\(\int \frac {2 x \log (x)+(-216 x^3+216 x^4-72 x^5+8 x^6) \log ^3(x)+(6-2 x+(6-4 x) \log (x)) \log (9-6 x+x^2)}{(-27 x^3+27 x^4-9 x^5+x^6) \log ^3(x)} \, dx\) [6477]

Optimal result

Integrand size = 81, antiderivative size = 23 \[ \int \frac {2 x \log (x)+\left (-216 x^3+216 x^4-72 x^5+8 x^6\right ) \log ^3(x)+(6-2 x+(6-4 x) \log (x)) \log \left (9-6 x+x^2\right )}{\left (-27 x^3+27 x^4-9 x^5+x^6\right ) \log ^3(x)} \, dx=8 x+\frac {\log \left ((-3+x)^2\right )}{(-3+x)^2 x^2 \log ^2(x)} \]

[Out]

ln((-3+x)^2)/ln(x)^2/x^2/(-3+x)^2+8*x

Rubi [F]

\[ \int \frac {2 x \log (x)+\left (-216 x^3+216 x^4-72 x^5+8 x^6\right ) \log ^3(x)+(6-2 x+(6-4 x) \log (x)) \log \left (9-6 x+x^2\right )}{\left (-27 x^3+27 x^4-9 x^5+x^6\right ) \log ^3(x)} \, dx=\int \frac {2 x \log (x)+\left (-216 x^3+216 x^4-72 x^5+8 x^6\right ) \log ^3(x)+(6-2 x+(6-4 x) \log (x)) \log \left (9-6 x+x^2\right )}{\left (-27 x^3+27 x^4-9 x^5+x^6\right ) \log ^3(x)} \, dx \]

[In]

Int[(2*x*Log[x] + (-216*x^3 + 216*x^4 - 72*x^5 + 8*x^6)*Log[x]^3 + (6 - 2*x + (6 - 4*x)*Log[x])*Log[9 - 6*x +
x^2])/((-27*x^3 + 27*x^4 - 9*x^5 + x^6)*Log[x]^3),x]

[Out]

8*x - (2*Defer[Int][Log[(-3 + x)^2]/((-3 + x)^2*Log[x]^3), x])/27 + (2*Defer[Int][Log[(-3 + x)^2]/((-3 + x)*Lo
g[x]^3), x])/27 - (2*Defer[Int][Log[(-3 + x)^2]/(x^3*Log[x]^3), x])/9 - (4*Defer[Int][Log[(-3 + x)^2]/(x^2*Log
[x]^3), x])/27 - (2*Defer[Int][Log[(-3 + x)^2]/(x*Log[x]^3), x])/27 + 2*Defer[Int][1/((-3 + x)^3*x^2*Log[x]^2)
, x] - (2*Defer[Int][Log[(-3 + x)^2]/((-3 + x)^3*Log[x]^2), x])/9 + (2*Defer[Int][Log[(-3 + x)^2]/((-3 + x)^2*
Log[x]^2), x])/27 - (2*Defer[Int][Log[(-3 + x)^2]/(x^3*Log[x]^2), x])/9 - (2*Defer[Int][Log[(-3 + x)^2]/(x^2*L
og[x]^2), x])/27

Rubi steps \begin{align*} \text {integral}& = \int \left (8+\frac {2}{(-3+x)^3 x^2 \log ^2(x)}-\frac {2 \log \left ((-3+x)^2\right ) (-3+x+(-3+2 x) \log (x))}{(-3+x)^3 x^3 \log ^3(x)}\right ) \, dx \\ & = 8 x+2 \int \frac {1}{(-3+x)^3 x^2 \log ^2(x)} \, dx-2 \int \frac {\log \left ((-3+x)^2\right ) (-3+x+(-3+2 x) \log (x))}{(-3+x)^3 x^3 \log ^3(x)} \, dx \\ & = 8 x-2 \int \left (\frac {\log \left ((-3+x)^2\right )}{(-3+x)^2 x^3 \log ^3(x)}+\frac {(-3+2 x) \log \left ((-3+x)^2\right )}{(-3+x)^3 x^3 \log ^2(x)}\right ) \, dx+2 \int \frac {1}{(-3+x)^3 x^2 \log ^2(x)} \, dx \\ & = 8 x-2 \int \frac {\log \left ((-3+x)^2\right )}{(-3+x)^2 x^3 \log ^3(x)} \, dx+2 \int \frac {1}{(-3+x)^3 x^2 \log ^2(x)} \, dx-2 \int \frac {(-3+2 x) \log \left ((-3+x)^2\right )}{(-3+x)^3 x^3 \log ^2(x)} \, dx \\ & = 8 x-2 \int \left (\frac {\log \left ((-3+x)^2\right )}{27 (-3+x)^2 \log ^3(x)}-\frac {\log \left ((-3+x)^2\right )}{27 (-3+x) \log ^3(x)}+\frac {\log \left ((-3+x)^2\right )}{9 x^3 \log ^3(x)}+\frac {2 \log \left ((-3+x)^2\right )}{27 x^2 \log ^3(x)}+\frac {\log \left ((-3+x)^2\right )}{27 x \log ^3(x)}\right ) \, dx-2 \int \left (\frac {\log \left ((-3+x)^2\right )}{9 (-3+x)^3 \log ^2(x)}-\frac {\log \left ((-3+x)^2\right )}{27 (-3+x)^2 \log ^2(x)}+\frac {\log \left ((-3+x)^2\right )}{9 x^3 \log ^2(x)}+\frac {\log \left ((-3+x)^2\right )}{27 x^2 \log ^2(x)}\right ) \, dx+2 \int \frac {1}{(-3+x)^3 x^2 \log ^2(x)} \, dx \\ & = 8 x-\frac {2}{27} \int \frac {\log \left ((-3+x)^2\right )}{(-3+x)^2 \log ^3(x)} \, dx+\frac {2}{27} \int \frac {\log \left ((-3+x)^2\right )}{(-3+x) \log ^3(x)} \, dx-\frac {2}{27} \int \frac {\log \left ((-3+x)^2\right )}{x \log ^3(x)} \, dx+\frac {2}{27} \int \frac {\log \left ((-3+x)^2\right )}{(-3+x)^2 \log ^2(x)} \, dx-\frac {2}{27} \int \frac {\log \left ((-3+x)^2\right )}{x^2 \log ^2(x)} \, dx-\frac {4}{27} \int \frac {\log \left ((-3+x)^2\right )}{x^2 \log ^3(x)} \, dx-\frac {2}{9} \int \frac {\log \left ((-3+x)^2\right )}{x^3 \log ^3(x)} \, dx-\frac {2}{9} \int \frac {\log \left ((-3+x)^2\right )}{(-3+x)^3 \log ^2(x)} \, dx-\frac {2}{9} \int \frac {\log \left ((-3+x)^2\right )}{x^3 \log ^2(x)} \, dx+2 \int \frac {1}{(-3+x)^3 x^2 \log ^2(x)} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 4.21 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {2 x \log (x)+\left (-216 x^3+216 x^4-72 x^5+8 x^6\right ) \log ^3(x)+(6-2 x+(6-4 x) \log (x)) \log \left (9-6 x+x^2\right )}{\left (-27 x^3+27 x^4-9 x^5+x^6\right ) \log ^3(x)} \, dx=8 x+\frac {\log \left ((-3+x)^2\right )}{(-3+x)^2 x^2 \log ^2(x)} \]

[In]

Integrate[(2*x*Log[x] + (-216*x^3 + 216*x^4 - 72*x^5 + 8*x^6)*Log[x]^3 + (6 - 2*x + (6 - 4*x)*Log[x])*Log[9 -
6*x + x^2])/((-27*x^3 + 27*x^4 - 9*x^5 + x^6)*Log[x]^3),x]

[Out]

8*x + Log[(-3 + x)^2]/((-3 + x)^2*x^2*Log[x]^2)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(62\) vs. \(2(23)=46\).

Time = 5.42 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.74

method	result	size
parallelrisch	\(-\frac {-864 x^{5} \ln \left (x \right )^{2}+3456 x^{4} \ln \left (x \right )^{2}+2592 x^{3} \ln \left (x \right )^{2}-15552 x^{2} \ln \left (x \right )^{2}-108 \ln \left (x^{2}-6 x +9\right )}{108 x^{2} \ln \left (x \right )^{2} \left (-3+x \right )^{2}}\)	\(63\)
risch	\(\frac {2 \ln \left (-3+x \right )}{x^{2} \left (x^{2}-6 x +9\right ) \ln \left (x \right )^{2}}+\frac {16 x^{5} \ln \left (x \right )^{2}-96 x^{4} \ln \left (x \right )^{2}-i \pi \operatorname {csgn}\left (i \left (-3+x \right )^{2}\right )^{3}+2 i \pi \operatorname {csgn}\left (i \left (-3+x \right )^{2}\right )^{2} \operatorname {csgn}\left (i \left (-3+x \right )\right )-i \pi \,\operatorname {csgn}\left (i \left (-3+x \right )^{2}\right ) \operatorname {csgn}\left (i \left (-3+x \right )\right )^{2}+144 x^{3} \ln \left (x \right )^{2}}{2 x^{2} \left (x^{2}-6 x +9\right ) \ln \left (x \right )^{2}}\)	\(131\)

[In]

int((((6-4*x)*ln(x)+6-2*x)*ln(x^2-6*x+9)+(8*x^6-72*x^5+216*x^4-216*x^3)*ln(x)^3+2*x*ln(x))/(x^6-9*x^5+27*x^4-2
7*x^3)/ln(x)^3,x,method=_RETURNVERBOSE)

[Out]

-1/108*(-864*x^5*ln(x)^2+3456*x^4*ln(x)^2+2592*x^3*ln(x)^2-15552*x^2*ln(x)^2-108*ln(x^2-6*x+9))/x^2/ln(x)^2/(-
3+x)^2

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (23) = 46\).

Time = 0.26 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.22 \[ \int \frac {2 x \log (x)+\left (-216 x^3+216 x^4-72 x^5+8 x^6\right ) \log ^3(x)+(6-2 x+(6-4 x) \log (x)) \log \left (9-6 x+x^2\right )}{\left (-27 x^3+27 x^4-9 x^5+x^6\right ) \log ^3(x)} \, dx=\frac {8 \, {\left (x^{5} - 6 \, x^{4} + 9 \, x^{3}\right )} \log \left (x\right )^{2} + \log \left (x^{2} - 6 \, x + 9\right )}{{\left (x^{4} - 6 \, x^{3} + 9 \, x^{2}\right )} \log \left (x\right )^{2}} \]

[In]

integrate((((6-4*x)*log(x)+6-2*x)*log(x^2-6*x+9)+(8*x^6-72*x^5+216*x^4-216*x^3)*log(x)^3+2*x*log(x))/(x^6-9*x^
5+27*x^4-27*x^3)/log(x)^3,x, algorithm="fricas")

[Out]

(8*(x^5 - 6*x^4 + 9*x^3)*log(x)^2 + log(x^2 - 6*x + 9))/((x^4 - 6*x^3 + 9*x^2)*log(x)^2)

Sympy [F(-2)]

Exception generated. \[ \int \frac {2 x \log (x)+\left (-216 x^3+216 x^4-72 x^5+8 x^6\right ) \log ^3(x)+(6-2 x+(6-4 x) \log (x)) \log \left (9-6 x+x^2\right )}{\left (-27 x^3+27 x^4-9 x^5+x^6\right ) \log ^3(x)} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((((6-4*x)*ln(x)+6-2*x)*ln(x**2-6*x+9)+(8*x**6-72*x**5+216*x**4-216*x**3)*ln(x)**3+2*x*ln(x))/(x**6-9
*x**5+27*x**4-27*x**3)/ln(x)**3,x)

[Out]

Exception raised: TypeError >> '>' not supported between instances of 'Poly' and 'int'

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 47 vs. \(2 (23) = 46\).

Time = 0.22 (sec) , antiderivative size = 47, normalized size of antiderivative = 2.04 \[ \int \frac {2 x \log (x)+\left (-216 x^3+216 x^4-72 x^5+8 x^6\right ) \log ^3(x)+(6-2 x+(6-4 x) \log (x)) \log \left (9-6 x+x^2\right )}{\left (-27 x^3+27 x^4-9 x^5+x^6\right ) \log ^3(x)} \, dx=\frac {2 \, {\left (4 \, {\left (x^{5} - 6 \, x^{4} + 9 \, x^{3}\right )} \log \left (x\right )^{2} + \log \left (x - 3\right )\right )}}{{\left (x^{4} - 6 \, x^{3} + 9 \, x^{2}\right )} \log \left (x\right )^{2}} \]

[In]

integrate((((6-4*x)*log(x)+6-2*x)*log(x^2-6*x+9)+(8*x^6-72*x^5+216*x^4-216*x^3)*log(x)^3+2*x*log(x))/(x^6-9*x^
5+27*x^4-27*x^3)/log(x)^3,x, algorithm="maxima")

[Out]

2*(4*(x^5 - 6*x^4 + 9*x^3)*log(x)^2 + log(x - 3))/((x^4 - 6*x^3 + 9*x^2)*log(x)^2)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.87 \[ \int \frac {2 x \log (x)+\left (-216 x^3+216 x^4-72 x^5+8 x^6\right ) \log ^3(x)+(6-2 x+(6-4 x) \log (x)) \log \left (9-6 x+x^2\right )}{\left (-27 x^3+27 x^4-9 x^5+x^6\right ) \log ^3(x)} \, dx=8 \, x + \frac {\log \left (x^{2} - 6 \, x + 9\right )}{x^{4} \log \left (x\right )^{2} - 6 \, x^{3} \log \left (x\right )^{2} + 9 \, x^{2} \log \left (x\right )^{2}} \]

[In]

integrate((((6-4*x)*log(x)+6-2*x)*log(x^2-6*x+9)+(8*x^6-72*x^5+216*x^4-216*x^3)*log(x)^3+2*x*log(x))/(x^6-9*x^
5+27*x^4-27*x^3)/log(x)^3,x, algorithm="giac")

[Out]

8*x + log(x^2 - 6*x + 9)/(x^4*log(x)^2 - 6*x^3*log(x)^2 + 9*x^2*log(x)^2)

Mupad [B] (verification not implemented)

Time = 13.78 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.13 \[ \int \frac {2 x \log (x)+\left (-216 x^3+216 x^4-72 x^5+8 x^6\right ) \log ^3(x)+(6-2 x+(6-4 x) \log (x)) \log \left (9-6 x+x^2\right )}{\left (-27 x^3+27 x^4-9 x^5+x^6\right ) \log ^3(x)} \, dx=8\,x+\frac {\ln \left (x^2-6\,x+9\right )}{x^2\,{\ln \left (x\right )}^2\,{\left (x-3\right )}^2} \]

[In]

int((log(x^2 - 6*x + 9)*(2*x + log(x)*(4*x - 6) - 6) - 2*x*log(x) + log(x)^3*(216*x^3 - 216*x^4 + 72*x^5 - 8*x
^6))/(log(x)^3*(27*x^3 - 27*x^4 + 9*x^5 - x^6)),x)

[Out]

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