\(\int \frac {500-5 x^2+125 \log (3)+(-625-125 \log (3)) \log (x)+(-125+125 \log (x)) \log (\log (x))}{25 x^2+10 x^3+x^4+(-1250 x-250 x^2+(-250 x-50 x^2) \log (3)) \log (x)+(15625+6250 \log (3)+625 \log ^2(3)) \log ^2(x)+((250 x+50 x^2) \log (x)+(-6250-1250 \log (3)) \log ^2(x)) \log (\log (x))+625 \log ^2(x) \log ^2(\log (x))} \, dx\) [5022]

Optimal result

Integrand size = 127, antiderivative size = 26 \[ \int \frac {500-5 x^2+125 \log (3)+(-625-125 \log (3)) \log (x)+(-125+125 \log (x)) \log (\log (x))}{25 x^2+10 x^3+x^4+\left (-1250 x-250 x^2+\left (-250 x-50 x^2\right ) \log (3)\right ) \log (x)+\left (15625+6250 \log (3)+625 \log ^2(3)\right ) \log ^2(x)+\left (\left (250 x+50 x^2\right ) \log (x)+(-6250-1250 \log (3)) \log ^2(x)\right ) \log (\log (x))+625 \log ^2(x) \log ^2(\log (x))} \, dx=\frac {x}{\frac {1}{5} x (5+x)+5 \log (x) (-5-\log (3)+\log (\log (x)))} \]

[Out]

x/(x*(1+1/5*x)+5*(ln(ln(x))-5-ln(3))*ln(x))

Rubi [F]

\[ \int \frac {500-5 x^2+125 \log (3)+(-625-125 \log (3)) \log (x)+(-125+125 \log (x)) \log (\log (x))}{25 x^2+10 x^3+x^4+\left (-1250 x-250 x^2+\left (-250 x-50 x^2\right ) \log (3)\right ) \log (x)+\left (15625+6250 \log (3)+625 \log ^2(3)\right ) \log ^2(x)+\left (\left (250 x+50 x^2\right ) \log (x)+(-6250-1250 \log (3)) \log ^2(x)\right ) \log (\log (x))+625 \log ^2(x) \log ^2(\log (x))} \, dx=\int \frac {500-5 x^2+125 \log (3)+(-625-125 \log (3)) \log (x)+(-125+125 \log (x)) \log (\log (x))}{25 x^2+10 x^3+x^4+\left (-1250 x-250 x^2+\left (-250 x-50 x^2\right ) \log (3)\right ) \log (x)+\left (15625+6250 \log (3)+625 \log ^2(3)\right ) \log ^2(x)+\left (\left (250 x+50 x^2\right ) \log (x)+(-6250-1250 \log (3)) \log ^2(x)\right ) \log (\log (x))+625 \log ^2(x) \log ^2(\log (x))} \, dx \]

[In]

Int[(500 - 5*x^2 + 125*Log[3] + (-625 - 125*Log[3])*Log[x] + (-125 + 125*Log[x])*Log[Log[x]])/(25*x^2 + 10*x^3
+ x^4 + (-1250*x - 250*x^2 + (-250*x - 50*x^2)*Log[3])*Log[x] + (15625 + 6250*Log[3] + 625*Log[3]^2)*Log[x]^2
+ ((250*x + 50*x^2)*Log[x] + (-6250 - 1250*Log[3])*Log[x]^2)*Log[Log[x]] + 625*Log[x]^2*Log[Log[x]]^2),x]

[Out]

125*(4 + Log[3])*Defer[Int][(5*x + x^2 - 125*Log[x] + 25*Log[x]*Log[Log[x]/3])^(-2), x] - 5*Defer[Int][x^2/(5*
x + x^2 - 125*Log[x] + 25*Log[x]*Log[Log[x]/3])^2, x] - 125*(5 + Log[3])*Defer[Int][Log[x]/(5*x + x^2 - 125*Lo
g[x] + 25*Log[x]*Log[Log[x]/3])^2, x] - 125*Defer[Int][Log[Log[x]]/(5*x + x^2 - 125*Log[x] + 25*Log[x]*Log[Log
[x]/3])^2, x] + 125*Defer[Int][(Log[x]*Log[Log[x]])/(5*x + x^2 - 125*Log[x] + 25*Log[x]*Log[Log[x]/3])^2, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {5 \left (-x^2+100 \left (1+\frac {\log (3)}{4}\right )-25 \log (x) (5+\log (3)-\log (\log (x)))-25 \log (\log (x))\right )}{\left (x (5+x)+25 \log (x) \left (-5+\log \left (\frac {\log (x)}{3}\right )\right )\right )^2} \, dx \\ & = 5 \int \frac {-x^2+100 \left (1+\frac {\log (3)}{4}\right )-25 \log (x) (5+\log (3)-\log (\log (x)))-25 \log (\log (x))}{\left (x (5+x)+25 \log (x) \left (-5+\log \left (\frac {\log (x)}{3}\right )\right )\right )^2} \, dx \\ & = 5 \int \left (-\frac {x^2}{\left (5 x+x^2-125 \log (x)+25 \log (x) \log \left (\frac {\log (x)}{3}\right )\right )^2}+\frac {25 (4+\log (3))}{\left (5 x+x^2-125 \log (x)+25 \log (x) \log \left (\frac {\log (x)}{3}\right )\right )^2}-\frac {125 \left (1+\frac {\log (3)}{5}\right ) \log (x)}{\left (5 x+x^2-125 \log (x)+25 \log (x) \log \left (\frac {\log (x)}{3}\right )\right )^2}+\frac {25 (-1+\log (x)) \log (\log (x))}{\left (5 x+x^2-125 \log (x)+25 \log (x) \log \left (\frac {\log (x)}{3}\right )\right )^2}\right ) \, dx \\ & = -\left (5 \int \frac {x^2}{\left (5 x+x^2-125 \log (x)+25 \log (x) \log \left (\frac {\log (x)}{3}\right )\right )^2} \, dx\right )+125 \int \frac {(-1+\log (x)) \log (\log (x))}{\left (5 x+x^2-125 \log (x)+25 \log (x) \log \left (\frac {\log (x)}{3}\right )\right )^2} \, dx+(125 (4+\log (3))) \int \frac {1}{\left (5 x+x^2-125 \log (x)+25 \log (x) \log \left (\frac {\log (x)}{3}\right )\right )^2} \, dx-(125 (5+\log (3))) \int \frac {\log (x)}{\left (5 x+x^2-125 \log (x)+25 \log (x) \log \left (\frac {\log (x)}{3}\right )\right )^2} \, dx \\ & = -\left (5 \int \frac {x^2}{\left (5 x+x^2-125 \log (x)+25 \log (x) \log \left (\frac {\log (x)}{3}\right )\right )^2} \, dx\right )+125 \int \left (-\frac {\log (\log (x))}{\left (5 x+x^2-125 \log (x)+25 \log (x) \log \left (\frac {\log (x)}{3}\right )\right )^2}+\frac {\log (x) \log (\log (x))}{\left (5 x+x^2-125 \log (x)+25 \log (x) \log \left (\frac {\log (x)}{3}\right )\right )^2}\right ) \, dx+(125 (4+\log (3))) \int \frac {1}{\left (5 x+x^2-125 \log (x)+25 \log (x) \log \left (\frac {\log (x)}{3}\right )\right )^2} \, dx-(125 (5+\log (3))) \int \frac {\log (x)}{\left (5 x+x^2-125 \log (x)+25 \log (x) \log \left (\frac {\log (x)}{3}\right )\right )^2} \, dx \\ & = -\left (5 \int \frac {x^2}{\left (5 x+x^2-125 \log (x)+25 \log (x) \log \left (\frac {\log (x)}{3}\right )\right )^2} \, dx\right )-125 \int \frac {\log (\log (x))}{\left (5 x+x^2-125 \log (x)+25 \log (x) \log \left (\frac {\log (x)}{3}\right )\right )^2} \, dx+125 \int \frac {\log (x) \log (\log (x))}{\left (5 x+x^2-125 \log (x)+25 \log (x) \log \left (\frac {\log (x)}{3}\right )\right )^2} \, dx+(125 (4+\log (3))) \int \frac {1}{\left (5 x+x^2-125 \log (x)+25 \log (x) \log \left (\frac {\log (x)}{3}\right )\right )^2} \, dx-(125 (5+\log (3))) \int \frac {\log (x)}{\left (5 x+x^2-125 \log (x)+25 \log (x) \log \left (\frac {\log (x)}{3}\right )\right )^2} \, dx \\ \end{align*}

Mathematica [F]

\[ \int \frac {500-5 x^2+125 \log (3)+(-625-125 \log (3)) \log (x)+(-125+125 \log (x)) \log (\log (x))}{25 x^2+10 x^3+x^4+\left (-1250 x-250 x^2+\left (-250 x-50 x^2\right ) \log (3)\right ) \log (x)+\left (15625+6250 \log (3)+625 \log ^2(3)\right ) \log ^2(x)+\left (\left (250 x+50 x^2\right ) \log (x)+(-6250-1250 \log (3)) \log ^2(x)\right ) \log (\log (x))+625 \log ^2(x) \log ^2(\log (x))} \, dx=\int \frac {500-5 x^2+125 \log (3)+(-625-125 \log (3)) \log (x)+(-125+125 \log (x)) \log (\log (x))}{25 x^2+10 x^3+x^4+\left (-1250 x-250 x^2+\left (-250 x-50 x^2\right ) \log (3)\right ) \log (x)+\left (15625+6250 \log (3)+625 \log ^2(3)\right ) \log ^2(x)+\left (\left (250 x+50 x^2\right ) \log (x)+(-6250-1250 \log (3)) \log ^2(x)\right ) \log (\log (x))+625 \log ^2(x) \log ^2(\log (x))} \, dx \]

[In]

Integrate[(500 - 5*x^2 + 125*Log[3] + (-625 - 125*Log[3])*Log[x] + (-125 + 125*Log[x])*Log[Log[x]])/(25*x^2 +
10*x^3 + x^4 + (-1250*x - 250*x^2 + (-250*x - 50*x^2)*Log[3])*Log[x] + (15625 + 6250*Log[3] + 625*Log[3]^2)*Lo
g[x]^2 + ((250*x + 50*x^2)*Log[x] + (-6250 - 1250*Log[3])*Log[x]^2)*Log[Log[x]] + 625*Log[x]^2*Log[Log[x]]^2),
x]

[Out]

Integrate[(500 - 5*x^2 + 125*Log[3] + (-625 - 125*Log[3])*Log[x] + (-125 + 125*Log[x])*Log[Log[x]])/(25*x^2 +
10*x^3 + x^4 + (-1250*x - 250*x^2 + (-250*x - 50*x^2)*Log[3])*Log[x] + (15625 + 6250*Log[3] + 625*Log[3]^2)*Lo
g[x]^2 + ((250*x + 50*x^2)*Log[x] + (-6250 - 1250*Log[3])*Log[x]^2)*Log[Log[x]] + 625*Log[x]^2*Log[Log[x]]^2),
x]

Maple [A] (verified)

Time = 0.90 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.23

method	result	size
default	\(-\frac {5 x}{25 \ln \left (3\right ) \ln \left (x \right )-25 \ln \left (x \right ) \ln \left (\ln \left (x \right )\right )-x^{2}+125 \ln \left (x \right )-5 x}\)	\(32\)
risch	\(-\frac {5 x}{25 \ln \left (3\right ) \ln \left (x \right )-25 \ln \left (x \right ) \ln \left (\ln \left (x \right )\right )-x^{2}+125 \ln \left (x \right )-5 x}\)	\(32\)
parallelrisch	\(-\frac {5 x}{25 \ln \left (3\right ) \ln \left (x \right )-25 \ln \left (x \right ) \ln \left (\ln \left (x \right )\right )-x^{2}+125 \ln \left (x \right )-5 x}\)	\(32\)

[In]

int(((125*ln(x)-125)*ln(ln(x))+(-125*ln(3)-625)*ln(x)+125*ln(3)-5*x^2+500)/(625*ln(x)^2*ln(ln(x))^2+((-1250*ln
(3)-6250)*ln(x)^2+(50*x^2+250*x)*ln(x))*ln(ln(x))+(625*ln(3)^2+6250*ln(3)+15625)*ln(x)^2+((-50*x^2-250*x)*ln(3
)-250*x^2-1250*x)*ln(x)+x^4+10*x^3+25*x^2),x,method=_RETURNVERBOSE)

[Out]

-5*x/(25*ln(3)*ln(x)-25*ln(x)*ln(ln(x))-x^2+125*ln(x)-5*x)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.04 \[ \int \frac {500-5 x^2+125 \log (3)+(-625-125 \log (3)) \log (x)+(-125+125 \log (x)) \log (\log (x))}{25 x^2+10 x^3+x^4+\left (-1250 x-250 x^2+\left (-250 x-50 x^2\right ) \log (3)\right ) \log (x)+\left (15625+6250 \log (3)+625 \log ^2(3)\right ) \log ^2(x)+\left (\left (250 x+50 x^2\right ) \log (x)+(-6250-1250 \log (3)) \log ^2(x)\right ) \log (\log (x))+625 \log ^2(x) \log ^2(\log (x))} \, dx=\frac {5 \, x}{x^{2} - 25 \, {\left (\log \left (3\right ) + 5\right )} \log \left (x\right ) + 25 \, \log \left (x\right ) \log \left (\log \left (x\right )\right ) + 5 \, x} \]

[In]

integrate(((125*log(x)-125)*log(log(x))+(-125*log(3)-625)*log(x)+125*log(3)-5*x^2+500)/(625*log(x)^2*log(log(x
))^2+((-1250*log(3)-6250)*log(x)^2+(50*x^2+250*x)*log(x))*log(log(x))+(625*log(3)^2+6250*log(3)+15625)*log(x)^
2+((-50*x^2-250*x)*log(3)-250*x^2-1250*x)*log(x)+x^4+10*x^3+25*x^2),x, algorithm="fricas")

[Out]

5*x/(x^2 - 25*(log(3) + 5)*log(x) + 25*log(x)*log(log(x)) + 5*x)

Sympy [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.23 \[ \int \frac {500-5 x^2+125 \log (3)+(-625-125 \log (3)) \log (x)+(-125+125 \log (x)) \log (\log (x))}{25 x^2+10 x^3+x^4+\left (-1250 x-250 x^2+\left (-250 x-50 x^2\right ) \log (3)\right ) \log (x)+\left (15625+6250 \log (3)+625 \log ^2(3)\right ) \log ^2(x)+\left (\left (250 x+50 x^2\right ) \log (x)+(-6250-1250 \log (3)) \log ^2(x)\right ) \log (\log (x))+625 \log ^2(x) \log ^2(\log (x))} \, dx=\frac {5 x}{x^{2} + 5 x + 25 \log {\left (x \right )} \log {\left (\log {\left (x \right )} \right )} - 125 \log {\left (x \right )} - 25 \log {\left (3 \right )} \log {\left (x \right )}} \]

[In]

integrate(((125*ln(x)-125)*ln(ln(x))+(-125*ln(3)-625)*ln(x)+125*ln(3)-5*x**2+500)/(625*ln(x)**2*ln(ln(x))**2+(
(-1250*ln(3)-6250)*ln(x)**2+(50*x**2+250*x)*ln(x))*ln(ln(x))+(625*ln(3)**2+6250*ln(3)+15625)*ln(x)**2+((-50*x*
*2-250*x)*ln(3)-250*x**2-1250*x)*ln(x)+x**4+10*x**3+25*x**2),x)

[Out]

5*x/(x**2 + 5*x + 25*log(x)*log(log(x)) - 125*log(x) - 25*log(3)*log(x))

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.04 \[ \int \frac {500-5 x^2+125 \log (3)+(-625-125 \log (3)) \log (x)+(-125+125 \log (x)) \log (\log (x))}{25 x^2+10 x^3+x^4+\left (-1250 x-250 x^2+\left (-250 x-50 x^2\right ) \log (3)\right ) \log (x)+\left (15625+6250 \log (3)+625 \log ^2(3)\right ) \log ^2(x)+\left (\left (250 x+50 x^2\right ) \log (x)+(-6250-1250 \log (3)) \log ^2(x)\right ) \log (\log (x))+625 \log ^2(x) \log ^2(\log (x))} \, dx=\frac {5 \, x}{x^{2} - 25 \, {\left (\log \left (3\right ) + 5\right )} \log \left (x\right ) + 25 \, \log \left (x\right ) \log \left (\log \left (x\right )\right ) + 5 \, x} \]

[In]

integrate(((125*log(x)-125)*log(log(x))+(-125*log(3)-625)*log(x)+125*log(3)-5*x^2+500)/(625*log(x)^2*log(log(x
))^2+((-1250*log(3)-6250)*log(x)^2+(50*x^2+250*x)*log(x))*log(log(x))+(625*log(3)^2+6250*log(3)+15625)*log(x)^
2+((-50*x^2-250*x)*log(3)-250*x^2-1250*x)*log(x)+x^4+10*x^3+25*x^2),x, algorithm="maxima")

[Out]

5*x/(x^2 - 25*(log(3) + 5)*log(x) + 25*log(x)*log(log(x)) + 5*x)

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12 \[ \int \frac {500-5 x^2+125 \log (3)+(-625-125 \log (3)) \log (x)+(-125+125 \log (x)) \log (\log (x))}{25 x^2+10 x^3+x^4+\left (-1250 x-250 x^2+\left (-250 x-50 x^2\right ) \log (3)\right ) \log (x)+\left (15625+6250 \log (3)+625 \log ^2(3)\right ) \log ^2(x)+\left (\left (250 x+50 x^2\right ) \log (x)+(-6250-1250 \log (3)) \log ^2(x)\right ) \log (\log (x))+625 \log ^2(x) \log ^2(\log (x))} \, dx=\frac {5 \, x}{x^{2} - 25 \, \log \left (3\right ) \log \left (x\right ) + 25 \, \log \left (x\right ) \log \left (\log \left (x\right )\right ) + 5 \, x - 125 \, \log \left (x\right )} \]

[In]

integrate(((125*log(x)-125)*log(log(x))+(-125*log(3)-625)*log(x)+125*log(3)-5*x^2+500)/(625*log(x)^2*log(log(x
))^2+((-1250*log(3)-6250)*log(x)^2+(50*x^2+250*x)*log(x))*log(log(x))+(625*log(3)^2+6250*log(3)+15625)*log(x)^
2+((-50*x^2-250*x)*log(3)-250*x^2-1250*x)*log(x)+x^4+10*x^3+25*x^2),x, algorithm="giac")

[Out]

5*x/(x^2 - 25*log(3)*log(x) + 25*log(x)*log(log(x)) + 5*x - 125*log(x))

Mupad [F(-1)]

Timed out. \[ \int \frac {500-5 x^2+125 \log (3)+(-625-125 \log (3)) \log (x)+(-125+125 \log (x)) \log (\log (x))}{25 x^2+10 x^3+x^4+\left (-1250 x-250 x^2+\left (-250 x-50 x^2\right ) \log (3)\right ) \log (x)+\left (15625+6250 \log (3)+625 \log ^2(3)\right ) \log ^2(x)+\left (\left (250 x+50 x^2\right ) \log (x)+(-6250-1250 \log (3)) \log ^2(x)\right ) \log (\log (x))+625 \log ^2(x) \log ^2(\log (x))} \, dx=\int \frac {125\,\ln \left (3\right )+\ln \left (\ln \left (x\right )\right )\,\left (125\,\ln \left (x\right )-125\right )-\ln \left (x\right )\,\left (125\,\ln \left (3\right )+625\right )-5\,x^2+500}{625\,{\ln \left (\ln \left (x\right )\right )}^2\,{\ln \left (x\right )}^2+\ln \left (\ln \left (x\right )\right )\,\left (\ln \left (x\right )\,\left (50\,x^2+250\,x\right )-{\ln \left (x\right )}^2\,\left (1250\,\ln \left (3\right )+6250\right )\right )+25\,x^2+10\,x^3+x^4-\ln \left (x\right )\,\left (1250\,x+\ln \left (3\right )\,\left (50\,x^2+250\,x\right )+250\,x^2\right )+{\ln \left (x\right )}^2\,\left (6250\,\ln \left (3\right )+625\,{\ln \left (3\right )}^2+15625\right )} \,d x \]

[In]

int((125*log(3) + log(log(x))*(125*log(x) - 125) - log(x)*(125*log(3) + 625) - 5*x^2 + 500)/(625*log(log(x))^2
*log(x)^2 + log(log(x))*(log(x)*(250*x + 50*x^2) - log(x)^2*(1250*log(3) + 6250)) + 25*x^2 + 10*x^3 + x^4 - lo
g(x)*(1250*x + log(3)*(250*x + 50*x^2) + 250*x^2) + log(x)^2*(6250*log(3) + 625*log(3)^2 + 15625)),x)

[Out]

int((125*log(3) + log(log(x))*(125*log(x) - 125) - log(x)*(125*log(3) + 625) - 5*x^2 + 500)/(625*log(log(x))^2
*log(x)^2 + log(log(x))*(log(x)*(250*x + 50*x^2) - log(x)^2*(1250*log(3) + 6250)) + 25*x^2 + 10*x^3 + x^4 - lo
g(x)*(1250*x + log(3)*(250*x + 50*x^2) + 250*x^2) + log(x)^2*(6250*log(3) + 625*log(3)^2 + 15625)), x)