\(\int \frac {-4 x-4 x^3+6 x^8+(4 x+18 x^6) \log (x)+18 x^4 \log ^2(x)+6 x^2 \log ^3(x)}{3 x^6+9 x^4 \log (x)+9 x^2 \log ^2(x)+3 \log ^3(x)} \, dx\) [1295]

Optimal result

Integrand size = 75, antiderivative size = 17 \[ \int \frac {-4 x-4 x^3+6 x^8+\left (4 x+18 x^6\right ) \log (x)+18 x^4 \log ^2(x)+6 x^2 \log ^3(x)}{3 x^6+9 x^4 \log (x)+9 x^2 \log ^2(x)+3 \log ^3(x)} \, dx=\frac {2}{3} x^2 \left (x+\frac {1}{\left (x^2+\log (x)\right )^2}\right ) \]

[Out]

2/3*x^2*(x+1/(ln(x)+x^2)^2)

Rubi [F]

\[ \int \frac {-4 x-4 x^3+6 x^8+\left (4 x+18 x^6\right ) \log (x)+18 x^4 \log ^2(x)+6 x^2 \log ^3(x)}{3 x^6+9 x^4 \log (x)+9 x^2 \log ^2(x)+3 \log ^3(x)} \, dx=\int \frac {-4 x-4 x^3+6 x^8+\left (4 x+18 x^6\right ) \log (x)+18 x^4 \log ^2(x)+6 x^2 \log ^3(x)}{3 x^6+9 x^4 \log (x)+9 x^2 \log ^2(x)+3 \log ^3(x)} \, dx \]

[In]

Int[(-4*x - 4*x^3 + 6*x^8 + (4*x + 18*x^6)*Log[x] + 18*x^4*Log[x]^2 + 6*x^2*Log[x]^3)/(3*x^6 + 9*x^4*Log[x] +
9*x^2*Log[x]^2 + 3*Log[x]^3),x]

[Out]

(2*x^3)/3 - (4*Defer[Int][x/(x^2 + Log[x])^3, x])/3 - (8*Defer[Int][x^3/(x^2 + Log[x])^3, x])/3 + (4*Defer[Int
][x/(x^2 + Log[x])^2, x])/3

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

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.18 \[ \int \frac {-4 x-4 x^3+6 x^8+\left (4 x+18 x^6\right ) \log (x)+18 x^4 \log ^2(x)+6 x^2 \log ^3(x)}{3 x^6+9 x^4 \log (x)+9 x^2 \log ^2(x)+3 \log ^3(x)} \, dx=\frac {2}{3} \left (x^3+\frac {x^2}{\left (x^2+\log (x)\right )^2}\right ) \]

[In]

Integrate[(-4*x - 4*x^3 + 6*x^8 + (4*x + 18*x^6)*Log[x] + 18*x^4*Log[x]^2 + 6*x^2*Log[x]^3)/(3*x^6 + 9*x^4*Log
[x] + 9*x^2*Log[x]^2 + 3*Log[x]^3),x]

[Out]

(2*(x^3 + x^2/(x^2 + Log[x])^2))/3

Maple [A] (verified)

Time = 0.24 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.18

method	result	size
default	\(\frac {2 x^{3}}{3}+\frac {2 x^{2}}{3 \left (\ln \left (x \right )+x^{2}\right )^{2}}\)	\(20\)
risch	\(\frac {2 x^{3}}{3}+\frac {2 x^{2}}{3 \left (\ln \left (x \right )+x^{2}\right )^{2}}\)	\(20\)
parallelrisch	\(\frac {2 x^{7}+4 x^{5} \ln \left (x \right )+2 x^{3} \ln \left (x \right )^{2}+2 x^{2}}{3 x^{4}+6 x^{2} \ln \left (x \right )+3 \ln \left (x \right )^{2}}\)	\(47\)

[In]

int((6*x^2*ln(x)^3+18*x^4*ln(x)^2+(18*x^6+4*x)*ln(x)+6*x^8-4*x^3-4*x)/(3*ln(x)^3+9*x^2*ln(x)^2+9*x^4*ln(x)+3*x
^6),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 41 vs. \(2 (15) = 30\).

Time = 0.25 (sec) , antiderivative size = 41, normalized size of antiderivative = 2.41 \[ \int \frac {-4 x-4 x^3+6 x^8+\left (4 x+18 x^6\right ) \log (x)+18 x^4 \log ^2(x)+6 x^2 \log ^3(x)}{3 x^6+9 x^4 \log (x)+9 x^2 \log ^2(x)+3 \log ^3(x)} \, dx=\frac {2 \, {\left (x^{7} + 2 \, x^{5} \log \left (x\right ) + x^{3} \log \left (x\right )^{2} + x^{2}\right )}}{3 \, {\left (x^{4} + 2 \, x^{2} \log \left (x\right ) + \log \left (x\right )^{2}\right )}} \]

[In]

integrate((6*x^2*log(x)^3+18*x^4*log(x)^2+(18*x^6+4*x)*log(x)+6*x^8-4*x^3-4*x)/(3*log(x)^3+9*x^2*log(x)^2+9*x^
4*log(x)+3*x^6),x, algorithm="fricas")

[Out]

2/3*(x^7 + 2*x^5*log(x) + x^3*log(x)^2 + x^2)/(x^4 + 2*x^2*log(x) + log(x)^2)

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.82 \[ \int \frac {-4 x-4 x^3+6 x^8+\left (4 x+18 x^6\right ) \log (x)+18 x^4 \log ^2(x)+6 x^2 \log ^3(x)}{3 x^6+9 x^4 \log (x)+9 x^2 \log ^2(x)+3 \log ^3(x)} \, dx=\frac {2 x^{3}}{3} + \frac {2 x^{2}}{3 x^{4} + 6 x^{2} \log {\left (x \right )} + 3 \log {\left (x \right )}^{2}} \]

[In]

integrate((6*x**2*ln(x)**3+18*x**4*ln(x)**2+(18*x**6+4*x)*ln(x)+6*x**8-4*x**3-4*x)/(3*ln(x)**3+9*x**2*ln(x)**2
+9*x**4*ln(x)+3*x**6),x)

[Out]

2*x**3/3 + 2*x**2/(3*x**4 + 6*x**2*log(x) + 3*log(x)**2)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 41 vs. \(2 (15) = 30\).

Time = 0.21 (sec) , antiderivative size = 41, normalized size of antiderivative = 2.41 \[ \int \frac {-4 x-4 x^3+6 x^8+\left (4 x+18 x^6\right ) \log (x)+18 x^4 \log ^2(x)+6 x^2 \log ^3(x)}{3 x^6+9 x^4 \log (x)+9 x^2 \log ^2(x)+3 \log ^3(x)} \, dx=\frac {2 \, {\left (x^{7} + 2 \, x^{5} \log \left (x\right ) + x^{3} \log \left (x\right )^{2} + x^{2}\right )}}{3 \, {\left (x^{4} + 2 \, x^{2} \log \left (x\right ) + \log \left (x\right )^{2}\right )}} \]

[In]

integrate((6*x^2*log(x)^3+18*x^4*log(x)^2+(18*x^6+4*x)*log(x)+6*x^8-4*x^3-4*x)/(3*log(x)^3+9*x^2*log(x)^2+9*x^
4*log(x)+3*x^6),x, algorithm="maxima")

[Out]

2/3*(x^7 + 2*x^5*log(x) + x^3*log(x)^2 + x^2)/(x^4 + 2*x^2*log(x) + log(x)^2)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 55 vs. \(2 (15) = 30\).

Time = 0.25 (sec) , antiderivative size = 55, normalized size of antiderivative = 3.24 \[ \int \frac {-4 x-4 x^3+6 x^8+\left (4 x+18 x^6\right ) \log (x)+18 x^4 \log ^2(x)+6 x^2 \log ^3(x)}{3 x^6+9 x^4 \log (x)+9 x^2 \log ^2(x)+3 \log ^3(x)} \, dx=\frac {2}{3} \, x^{3} + \frac {2 \, {\left (2 \, x^{4} + x^{2}\right )}}{3 \, {\left (2 \, x^{6} + 4 \, x^{4} \log \left (x\right ) + x^{4} + 2 \, x^{2} \log \left (x\right )^{2} + 2 \, x^{2} \log \left (x\right ) + \log \left (x\right )^{2}\right )}} \]

[In]

integrate((6*x^2*log(x)^3+18*x^4*log(x)^2+(18*x^6+4*x)*log(x)+6*x^8-4*x^3-4*x)/(3*log(x)^3+9*x^2*log(x)^2+9*x^
4*log(x)+3*x^6),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 8.73 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.12 \[ \int \frac {-4 x-4 x^3+6 x^8+\left (4 x+18 x^6\right ) \log (x)+18 x^4 \log ^2(x)+6 x^2 \log ^3(x)}{3 x^6+9 x^4 \log (x)+9 x^2 \log ^2(x)+3 \log ^3(x)} \, dx=\frac {2\,x^2}{3\,{\left (\ln \left (x\right )+x^2\right )}^2}+\frac {2\,x^3}{3} \]

[In]

int((6*x^2*log(x)^3 - 4*x + 18*x^4*log(x)^2 + log(x)*(4*x + 18*x^6) - 4*x^3 + 6*x^8)/(9*x^4*log(x) + 3*log(x)^
3 + 9*x^2*log(x)^2 + 3*x^6),x)

[Out]

(2*x^2)/(3*(log(x) + x^2)^2) + (2*x^3)/3