\(\int \frac {(100-100 x) \log ^2(x)+100 x \log ^3(x)+((200-200 x) \log (x)+100 x \log ^2(x)) \log ^2(\log (4))}{(x-2 x^2+x^3) \log ^2(x)+(2 x-4 x^2+2 x^3) \log (x) \log ^2(\log (4))+(x-2 x^2+x^3) \log ^4(\log (4))} \, dx\) [5241]

Optimal result

Integrand size = 97, antiderivative size = 23 \[ \int \frac {(100-100 x) \log ^2(x)+100 x \log ^3(x)+\left ((200-200 x) \log (x)+100 x \log ^2(x)\right ) \log ^2(\log (4))}{\left (x-2 x^2+x^3\right ) \log ^2(x)+\left (2 x-4 x^2+2 x^3\right ) \log (x) \log ^2(\log (4))+\left (x-2 x^2+x^3\right ) \log ^4(\log (4))} \, dx=\frac {100 \log ^2(x)}{(1-x) \left (\log (x)+\log ^2(\log (4))\right )} \]

[Out]

100/(1-x)*ln(x)^2/(ln(x)+ln(2*ln(2))^2)

Rubi [F]

\[ \int \frac {(100-100 x) \log ^2(x)+100 x \log ^3(x)+\left ((200-200 x) \log (x)+100 x \log ^2(x)\right ) \log ^2(\log (4))}{\left (x-2 x^2+x^3\right ) \log ^2(x)+\left (2 x-4 x^2+2 x^3\right ) \log (x) \log ^2(\log (4))+\left (x-2 x^2+x^3\right ) \log ^4(\log (4))} \, dx=\int \frac {(100-100 x) \log ^2(x)+100 x \log ^3(x)+\left ((200-200 x) \log (x)+100 x \log ^2(x)\right ) \log ^2(\log (4))}{\left (x-2 x^2+x^3\right ) \log ^2(x)+\left (2 x-4 x^2+2 x^3\right ) \log (x) \log ^2(\log (4))+\left (x-2 x^2+x^3\right ) \log ^4(\log (4))} \, dx \]

[In]

Int[((100 - 100*x)*Log[x]^2 + 100*x*Log[x]^3 + ((200 - 200*x)*Log[x] + 100*x*Log[x]^2)*Log[Log[4]]^2)/((x - 2*
x^2 + x^3)*Log[x]^2 + (2*x - 4*x^2 + 2*x^3)*Log[x]*Log[Log[4]]^2 + (x - 2*x^2 + x^3)*Log[Log[4]]^4),x]

[Out]

100*Log[x] + (100*x*Log[x])/(1 - x) - (100*Log[Log[4]]^2)/(1 - x) + 100*Log[Log[4]]^4*Defer[Int][1/((-1 + x)*x
*(Log[x] + Log[Log[4]]^2)^2), x] + 100*Log[Log[4]]^4*Defer[Int][1/((-1 + x)^2*(Log[x] + Log[Log[4]]^2)), x]

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

Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {(100-100 x) \log ^2(x)+100 x \log ^3(x)+\left ((200-200 x) \log (x)+100 x \log ^2(x)\right ) \log ^2(\log (4))}{\left (x-2 x^2+x^3\right ) \log ^2(x)+\left (2 x-4 x^2+2 x^3\right ) \log (x) \log ^2(\log (4))+\left (x-2 x^2+x^3\right ) \log ^4(\log (4))} \, dx=-\frac {100 \log ^2(x)}{(-1+x) \left (\log (x)+\log ^2(\log (4))\right )} \]

[In]

Integrate[((100 - 100*x)*Log[x]^2 + 100*x*Log[x]^3 + ((200 - 200*x)*Log[x] + 100*x*Log[x]^2)*Log[Log[4]]^2)/((
x - 2*x^2 + x^3)*Log[x]^2 + (2*x - 4*x^2 + 2*x^3)*Log[x]*Log[Log[4]]^2 + (x - 2*x^2 + x^3)*Log[Log[4]]^4),x]

[Out]

(-100*Log[x]^2)/((-1 + x)*(Log[x] + Log[Log[4]]^2))

Maple [A] (verified)

Time = 1.38 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04

method	result	size
norman	\(-\frac {100 \ln \left (x \right )^{2}}{\left (-1+x \right ) \left (\ln \left (x \right )+\ln \left (2 \ln \left (2\right )\right )^{2}\right )}\)	\(24\)
parallelrisch	\(-\frac {100 \ln \left (x \right )^{2}}{\left (-1+x \right ) \left (\ln \left (x \right )+\ln \left (2 \ln \left (2\right )\right )^{2}\right )}\)	\(24\)
default	\(-\frac {100 \ln \left (x \right )^{2}}{\left (-1+x \right ) \left (\ln \left (2\right )^{2}+2 \ln \left (2\right ) \ln \left (\ln \left (2\right )\right )+\ln \left (\ln \left (2\right )\right )^{2}+\ln \left (x \right )\right )}\)	\(33\)
risch	\(-\frac {100 \ln \left (x \right )}{-1+x}+\frac {100 \ln \left (2\right )^{2}}{-1+x}+\frac {200 \ln \left (2\right ) \ln \left (\ln \left (2\right )\right )}{-1+x}+\frac {100 \ln \left (\ln \left (2\right )\right )^{2}}{-1+x}-\frac {100 \left (\ln \left (2\right )^{4}+4 \ln \left (2\right )^{3} \ln \left (\ln \left (2\right )\right )+6 \ln \left (2\right )^{2} \ln \left (\ln \left (2\right )\right )^{2}+4 \ln \left (2\right ) \ln \left (\ln \left (2\right )\right )^{3}+\ln \left (\ln \left (2\right )\right )^{4}\right )}{\left (-1+x \right ) \left (\ln \left (2\right )^{2}+2 \ln \left (2\right ) \ln \left (\ln \left (2\right )\right )+\ln \left (\ln \left (2\right )\right )^{2}+\ln \left (x \right )\right )}\)	\(113\)

[In]

int(((100*x*ln(x)^2+(-200*x+200)*ln(x))*ln(2*ln(2))^2+100*x*ln(x)^3+(-100*x+100)*ln(x)^2)/((x^3-2*x^2+x)*ln(2*
ln(2))^4+(2*x^3-4*x^2+2*x)*ln(x)*ln(2*ln(2))^2+(x^3-2*x^2+x)*ln(x)^2),x,method=_RETURNVERBOSE)

[Out]

-100*ln(x)^2/(-1+x)/(ln(x)+ln(2*ln(2))^2)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.13 \[ \int \frac {(100-100 x) \log ^2(x)+100 x \log ^3(x)+\left ((200-200 x) \log (x)+100 x \log ^2(x)\right ) \log ^2(\log (4))}{\left (x-2 x^2+x^3\right ) \log ^2(x)+\left (2 x-4 x^2+2 x^3\right ) \log (x) \log ^2(\log (4))+\left (x-2 x^2+x^3\right ) \log ^4(\log (4))} \, dx=-\frac {100 \, \log \left (x\right )^{2}}{{\left (x - 1\right )} \log \left (2 \, \log \left (2\right )\right )^{2} + {\left (x - 1\right )} \log \left (x\right )} \]

[In]

integrate(((100*x*log(x)^2+(-200*x+200)*log(x))*log(2*log(2))^2+100*x*log(x)^3+(-100*x+100)*log(x)^2)/((x^3-2*
x^2+x)*log(2*log(2))^4+(2*x^3-4*x^2+2*x)*log(x)*log(2*log(2))^2+(x^3-2*x^2+x)*log(x)^2),x, algorithm="fricas")

[Out]

-100*log(x)^2/((x - 1)*log(2*log(2))^2 + (x - 1)*log(x))

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 148 vs. \(2 (20) = 40\).

Time = 0.15 (sec) , antiderivative size = 148, normalized size of antiderivative = 6.43 \[ \int \frac {(100-100 x) \log ^2(x)+100 x \log ^3(x)+\left ((200-200 x) \log (x)+100 x \log ^2(x)\right ) \log ^2(\log (4))}{\left (x-2 x^2+x^3\right ) \log ^2(x)+\left (2 x-4 x^2+2 x^3\right ) \log (x) \log ^2(\log (4))+\left (x-2 x^2+x^3\right ) \log ^4(\log (4))} \, dx=\frac {- 600 \log {\left (2 \right )}^{2} \log {\left (\log {\left (2 \right )} \right )}^{2} - 100 \log {\left (2 \right )}^{4} - 100 \log {\left (\log {\left (2 \right )} \right )}^{4} - 400 \log {\left (2 \right )} \log {\left (\log {\left (2 \right )} \right )}^{3} - 400 \log {\left (2 \right )}^{3} \log {\left (\log {\left (2 \right )} \right )}}{2 x \log {\left (2 \right )} \log {\left (\log {\left (2 \right )} \right )} + x \log {\left (\log {\left (2 \right )} \right )}^{2} + x \log {\left (2 \right )}^{2} + \left (x - 1\right ) \log {\left (x \right )} - \log {\left (2 \right )}^{2} - \log {\left (\log {\left (2 \right )} \right )}^{2} - 2 \log {\left (2 \right )} \log {\left (\log {\left (2 \right )} \right )}} - \frac {100 \log {\left (x \right )}}{x - 1} - \frac {- 100 \log {\left (2 \right )}^{2} - 100 \log {\left (\log {\left (2 \right )} \right )}^{2} - 200 \log {\left (2 \right )} \log {\left (\log {\left (2 \right )} \right )}}{x - 1} \]

[In]

integrate(((100*x*ln(x)**2+(-200*x+200)*ln(x))*ln(2*ln(2))**2+100*x*ln(x)**3+(-100*x+100)*ln(x)**2)/((x**3-2*x
**2+x)*ln(2*ln(2))**4+(2*x**3-4*x**2+2*x)*ln(x)*ln(2*ln(2))**2+(x**3-2*x**2+x)*ln(x)**2),x)

[Out]

(-600*log(2)**2*log(log(2))**2 - 100*log(2)**4 - 100*log(log(2))**4 - 400*log(2)*log(log(2))**3 - 400*log(2)**
3*log(log(2)))/(2*x*log(2)*log(log(2)) + x*log(log(2))**2 + x*log(2)**2 + (x - 1)*log(x) - log(2)**2 - log(log
(2))**2 - 2*log(2)*log(log(2))) - 100*log(x)/(x - 1) - (-100*log(2)**2 - 100*log(log(2))**2 - 200*log(2)*log(l
og(2)))/(x - 1)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (23) = 46\).

Time = 0.32 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.35 \[ \int \frac {(100-100 x) \log ^2(x)+100 x \log ^3(x)+\left ((200-200 x) \log (x)+100 x \log ^2(x)\right ) \log ^2(\log (4))}{\left (x-2 x^2+x^3\right ) \log ^2(x)+\left (2 x-4 x^2+2 x^3\right ) \log (x) \log ^2(\log (4))+\left (x-2 x^2+x^3\right ) \log ^4(\log (4))} \, dx=-\frac {100 \, \log \left (x\right )^{2}}{{\left (\log \left (2\right )^{2} + 2 \, \log \left (2\right ) \log \left (\log \left (2\right )\right ) + \log \left (\log \left (2\right )\right )^{2}\right )} x - \log \left (2\right )^{2} + {\left (x - 1\right )} \log \left (x\right ) - 2 \, \log \left (2\right ) \log \left (\log \left (2\right )\right ) - \log \left (\log \left (2\right )\right )^{2}} \]

[In]

integrate(((100*x*log(x)^2+(-200*x+200)*log(x))*log(2*log(2))^2+100*x*log(x)^3+(-100*x+100)*log(x)^2)/((x^3-2*
x^2+x)*log(2*log(2))^4+(2*x^3-4*x^2+2*x)*log(x)*log(2*log(2))^2+(x^3-2*x^2+x)*log(x)^2),x, algorithm="maxima")

[Out]

-100*log(x)^2/((log(2)^2 + 2*log(2)*log(log(2)) + log(log(2))^2)*x - log(2)^2 + (x - 1)*log(x) - 2*log(2)*log(
log(2)) - log(log(2))^2)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 127 vs. \(2 (23) = 46\).

Time = 0.29 (sec) , antiderivative size = 127, normalized size of antiderivative = 5.52 \[ \int \frac {(100-100 x) \log ^2(x)+100 x \log ^3(x)+\left ((200-200 x) \log (x)+100 x \log ^2(x)\right ) \log ^2(\log (4))}{\left (x-2 x^2+x^3\right ) \log ^2(x)+\left (2 x-4 x^2+2 x^3\right ) \log (x) \log ^2(\log (4))+\left (x-2 x^2+x^3\right ) \log ^4(\log (4))} \, dx=-\frac {100 \, {\left (\log \left (2\right )^{4} + 4 \, \log \left (2\right )^{3} \log \left (\log \left (2\right )\right ) + 6 \, \log \left (2\right )^{2} \log \left (\log \left (2\right )\right )^{2} + 4 \, \log \left (2\right ) \log \left (\log \left (2\right )\right )^{3} + \log \left (\log \left (2\right )\right )^{4}\right )}}{x \log \left (2\right )^{2} + 2 \, x \log \left (2\right ) \log \left (\log \left (2\right )\right ) + x \log \left (\log \left (2\right )\right )^{2} - \log \left (2\right )^{2} + x \log \left (x\right ) - 2 \, \log \left (2\right ) \log \left (\log \left (2\right )\right ) - \log \left (\log \left (2\right )\right )^{2} - \log \left (x\right )} + \frac {100 \, {\left (\log \left (2\right )^{2} + 2 \, \log \left (2\right ) \log \left (\log \left (2\right )\right ) + \log \left (\log \left (2\right )\right )^{2}\right )}}{x - 1} - \frac {100 \, \log \left (x\right )}{x - 1} \]

[In]

integrate(((100*x*log(x)^2+(-200*x+200)*log(x))*log(2*log(2))^2+100*x*log(x)^3+(-100*x+100)*log(x)^2)/((x^3-2*
x^2+x)*log(2*log(2))^4+(2*x^3-4*x^2+2*x)*log(x)*log(2*log(2))^2+(x^3-2*x^2+x)*log(x)^2),x, algorithm="giac")

[Out]

-100*(log(2)^4 + 4*log(2)^3*log(log(2)) + 6*log(2)^2*log(log(2))^2 + 4*log(2)*log(log(2))^3 + log(log(2))^4)/(
x*log(2)^2 + 2*x*log(2)*log(log(2)) + x*log(log(2))^2 - log(2)^2 + x*log(x) - 2*log(2)*log(log(2)) - log(log(2
))^2 - log(x)) + 100*(log(2)^2 + 2*log(2)*log(log(2)) + log(log(2))^2)/(x - 1) - 100*log(x)/(x - 1)

Mupad [B] (verification not implemented)

Time = 10.64 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {(100-100 x) \log ^2(x)+100 x \log ^3(x)+\left ((200-200 x) \log (x)+100 x \log ^2(x)\right ) \log ^2(\log (4))}{\left (x-2 x^2+x^3\right ) \log ^2(x)+\left (2 x-4 x^2+2 x^3\right ) \log (x) \log ^2(\log (4))+\left (x-2 x^2+x^3\right ) \log ^4(\log (4))} \, dx=-\frac {100\,{\ln \left (x\right )}^2}{\left (\ln \left (x\right )+{\ln \left (\ln \left (4\right )\right )}^2\right )\,\left (x-1\right )} \]

[In]

int((100*x*log(x)^3 - log(x)^2*(100*x - 100) + log(2*log(2))^2*(100*x*log(x)^2 - log(x)*(200*x - 200)))/(log(x
)^2*(x - 2*x^2 + x^3) + log(2*log(2))^4*(x - 2*x^2 + x^3) + log(2*log(2))^2*log(x)*(2*x - 4*x^2 + 2*x^3)),x)

[Out]

-(100*log(x)^2)/((log(x) + log(log(4))^2)*(x - 1))