\(\int \frac {(-125-55 x-6 x^2+50 x^4+20 x^5+2 x^6) \log (4)+(-5 x+200 x^4+80 x^5+8 x^6) \log (4) \log (x) \log (\frac {2}{\log (x)})}{(25 x+10 x^2+x^3) \log (x) \log ^2(\frac {2}{\log (x)})} \, dx\) [9859]

Optimal result

Integrand size = 88, antiderivative size = 27 \[ \int \frac {\left (-125-55 x-6 x^2+50 x^4+20 x^5+2 x^6\right ) \log (4)+\left (-5 x+200 x^4+80 x^5+8 x^6\right ) \log (4) \log (x) \log \left (\frac {2}{\log (x)}\right )}{\left (25 x+10 x^2+x^3\right ) \log (x) \log ^2\left (\frac {2}{\log (x)}\right )} \, dx=\frac {\left (-5+2 x^4-\frac {x}{5+x}\right ) \log (4)}{\log \left (\frac {2}{\log (x)}\right )} \]

[Out]

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

Rubi [F]

\[ \int \frac {\left (-125-55 x-6 x^2+50 x^4+20 x^5+2 x^6\right ) \log (4)+\left (-5 x+200 x^4+80 x^5+8 x^6\right ) \log (4) \log (x) \log \left (\frac {2}{\log (x)}\right )}{\left (25 x+10 x^2+x^3\right ) \log (x) \log ^2\left (\frac {2}{\log (x)}\right )} \, dx=\int \frac {\left (-125-55 x-6 x^2+50 x^4+20 x^5+2 x^6\right ) \log (4)+\left (-5 x+200 x^4+80 x^5+8 x^6\right ) \log (4) \log (x) \log \left (\frac {2}{\log (x)}\right )}{\left (25 x+10 x^2+x^3\right ) \log (x) \log ^2\left (\frac {2}{\log (x)}\right )} \, dx \]

[In]

Int[((-125 - 55*x - 6*x^2 + 50*x^4 + 20*x^5 + 2*x^6)*Log[4] + (-5*x + 200*x^4 + 80*x^5 + 8*x^6)*Log[4]*Log[x]*
Log[2/Log[x]])/((25*x + 10*x^2 + x^3)*Log[x]*Log[2/Log[x]]^2),x]

[Out]

(-5*Log[4])/Log[2/Log[x]] + 2*Log[4]*Defer[Int][x^3/(Log[x]*Log[2/Log[x]]^2), x] - Log[4]*Defer[Int][1/((5 + x
)*Log[x]*Log[2/Log[x]]^2), x] + 8*Log[4]*Defer[Int][x^3/Log[2/Log[x]], x] - 5*Log[4]*Defer[Int][1/((5 + x)^2*L
og[2/Log[x]]), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\left (-125-55 x-6 x^2+50 x^4+20 x^5+2 x^6\right ) \log (4)+\left (-5 x+200 x^4+80 x^5+8 x^6\right ) \log (4) \log (x) \log \left (\frac {2}{\log (x)}\right )}{x \left (25+10 x+x^2\right ) \log (x) \log ^2\left (\frac {2}{\log (x)}\right )} \, dx \\ & = \int \frac {\left (-125-55 x-6 x^2+50 x^4+20 x^5+2 x^6\right ) \log (4)+\left (-5 x+200 x^4+80 x^5+8 x^6\right ) \log (4) \log (x) \log \left (\frac {2}{\log (x)}\right )}{x (5+x)^2 \log (x) \log ^2\left (\frac {2}{\log (x)}\right )} \, dx \\ & = \int \left (\frac {\left (-25-6 x+10 x^4+2 x^5\right ) \log (4)}{x (5+x) \log (x) \log ^2\left (\frac {2}{\log (x)}\right )}+\frac {\left (-5+200 x^3+80 x^4+8 x^5\right ) \log (4)}{(5+x)^2 \log \left (\frac {2}{\log (x)}\right )}\right ) \, dx \\ & = \log (4) \int \frac {-25-6 x+10 x^4+2 x^5}{x (5+x) \log (x) \log ^2\left (\frac {2}{\log (x)}\right )} \, dx+\log (4) \int \frac {-5+200 x^3+80 x^4+8 x^5}{(5+x)^2 \log \left (\frac {2}{\log (x)}\right )} \, dx \\ & = \log (4) \int \left (-\frac {5}{x \log (x) \log ^2\left (\frac {2}{\log (x)}\right )}+\frac {2 x^3}{\log (x) \log ^2\left (\frac {2}{\log (x)}\right )}-\frac {1}{(5+x) \log (x) \log ^2\left (\frac {2}{\log (x)}\right )}\right ) \, dx+\log (4) \int \left (\frac {8 x^3}{\log \left (\frac {2}{\log (x)}\right )}-\frac {5}{(5+x)^2 \log \left (\frac {2}{\log (x)}\right )}\right ) \, dx \\ & = -\left (\log (4) \int \frac {1}{(5+x) \log (x) \log ^2\left (\frac {2}{\log (x)}\right )} \, dx\right )+(2 \log (4)) \int \frac {x^3}{\log (x) \log ^2\left (\frac {2}{\log (x)}\right )} \, dx-(5 \log (4)) \int \frac {1}{x \log (x) \log ^2\left (\frac {2}{\log (x)}\right )} \, dx-(5 \log (4)) \int \frac {1}{(5+x)^2 \log \left (\frac {2}{\log (x)}\right )} \, dx+(8 \log (4)) \int \frac {x^3}{\log \left (\frac {2}{\log (x)}\right )} \, dx \\ & = -\left (\log (4) \int \frac {1}{(5+x) \log (x) \log ^2\left (\frac {2}{\log (x)}\right )} \, dx\right )+(2 \log (4)) \int \frac {x^3}{\log (x) \log ^2\left (\frac {2}{\log (x)}\right )} \, dx-(5 \log (4)) \int \frac {1}{(5+x)^2 \log \left (\frac {2}{\log (x)}\right )} \, dx-(5 \log (4)) \text {Subst}\left (\int \frac {1}{x \log ^2\left (\frac {2}{x}\right )} \, dx,x,\log (x)\right )+(8 \log (4)) \int \frac {x^3}{\log \left (\frac {2}{\log (x)}\right )} \, dx \\ & = -\left (\log (4) \int \frac {1}{(5+x) \log (x) \log ^2\left (\frac {2}{\log (x)}\right )} \, dx\right )+(2 \log (4)) \int \frac {x^3}{\log (x) \log ^2\left (\frac {2}{\log (x)}\right )} \, dx-(5 \log (4)) \int \frac {1}{(5+x)^2 \log \left (\frac {2}{\log (x)}\right )} \, dx+(5 \log (4)) \text {Subst}\left (\int \frac {1}{x^2} \, dx,x,\log \left (\frac {2}{\log (x)}\right )\right )+(8 \log (4)) \int \frac {x^3}{\log \left (\frac {2}{\log (x)}\right )} \, dx \\ & = -\frac {5 \log (4)}{\log \left (\frac {2}{\log (x)}\right )}-\log (4) \int \frac {1}{(5+x) \log (x) \log ^2\left (\frac {2}{\log (x)}\right )} \, dx+(2 \log (4)) \int \frac {x^3}{\log (x) \log ^2\left (\frac {2}{\log (x)}\right )} \, dx-(5 \log (4)) \int \frac {1}{(5+x)^2 \log \left (\frac {2}{\log (x)}\right )} \, dx+(8 \log (4)) \int \frac {x^3}{\log \left (\frac {2}{\log (x)}\right )} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.19 \[ \int \frac {\left (-125-55 x-6 x^2+50 x^4+20 x^5+2 x^6\right ) \log (4)+\left (-5 x+200 x^4+80 x^5+8 x^6\right ) \log (4) \log (x) \log \left (\frac {2}{\log (x)}\right )}{\left (25 x+10 x^2+x^3\right ) \log (x) \log ^2\left (\frac {2}{\log (x)}\right )} \, dx=\frac {\left (-25-6 x+10 x^4+2 x^5\right ) \log (4)}{(5+x) \log \left (\frac {2}{\log (x)}\right )} \]

[In]

Integrate[((-125 - 55*x - 6*x^2 + 50*x^4 + 20*x^5 + 2*x^6)*Log[4] + (-5*x + 200*x^4 + 80*x^5 + 8*x^6)*Log[4]*L
og[x]*Log[2/Log[x]])/((25*x + 10*x^2 + x^3)*Log[x]*Log[2/Log[x]]^2),x]

[Out]

((-25 - 6*x + 10*x^4 + 2*x^5)*Log[4])/((5 + x)*Log[2/Log[x]])

Maple [A] (verified)

Time = 5.45 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.37

method	result	size
risch	\(\frac {4 \ln \left (2\right ) \left (2 x^{5}+10 x^{4}-6 x -25\right )}{\left (5+x \right ) \left (2 \ln \left (2\right )-2 \ln \left (\ln \left (x \right )\right )\right )}\)	\(37\)
parallelrisch	\(\frac {4 x^{5} \ln \left (2\right )+20 x^{4} \ln \left (2\right )-12 x \ln \left (2\right )-50 \ln \left (2\right )}{\ln \left (\frac {2}{\ln \left (x \right )}\right ) \left (5+x \right )}\)	\(40\)

[In]

int((2*(8*x^6+80*x^5+200*x^4-5*x)*ln(2)*ln(x)*ln(2/ln(x))+2*(2*x^6+20*x^5+50*x^4-6*x^2-55*x-125)*ln(2))/(x^3+1
0*x^2+25*x)/ln(x)/ln(2/ln(x))^2,x,method=_RETURNVERBOSE)

[Out]

4*ln(2)*(2*x^5+10*x^4-6*x-25)/(5+x)/(2*ln(2)-2*ln(ln(x)))

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.22 \[ \int \frac {\left (-125-55 x-6 x^2+50 x^4+20 x^5+2 x^6\right ) \log (4)+\left (-5 x+200 x^4+80 x^5+8 x^6\right ) \log (4) \log (x) \log \left (\frac {2}{\log (x)}\right )}{\left (25 x+10 x^2+x^3\right ) \log (x) \log ^2\left (\frac {2}{\log (x)}\right )} \, dx=\frac {2 \, {\left (2 \, x^{5} + 10 \, x^{4} - 6 \, x - 25\right )} \log \left (2\right )}{{\left (x + 5\right )} \log \left (\frac {2}{\log \left (x\right )}\right )} \]

[In]

integrate((2*(8*x^6+80*x^5+200*x^4-5*x)*log(2)*log(x)*log(2/log(x))+2*(2*x^6+20*x^5+50*x^4-6*x^2-55*x-125)*log
(2))/(x^3+10*x^2+25*x)/log(x)/log(2/log(x))^2,x, algorithm="fricas")

[Out]

2*(2*x^5 + 10*x^4 - 6*x - 25)*log(2)/((x + 5)*log(2/log(x)))

Sympy [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.37 \[ \int \frac {\left (-125-55 x-6 x^2+50 x^4+20 x^5+2 x^6\right ) \log (4)+\left (-5 x+200 x^4+80 x^5+8 x^6\right ) \log (4) \log (x) \log \left (\frac {2}{\log (x)}\right )}{\left (25 x+10 x^2+x^3\right ) \log (x) \log ^2\left (\frac {2}{\log (x)}\right )} \, dx=\frac {4 x^{5} \log {\left (2 \right )} + 20 x^{4} \log {\left (2 \right )} - 12 x \log {\left (2 \right )} - 50 \log {\left (2 \right )}}{\left (x + 5\right ) \log {\left (\frac {2}{\log {\left (x \right )}} \right )}} \]

[In]

integrate((2*(8*x**6+80*x**5+200*x**4-5*x)*ln(2)*ln(x)*ln(2/ln(x))+2*(2*x**6+20*x**5+50*x**4-6*x**2-55*x-125)*
ln(2))/(x**3+10*x**2+25*x)/ln(x)/ln(2/ln(x))**2,x)

[Out]

(4*x**5*log(2) + 20*x**4*log(2) - 12*x*log(2) - 50*log(2))/((x + 5)*log(2/log(x)))

Maxima [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.67 \[ \int \frac {\left (-125-55 x-6 x^2+50 x^4+20 x^5+2 x^6\right ) \log (4)+\left (-5 x+200 x^4+80 x^5+8 x^6\right ) \log (4) \log (x) \log \left (\frac {2}{\log (x)}\right )}{\left (25 x+10 x^2+x^3\right ) \log (x) \log ^2\left (\frac {2}{\log (x)}\right )} \, dx=\frac {2 \, {\left (2 \, x^{5} \log \left (2\right ) + 10 \, x^{4} \log \left (2\right ) - 6 \, x \log \left (2\right ) - 25 \, \log \left (2\right )\right )}}{x \log \left (2\right ) - {\left (x + 5\right )} \log \left (\log \left (x\right )\right ) + 5 \, \log \left (2\right )} \]

[In]

integrate((2*(8*x^6+80*x^5+200*x^4-5*x)*log(2)*log(x)*log(2/log(x))+2*(2*x^6+20*x^5+50*x^4-6*x^2-55*x-125)*log
(2))/(x^3+10*x^2+25*x)/log(x)/log(2/log(x))^2,x, algorithm="maxima")

[Out]

2*(2*x^5*log(2) + 10*x^4*log(2) - 6*x*log(2) - 25*log(2))/(x*log(2) - (x + 5)*log(log(x)) + 5*log(2))

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.78 \[ \int \frac {\left (-125-55 x-6 x^2+50 x^4+20 x^5+2 x^6\right ) \log (4)+\left (-5 x+200 x^4+80 x^5+8 x^6\right ) \log (4) \log (x) \log \left (\frac {2}{\log (x)}\right )}{\left (25 x+10 x^2+x^3\right ) \log (x) \log ^2\left (\frac {2}{\log (x)}\right )} \, dx=\frac {2 \, {\left (2 \, x^{5} \log \left (2\right ) + 10 \, x^{4} \log \left (2\right ) - 6 \, x \log \left (2\right ) - 25 \, \log \left (2\right )\right )}}{x \log \left (2\right ) - x \log \left (\log \left (x\right )\right ) + 5 \, \log \left (2\right ) - 5 \, \log \left (\log \left (x\right )\right )} \]

[In]

integrate((2*(8*x^6+80*x^5+200*x^4-5*x)*log(2)*log(x)*log(2/log(x))+2*(2*x^6+20*x^5+50*x^4-6*x^2-55*x-125)*log
(2))/(x^3+10*x^2+25*x)/log(x)/log(2/log(x))^2,x, algorithm="giac")

[Out]

2*(2*x^5*log(2) + 10*x^4*log(2) - 6*x*log(2) - 25*log(2))/(x*log(2) - x*log(log(x)) + 5*log(2) - 5*log(log(x))
)

Mupad [B] (verification not implemented)

Time = 15.73 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.22 \[ \int \frac {\left (-125-55 x-6 x^2+50 x^4+20 x^5+2 x^6\right ) \log (4)+\left (-5 x+200 x^4+80 x^5+8 x^6\right ) \log (4) \log (x) \log \left (\frac {2}{\log (x)}\right )}{\left (25 x+10 x^2+x^3\right ) \log (x) \log ^2\left (\frac {2}{\log (x)}\right )} \, dx=-\frac {2\,\ln \left (2\right )\,\left (-2\,x^5-10\,x^4+6\,x+25\right )}{\ln \left (\frac {2}{\ln \left (x\right )}\right )\,\left (x+5\right )} \]

[In]

int(-(2*log(2)*(55*x + 6*x^2 - 50*x^4 - 20*x^5 - 2*x^6 + 125) - 2*log(2/log(x))*log(2)*log(x)*(200*x^4 - 5*x +
80*x^5 + 8*x^6))/(log(2/log(x))^2*log(x)*(25*x + 10*x^2 + x^3)),x)

[Out]

-(2*log(2)*(6*x - 10*x^4 - 2*x^5 + 25))/(log(2/log(x))*(x + 5))