\(\int \frac {216-24 x-24 x^2-4 x^3-4 x^4+(-24 x-2 x^3) \log (x)}{-9 x^3+x^4+x^5+x^4 \log (x)} \, dx\) [7344]

Optimal result

Integrand size = 53, antiderivative size = 25 \[ \int \frac {216-24 x-24 x^2-4 x^3-4 x^4+\left (-24 x-2 x^3\right ) \log (x)}{-9 x^3+x^4+x^5+x^4 \log (x)} \, dx=\frac {12}{x^2}-\log \left (9 \left (-3+\frac {1}{3} x (1+x+\log (x))\right )^2\right ) \]

[Out]

12/x^2-ln(3*(1/3*(1+x+ln(x))*x-3)*((1+x+ln(x))*x-9))

Rubi [F]

\[ \int \frac {216-24 x-24 x^2-4 x^3-4 x^4+\left (-24 x-2 x^3\right ) \log (x)}{-9 x^3+x^4+x^5+x^4 \log (x)} \, dx=\int \frac {216-24 x-24 x^2-4 x^3-4 x^4+\left (-24 x-2 x^3\right ) \log (x)}{-9 x^3+x^4+x^5+x^4 \log (x)} \, dx \]

[In]

Int[(216 - 24*x - 24*x^2 - 4*x^3 - 4*x^4 + (-24*x - 2*x^3)*Log[x])/(-9*x^3 + x^4 + x^5 + x^4*Log[x]),x]

[Out]

12/x^2 - 2*Log[x] + 2*Defer[Int][(9 - x - x^2 - x*Log[x])^(-1), x] - 18*Defer[Int][1/(x*(-9 + x + x^2 + x*Log[
x])), x] - 2*Defer[Int][x/(-9 + x + x^2 + x*Log[x]), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {2 \left (-108+12 x+12 x^2+2 x^3+2 x^4+12 x \log (x)+x^3 \log (x)\right )}{9 x^3-x^4-x^5-x^4 \log (x)} \, dx \\ & = 2 \int \frac {-108+12 x+12 x^2+2 x^3+2 x^4+12 x \log (x)+x^3 \log (x)}{9 x^3-x^4-x^5-x^4 \log (x)} \, dx \\ & = 2 \int \left (\frac {-12-x^2}{x^3}+\frac {-9-x-x^2}{x \left (-9+x+x^2+x \log (x)\right )}\right ) \, dx \\ & = 2 \int \frac {-12-x^2}{x^3} \, dx+2 \int \frac {-9-x-x^2}{x \left (-9+x+x^2+x \log (x)\right )} \, dx \\ & = 2 \int \left (-\frac {12}{x^3}-\frac {1}{x}\right ) \, dx+2 \int \left (\frac {1}{9-x-x^2-x \log (x)}-\frac {9}{x \left (-9+x+x^2+x \log (x)\right )}-\frac {x}{-9+x+x^2+x \log (x)}\right ) \, dx \\ & = \frac {12}{x^2}-2 \log (x)+2 \int \frac {1}{9-x-x^2-x \log (x)} \, dx-2 \int \frac {x}{-9+x+x^2+x \log (x)} \, dx-18 \int \frac {1}{x \left (-9+x+x^2+x \log (x)\right )} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {216-24 x-24 x^2-4 x^3-4 x^4+\left (-24 x-2 x^3\right ) \log (x)}{-9 x^3+x^4+x^5+x^4 \log (x)} \, dx=-2 \left (-\frac {6}{x^2}+\log \left (9-x-x^2-x \log (x)\right )\right ) \]

[In]

Integrate[(216 - 24*x - 24*x^2 - 4*x^3 - 4*x^4 + (-24*x - 2*x^3)*Log[x])/(-9*x^3 + x^4 + x^5 + x^4*Log[x]),x]

[Out]

-2*(-6/x^2 + Log[9 - x - x^2 - x*Log[x]])

Maple [A] (verified)

Time = 1.04 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80

method	result	size
default	\(\frac {12}{x^{2}}-2 \ln \left (x \ln \left (x \right )+x^{2}+x -9\right )\)	\(20\)
norman	\(\frac {12}{x^{2}}-2 \ln \left (x \ln \left (x \right )+x^{2}+x -9\right )\)	\(20\)
parallelrisch	\(\frac {-2 \ln \left (x \ln \left (x \right )+x^{2}+x -9\right ) x^{2}+12}{x^{2}}\)	\(23\)
risch	\(-\frac {2 \left (x^{2} \ln \left (x \right )-6\right )}{x^{2}}-2 \ln \left (\ln \left (x \right )+\frac {x^{2}+x -9}{x}\right )\)	\(31\)

[In]

int(((-2*x^3-24*x)*ln(x)-4*x^4-4*x^3-24*x^2-24*x+216)/(x^4*ln(x)+x^5+x^4-9*x^3),x,method=_RETURNVERBOSE)

[Out]

12/x^2-2*ln(x*ln(x)+x^2+x-9)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.28 \[ \int \frac {216-24 x-24 x^2-4 x^3-4 x^4+\left (-24 x-2 x^3\right ) \log (x)}{-9 x^3+x^4+x^5+x^4 \log (x)} \, dx=-\frac {2 \, {\left (x^{2} \log \left (x\right ) + x^{2} \log \left (\frac {x^{2} + x \log \left (x\right ) + x - 9}{x}\right ) - 6\right )}}{x^{2}} \]

[In]

integrate(((-2*x^3-24*x)*log(x)-4*x^4-4*x^3-24*x^2-24*x+216)/(x^4*log(x)+x^5+x^4-9*x^3),x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {216-24 x-24 x^2-4 x^3-4 x^4+\left (-24 x-2 x^3\right ) \log (x)}{-9 x^3+x^4+x^5+x^4 \log (x)} \, dx=- 2 \log {\left (x \right )} - 2 \log {\left (\log {\left (x \right )} + \frac {x^{2} + x - 9}{x} \right )} + \frac {12}{x^{2}} \]

[In]

integrate(((-2*x**3-24*x)*ln(x)-4*x**4-4*x**3-24*x**2-24*x+216)/(x**4*ln(x)+x**5+x**4-9*x**3),x)

[Out]

-2*log(x) - 2*log(log(x) + (x**2 + x - 9)/x) + 12/x**2

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {216-24 x-24 x^2-4 x^3-4 x^4+\left (-24 x-2 x^3\right ) \log (x)}{-9 x^3+x^4+x^5+x^4 \log (x)} \, dx=\frac {12}{x^{2}} - 2 \, \log \left (x\right ) - 2 \, \log \left (\frac {x^{2} + x \log \left (x\right ) + x - 9}{x}\right ) \]

[In]

integrate(((-2*x^3-24*x)*log(x)-4*x^4-4*x^3-24*x^2-24*x+216)/(x^4*log(x)+x^5+x^4-9*x^3),x, algorithm="maxima")

[Out]

12/x^2 - 2*log(x) - 2*log((x^2 + x*log(x) + x - 9)/x)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int \frac {216-24 x-24 x^2-4 x^3-4 x^4+\left (-24 x-2 x^3\right ) \log (x)}{-9 x^3+x^4+x^5+x^4 \log (x)} \, dx=\frac {12}{x^{2}} - 2 \, \log \left (x^{2} + x \log \left (x\right ) + x - 9\right ) \]

[In]

integrate(((-2*x^3-24*x)*log(x)-4*x^4-4*x^3-24*x^2-24*x+216)/(x^4*log(x)+x^5+x^4-9*x^3),x, algorithm="giac")

[Out]

12/x^2 - 2*log(x^2 + x*log(x) + x - 9)

Mupad [B] (verification not implemented)

Time = 14.44 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {216-24 x-24 x^2-4 x^3-4 x^4+\left (-24 x-2 x^3\right ) \log (x)}{-9 x^3+x^4+x^5+x^4 \log (x)} \, dx=\frac {12}{x^2}-2\,\ln \left (x\right )-2\,\ln \left (\frac {x+x\,\ln \left (x\right )+x^2-9}{x}\right ) \]

[In]

int(-(24*x + log(x)*(24*x + 2*x^3) + 24*x^2 + 4*x^3 + 4*x^4 - 216)/(x^4*log(x) - 9*x^3 + x^4 + x^5),x)

[Out]

12/x^2 - 2*log(x) - 2*log((x + x*log(x) + x^2 - 9)/x)