Integrand size = 122, antiderivative size = 32 \[ \int \frac {-20+6 x^2+\left (20-6 x^2\right ) \log (x) \log (4 \log (x))+\left (-10-3 x^2\right ) \log (x) \log (4 \log (x)) \log \left (\frac {\log (4 \log (x))}{2 x}\right ) \log \left (\log ^2\left (\frac {\log (4 \log (x))}{2 x}\right )\right )}{\left (-10 x+3 x^3\right ) \log (x) \log (4 \log (x)) \log \left (\frac {\log (4 \log (x))}{2 x}\right ) \log \left (\log ^2\left (\frac {\log (4 \log (x))}{2 x}\right )\right )} \, dx=\log \left (\frac {4 \log \left (\log ^2\left (\frac {\log (4 \log (x))}{2 x}\right )\right )}{-\frac {2}{x}+\frac {3 x}{5}}\right ) \]
Time = 0.11 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.94 \[ \int \frac {-20+6 x^2+\left (20-6 x^2\right ) \log (x) \log (4 \log (x))+\left (-10-3 x^2\right ) \log (x) \log (4 \log (x)) \log \left (\frac {\log (4 \log (x))}{2 x}\right ) \log \left (\log ^2\left (\frac {\log (4 \log (x))}{2 x}\right )\right )}{\left (-10 x+3 x^3\right ) \log (x) \log (4 \log (x)) \log \left (\frac {\log (4 \log (x))}{2 x}\right ) \log \left (\log ^2\left (\frac {\log (4 \log (x))}{2 x}\right )\right )} \, dx=\log (x)-\log \left (10-3 x^2\right )+\log \left (\log \left (\log ^2\left (\frac {\log (4 \log (x))}{2 x}\right )\right )\right ) \]
Integrate[(-20 + 6*x^2 + (20 - 6*x^2)*Log[x]*Log[4*Log[x]] + (-10 - 3*x^2) *Log[x]*Log[4*Log[x]]*Log[Log[4*Log[x]]/(2*x)]*Log[Log[Log[4*Log[x]]/(2*x) ]^2])/((-10*x + 3*x^3)*Log[x]*Log[4*Log[x]]*Log[Log[4*Log[x]]/(2*x)]*Log[L og[Log[4*Log[x]]/(2*x)]^2]),x]
Time = 3.07 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.94, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.025, Rules used = {2026, 7276, 2009}
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.
\(\displaystyle \int \frac {6 x^2+\left (-3 x^2-10\right ) \log (x) \log (4 \log (x)) \log \left (\frac {\log (4 \log (x))}{2 x}\right ) \log \left (\log ^2\left (\frac {\log (4 \log (x))}{2 x}\right )\right )+\left (20-6 x^2\right ) \log (x) \log (4 \log (x))-20}{\left (3 x^3-10 x\right ) \log (x) \log (4 \log (x)) \log \left (\frac {\log (4 \log (x))}{2 x}\right ) \log \left (\log ^2\left (\frac {\log (4 \log (x))}{2 x}\right )\right )} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {6 x^2+\left (-3 x^2-10\right ) \log (x) \log (4 \log (x)) \log \left (\frac {\log (4 \log (x))}{2 x}\right ) \log \left (\log ^2\left (\frac {\log (4 \log (x))}{2 x}\right )\right )+\left (20-6 x^2\right ) \log (x) \log (4 \log (x))-20}{x \left (3 x^2-10\right ) \log (x) \log (4 \log (x)) \log \left (\frac {\log (4 \log (x))}{2 x}\right ) \log \left (\log ^2\left (\frac {\log (4 \log (x))}{2 x}\right )\right )}dx\) |
\(\Big \downarrow \) 7276 |
\(\displaystyle \int \left (\frac {-3 x^2-10}{x \left (3 x^2-10\right )}-\frac {2 (\log (x) \log (4 \log (x))-1)}{x \log (x) \log (4 \log (x)) \log \left (\frac {\log (4 \log (x))}{2 x}\right ) \log \left (\log ^2\left (\frac {\log (4 \log (x))}{2 x}\right )\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\log \left (10-3 x^2\right )+\log \left (\log \left (\log ^2\left (\frac {\log (4 \log (x))}{2 x}\right )\right )\right )+\log (x)\) |
Int[(-20 + 6*x^2 + (20 - 6*x^2)*Log[x]*Log[4*Log[x]] + (-10 - 3*x^2)*Log[x ]*Log[4*Log[x]]*Log[Log[4*Log[x]]/(2*x)]*Log[Log[Log[4*Log[x]]/(2*x)]^2])/ ((-10*x + 3*x^3)*Log[x]*Log[4*Log[x]]*Log[Log[4*Log[x]]/(2*x)]*Log[Log[Log [4*Log[x]]/(2*x)]^2]),x]
3.12.41.3.1 Defintions of rubi rules used
Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p *r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ erQ[p] && !MonomialQ[Px, x] && (ILtQ[p, 0] || !PolyQ[u, x])
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Time = 198.05 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.84
method | result | size |
parallelrisch | \(\ln \left (x \right )+\ln \left (\ln \left (\ln \left (\frac {\ln \left (4 \ln \left (x \right )\right )}{2 x}\right )^{2}\right )\right )-\ln \left (x^{2}-\frac {10}{3}\right )\) | \(27\) |
risch | \(\text {Expression too large to display}\) | \(892\) |
int(((-3*x^2-10)*ln(x)*ln(4*ln(x))*ln(1/2*ln(4*ln(x))/x)*ln(ln(1/2*ln(4*ln (x))/x)^2)+(-6*x^2+20)*ln(x)*ln(4*ln(x))+6*x^2-20)/(3*x^3-10*x)/ln(x)/ln(4 *ln(x))/ln(1/2*ln(4*ln(x))/x)/ln(ln(1/2*ln(4*ln(x))/x)^2),x,method=_RETURN VERBOSE)
Time = 0.26 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.88 \[ \int \frac {-20+6 x^2+\left (20-6 x^2\right ) \log (x) \log (4 \log (x))+\left (-10-3 x^2\right ) \log (x) \log (4 \log (x)) \log \left (\frac {\log (4 \log (x))}{2 x}\right ) \log \left (\log ^2\left (\frac {\log (4 \log (x))}{2 x}\right )\right )}{\left (-10 x+3 x^3\right ) \log (x) \log (4 \log (x)) \log \left (\frac {\log (4 \log (x))}{2 x}\right ) \log \left (\log ^2\left (\frac {\log (4 \log (x))}{2 x}\right )\right )} \, dx=-\log \left (3 \, x^{2} - 10\right ) + \log \left (x\right ) + \log \left (\log \left (\log \left (\frac {\log \left (4 \, \log \left (x\right )\right )}{2 \, x}\right )^{2}\right )\right ) \]
integrate(((-3*x^2-10)*log(x)*log(4*log(x))*log(1/2*log(4*log(x))/x)*log(l og(1/2*log(4*log(x))/x)^2)+(-6*x^2+20)*log(x)*log(4*log(x))+6*x^2-20)/(3*x ^3-10*x)/log(x)/log(4*log(x))/log(1/2*log(4*log(x))/x)/log(log(1/2*log(4*l og(x))/x)^2),x, algorithm=\
Time = 0.49 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.84 \[ \int \frac {-20+6 x^2+\left (20-6 x^2\right ) \log (x) \log (4 \log (x))+\left (-10-3 x^2\right ) \log (x) \log (4 \log (x)) \log \left (\frac {\log (4 \log (x))}{2 x}\right ) \log \left (\log ^2\left (\frac {\log (4 \log (x))}{2 x}\right )\right )}{\left (-10 x+3 x^3\right ) \log (x) \log (4 \log (x)) \log \left (\frac {\log (4 \log (x))}{2 x}\right ) \log \left (\log ^2\left (\frac {\log (4 \log (x))}{2 x}\right )\right )} \, dx=\log {\left (x \right )} - \log {\left (3 x^{2} - 10 \right )} + \log {\left (\log {\left (\log {\left (\frac {\log {\left (4 \log {\left (x \right )} \right )}}{2 x} \right )}^{2} \right )} \right )} \]
integrate(((-3*x**2-10)*ln(x)*ln(4*ln(x))*ln(1/2*ln(4*ln(x))/x)*ln(ln(1/2* ln(4*ln(x))/x)**2)+(-6*x**2+20)*ln(x)*ln(4*ln(x))+6*x**2-20)/(3*x**3-10*x) /ln(x)/ln(4*ln(x))/ln(1/2*ln(4*ln(x))/x)/ln(ln(1/2*ln(4*ln(x))/x)**2),x)
Time = 0.35 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.97 \[ \int \frac {-20+6 x^2+\left (20-6 x^2\right ) \log (x) \log (4 \log (x))+\left (-10-3 x^2\right ) \log (x) \log (4 \log (x)) \log \left (\frac {\log (4 \log (x))}{2 x}\right ) \log \left (\log ^2\left (\frac {\log (4 \log (x))}{2 x}\right )\right )}{\left (-10 x+3 x^3\right ) \log (x) \log (4 \log (x)) \log \left (\frac {\log (4 \log (x))}{2 x}\right ) \log \left (\log ^2\left (\frac {\log (4 \log (x))}{2 x}\right )\right )} \, dx=-\log \left (3 \, x^{2} - 10\right ) + \log \left (x\right ) + \log \left (\log \left (\log \left (2\right ) + \log \left (x\right ) - \log \left (2 \, \log \left (2\right ) + \log \left (\log \left (x\right )\right )\right )\right )\right ) \]
integrate(((-3*x^2-10)*log(x)*log(4*log(x))*log(1/2*log(4*log(x))/x)*log(l og(1/2*log(4*log(x))/x)^2)+(-6*x^2+20)*log(x)*log(4*log(x))+6*x^2-20)/(3*x ^3-10*x)/log(x)/log(4*log(x))/log(1/2*log(4*log(x))/x)/log(log(1/2*log(4*l og(x))/x)^2),x, algorithm=\
Timed out. \[ \int \frac {-20+6 x^2+\left (20-6 x^2\right ) \log (x) \log (4 \log (x))+\left (-10-3 x^2\right ) \log (x) \log (4 \log (x)) \log \left (\frac {\log (4 \log (x))}{2 x}\right ) \log \left (\log ^2\left (\frac {\log (4 \log (x))}{2 x}\right )\right )}{\left (-10 x+3 x^3\right ) \log (x) \log (4 \log (x)) \log \left (\frac {\log (4 \log (x))}{2 x}\right ) \log \left (\log ^2\left (\frac {\log (4 \log (x))}{2 x}\right )\right )} \, dx=\text {Timed out} \]
integrate(((-3*x^2-10)*log(x)*log(4*log(x))*log(1/2*log(4*log(x))/x)*log(l og(1/2*log(4*log(x))/x)^2)+(-6*x^2+20)*log(x)*log(4*log(x))+6*x^2-20)/(3*x ^3-10*x)/log(x)/log(4*log(x))/log(1/2*log(4*log(x))/x)/log(log(1/2*log(4*l og(x))/x)^2),x, algorithm=\
Time = 16.70 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.81 \[ \int \frac {-20+6 x^2+\left (20-6 x^2\right ) \log (x) \log (4 \log (x))+\left (-10-3 x^2\right ) \log (x) \log (4 \log (x)) \log \left (\frac {\log (4 \log (x))}{2 x}\right ) \log \left (\log ^2\left (\frac {\log (4 \log (x))}{2 x}\right )\right )}{\left (-10 x+3 x^3\right ) \log (x) \log (4 \log (x)) \log \left (\frac {\log (4 \log (x))}{2 x}\right ) \log \left (\log ^2\left (\frac {\log (4 \log (x))}{2 x}\right )\right )} \, dx=\ln \left (\ln \left ({\ln \left (\frac {\ln \left (4\,\ln \left (x\right )\right )}{2\,x}\right )}^2\right )\right )-\ln \left (x^2-\frac {10}{3}\right )+\ln \left (x\right ) \]