Integrand size = 69, antiderivative size = 30 \[ \int \frac {-20 x^2+\left (6 x-3 x^2-x^4\right ) \log (4 x)+\left (-100+10 x^2 \log (4 x)\right ) \log \left (\log ^2(4 x)\right )+75 \log (4 x) \log ^2\left (\log ^2(4 x)\right )}{x^4 \log (4 x)} \, dx=\frac {3-\frac {3}{x}+x-\left (x+\frac {5 \log \left (\log ^2(4 x)\right )}{x}\right )^2}{x} \]
Time = 0.18 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.33 \[ \int \frac {-20 x^2+\left (6 x-3 x^2-x^4\right ) \log (4 x)+\left (-100+10 x^2 \log (4 x)\right ) \log \left (\log ^2(4 x)\right )+75 \log (4 x) \log ^2\left (\log ^2(4 x)\right )}{x^4 \log (4 x)} \, dx=-\frac {3}{x^2}+\frac {3}{x}-x-\frac {10 \log \left (\log ^2(4 x)\right )}{x}-\frac {25 \log ^2\left (\log ^2(4 x)\right )}{x^3} \]
Integrate[(-20*x^2 + (6*x - 3*x^2 - x^4)*Log[4*x] + (-100 + 10*x^2*Log[4*x ])*Log[Log[4*x]^2] + 75*Log[4*x]*Log[Log[4*x]^2]^2)/(x^4*Log[4*x]),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.
\(\displaystyle \int \frac {-20 x^2+\left (10 x^2 \log (4 x)-100\right ) \log \left (\log ^2(4 x)\right )+\left (-x^4-3 x^2+6 x\right ) \log (4 x)+75 \log (4 x) \log ^2\left (\log ^2(4 x)\right )}{x^4 \log (4 x)} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {75 \log ^2\left (\log ^2(4 x)\right )}{x^4}+\frac {x^3 (-\log (4 x))-20 x-3 x \log (4 x)+6 \log (4 x)}{x^3 \log (4 x)}+\frac {10 \left (x^2 \log (4 x)-10\right ) \log \left (\log ^2(4 x)\right )}{x^4 \log (4 x)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -100 \int \frac {\log \left (\log ^2(4 x)\right )}{x^4 \log (4 x)}dx+75 \int \frac {\log ^2\left (\log ^2(4 x)\right )}{x^4}dx-\frac {3}{x^2}-x+\frac {3}{x}-\frac {10 \log \left (\log ^2(4 x)\right )}{x}\) |
Int[(-20*x^2 + (6*x - 3*x^2 - x^4)*Log[4*x] + (-100 + 10*x^2*Log[4*x])*Log [Log[4*x]^2] + 75*Log[4*x]*Log[Log[4*x]^2]^2)/(x^4*Log[4*x]),x]
3.27.65.3.1 Defintions of rubi rules used
Time = 9.56 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.40
method | result | size |
parallelrisch | \(\frac {-x^{4}-10 x^{2} \ln \left (\ln \left (4 x \right )^{2}\right )+3 x^{2}-25 \ln \left (\ln \left (4 x \right )^{2}\right )^{2}-3 x}{x^{3}}\) | \(42\) |
risch | \(-\frac {100 \ln \left (\ln \left (4 x \right )\right )^{2}}{x^{3}}-\frac {10 \left (-5 i \pi \operatorname {csgn}\left (i \ln \left (4 x \right )\right )^{2} \operatorname {csgn}\left (i \ln \left (4 x \right )^{2}\right )+10 i \pi \,\operatorname {csgn}\left (i \ln \left (4 x \right )\right ) \operatorname {csgn}\left (i \ln \left (4 x \right )^{2}\right )^{2}-5 i \pi \operatorname {csgn}\left (i \ln \left (4 x \right )^{2}\right )^{3}+2 x^{2}\right ) \ln \left (\ln \left (4 x \right )\right )}{x^{3}}-\frac {-25 \pi ^{2} \operatorname {csgn}\left (i \ln \left (4 x \right )\right )^{4} \operatorname {csgn}\left (i \ln \left (4 x \right )^{2}\right )^{2}+100 \pi ^{2} \operatorname {csgn}\left (i \ln \left (4 x \right )\right )^{3} \operatorname {csgn}\left (i \ln \left (4 x \right )^{2}\right )^{3}-150 \pi ^{2} \operatorname {csgn}\left (i \ln \left (4 x \right )\right )^{2} \operatorname {csgn}\left (i \ln \left (4 x \right )^{2}\right )^{4}+100 \pi ^{2} \operatorname {csgn}\left (i \ln \left (4 x \right )\right ) \operatorname {csgn}\left (i \ln \left (4 x \right )^{2}\right )^{5}-25 \pi ^{2} \operatorname {csgn}\left (i \ln \left (4 x \right )^{2}\right )^{6}-20 i \pi \,x^{2} \operatorname {csgn}\left (i \ln \left (4 x \right )\right )^{2} \operatorname {csgn}\left (i \ln \left (4 x \right )^{2}\right )+40 i \pi \,x^{2} \operatorname {csgn}\left (i \ln \left (4 x \right )\right ) \operatorname {csgn}\left (i \ln \left (4 x \right )^{2}\right )^{2}-20 i \pi \,x^{2} \operatorname {csgn}\left (i \ln \left (4 x \right )^{2}\right )^{3}+4 x^{4}-12 x^{2}+12 x}{4 x^{3}}\) | \(309\) |
int((75*ln(4*x)*ln(ln(4*x)^2)^2+(10*x^2*ln(4*x)-100)*ln(ln(4*x)^2)+(-x^4-3 *x^2+6*x)*ln(4*x)-20*x^2)/x^4/ln(4*x),x,method=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.33 \[ \int \frac {-20 x^2+\left (6 x-3 x^2-x^4\right ) \log (4 x)+\left (-100+10 x^2 \log (4 x)\right ) \log \left (\log ^2(4 x)\right )+75 \log (4 x) \log ^2\left (\log ^2(4 x)\right )}{x^4 \log (4 x)} \, dx=-\frac {x^{4} + 10 \, x^{2} \log \left (\log \left (4 \, x\right )^{2}\right ) - 3 \, x^{2} + 25 \, \log \left (\log \left (4 \, x\right )^{2}\right )^{2} + 3 \, x}{x^{3}} \]
integrate((75*log(4*x)*log(log(4*x)^2)^2+(10*x^2*log(4*x)-100)*log(log(4*x )^2)+(-x^4-3*x^2+6*x)*log(4*x)-20*x^2)/x^4/log(4*x),x, algorithm=\
Time = 0.18 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.23 \[ \int \frac {-20 x^2+\left (6 x-3 x^2-x^4\right ) \log (4 x)+\left (-100+10 x^2 \log (4 x)\right ) \log \left (\log ^2(4 x)\right )+75 \log (4 x) \log ^2\left (\log ^2(4 x)\right )}{x^4 \log (4 x)} \, dx=- x - \frac {10 \log {\left (\log {\left (4 x \right )}^{2} \right )}}{x} - \frac {3 - 3 x}{x^{2}} - \frac {25 \log {\left (\log {\left (4 x \right )}^{2} \right )}^{2}}{x^{3}} \]
integrate((75*ln(4*x)*ln(ln(4*x)**2)**2+(10*x**2*ln(4*x)-100)*ln(ln(4*x)** 2)+(-x**4-3*x**2+6*x)*ln(4*x)-20*x**2)/x**4/ln(4*x),x)
Time = 0.36 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.37 \[ \int \frac {-20 x^2+\left (6 x-3 x^2-x^4\right ) \log (4 x)+\left (-100+10 x^2 \log (4 x)\right ) \log \left (\log ^2(4 x)\right )+75 \log (4 x) \log ^2\left (\log ^2(4 x)\right )}{x^4 \log (4 x)} \, dx=-x - \frac {10 \, \log \left (\log \left (4 \, x\right )^{2}\right )}{x} + \frac {3}{x} - \frac {100 \, \log \left (2 \, \log \left (2\right ) + \log \left (x\right )\right )^{2}}{x^{3}} - \frac {3}{x^{2}} \]
integrate((75*log(4*x)*log(log(4*x)^2)^2+(10*x^2*log(4*x)-100)*log(log(4*x )^2)+(-x^4-3*x^2+6*x)*log(4*x)-20*x^2)/x^4/log(4*x),x, algorithm=\
Time = 0.38 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.27 \[ \int \frac {-20 x^2+\left (6 x-3 x^2-x^4\right ) \log (4 x)+\left (-100+10 x^2 \log (4 x)\right ) \log \left (\log ^2(4 x)\right )+75 \log (4 x) \log ^2\left (\log ^2(4 x)\right )}{x^4 \log (4 x)} \, dx=-x - \frac {10 \, \log \left (\log \left (4 \, x\right )^{2}\right )}{x} + \frac {3 \, {\left (x - 1\right )}}{x^{2}} - \frac {25 \, \log \left (\log \left (4 \, x\right )^{2}\right )^{2}}{x^{3}} \]
integrate((75*log(4*x)*log(log(4*x)^2)^2+(10*x^2*log(4*x)-100)*log(log(4*x )^2)+(-x^4-3*x^2+6*x)*log(4*x)-20*x^2)/x^4/log(4*x),x, algorithm=\
Time = 9.23 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.30 \[ \int \frac {-20 x^2+\left (6 x-3 x^2-x^4\right ) \log (4 x)+\left (-100+10 x^2 \log (4 x)\right ) \log \left (\log ^2(4 x)\right )+75 \log (4 x) \log ^2\left (\log ^2(4 x)\right )}{x^4 \log (4 x)} \, dx=\frac {3\,x-3}{x^2}-\frac {10\,\ln \left ({\ln \left (4\,x\right )}^2\right )}{x}-x-\frac {25\,{\ln \left ({\ln \left (4\,x\right )}^2\right )}^2}{x^3} \]