\(\int \frac {-5 x+6 x^3-x^3 \log (x)+(-30 x+5 x \log (x)) \log (6-\log (x))+(-12 x^3+2 x^3 \log (x)+(-60 x+10 x \log (x)) \log (6-\log (x))) \log (\frac {x^2+5 \log (6-\log (x))}{x})}{(-6 x^2+x^2 \log (x)+(-30+5 \log (x)) \log (6-\log (x))) \log ^2(\frac {x^2+5 \log (6-\log (x))}{x})} \, dx\) [8516]

Optimal result

Integrand size = 131, antiderivative size = 21 \[ \int \frac {-5 x+6 x^3-x^3 \log (x)+(-30 x+5 x \log (x)) \log (6-\log (x))+\left (-12 x^3+2 x^3 \log (x)+(-60 x+10 x \log (x)) \log (6-\log (x))\right ) \log \left (\frac {x^2+5 \log (6-\log (x))}{x}\right )}{\left (-6 x^2+x^2 \log (x)+(-30+5 \log (x)) \log (6-\log (x))\right ) \log ^2\left (\frac {x^2+5 \log (6-\log (x))}{x}\right )} \, dx=\frac {x^2}{\log \left (x+\frac {5 \log (6-\log (x))}{x}\right )} \]

[Out]

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

Rubi [F]

\[ \int \frac {-5 x+6 x^3-x^3 \log (x)+(-30 x+5 x \log (x)) \log (6-\log (x))+\left (-12 x^3+2 x^3 \log (x)+(-60 x+10 x \log (x)) \log (6-\log (x))\right ) \log \left (\frac {x^2+5 \log (6-\log (x))}{x}\right )}{\left (-6 x^2+x^2 \log (x)+(-30+5 \log (x)) \log (6-\log (x))\right ) \log ^2\left (\frac {x^2+5 \log (6-\log (x))}{x}\right )} \, dx=\int \frac {-5 x+6 x^3-x^3 \log (x)+(-30 x+5 x \log (x)) \log (6-\log (x))+\left (-12 x^3+2 x^3 \log (x)+(-60 x+10 x \log (x)) \log (6-\log (x))\right ) \log \left (\frac {x^2+5 \log (6-\log (x))}{x}\right )}{\left (-6 x^2+x^2 \log (x)+(-30+5 \log (x)) \log (6-\log (x))\right ) \log ^2\left (\frac {x^2+5 \log (6-\log (x))}{x}\right )} \, dx \]

[In]

Int[(-5*x + 6*x^3 - x^3*Log[x] + (-30*x + 5*x*Log[x])*Log[6 - Log[x]] + (-12*x^3 + 2*x^3*Log[x] + (-60*x + 10*
x*Log[x])*Log[6 - Log[x]])*Log[(x^2 + 5*Log[6 - Log[x]])/x])/((-6*x^2 + x^2*Log[x] + (-30 + 5*Log[x])*Log[6 -
Log[x]])*Log[(x^2 + 5*Log[6 - Log[x]])/x]^2),x]

[Out]

-5*Defer[Int][x/((-6 + Log[x])*(x^2 + 5*Log[6 - Log[x]])*Log[x + (5*Log[6 - Log[x]])/x]^2), x] + 6*Defer[Int][
x^3/((-6 + Log[x])*(x^2 + 5*Log[6 - Log[x]])*Log[x + (5*Log[6 - Log[x]])/x]^2), x] - Defer[Int][(x^3*Log[x])/(
(-6 + Log[x])*(x^2 + 5*Log[6 - Log[x]])*Log[x + (5*Log[6 - Log[x]])/x]^2), x] - 30*Defer[Int][(x*Log[6 - Log[x
]])/((-6 + Log[x])*(x^2 + 5*Log[6 - Log[x]])*Log[x + (5*Log[6 - Log[x]])/x]^2), x] + 5*Defer[Int][(x*Log[x]*Lo
g[6 - Log[x]])/((-6 + Log[x])*(x^2 + 5*Log[6 - Log[x]])*Log[x + (5*Log[6 - Log[x]])/x]^2), x] + 2*Defer[Int][x
/Log[x + (5*Log[6 - Log[x]])/x], x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {5 x-6 x^3+x^3 \log (x)-(-30 x+5 x \log (x)) \log (6-\log (x))-\left (-12 x^3+2 x^3 \log (x)+(-60 x+10 x \log (x)) \log (6-\log (x))\right ) \log \left (\frac {x^2+5 \log (6-\log (x))}{x}\right )}{(6-\log (x)) \left (x^2+5 \log (6-\log (x))\right ) \log ^2\left (x+\frac {5 \log (6-\log (x))}{x}\right )} \, dx \\ & = \int \frac {x \left (5-6 x^2+x^2 \log (x)-5 (-6+\log (x)) \log (6-\log (x))-2 (-6+\log (x)) \left (x^2+5 \log (6-\log (x))\right ) \log \left (x+\frac {5 \log (6-\log (x))}{x}\right )\right )}{(6-\log (x)) \left (x^2+5 \log (6-\log (x))\right ) \log ^2\left (x+\frac {5 \log (6-\log (x))}{x}\right )} \, dx \\ & = \int \left (-\frac {x \left (5-6 x^2+x^2 \log (x)+30 \log (6-\log (x))-5 \log (x) \log (6-\log (x))\right )}{(-6+\log (x)) \left (x^2+5 \log (6-\log (x))\right ) \log ^2\left (x+\frac {5 \log (6-\log (x))}{x}\right )}+\frac {2 x}{\log \left (x+\frac {5 \log (6-\log (x))}{x}\right )}\right ) \, dx \\ & = 2 \int \frac {x}{\log \left (x+\frac {5 \log (6-\log (x))}{x}\right )} \, dx-\int \frac {x \left (5-6 x^2+x^2 \log (x)+30 \log (6-\log (x))-5 \log (x) \log (6-\log (x))\right )}{(-6+\log (x)) \left (x^2+5 \log (6-\log (x))\right ) \log ^2\left (x+\frac {5 \log (6-\log (x))}{x}\right )} \, dx \\ & = 2 \int \frac {x}{\log \left (x+\frac {5 \log (6-\log (x))}{x}\right )} \, dx-\int \left (\frac {5 x}{(-6+\log (x)) \left (x^2+5 \log (6-\log (x))\right ) \log ^2\left (x+\frac {5 \log (6-\log (x))}{x}\right )}-\frac {6 x^3}{(-6+\log (x)) \left (x^2+5 \log (6-\log (x))\right ) \log ^2\left (x+\frac {5 \log (6-\log (x))}{x}\right )}+\frac {x^3 \log (x)}{(-6+\log (x)) \left (x^2+5 \log (6-\log (x))\right ) \log ^2\left (x+\frac {5 \log (6-\log (x))}{x}\right )}+\frac {30 x \log (6-\log (x))}{(-6+\log (x)) \left (x^2+5 \log (6-\log (x))\right ) \log ^2\left (x+\frac {5 \log (6-\log (x))}{x}\right )}-\frac {5 x \log (x) \log (6-\log (x))}{(-6+\log (x)) \left (x^2+5 \log (6-\log (x))\right ) \log ^2\left (x+\frac {5 \log (6-\log (x))}{x}\right )}\right ) \, dx \\ & = 2 \int \frac {x}{\log \left (x+\frac {5 \log (6-\log (x))}{x}\right )} \, dx-5 \int \frac {x}{(-6+\log (x)) \left (x^2+5 \log (6-\log (x))\right ) \log ^2\left (x+\frac {5 \log (6-\log (x))}{x}\right )} \, dx+5 \int \frac {x \log (x) \log (6-\log (x))}{(-6+\log (x)) \left (x^2+5 \log (6-\log (x))\right ) \log ^2\left (x+\frac {5 \log (6-\log (x))}{x}\right )} \, dx+6 \int \frac {x^3}{(-6+\log (x)) \left (x^2+5 \log (6-\log (x))\right ) \log ^2\left (x+\frac {5 \log (6-\log (x))}{x}\right )} \, dx-30 \int \frac {x \log (6-\log (x))}{(-6+\log (x)) \left (x^2+5 \log (6-\log (x))\right ) \log ^2\left (x+\frac {5 \log (6-\log (x))}{x}\right )} \, dx-\int \frac {x^3 \log (x)}{(-6+\log (x)) \left (x^2+5 \log (6-\log (x))\right ) \log ^2\left (x+\frac {5 \log (6-\log (x))}{x}\right )} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {-5 x+6 x^3-x^3 \log (x)+(-30 x+5 x \log (x)) \log (6-\log (x))+\left (-12 x^3+2 x^3 \log (x)+(-60 x+10 x \log (x)) \log (6-\log (x))\right ) \log \left (\frac {x^2+5 \log (6-\log (x))}{x}\right )}{\left (-6 x^2+x^2 \log (x)+(-30+5 \log (x)) \log (6-\log (x))\right ) \log ^2\left (\frac {x^2+5 \log (6-\log (x))}{x}\right )} \, dx=\frac {x^2}{\log \left (x+\frac {5 \log (6-\log (x))}{x}\right )} \]

[In]

Integrate[(-5*x + 6*x^3 - x^3*Log[x] + (-30*x + 5*x*Log[x])*Log[6 - Log[x]] + (-12*x^3 + 2*x^3*Log[x] + (-60*x
+ 10*x*Log[x])*Log[6 - Log[x]])*Log[(x^2 + 5*Log[6 - Log[x]])/x])/((-6*x^2 + x^2*Log[x] + (-30 + 5*Log[x])*Lo
g[6 - Log[x]])*Log[(x^2 + 5*Log[6 - Log[x]])/x]^2),x]

[Out]

x^2/Log[x + (5*Log[6 - Log[x]])/x]

Maple [A] (verified)

Time = 11.98 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.19

method	result	size
parallelrisch	\(\frac {x^{2}}{\ln \left (\frac {5 \ln \left (-\ln \left (x \right )+6\right )+x^{2}}{x}\right )}\)	\(25\)
risch	\(-\frac {2 i x^{2}}{\pi \,\operatorname {csgn}\left (i \left (5 \ln \left (-\ln \left (x \right )+6\right )+x^{2}\right )\right ) {\operatorname {csgn}\left (\frac {i \left (5 \ln \left (-\ln \left (x \right )+6\right )+x^{2}\right )}{x}\right )}^{2}-\pi \,\operatorname {csgn}\left (i \left (5 \ln \left (-\ln \left (x \right )+6\right )+x^{2}\right )\right ) \operatorname {csgn}\left (\frac {i}{x}\right ) \operatorname {csgn}\left (\frac {i \left (5 \ln \left (-\ln \left (x \right )+6\right )+x^{2}\right )}{x}\right )-\pi {\operatorname {csgn}\left (\frac {i \left (5 \ln \left (-\ln \left (x \right )+6\right )+x^{2}\right )}{x}\right )}^{3}+\pi \,\operatorname {csgn}\left (\frac {i}{x}\right ) {\operatorname {csgn}\left (\frac {i \left (5 \ln \left (-\ln \left (x \right )+6\right )+x^{2}\right )}{x}\right )}^{2}+2 i \ln \left (x \right )-2 i \ln \left (5 \ln \left (-\ln \left (x \right )+6\right )+x^{2}\right )}\)	\(176\)

[In]

int((((10*x*ln(x)-60*x)*ln(-ln(x)+6)+2*x^3*ln(x)-12*x^3)*ln((5*ln(-ln(x)+6)+x^2)/x)+(5*x*ln(x)-30*x)*ln(-ln(x)
+6)-x^3*ln(x)+6*x^3-5*x)/((5*ln(x)-30)*ln(-ln(x)+6)+x^2*ln(x)-6*x^2)/ln((5*ln(-ln(x)+6)+x^2)/x)^2,x,method=_RE
TURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.14 \[ \int \frac {-5 x+6 x^3-x^3 \log (x)+(-30 x+5 x \log (x)) \log (6-\log (x))+\left (-12 x^3+2 x^3 \log (x)+(-60 x+10 x \log (x)) \log (6-\log (x))\right ) \log \left (\frac {x^2+5 \log (6-\log (x))}{x}\right )}{\left (-6 x^2+x^2 \log (x)+(-30+5 \log (x)) \log (6-\log (x))\right ) \log ^2\left (\frac {x^2+5 \log (6-\log (x))}{x}\right )} \, dx=\frac {x^{2}}{\log \left (\frac {x^{2} + 5 \, \log \left (-\log \left (x\right ) + 6\right )}{x}\right )} \]

[In]

integrate((((10*x*log(x)-60*x)*log(-log(x)+6)+2*x^3*log(x)-12*x^3)*log((5*log(-log(x)+6)+x^2)/x)+(5*x*log(x)-3
0*x)*log(-log(x)+6)-x^3*log(x)+6*x^3-5*x)/((5*log(x)-30)*log(-log(x)+6)+x^2*log(x)-6*x^2)/log((5*log(-log(x)+6
)+x^2)/x)^2,x, algorithm="fricas")

[Out]

x^2/log((x^2 + 5*log(-log(x) + 6))/x)

Sympy [A] (verification not implemented)

Time = 0.44 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int \frac {-5 x+6 x^3-x^3 \log (x)+(-30 x+5 x \log (x)) \log (6-\log (x))+\left (-12 x^3+2 x^3 \log (x)+(-60 x+10 x \log (x)) \log (6-\log (x))\right ) \log \left (\frac {x^2+5 \log (6-\log (x))}{x}\right )}{\left (-6 x^2+x^2 \log (x)+(-30+5 \log (x)) \log (6-\log (x))\right ) \log ^2\left (\frac {x^2+5 \log (6-\log (x))}{x}\right )} \, dx=\frac {x^{2}}{\log {\left (\frac {x^{2} + 5 \log {\left (6 - \log {\left (x \right )} \right )}}{x} \right )}} \]

[In]

integrate((((10*x*ln(x)-60*x)*ln(-ln(x)+6)+2*x**3*ln(x)-12*x**3)*ln((5*ln(-ln(x)+6)+x**2)/x)+(5*x*ln(x)-30*x)*
ln(-ln(x)+6)-x**3*ln(x)+6*x**3-5*x)/((5*ln(x)-30)*ln(-ln(x)+6)+x**2*ln(x)-6*x**2)/ln((5*ln(-ln(x)+6)+x**2)/x)*
*2,x)

[Out]

x**2/log((x**2 + 5*log(6 - log(x)))/x)

Maxima [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.19 \[ \int \frac {-5 x+6 x^3-x^3 \log (x)+(-30 x+5 x \log (x)) \log (6-\log (x))+\left (-12 x^3+2 x^3 \log (x)+(-60 x+10 x \log (x)) \log (6-\log (x))\right ) \log \left (\frac {x^2+5 \log (6-\log (x))}{x}\right )}{\left (-6 x^2+x^2 \log (x)+(-30+5 \log (x)) \log (6-\log (x))\right ) \log ^2\left (\frac {x^2+5 \log (6-\log (x))}{x}\right )} \, dx=\frac {x^{2}}{\log \left (x^{2} + 5 \, \log \left (-\log \left (x\right ) + 6\right )\right ) - \log \left (x\right )} \]

[In]

integrate((((10*x*log(x)-60*x)*log(-log(x)+6)+2*x^3*log(x)-12*x^3)*log((5*log(-log(x)+6)+x^2)/x)+(5*x*log(x)-3
0*x)*log(-log(x)+6)-x^3*log(x)+6*x^3-5*x)/((5*log(x)-30)*log(-log(x)+6)+x^2*log(x)-6*x^2)/log((5*log(-log(x)+6
)+x^2)/x)^2,x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.41 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.19 \[ \int \frac {-5 x+6 x^3-x^3 \log (x)+(-30 x+5 x \log (x)) \log (6-\log (x))+\left (-12 x^3+2 x^3 \log (x)+(-60 x+10 x \log (x)) \log (6-\log (x))\right ) \log \left (\frac {x^2+5 \log (6-\log (x))}{x}\right )}{\left (-6 x^2+x^2 \log (x)+(-30+5 \log (x)) \log (6-\log (x))\right ) \log ^2\left (\frac {x^2+5 \log (6-\log (x))}{x}\right )} \, dx=\frac {x^{2}}{\log \left (x^{2} + 5 \, \log \left (-\log \left (x\right ) + 6\right )\right ) - \log \left (x\right )} \]

[In]

integrate((((10*x*log(x)-60*x)*log(-log(x)+6)+2*x^3*log(x)-12*x^3)*log((5*log(-log(x)+6)+x^2)/x)+(5*x*log(x)-3
0*x)*log(-log(x)+6)-x^3*log(x)+6*x^3-5*x)/((5*log(x)-30)*log(-log(x)+6)+x^2*log(x)-6*x^2)/log((5*log(-log(x)+6
)+x^2)/x)^2,x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {-5 x+6 x^3-x^3 \log (x)+(-30 x+5 x \log (x)) \log (6-\log (x))+\left (-12 x^3+2 x^3 \log (x)+(-60 x+10 x \log (x)) \log (6-\log (x))\right ) \log \left (\frac {x^2+5 \log (6-\log (x))}{x}\right )}{\left (-6 x^2+x^2 \log (x)+(-30+5 \log (x)) \log (6-\log (x))\right ) \log ^2\left (\frac {x^2+5 \log (6-\log (x))}{x}\right )} \, dx=-\int \frac {5\,x+x^3\,\ln \left (x\right )+\ln \left (6-\ln \left (x\right )\right )\,\left (30\,x-5\,x\,\ln \left (x\right )\right )+\ln \left (\frac {5\,\ln \left (6-\ln \left (x\right )\right )+x^2}{x}\right )\,\left (\ln \left (6-\ln \left (x\right )\right )\,\left (60\,x-10\,x\,\ln \left (x\right )\right )-2\,x^3\,\ln \left (x\right )+12\,x^3\right )-6\,x^3}{{\ln \left (\frac {5\,\ln \left (6-\ln \left (x\right )\right )+x^2}{x}\right )}^2\,\left (\ln \left (6-\ln \left (x\right )\right )\,\left (5\,\ln \left (x\right )-30\right )+x^2\,\ln \left (x\right )-6\,x^2\right )} \,d x \]

[In]

int(-(5*x + x^3*log(x) + log(6 - log(x))*(30*x - 5*x*log(x)) + log((5*log(6 - log(x)) + x^2)/x)*(log(6 - log(x
))*(60*x - 10*x*log(x)) - 2*x^3*log(x) + 12*x^3) - 6*x^3)/(log((5*log(6 - log(x)) + x^2)/x)^2*(log(6 - log(x))
*(5*log(x) - 30) + x^2*log(x) - 6*x^2)),x)

[Out]

-int((5*x + x^3*log(x) + log(6 - log(x))*(30*x - 5*x*log(x)) + log((5*log(6 - log(x)) + x^2)/x)*(log(6 - log(x
))*(60*x - 10*x*log(x)) - 2*x^3*log(x) + 12*x^3) - 6*x^3)/(log((5*log(6 - log(x)) + x^2)/x)^2*(log(6 - log(x))
*(5*log(x) - 30) + x^2*log(x) - 6*x^2)), x)