\(\int \frac {-5 \log (3)+(x-12 x^6-9 x^9) \log ^2(x)}{-5 x \log (3) \log (x)+(-x^2+2 x^7+x^{10}) \log ^2(x)} \, dx\) [3023]

Optimal result

Integrand size = 52, antiderivative size = 35 \[ \int \frac {-5 \log (3)+\left (x-12 x^6-9 x^9\right ) \log ^2(x)}{-5 x \log (3) \log (x)+\left (-x^2+2 x^7+x^{10}\right ) \log ^2(x)} \, dx=\log \left (\frac {4}{\frac {x-x \left (-x^2+\left (x+x^4\right )^2\right )}{\log (3)}+\frac {5}{\log (x)}}\right ) \]

[Out]

ln(4/(5/ln(x)+(x-((x^4+x)^2-x^2)*x)/ln(3)))

Rubi [F]

\[ \int \frac {-5 \log (3)+\left (x-12 x^6-9 x^9\right ) \log ^2(x)}{-5 x \log (3) \log (x)+\left (-x^2+2 x^7+x^{10}\right ) \log ^2(x)} \, dx=\int \frac {-5 \log (3)+\left (x-12 x^6-9 x^9\right ) \log ^2(x)}{-5 x \log (3) \log (x)+\left (-x^2+2 x^7+x^{10}\right ) \log ^2(x)} \, dx \]

[In]

Int[(-5*Log[3] + (x - 12*x^6 - 9*x^9)*Log[x]^2)/(-5*x*Log[3]*Log[x] + (-x^2 + 2*x^7 + x^10)*Log[x]^2),x]

[Out]

-Log[-(x*(1 - 2*x^5 - x^8))] + Log[Log[x]] - Defer[Int][(5*Log[3] + x*Log[x] - 2*x^6*Log[x] - x^9*Log[x])^(-1)
, x] - 5*Log[3]*Defer[Int][1/(x*(-5*Log[3] - x*Log[x] + 2*x^6*Log[x] + x^9*Log[x])), x] - 2*Defer[Int][x^5/(-5
*Log[3] - x*Log[x] + 2*x^6*Log[x] + x^9*Log[x]), x] - Defer[Int][x^8/(-5*Log[3] - x*Log[x] + 2*x^6*Log[x] + x^
9*Log[x]), x] - 50*Log[3]*Defer[Int][x^4/((-1 + 2*x^5 + x^8)*(-5*Log[3] - x*Log[x] + 2*x^6*Log[x] + x^9*Log[x]
)), x] - 40*Log[3]*Defer[Int][x^7/((-1 + 2*x^5 + x^8)*(-5*Log[3] - x*Log[x] + 2*x^6*Log[x] + x^9*Log[x])), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {5 \log (3)-\left (x-12 x^6-9 x^9\right ) \log ^2(x)}{x \log (x) \left (5 \log (3)+x \log (x)-2 x^6 \log (x)-x^9 \log (x)\right )} \, dx \\ & = \int \left (\frac {1-12 x^5-9 x^8}{x \left (-1+2 x^5+x^8\right )}+\frac {1}{x \log (x)}+\frac {1-2 x^5-x^8}{-5 \log (3)-x \log (x)+2 x^6 \log (x)+x^9 \log (x)}-\frac {5 \left (-1+12 x^5+9 x^8\right ) \log (3)}{x \left (-1+2 x^5+x^8\right ) \left (-5 \log (3)-x \log (x)+2 x^6 \log (x)+x^9 \log (x)\right )}\right ) \, dx \\ & = -\left ((5 \log (3)) \int \frac {-1+12 x^5+9 x^8}{x \left (-1+2 x^5+x^8\right ) \left (-5 \log (3)-x \log (x)+2 x^6 \log (x)+x^9 \log (x)\right )} \, dx\right )+\int \frac {1-12 x^5-9 x^8}{x \left (-1+2 x^5+x^8\right )} \, dx+\int \frac {1}{x \log (x)} \, dx+\int \frac {1-2 x^5-x^8}{-5 \log (3)-x \log (x)+2 x^6 \log (x)+x^9 \log (x)} \, dx \\ & = -\log \left (-x \left (1-2 x^5-x^8\right )\right )-(5 \log (3)) \int \left (\frac {1}{x \left (-5 \log (3)-x \log (x)+2 x^6 \log (x)+x^9 \log (x)\right )}+\frac {2 x^4 \left (5+4 x^3\right )}{\left (-1+2 x^5+x^8\right ) \left (-5 \log (3)-x \log (x)+2 x^6 \log (x)+x^9 \log (x)\right )}\right ) \, dx+\int \left (-\frac {1}{5 \log (3)+x \log (x)-2 x^6 \log (x)-x^9 \log (x)}-\frac {2 x^5}{-5 \log (3)-x \log (x)+2 x^6 \log (x)+x^9 \log (x)}-\frac {x^8}{-5 \log (3)-x \log (x)+2 x^6 \log (x)+x^9 \log (x)}\right ) \, dx+\text {Subst}\left (\int \frac {1}{x} \, dx,x,\log (x)\right ) \\ & = -\log \left (-x \left (1-2 x^5-x^8\right )\right )+\log (\log (x))-2 \int \frac {x^5}{-5 \log (3)-x \log (x)+2 x^6 \log (x)+x^9 \log (x)} \, dx-(5 \log (3)) \int \frac {1}{x \left (-5 \log (3)-x \log (x)+2 x^6 \log (x)+x^9 \log (x)\right )} \, dx-(10 \log (3)) \int \frac {x^4 \left (5+4 x^3\right )}{\left (-1+2 x^5+x^8\right ) \left (-5 \log (3)-x \log (x)+2 x^6 \log (x)+x^9 \log (x)\right )} \, dx-\int \frac {1}{5 \log (3)+x \log (x)-2 x^6 \log (x)-x^9 \log (x)} \, dx-\int \frac {x^8}{-5 \log (3)-x \log (x)+2 x^6 \log (x)+x^9 \log (x)} \, dx \\ & = -\log \left (-x \left (1-2 x^5-x^8\right )\right )+\log (\log (x))-2 \int \frac {x^5}{-5 \log (3)-x \log (x)+2 x^6 \log (x)+x^9 \log (x)} \, dx-(5 \log (3)) \int \frac {1}{x \left (-5 \log (3)-x \log (x)+2 x^6 \log (x)+x^9 \log (x)\right )} \, dx-(10 \log (3)) \int \left (\frac {5 x^4}{\left (-1+2 x^5+x^8\right ) \left (-5 \log (3)-x \log (x)+2 x^6 \log (x)+x^9 \log (x)\right )}+\frac {4 x^7}{\left (-1+2 x^5+x^8\right ) \left (-5 \log (3)-x \log (x)+2 x^6 \log (x)+x^9 \log (x)\right )}\right ) \, dx-\int \frac {1}{5 \log (3)+x \log (x)-2 x^6 \log (x)-x^9 \log (x)} \, dx-\int \frac {x^8}{-5 \log (3)-x \log (x)+2 x^6 \log (x)+x^9 \log (x)} \, dx \\ & = -\log \left (-x \left (1-2 x^5-x^8\right )\right )+\log (\log (x))-2 \int \frac {x^5}{-5 \log (3)-x \log (x)+2 x^6 \log (x)+x^9 \log (x)} \, dx-(5 \log (3)) \int \frac {1}{x \left (-5 \log (3)-x \log (x)+2 x^6 \log (x)+x^9 \log (x)\right )} \, dx-(40 \log (3)) \int \frac {x^7}{\left (-1+2 x^5+x^8\right ) \left (-5 \log (3)-x \log (x)+2 x^6 \log (x)+x^9 \log (x)\right )} \, dx-(50 \log (3)) \int \frac {x^4}{\left (-1+2 x^5+x^8\right ) \left (-5 \log (3)-x \log (x)+2 x^6 \log (x)+x^9 \log (x)\right )} \, dx-\int \frac {1}{5 \log (3)+x \log (x)-2 x^6 \log (x)-x^9 \log (x)} \, dx-\int \frac {x^8}{-5 \log (3)-x \log (x)+2 x^6 \log (x)+x^9 \log (x)} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.48 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.83 \[ \int \frac {-5 \log (3)+\left (x-12 x^6-9 x^9\right ) \log ^2(x)}{-5 x \log (3) \log (x)+\left (-x^2+2 x^7+x^{10}\right ) \log ^2(x)} \, dx=-\log (x)-\log \left (1-2 x^5-x^8\right )+\log \left (x \left (1-2 x^5-x^8\right )\right )+\log (\log (x))-\log \left (5 \log (3)+x \log (x)-2 x^6 \log (x)-x^9 \log (x)\right ) \]

[In]

Integrate[(-5*Log[3] + (x - 12*x^6 - 9*x^9)*Log[x]^2)/(-5*x*Log[3]*Log[x] + (-x^2 + 2*x^7 + x^10)*Log[x]^2),x]

[Out]

-Log[x] - Log[1 - 2*x^5 - x^8] + Log[x*(1 - 2*x^5 - x^8)] + Log[Log[x]] - Log[5*Log[3] + x*Log[x] - 2*x^6*Log[
x] - x^9*Log[x]]

Maple [A] (verified)

Time = 1.15 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.69

method	result	size
default	\(-\ln \left (x^{9}+2 x^{6}-x -\frac {5 \ln \left (3\right )}{\ln \left (x \right )}\right )\)	\(24\)
parallelrisch	\(\ln \left (\ln \left (x \right )\right )-\ln \left (x^{9} \ln \left (x \right )+2 x^{6} \ln \left (x \right )-x \ln \left (x \right )-5 \ln \left (3\right )\right )\)	\(31\)
risch	\(-\ln \left (x^{9}+2 x^{6}-x \right )+\ln \left (\ln \left (x \right )\right )-\ln \left (\ln \left (x \right )-\frac {5 \ln \left (3\right )}{x \left (x^{8}+2 x^{5}-1\right )}\right )\)	\(45\)

[In]

int(((-9*x^9-12*x^6+x)*ln(x)^2-5*ln(3))/((x^10+2*x^7-x^2)*ln(x)^2-5*x*ln(3)*ln(x)),x,method=_RETURNVERBOSE)

[Out]

-ln(x^9+2*x^6-x-5*ln(3)/ln(x))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.63 \[ \int \frac {-5 \log (3)+\left (x-12 x^6-9 x^9\right ) \log ^2(x)}{-5 x \log (3) \log (x)+\left (-x^2+2 x^7+x^{10}\right ) \log ^2(x)} \, dx=-\log \left (x^{9} + 2 \, x^{6} - x\right ) - \log \left (\frac {{\left (x^{9} + 2 \, x^{6} - x\right )} \log \left (x\right ) - 5 \, \log \left (3\right )}{x^{9} + 2 \, x^{6} - x}\right ) + \log \left (\log \left (x\right )\right ) \]

[In]

integrate(((-9*x^9-12*x^6+x)*log(x)^2-5*log(3))/((x^10+2*x^7-x^2)*log(x)^2-5*x*log(3)*log(x)),x, algorithm="fr
icas")

[Out]

-log(x^9 + 2*x^6 - x) - log(((x^9 + 2*x^6 - x)*log(x) - 5*log(3))/(x^9 + 2*x^6 - x)) + log(log(x))

Sympy [F(-2)]

Exception generated. \[ \int \frac {-5 \log (3)+\left (x-12 x^6-9 x^9\right ) \log ^2(x)}{-5 x \log (3) \log (x)+\left (-x^2+2 x^7+x^{10}\right ) \log ^2(x)} \, dx=\text {Exception raised: PolynomialError} \]

[In]

integrate(((-9*x**9-12*x**6+x)*ln(x)**2-5*ln(3))/((x**10+2*x**7-x**2)*ln(x)**2-5*x*ln(3)*ln(x)),x)

[Out]

Exception raised: PolynomialError >> 1/(x**18 + 4*x**15 + 4*x**12 - 2*x**10 - 4*x**7 + x**2) contains an eleme
nt of the set of generators.

Maxima [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.69 \[ \int \frac {-5 \log (3)+\left (x-12 x^6-9 x^9\right ) \log ^2(x)}{-5 x \log (3) \log (x)+\left (-x^2+2 x^7+x^{10}\right ) \log ^2(x)} \, dx=-\log \left (x^{8} + 2 \, x^{5} - 1\right ) - \log \left (x\right ) - \log \left (\frac {{\left (x^{9} + 2 \, x^{6} - x\right )} \log \left (x\right ) - 5 \, \log \left (3\right )}{x^{9} + 2 \, x^{6} - x}\right ) + \log \left (\log \left (x\right )\right ) \]

[In]

integrate(((-9*x^9-12*x^6+x)*log(x)^2-5*log(3))/((x^10+2*x^7-x^2)*log(x)^2-5*x*log(3)*log(x)),x, algorithm="ma
xima")

[Out]

-log(x^8 + 2*x^5 - 1) - log(x) - log(((x^9 + 2*x^6 - x)*log(x) - 5*log(3))/(x^9 + 2*x^6 - x)) + log(log(x))

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.86 \[ \int \frac {-5 \log (3)+\left (x-12 x^6-9 x^9\right ) \log ^2(x)}{-5 x \log (3) \log (x)+\left (-x^2+2 x^7+x^{10}\right ) \log ^2(x)} \, dx=-\log \left (-x^{9} \log \left (x\right ) - 2 \, x^{6} \log \left (x\right ) + x \log \left (x\right ) + 5 \, \log \left (3\right )\right ) + \log \left (\log \left (x\right )\right ) \]

[In]

integrate(((-9*x^9-12*x^6+x)*log(x)^2-5*log(3))/((x^10+2*x^7-x^2)*log(x)^2-5*x*log(3)*log(x)),x, algorithm="gi
ac")

[Out]

-log(-x^9*log(x) - 2*x^6*log(x) + x*log(x) + 5*log(3)) + log(log(x))

Mupad [B] (verification not implemented)

Time = 38.39 (sec) , antiderivative size = 186, normalized size of antiderivative = 5.31 \[ \int \frac {-5 \log (3)+\left (x-12 x^6-9 x^9\right ) \log ^2(x)}{-5 x \log (3) \log (x)+\left (-x^2+2 x^7+x^{10}\right ) \log ^2(x)} \, dx=\ln \left (8\,x^6\,\ln \left (x\right )+4\,x^9\,\ln \left (x\right )-8\,x^{11}\,\ln \left (x\right )-8\,x^{14}\,\ln \left (x\right )-2\,x^{17}\,\ln \left (x\right )+10\,\ln \left (3\right )\,\ln \left (x\right )-2\,x\,\ln \left (x\right )-120\,x^5\,\ln \left (3\right )\,\ln \left (x\right )-90\,x^8\,\ln \left (3\right )\,\ln \left (x\right )\right )-\ln \left (10\,\ln \left (3\right )-8\,x^6\,\ln \left (x\right )-4\,x^9\,\ln \left (x\right )+8\,x^{11}\,\ln \left (x\right )+8\,x^{14}\,\ln \left (x\right )+2\,x^{17}\,\ln \left (x\right )-20\,x^5\,\ln \left (3\right )-10\,x^8\,\ln \left (3\right )+2\,x\,\ln \left (x\right )\right )-\ln \left (x^{17}+4\,x^{14}+4\,x^{11}-2\,x^9+45\,\ln \left (3\right )\,x^8-4\,x^6+60\,\ln \left (3\right )\,x^5+x-5\,\ln \left (3\right )\right )+\ln \left (x^8+2\,x^5-1\right ) \]

[In]

int(-(5*log(3) + log(x)^2*(12*x^6 - x + 9*x^9))/(log(x)^2*(2*x^7 - x^2 + x^10) - 5*x*log(3)*log(x)),x)

[Out]

log(8*x^6*log(x) + 4*x^9*log(x) - 8*x^11*log(x) - 8*x^14*log(x) - 2*x^17*log(x) + 10*log(3)*log(x) - 2*x*log(x
) - 120*x^5*log(3)*log(x) - 90*x^8*log(3)*log(x)) - log(10*log(3) - 8*x^6*log(x) - 4*x^9*log(x) + 8*x^11*log(x
) + 8*x^14*log(x) + 2*x^17*log(x) - 20*x^5*log(3) - 10*x^8*log(3) + 2*x*log(x)) - log(x - 5*log(3) + 60*x^5*lo
g(3) + 45*x^8*log(3) - 4*x^6 - 2*x^9 + 4*x^11 + 4*x^14 + x^17) + log(2*x^5 + x^8 - 1)