\(\int \frac {-300+170 x-10 x^2-130 x \log (2 x) \log (\log (2 x)) \log (\log (\log (2 x)))+(-900 x+120 x^2-4 x^3) \log (2 x) \log (\log (2 x)) \log ^2(\log (\log (2 x)))}{(225 x-30 x^2+x^3) \log (2 x) \log (\log (2 x)) \log ^2(\log (\log (2 x)))} \, dx\) [4357]

Optimal result

Integrand size = 96, antiderivative size = 29 \[ \int \frac {-300+170 x-10 x^2-130 x \log (2 x) \log (\log (2 x)) \log (\log (\log (2 x)))+\left (-900 x+120 x^2-4 x^3\right ) \log (2 x) \log (\log (2 x)) \log ^2(\log (\log (2 x)))}{\left (225 x-30 x^2+x^3\right ) \log (2 x) \log (\log (2 x)) \log ^2(\log (\log (2 x)))} \, dx=x+(2-x) \left (5+\frac {2}{\left (3-\frac {x}{5}\right ) \log (\log (\log (2 x)))}\right ) \]

[Out]

(2-x)*(5+2/(3-1/5*x)/ln(ln(ln(2*x))))+x

Rubi [F]

\[ \int \frac {-300+170 x-10 x^2-130 x \log (2 x) \log (\log (2 x)) \log (\log (\log (2 x)))+\left (-900 x+120 x^2-4 x^3\right ) \log (2 x) \log (\log (2 x)) \log ^2(\log (\log (2 x)))}{\left (225 x-30 x^2+x^3\right ) \log (2 x) \log (\log (2 x)) \log ^2(\log (\log (2 x)))} \, dx=\int \frac {-300+170 x-10 x^2-130 x \log (2 x) \log (\log (2 x)) \log (\log (\log (2 x)))+\left (-900 x+120 x^2-4 x^3\right ) \log (2 x) \log (\log (2 x)) \log ^2(\log (\log (2 x)))}{\left (225 x-30 x^2+x^3\right ) \log (2 x) \log (\log (2 x)) \log ^2(\log (\log (2 x)))} \, dx \]

[In]

Int[(-300 + 170*x - 10*x^2 - 130*x*Log[2*x]*Log[Log[2*x]]*Log[Log[Log[2*x]]] + (-900*x + 120*x^2 - 4*x^3)*Log[
2*x]*Log[Log[2*x]]*Log[Log[Log[2*x]]]^2)/((225*x - 30*x^2 + x^3)*Log[2*x]*Log[Log[2*x]]*Log[Log[Log[2*x]]]^2),
x]

[Out]

-4*x + 4/(3*Log[Log[Log[2*x]]]) - (26*Defer[Int][1/((-15 + x)*Log[2*x]*Log[Log[2*x]]*Log[Log[Log[2*x]]]^2), x]
)/3 - 130*Defer[Int][1/((-15 + x)^2*Log[Log[Log[2*x]]]), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {-300+170 x-10 x^2-130 x \log (2 x) \log (\log (2 x)) \log (\log (\log (2 x)))+\left (-900 x+120 x^2-4 x^3\right ) \log (2 x) \log (\log (2 x)) \log ^2(\log (\log (2 x)))}{x \left (225-30 x+x^2\right ) \log (2 x) \log (\log (2 x)) \log ^2(\log (\log (2 x)))} \, dx \\ & = \int \frac {-300+170 x-10 x^2-130 x \log (2 x) \log (\log (2 x)) \log (\log (\log (2 x)))+\left (-900 x+120 x^2-4 x^3\right ) \log (2 x) \log (\log (2 x)) \log ^2(\log (\log (2 x)))}{(-15+x)^2 x \log (2 x) \log (\log (2 x)) \log ^2(\log (\log (2 x)))} \, dx \\ & = \int \left (-4-\frac {10 (-2+x)}{(-15+x) x \log (2 x) \log (\log (2 x)) \log ^2(\log (\log (2 x)))}-\frac {130}{(-15+x)^2 \log (\log (\log (2 x)))}\right ) \, dx \\ & = -4 x-10 \int \frac {-2+x}{(-15+x) x \log (2 x) \log (\log (2 x)) \log ^2(\log (\log (2 x)))} \, dx-130 \int \frac {1}{(-15+x)^2 \log (\log (\log (2 x)))} \, dx \\ & = -4 x-10 \int \left (\frac {13}{15 (-15+x) \log (2 x) \log (\log (2 x)) \log ^2(\log (\log (2 x)))}+\frac {2}{15 x \log (2 x) \log (\log (2 x)) \log ^2(\log (\log (2 x)))}\right ) \, dx-130 \int \frac {1}{(-15+x)^2 \log (\log (\log (2 x)))} \, dx \\ & = -4 x-\frac {4}{3} \int \frac {1}{x \log (2 x) \log (\log (2 x)) \log ^2(\log (\log (2 x)))} \, dx-\frac {26}{3} \int \frac {1}{(-15+x) \log (2 x) \log (\log (2 x)) \log ^2(\log (\log (2 x)))} \, dx-130 \int \frac {1}{(-15+x)^2 \log (\log (\log (2 x)))} \, dx \\ & = -4 x-\frac {4}{3} \text {Subst}\left (\int \frac {1}{x \log (x) \log ^2(\log (x))} \, dx,x,\log (2 x)\right )-\frac {26}{3} \int \frac {1}{(-15+x) \log (2 x) \log (\log (2 x)) \log ^2(\log (\log (2 x)))} \, dx-130 \int \frac {1}{(-15+x)^2 \log (\log (\log (2 x)))} \, dx \\ & = -4 x-\frac {4}{3} \text {Subst}\left (\int \frac {1}{x \log ^2(x)} \, dx,x,\log (\log (2 x))\right )-\frac {26}{3} \int \frac {1}{(-15+x) \log (2 x) \log (\log (2 x)) \log ^2(\log (\log (2 x)))} \, dx-130 \int \frac {1}{(-15+x)^2 \log (\log (\log (2 x)))} \, dx \\ & = -4 x-\frac {4}{3} \text {Subst}\left (\int \frac {1}{x^2} \, dx,x,\log (\log (\log (2 x)))\right )-\frac {26}{3} \int \frac {1}{(-15+x) \log (2 x) \log (\log (2 x)) \log ^2(\log (\log (2 x)))} \, dx-130 \int \frac {1}{(-15+x)^2 \log (\log (\log (2 x)))} \, dx \\ & = -4 x+\frac {4}{3 \log (\log (\log (2 x)))}-\frac {26}{3} \int \frac {1}{(-15+x) \log (2 x) \log (\log (2 x)) \log ^2(\log (\log (2 x)))} \, dx-130 \int \frac {1}{(-15+x)^2 \log (\log (\log (2 x)))} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.83 \[ \int \frac {-300+170 x-10 x^2-130 x \log (2 x) \log (\log (2 x)) \log (\log (\log (2 x)))+\left (-900 x+120 x^2-4 x^3\right ) \log (2 x) \log (\log (2 x)) \log ^2(\log (\log (2 x)))}{\left (225 x-30 x^2+x^3\right ) \log (2 x) \log (\log (2 x)) \log ^2(\log (\log (2 x)))} \, dx=-2 \left (2 x-\frac {5 (-2+x)}{(-15+x) \log (\log (\log (2 x)))}\right ) \]

[In]

Integrate[(-300 + 170*x - 10*x^2 - 130*x*Log[2*x]*Log[Log[2*x]]*Log[Log[Log[2*x]]] + (-900*x + 120*x^2 - 4*x^3
)*Log[2*x]*Log[Log[2*x]]*Log[Log[Log[2*x]]]^2)/((225*x - 30*x^2 + x^3)*Log[2*x]*Log[Log[2*x]]*Log[Log[Log[2*x]
]]^2),x]

[Out]

-2*(2*x - (5*(-2 + x))/((-15 + x)*Log[Log[Log[2*x]]]))

Maple [A] (verified)

Time = 4.54 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.79

method	result	size
risch	\(-4 x +\frac {10 x -20}{\left (x -15\right ) \ln \left (\ln \left (\ln \left (2 x \right )\right )\right )}\)	\(23\)
parallelrisch	\(\frac {-300-60 \ln \left (\ln \left (\ln \left (2 x \right )\right )\right ) x^{2}+150 x +13500 \ln \left (\ln \left (\ln \left (2 x \right )\right )\right )}{15 \ln \left (\ln \left (\ln \left (2 x \right )\right )\right ) \left (x -15\right )}\)	\(40\)

[In]

int(((-4*x^3+120*x^2-900*x)*ln(2*x)*ln(ln(2*x))*ln(ln(ln(2*x)))^2-130*x*ln(2*x)*ln(ln(2*x))*ln(ln(ln(2*x)))-10
*x^2+170*x-300)/(x^3-30*x^2+225*x)/ln(2*x)/ln(ln(2*x))/ln(ln(ln(2*x)))^2,x,method=_RETURNVERBOSE)

[Out]

-4*x+10*(-2+x)/(x-15)/ln(ln(ln(2*x)))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.21 \[ \int \frac {-300+170 x-10 x^2-130 x \log (2 x) \log (\log (2 x)) \log (\log (\log (2 x)))+\left (-900 x+120 x^2-4 x^3\right ) \log (2 x) \log (\log (2 x)) \log ^2(\log (\log (2 x)))}{\left (225 x-30 x^2+x^3\right ) \log (2 x) \log (\log (2 x)) \log ^2(\log (\log (2 x)))} \, dx=-\frac {2 \, {\left (2 \, {\left (x^{2} - 15 \, x\right )} \log \left (\log \left (\log \left (2 \, x\right )\right )\right ) - 5 \, x + 10\right )}}{{\left (x - 15\right )} \log \left (\log \left (\log \left (2 \, x\right )\right )\right )} \]

[In]

integrate(((-4*x^3+120*x^2-900*x)*log(2*x)*log(log(2*x))*log(log(log(2*x)))^2-130*x*log(2*x)*log(log(2*x))*log
(log(log(2*x)))-10*x^2+170*x-300)/(x^3-30*x^2+225*x)/log(2*x)/log(log(2*x))/log(log(log(2*x)))^2,x, algorithm=
"fricas")

[Out]

-2*(2*(x^2 - 15*x)*log(log(log(2*x))) - 5*x + 10)/((x - 15)*log(log(log(2*x))))

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.66 \[ \int \frac {-300+170 x-10 x^2-130 x \log (2 x) \log (\log (2 x)) \log (\log (\log (2 x)))+\left (-900 x+120 x^2-4 x^3\right ) \log (2 x) \log (\log (2 x)) \log ^2(\log (\log (2 x)))}{\left (225 x-30 x^2+x^3\right ) \log (2 x) \log (\log (2 x)) \log ^2(\log (\log (2 x)))} \, dx=- 4 x + \frac {10 x - 20}{\left (x - 15\right ) \log {\left (\log {\left (\log {\left (2 x \right )} \right )} \right )}} \]

[In]

integrate(((-4*x**3+120*x**2-900*x)*ln(2*x)*ln(ln(2*x))*ln(ln(ln(2*x)))**2-130*x*ln(2*x)*ln(ln(2*x))*ln(ln(ln(
2*x)))-10*x**2+170*x-300)/(x**3-30*x**2+225*x)/ln(2*x)/ln(ln(2*x))/ln(ln(ln(2*x)))**2,x)

[Out]

-4*x + (10*x - 20)/((x - 15)*log(log(log(2*x))))

Maxima [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.28 \[ \int \frac {-300+170 x-10 x^2-130 x \log (2 x) \log (\log (2 x)) \log (\log (\log (2 x)))+\left (-900 x+120 x^2-4 x^3\right ) \log (2 x) \log (\log (2 x)) \log ^2(\log (\log (2 x)))}{\left (225 x-30 x^2+x^3\right ) \log (2 x) \log (\log (2 x)) \log ^2(\log (\log (2 x)))} \, dx=-\frac {2 \, {\left (2 \, {\left (x^{2} - 15 \, x\right )} \log \left (\log \left (\log \left (2\right ) + \log \left (x\right )\right )\right ) - 5 \, x + 10\right )}}{{\left (x - 15\right )} \log \left (\log \left (\log \left (2\right ) + \log \left (x\right )\right )\right )} \]

[In]

integrate(((-4*x^3+120*x^2-900*x)*log(2*x)*log(log(2*x))*log(log(log(2*x)))^2-130*x*log(2*x)*log(log(2*x))*log
(log(log(2*x)))-10*x^2+170*x-300)/(x^3-30*x^2+225*x)/log(2*x)/log(log(2*x))/log(log(log(2*x)))^2,x, algorithm=
"maxima")

[Out]

-2*(2*(x^2 - 15*x)*log(log(log(2) + log(x))) - 5*x + 10)/((x - 15)*log(log(log(2) + log(x))))

Giac [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.97 \[ \int \frac {-300+170 x-10 x^2-130 x \log (2 x) \log (\log (2 x)) \log (\log (\log (2 x)))+\left (-900 x+120 x^2-4 x^3\right ) \log (2 x) \log (\log (2 x)) \log ^2(\log (\log (2 x)))}{\left (225 x-30 x^2+x^3\right ) \log (2 x) \log (\log (2 x)) \log ^2(\log (\log (2 x)))} \, dx=-4 \, x + \frac {10 \, {\left (x - 2\right )}}{x \log \left (\log \left (\log \left (2 \, x\right )\right )\right ) - 15 \, \log \left (\log \left (\log \left (2 \, x\right )\right )\right )} \]

[In]

integrate(((-4*x^3+120*x^2-900*x)*log(2*x)*log(log(2*x))*log(log(log(2*x)))^2-130*x*log(2*x)*log(log(2*x))*log
(log(log(2*x)))-10*x^2+170*x-300)/(x^3-30*x^2+225*x)/log(2*x)/log(log(2*x))/log(log(log(2*x)))^2,x, algorithm=
"giac")

[Out]

-4*x + 10*(x - 2)/(x*log(log(log(2*x))) - 15*log(log(log(2*x))))

Mupad [B] (verification not implemented)

Time = 11.56 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.38 \[ \int \frac {-300+170 x-10 x^2-130 x \log (2 x) \log (\log (2 x)) \log (\log (\log (2 x)))+\left (-900 x+120 x^2-4 x^3\right ) \log (2 x) \log (\log (2 x)) \log ^2(\log (\log (2 x)))}{\left (225 x-30 x^2+x^3\right ) \log (2 x) \log (\log (2 x)) \log ^2(\log (\log (2 x)))} \, dx=\frac {2\,\left (5\,x-2\,x^2\,\ln \left (\ln \left (\ln \left (2\,x\right )\right )\right )+30\,x\,\ln \left (\ln \left (\ln \left (2\,x\right )\right )\right )-10\right )}{\ln \left (\ln \left (\ln \left (2\,x\right )\right )\right )\,\left (x-15\right )} \]

[In]

int(-(10*x^2 - 170*x + log(log(log(2*x)))^2*log(2*x)*log(log(2*x))*(900*x - 120*x^2 + 4*x^3) + 130*x*log(log(l
og(2*x)))*log(2*x)*log(log(2*x)) + 300)/(log(log(log(2*x)))^2*log(2*x)*log(log(2*x))*(225*x - 30*x^2 + x^3)),x
)

[Out]

(2*(5*x - 2*x^2*log(log(log(2*x))) + 30*x*log(log(log(2*x))) - 10))/(log(log(log(2*x)))*(x - 15))