\(\int \frac {-20 x^2+(6 x-3 x^2-x^4) \log (4 x)+(-100+10 x^2 \log (4 x)) \log (\log ^2(4 x))+75 \log (4 x) \log ^2(\log ^2(4 x))}{x^4 \log (4 x)} \, dx\) [2665]

Optimal result

Integrand size = 69, antiderivative size = 30 \[ \int \frac {-20 x^2+\left (6 x-3 x^2-x^4\right ) \log (4 x)+\left (-100+10 x^2 \log (4 x)\right ) \log \left (\log ^2(4 x)\right )+75 \log (4 x) \log ^2\left (\log ^2(4 x)\right )}{x^4 \log (4 x)} \, dx=\frac {3-\frac {3}{x}+x-\left (x+\frac {5 \log \left (\log ^2(4 x)\right )}{x}\right )^2}{x} \]

[Out]

(3+x-3/x-(5/x*ln(ln(4*x)^2)+x)^2)/x

Rubi [F]

\[ \int \frac {-20 x^2+\left (6 x-3 x^2-x^4\right ) \log (4 x)+\left (-100+10 x^2 \log (4 x)\right ) \log \left (\log ^2(4 x)\right )+75 \log (4 x) \log ^2\left (\log ^2(4 x)\right )}{x^4 \log (4 x)} \, dx=\int \frac {-20 x^2+\left (6 x-3 x^2-x^4\right ) \log (4 x)+\left (-100+10 x^2 \log (4 x)\right ) \log \left (\log ^2(4 x)\right )+75 \log (4 x) \log ^2\left (\log ^2(4 x)\right )}{x^4 \log (4 x)} \, dx \]

[In]

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]

[Out]

-3/x^2 + 3/x - x - (10*Log[Log[4*x]^2])/x - 100*Defer[Int][Log[Log[4*x]^2]/(x^4*Log[4*x]), x] + 75*Defer[Int][
Log[Log[4*x]^2]^2/x^4, x]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {-20 x+6 \log (4 x)-3 x \log (4 x)-x^3 \log (4 x)}{x^3 \log (4 x)}+\frac {10 \left (-10+x^2 \log (4 x)\right ) \log \left (\log ^2(4 x)\right )}{x^4 \log (4 x)}+\frac {75 \log ^2\left (\log ^2(4 x)\right )}{x^4}\right ) \, dx \\ & = 10 \int \frac {\left (-10+x^2 \log (4 x)\right ) \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+\int \frac {-20 x+6 \log (4 x)-3 x \log (4 x)-x^3 \log (4 x)}{x^3 \log (4 x)} \, dx \\ & = 10 \int \left (\frac {\log \left (\log ^2(4 x)\right )}{x^2}-\frac {10 \log \left (\log ^2(4 x)\right )}{x^4 \log (4 x)}\right ) \, dx+75 \int \frac {\log ^2\left (\log ^2(4 x)\right )}{x^4} \, dx+\int \left (-1+\frac {6}{x^3}-\frac {3}{x^2}-\frac {20}{x^2 \log (4 x)}\right ) \, dx \\ & = -\frac {3}{x^2}+\frac {3}{x}-x+10 \int \frac {\log \left (\log ^2(4 x)\right )}{x^2} \, dx-20 \int \frac {1}{x^2 \log (4 x)} \, dx+75 \int \frac {\log ^2\left (\log ^2(4 x)\right )}{x^4} \, dx-100 \int \frac {\log \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}+20 \int \frac {1}{x^2 \log (4 x)} \, dx+75 \int \frac {\log ^2\left (\log ^2(4 x)\right )}{x^4} \, dx-80 \text {Subst}\left (\int \frac {e^{-x}}{x} \, dx,x,\log (4 x)\right )-100 \int \frac {\log \left (\log ^2(4 x)\right )}{x^4 \log (4 x)} \, dx \\ & = -\frac {3}{x^2}+\frac {3}{x}-x-80 \text {Ei}(-\log (4 x))-\frac {10 \log \left (\log ^2(4 x)\right )}{x}+75 \int \frac {\log ^2\left (\log ^2(4 x)\right )}{x^4} \, dx+80 \text {Subst}\left (\int \frac {e^{-x}}{x} \, dx,x,\log (4 x)\right )-100 \int \frac {\log \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}+75 \int \frac {\log ^2\left (\log ^2(4 x)\right )}{x^4} \, dx-100 \int \frac {\log \left (\log ^2(4 x)\right )}{x^4 \log (4 x)} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.33 \[ \int \frac {-20 x^2+\left (6 x-3 x^2-x^4\right ) \log (4 x)+\left (-100+10 x^2 \log (4 x)\right ) \log \left (\log ^2(4 x)\right )+75 \log (4 x) \log ^2\left (\log ^2(4 x)\right )}{x^4 \log (4 x)} \, dx=-\frac {3}{x^2}+\frac {3}{x}-x-\frac {10 \log \left (\log ^2(4 x)\right )}{x}-\frac {25 \log ^2\left (\log ^2(4 x)\right )}{x^3} \]

[In]

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]

[Out]

-3/x^2 + 3/x - x - (10*Log[Log[4*x]^2])/x - (25*Log[Log[4*x]^2]^2)/x^3

Maple [A] (verified)

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	\(-\frac {100 \ln \left (\ln \left (4 x \right )\right )^{2}}{x^{3}}-\frac {10 \left (-5 i \pi \operatorname {csgn}\left (i \ln \left (4 x \right )\right )^{2} \operatorname {csgn}\left (i \ln \left (4 x \right )^{2}\right )+10 i \pi \,\operatorname {csgn}\left (i \ln \left (4 x \right )\right ) \operatorname {csgn}\left (i \ln \left (4 x \right )^{2}\right )^{2}-5 i \pi \operatorname {csgn}\left (i \ln \left (4 x \right )^{2}\right )^{3}+2 x^{2}\right ) \ln \left (\ln \left (4 x \right )\right )}{x^{3}}-\frac {-25 \pi ^{2} \operatorname {csgn}\left (i \ln \left (4 x \right )\right )^{4} \operatorname {csgn}\left (i \ln \left (4 x \right )^{2}\right )^{2}+100 \pi ^{2} \operatorname {csgn}\left (i \ln \left (4 x \right )\right )^{3} \operatorname {csgn}\left (i \ln \left (4 x \right )^{2}\right )^{3}-150 \pi ^{2} \operatorname {csgn}\left (i \ln \left (4 x \right )\right )^{2} \operatorname {csgn}\left (i \ln \left (4 x \right )^{2}\right )^{4}+100 \pi ^{2} \operatorname {csgn}\left (i \ln \left (4 x \right )\right ) \operatorname {csgn}\left (i \ln \left (4 x \right )^{2}\right )^{5}-25 \pi ^{2} \operatorname {csgn}\left (i \ln \left (4 x \right )^{2}\right )^{6}-20 i \pi \,x^{2} \operatorname {csgn}\left (i \ln \left (4 x \right )\right )^{2} \operatorname {csgn}\left (i \ln \left (4 x \right )^{2}\right )+40 i \pi \,x^{2} \operatorname {csgn}\left (i \ln \left (4 x \right )\right ) \operatorname {csgn}\left (i \ln \left (4 x \right )^{2}\right )^{2}-20 i \pi \,x^{2} \operatorname {csgn}\left (i \ln \left (4 x \right )^{2}\right )^{3}+4 x^{4}-12 x^{2}+12 x}{4 x^{3}}\)	\(309\)

[In]

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)

[Out]

1/x^3*(-x^4-10*x^2*ln(ln(4*x)^2)+3*x^2-25*ln(ln(4*x)^2)^2-3*x)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.33 \[ \int \frac {-20 x^2+\left (6 x-3 x^2-x^4\right ) \log (4 x)+\left (-100+10 x^2 \log (4 x)\right ) \log \left (\log ^2(4 x)\right )+75 \log (4 x) \log ^2\left (\log ^2(4 x)\right )}{x^4 \log (4 x)} \, dx=-\frac {x^{4} + 10 \, x^{2} \log \left (\log \left (4 \, x\right )^{2}\right ) - 3 \, x^{2} + 25 \, \log \left (\log \left (4 \, x\right )^{2}\right )^{2} + 3 \, x}{x^{3}} \]

[In]

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="fricas")

[Out]

-(x^4 + 10*x^2*log(log(4*x)^2) - 3*x^2 + 25*log(log(4*x)^2)^2 + 3*x)/x^3

Sympy [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.23 \[ \int \frac {-20 x^2+\left (6 x-3 x^2-x^4\right ) \log (4 x)+\left (-100+10 x^2 \log (4 x)\right ) \log \left (\log ^2(4 x)\right )+75 \log (4 x) \log ^2\left (\log ^2(4 x)\right )}{x^4 \log (4 x)} \, dx=- x - \frac {10 \log {\left (\log {\left (4 x \right )}^{2} \right )}}{x} - \frac {3 - 3 x}{x^{2}} - \frac {25 \log {\left (\log {\left (4 x \right )}^{2} \right )}^{2}}{x^{3}} \]

[In]

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)

[Out]

-x - 10*log(log(4*x)**2)/x - (3 - 3*x)/x**2 - 25*log(log(4*x)**2)**2/x**3

Maxima [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.37 \[ \int \frac {-20 x^2+\left (6 x-3 x^2-x^4\right ) \log (4 x)+\left (-100+10 x^2 \log (4 x)\right ) \log \left (\log ^2(4 x)\right )+75 \log (4 x) \log ^2\left (\log ^2(4 x)\right )}{x^4 \log (4 x)} \, dx=-x - \frac {10 \, \log \left (\log \left (4 \, x\right )^{2}\right )}{x} + \frac {3}{x} - \frac {100 \, \log \left (2 \, \log \left (2\right ) + \log \left (x\right )\right )^{2}}{x^{3}} - \frac {3}{x^{2}} \]

[In]

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="maxima")

[Out]

-x - 10*log(log(4*x)^2)/x + 3/x - 100*log(2*log(2) + log(x))^2/x^3 - 3/x^2

Giac [A] (verification not implemented)

none

Time = 0.38 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.27 \[ \int \frac {-20 x^2+\left (6 x-3 x^2-x^4\right ) \log (4 x)+\left (-100+10 x^2 \log (4 x)\right ) \log \left (\log ^2(4 x)\right )+75 \log (4 x) \log ^2\left (\log ^2(4 x)\right )}{x^4 \log (4 x)} \, dx=-x - \frac {10 \, \log \left (\log \left (4 \, x\right )^{2}\right )}{x} + \frac {3 \, {\left (x - 1\right )}}{x^{2}} - \frac {25 \, \log \left (\log \left (4 \, x\right )^{2}\right )^{2}}{x^{3}} \]

[In]

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="giac")

[Out]

-x - 10*log(log(4*x)^2)/x + 3*(x - 1)/x^2 - 25*log(log(4*x)^2)^2/x^3

Mupad [B] (verification not implemented)

Time = 9.23 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.30 \[ \int \frac {-20 x^2+\left (6 x-3 x^2-x^4\right ) \log (4 x)+\left (-100+10 x^2 \log (4 x)\right ) \log \left (\log ^2(4 x)\right )+75 \log (4 x) \log ^2\left (\log ^2(4 x)\right )}{x^4 \log (4 x)} \, dx=\frac {3\,x-3}{x^2}-\frac {10\,\ln \left ({\ln \left (4\,x\right )}^2\right )}{x}-x-\frac {25\,{\ln \left ({\ln \left (4\,x\right )}^2\right )}^2}{x^3} \]

[In]

int((75*log(log(4*x)^2)^2*log(4*x) + log(log(4*x)^2)*(10*x^2*log(4*x) - 100) - 20*x^2 - log(4*x)*(3*x^2 - 6*x
+ x^4))/(x^4*log(4*x)),x)

[Out]

(3*x - 3)/x^2 - (10*log(log(4*x)^2))/x - x - (25*log(log(4*x)^2)^2)/x^3