\(\int \frac {-4-4 x+(-4-6 x) \log (x)-1296 x^3 \log ^3(x)}{(81 x^3+162 x^4+81 x^5) \log ^3(x)} \, dx\) [2983]

Optimal result

Integrand size = 45, antiderivative size = 20 \[ \int \frac {-4-4 x+(-4-6 x) \log (x)-1296 x^3 \log ^3(x)}{\left (81 x^3+162 x^4+81 x^5\right ) \log ^3(x)} \, dx=\frac {2 \left (8+\frac {1}{81 x^2 \log ^2(x)}\right )}{1+x} \]

[Out]

2*(1/81/x^2/ln(x)^2+8)/(1+x)

Rubi [F]

\[ \int \frac {-4-4 x+(-4-6 x) \log (x)-1296 x^3 \log ^3(x)}{\left (81 x^3+162 x^4+81 x^5\right ) \log ^3(x)} \, dx=\int \frac {-4-4 x+(-4-6 x) \log (x)-1296 x^3 \log ^3(x)}{\left (81 x^3+162 x^4+81 x^5\right ) \log ^3(x)} \, dx \]

[In]

Int[(-4 - 4*x + (-4 - 6*x)*Log[x] - 1296*x^3*Log[x]^3)/((81*x^3 + 162*x^4 + 81*x^5)*Log[x]^3),x]

[Out]

16/(1 + x) - (4*Defer[Int][1/(x^3*(1 + x)*Log[x]^3), x])/81 - (2*Defer[Int][(2 + 3*x)/(x^3*(1 + x)^2*Log[x]^2)
, x])/81

Rubi steps \begin{align*} \text {integral}& = \int \frac {-4-4 x+(-4-6 x) \log (x)-1296 x^3 \log ^3(x)}{x^3 \left (81+162 x+81 x^2\right ) \log ^3(x)} \, dx \\ & = \int \frac {-4-4 x+(-4-6 x) \log (x)-1296 x^3 \log ^3(x)}{81 x^3 (1+x)^2 \log ^3(x)} \, dx \\ & = \frac {1}{81} \int \frac {-4-4 x+(-4-6 x) \log (x)-1296 x^3 \log ^3(x)}{x^3 (1+x)^2 \log ^3(x)} \, dx \\ & = \frac {1}{81} \int \left (-\frac {1296}{(1+x)^2}-\frac {4}{x^3 (1+x) \log ^3(x)}-\frac {2 (2+3 x)}{x^3 (1+x)^2 \log ^2(x)}\right ) \, dx \\ & = \frac {16}{1+x}-\frac {2}{81} \int \frac {2+3 x}{x^3 (1+x)^2 \log ^2(x)} \, dx-\frac {4}{81} \int \frac {1}{x^3 (1+x) \log ^3(x)} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95 \[ \int \frac {-4-4 x+(-4-6 x) \log (x)-1296 x^3 \log ^3(x)}{\left (81 x^3+162 x^4+81 x^5\right ) \log ^3(x)} \, dx=\frac {2 \left (648+\frac {1}{x^2 \log ^2(x)}\right )}{81 (1+x)} \]

[In]

Integrate[(-4 - 4*x + (-4 - 6*x)*Log[x] - 1296*x^3*Log[x]^3)/((81*x^3 + 162*x^4 + 81*x^5)*Log[x]^3),x]

[Out]

(2*(648 + 1/(x^2*Log[x]^2)))/(81*(1 + x))

Maple [A] (verified)

Time = 5.50 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.15

method	result	size
risch	\(\frac {16}{1+x}+\frac {2}{81 x^{2} \left (1+x \right ) \ln \left (x \right )^{2}}\)	\(23\)
norman	\(\frac {\frac {2}{81}+16 x^{2} \ln \left (x \right )^{2}}{x^{2} \left (1+x \right ) \ln \left (x \right )^{2}}\)	\(25\)
parallelrisch	\(\frac {2+1296 x^{2} \ln \left (x \right )^{2}}{81 x^{2} \ln \left (x \right )^{2} \left (1+x \right )}\)	\(26\)

[In]

int((-1296*x^3*ln(x)^3+(-4-6*x)*ln(x)-4*x-4)/(81*x^5+162*x^4+81*x^3)/ln(x)^3,x,method=_RETURNVERBOSE)

[Out]

16/(1+x)+2/81/x^2/(1+x)/ln(x)^2

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.30 \[ \int \frac {-4-4 x+(-4-6 x) \log (x)-1296 x^3 \log ^3(x)}{\left (81 x^3+162 x^4+81 x^5\right ) \log ^3(x)} \, dx=\frac {2 \, {\left (648 \, x^{2} \log \left (x\right )^{2} + 1\right )}}{81 \, {\left (x^{3} + x^{2}\right )} \log \left (x\right )^{2}} \]

[In]

integrate((-1296*x^3*log(x)^3+(-4-6*x)*log(x)-4*x-4)/(81*x^5+162*x^4+81*x^3)/log(x)^3,x, algorithm="fricas")

[Out]

2/81*(648*x^2*log(x)^2 + 1)/((x^3 + x^2)*log(x)^2)

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {-4-4 x+(-4-6 x) \log (x)-1296 x^3 \log ^3(x)}{\left (81 x^3+162 x^4+81 x^5\right ) \log ^3(x)} \, dx=\frac {2}{\left (81 x^{3} + 81 x^{2}\right ) \log {\left (x \right )}^{2}} + \frac {16}{x + 1} \]

[In]

integrate((-1296*x**3*ln(x)**3+(-4-6*x)*ln(x)-4*x-4)/(81*x**5+162*x**4+81*x**3)/ln(x)**3,x)

[Out]

2/((81*x**3 + 81*x**2)*log(x)**2) + 16/(x + 1)

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.30 \[ \int \frac {-4-4 x+(-4-6 x) \log (x)-1296 x^3 \log ^3(x)}{\left (81 x^3+162 x^4+81 x^5\right ) \log ^3(x)} \, dx=\frac {2 \, {\left (648 \, x^{2} \log \left (x\right )^{2} + 1\right )}}{81 \, {\left (x^{3} + x^{2}\right )} \log \left (x\right )^{2}} \]

[In]

integrate((-1296*x^3*log(x)^3+(-4-6*x)*log(x)-4*x-4)/(81*x^5+162*x^4+81*x^3)/log(x)^3,x, algorithm="maxima")

[Out]

2/81*(648*x^2*log(x)^2 + 1)/((x^3 + x^2)*log(x)^2)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.45 \[ \int \frac {-4-4 x+(-4-6 x) \log (x)-1296 x^3 \log ^3(x)}{\left (81 x^3+162 x^4+81 x^5\right ) \log ^3(x)} \, dx=\frac {2}{81 \, {\left (x^{3} \log \left (x\right )^{2} + x^{2} \log \left (x\right )^{2}\right )}} + \frac {16}{x + 1} \]

[In]

integrate((-1296*x^3*log(x)^3+(-4-6*x)*log(x)-4*x-4)/(81*x^5+162*x^4+81*x^3)/log(x)^3,x, algorithm="giac")

[Out]

2/81/(x^3*log(x)^2 + x^2*log(x)^2) + 16/(x + 1)

Mupad [B] (verification not implemented)

Time = 10.07 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.25 \[ \int \frac {-4-4 x+(-4-6 x) \log (x)-1296 x^3 \log ^3(x)}{\left (81 x^3+162 x^4+81 x^5\right ) \log ^3(x)} \, dx=-\frac {16\,x^3\,{\ln \left (x\right )}^2-\frac {2}{81}}{x^2\,{\ln \left (x\right )}^2\,\left (x+1\right )} \]

[In]

int(-(4*x + log(x)*(6*x + 4) + 1296*x^3*log(x)^3 + 4)/(log(x)^3*(81*x^3 + 162*x^4 + 81*x^5)),x)

[Out]

-(16*x^3*log(x)^2 - 2/81)/(x^2*log(x)^2*(x + 1))