ) \log \left (x\right ) \log \left (\log \left (4 \, x^{2}\right )^{2}\right )^{2} + x \log \left (x\right )^{2} \log \left (\log \left (4 \, x^{2}\right )^{2}\right )^{2}\right )}} \] Input:
integrate(((2*log(x)-2)*log(4*x^2)*log(log(4*x^2)^2)+16*log(x))/x^2/log(4* x^2)/log(log(4*x^2)^2)^3,x, algorithm="giac")
Output:
-1/4*(2*log(2)^2*log(4*x^2)*log(x)*log(log(4*x^2)^2) - log(2)*log(4*x^2)^2 *log(x)*log(log(4*x^2)^2) + 4*log(2)*log(4*x^2)*log(x)^2*log(log(4*x^2)^2) - log(4*x^2)^2*log(x)^2*log(log(4*x^2)^2) + 2*log(4*x^2)*log(x)^3*log(log (4*x^2)^2) - 2*log(2)^2*log(4*x^2)*log(log(4*x^2)^2) + log(2)*log(4*x^2)^2 *log(log(4*x^2)^2) - 2*log(2)*log(4*x^2)*log(x)*log(log(4*x^2)^2) + 4*log( 2)*log(4*x^2)*log(x) + 4*log(4*x^2)*log(x)^2)/(x*log(2)^2*log(log(4*x^2)^2 )^2 + 2*x*log(2)*log(x)*log(log(4*x^2)^2)^2 + x*log(x)^2*log(log(4*x^2)^2) ^2)
Time = 3.37 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.72 \[ \int \frac {16 \log (x)+(-2+2 \log (x)) \log \left (4 x^2\right ) \log \left (\log ^2\left (4 x^2\right )\right )}{x^2 \log \left (4 x^2\right ) \log ^3\left (\log ^2\left (4 x^2\right )\right )} \, dx=-\frac {2\,\ln \left (x\right )}{x\,{\ln \left ({\ln \left (x^2\right )}^2+4\,\ln \left (2\right )\,\ln \left (x^2\right )+4\,{\ln \left (2\right )}^2\right )}^2} \] Input:
int((16*log(x) + log(log(4*x^2)^2)*log(4*x^2)*(2*log(x) - 2))/(x^2*log(log (4*x^2)^2)^3*log(4*x^2)),x)
Output:
-(2*log(x))/(x*log(4*log(x^2)*log(2) + log(x^2)^2 + 4*log(2)^2)^2)
Time = 0.17 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {16 \log (x)+(-2+2 \log (x)) \log \left (4 x^2\right ) \log \left (\log ^2\left (4 x^2\right )\right )}{x^2 \log \left (4 x^2\right ) \log ^3\left (\log ^2\left (4 x^2\right )\right )} \, dx=-\frac {2 \,\mathrm {log}\left (x \right )}{{\mathrm {log}\left (\mathrm {log}\left (4 x^{2}\right )^{2}\right )}^{2} x} \] Input:
int(((2*log(x)-2)*log(4*x^2)*log(log(4*x^2)^2)+16*log(x))/x^2/log(4*x^2)/l og(log(4*x^2)^2)^3,x)
Output:
( - 2*log(x))/(log(log(4*x**2)**2)**2*x)