Integrand size = 151, antiderivative size = 28 \[ \int \frac {\left (-864-288 x^2-336 x^4+256 x^6+64 x^8+16 x^{10}\right ) \log \left (\frac {9+6 x^2+2 x^4+x^6}{6 x+2 x^3+x^5}\right )}{54 x+54 x^3+33 x^5+16 x^7+4 x^9+x^{11}+\left (216 x+216 x^3+132 x^5+64 x^7+16 x^9+4 x^{11}\right ) \log ^2\left (\frac {9+6 x^2+2 x^4+x^6}{6 x+2 x^3+x^5}\right )} \, dx=2 \log \left (\frac {1}{4}+\log ^2\left (x+\frac {9}{x \left (5+\left (1+x^2\right )^2\right )}\right )\right ) \]
Time = 0.06 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.43 \[ \int \frac {\left (-864-288 x^2-336 x^4+256 x^6+64 x^8+16 x^{10}\right ) \log \left (\frac {9+6 x^2+2 x^4+x^6}{6 x+2 x^3+x^5}\right )}{54 x+54 x^3+33 x^5+16 x^7+4 x^9+x^{11}+\left (216 x+216 x^3+132 x^5+64 x^7+16 x^9+4 x^{11}\right ) \log ^2\left (\frac {9+6 x^2+2 x^4+x^6}{6 x+2 x^3+x^5}\right )} \, dx=2 \log \left (1+4 \log ^2\left (\frac {9+6 x^2+2 x^4+x^6}{6 x+2 x^3+x^5}\right )\right ) \]
Integrate[((-864 - 288*x^2 - 336*x^4 + 256*x^6 + 64*x^8 + 16*x^10)*Log[(9 + 6*x^2 + 2*x^4 + x^6)/(6*x + 2*x^3 + x^5)])/(54*x + 54*x^3 + 33*x^5 + 16* x^7 + 4*x^9 + x^11 + (216*x + 216*x^3 + 132*x^5 + 64*x^7 + 16*x^9 + 4*x^11 )*Log[(9 + 6*x^2 + 2*x^4 + x^6)/(6*x + 2*x^3 + x^5)]^2),x]
Time = 36.28 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.43, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.046, Rules used = {7292, 27, 25, 2461, 7239, 7259, 16}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\left (16 x^{10}+64 x^8+256 x^6-336 x^4-288 x^2-864\right ) \log \left (\frac {x^6+2 x^4+6 x^2+9}{x^5+2 x^3+6 x}\right )}{x^{11}+4 x^9+16 x^7+33 x^5+54 x^3+\left (4 x^{11}+16 x^9+64 x^7+132 x^5+216 x^3+216 x\right ) \log ^2\left (\frac {x^6+2 x^4+6 x^2+9}{x^5+2 x^3+6 x}\right )+54 x} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {16 \left (x^{10}+4 x^8+16 x^6-21 x^4-18 x^2-54\right ) \log \left (\frac {x^6+2 x^4+6 x^2+9}{x \left (x^4+2 x^2+6\right )}\right )}{x \left (x^{10}+4 x^8+16 x^6+33 x^4+54 x^2+54\right ) \left (4 \log ^2\left (\frac {x^6+2 x^4+6 x^2+9}{x^5+2 x^3+6 x}\right )+1\right )}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 16 \int -\frac {\left (-x^{10}-4 x^8-16 x^6+21 x^4+18 x^2+54\right ) \log \left (\frac {x^6+2 x^4+6 x^2+9}{x \left (x^4+2 x^2+6\right )}\right )}{x \left (x^{10}+4 x^8+16 x^6+33 x^4+54 x^2+54\right ) \left (4 \log ^2\left (\frac {x^6+2 x^4+6 x^2+9}{x^5+2 x^3+6 x}\right )+1\right )}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -16 \int \frac {\left (-x^{10}-4 x^8-16 x^6+21 x^4+18 x^2+54\right ) \log \left (\frac {x^6+2 x^4+6 x^2+9}{x \left (x^4+2 x^2+6\right )}\right )}{x \left (x^{10}+4 x^8+16 x^6+33 x^4+54 x^2+54\right ) \left (4 \log ^2\left (\frac {x^6+2 x^4+6 x^2+9}{x^5+2 x^3+6 x}\right )+1\right )}dx\) |
\(\Big \downarrow \) 2461 |
\(\displaystyle -16 \int \left (\frac {\left (-x^{10}-4 x^8-16 x^6+21 x^4+18 x^2+54\right ) \log \left (\frac {x^6+2 x^4+6 x^2+9}{x \left (x^4+2 x^2+6\right )}\right )}{9 x \left (x^4+2 x^2+6\right ) \left (4 \log ^2\left (\frac {x^6+2 x^4+6 x^2+9}{x^5+2 x^3+6 x}\right )+1\right )}-\frac {x \left (-x^{10}-4 x^8-16 x^6+21 x^4+18 x^2+54\right ) \log \left (\frac {x^6+2 x^4+6 x^2+9}{x \left (x^4+2 x^2+6\right )}\right )}{9 \left (x^6+2 x^4+6 x^2+9\right ) \left (4 \log ^2\left (\frac {x^6+2 x^4+6 x^2+9}{x^5+2 x^3+6 x}\right )+1\right )}\right )dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle -16 \int \frac {\left (-x^{10}-4 x^8-16 x^6+21 x^4+18 x^2+54\right ) \log \left (\frac {x^6+2 x^4+6 x^2+9}{x \left (x^4+2 x^2+6\right )}\right )}{x \left (x^4+2 x^2+6\right ) \left (x^6+2 x^4+6 x^2+9\right ) \left (4 \log ^2\left (\frac {x^6+2 x^4+6 x^2+9}{x^5+2 x^3+6 x}\right )+1\right )}dx\) |
\(\Big \downarrow \) 7259 |
\(\displaystyle 2 \int \frac {1}{4 \log ^2\left (\frac {x^6+2 x^4+6 x^2+9}{x^5+2 x^3+6 x}\right )+1}d\left (4 \log ^2\left (\frac {x^6+2 x^4+6 x^2+9}{x^5+2 x^3+6 x}\right )\right )\) |
\(\Big \downarrow \) 16 |
\(\displaystyle 2 \log \left (4 \log ^2\left (\frac {x^6+2 x^4+6 x^2+9}{x^5+2 x^3+6 x}\right )+1\right )\) |
Int[((-864 - 288*x^2 - 336*x^4 + 256*x^6 + 64*x^8 + 16*x^10)*Log[(9 + 6*x^ 2 + 2*x^4 + x^6)/(6*x + 2*x^3 + x^5)])/(54*x + 54*x^3 + 33*x^5 + 16*x^7 + 4*x^9 + x^11 + (216*x + 216*x^3 + 132*x^5 + 64*x^7 + 16*x^9 + 4*x^11)*Log[ (9 + 6*x^2 + 2*x^4 + x^6)/(6*x + 2*x^3 + x^5)]^2),x]
3.11.16.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px /. x -> Sqrt[x]]}, Int[ExpandIntegrand[u, (Qx /. x -> x^2)^p, x], x] /; !SumQ[NonfreeFactors[ Qx, x]]] /; PolyQ[Px, x^2] && GtQ[Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Int[(u_)*((a_) + (b_.)*(v_)^(p_.)*(w_)^(p_.))^(m_.), x_Symbol] :> With[{c = Simplify[u/(w*D[v, x] + v*D[w, x])]}, Simp[c Subst[Int[(a + b*x^p)^m, x] , x, v*w], x] /; FreeQ[c, x]] /; FreeQ[{a, b, m, p}, x] && IntegerQ[p]
Time = 2.43 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.39
method | result | size |
risch | \(2 \ln \left (\ln \left (\frac {x^{6}+2 x^{4}+6 x^{2}+9}{x^{5}+2 x^{3}+6 x}\right )^{2}+\frac {1}{4}\right )\) | \(39\) |
parallelrisch | \(2 \ln \left (\ln \left (\frac {x^{6}+2 x^{4}+6 x^{2}+9}{x \left (x^{4}+2 x^{2}+6\right )}\right )^{2}+\frac {1}{4}\right )\) | \(40\) |
default | \(2 \ln \left (4 \ln \left (\frac {x^{6}+2 x^{4}+6 x^{2}+9}{x \left (x^{4}+2 x^{2}+6\right )}\right )^{2}+1\right )\) | \(42\) |
int((16*x^10+64*x^8+256*x^6-336*x^4-288*x^2-864)*ln((x^6+2*x^4+6*x^2+9)/(x ^5+2*x^3+6*x))/((4*x^11+16*x^9+64*x^7+132*x^5+216*x^3+216*x)*ln((x^6+2*x^4 +6*x^2+9)/(x^5+2*x^3+6*x))^2+x^11+4*x^9+16*x^7+33*x^5+54*x^3+54*x),x,metho d=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.43 \[ \int \frac {\left (-864-288 x^2-336 x^4+256 x^6+64 x^8+16 x^{10}\right ) \log \left (\frac {9+6 x^2+2 x^4+x^6}{6 x+2 x^3+x^5}\right )}{54 x+54 x^3+33 x^5+16 x^7+4 x^9+x^{11}+\left (216 x+216 x^3+132 x^5+64 x^7+16 x^9+4 x^{11}\right ) \log ^2\left (\frac {9+6 x^2+2 x^4+x^6}{6 x+2 x^3+x^5}\right )} \, dx=2 \, \log \left (4 \, \log \left (\frac {x^{6} + 2 \, x^{4} + 6 \, x^{2} + 9}{x^{5} + 2 \, x^{3} + 6 \, x}\right )^{2} + 1\right ) \]
integrate((16*x^10+64*x^8+256*x^6-336*x^4-288*x^2-864)*log((x^6+2*x^4+6*x^ 2+9)/(x^5+2*x^3+6*x))/((4*x^11+16*x^9+64*x^7+132*x^5+216*x^3+216*x)*log((x ^6+2*x^4+6*x^2+9)/(x^5+2*x^3+6*x))^2+x^11+4*x^9+16*x^7+33*x^5+54*x^3+54*x) ,x, algorithm=\
Time = 0.39 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.29 \[ \int \frac {\left (-864-288 x^2-336 x^4+256 x^6+64 x^8+16 x^{10}\right ) \log \left (\frac {9+6 x^2+2 x^4+x^6}{6 x+2 x^3+x^5}\right )}{54 x+54 x^3+33 x^5+16 x^7+4 x^9+x^{11}+\left (216 x+216 x^3+132 x^5+64 x^7+16 x^9+4 x^{11}\right ) \log ^2\left (\frac {9+6 x^2+2 x^4+x^6}{6 x+2 x^3+x^5}\right )} \, dx=2 \log {\left (\log {\left (\frac {x^{6} + 2 x^{4} + 6 x^{2} + 9}{x^{5} + 2 x^{3} + 6 x} \right )}^{2} + \frac {1}{4} \right )} \]
integrate((16*x**10+64*x**8+256*x**6-336*x**4-288*x**2-864)*ln((x**6+2*x** 4+6*x**2+9)/(x**5+2*x**3+6*x))/((4*x**11+16*x**9+64*x**7+132*x**5+216*x**3 +216*x)*ln((x**6+2*x**4+6*x**2+9)/(x**5+2*x**3+6*x))**2+x**11+4*x**9+16*x* *7+33*x**5+54*x**3+54*x),x)
Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (26) = 52\).
Time = 0.36 (sec) , antiderivative size = 87, normalized size of antiderivative = 3.11 \[ \int \frac {\left (-864-288 x^2-336 x^4+256 x^6+64 x^8+16 x^{10}\right ) \log \left (\frac {9+6 x^2+2 x^4+x^6}{6 x+2 x^3+x^5}\right )}{54 x+54 x^3+33 x^5+16 x^7+4 x^9+x^{11}+\left (216 x+216 x^3+132 x^5+64 x^7+16 x^9+4 x^{11}\right ) \log ^2\left (\frac {9+6 x^2+2 x^4+x^6}{6 x+2 x^3+x^5}\right )} \, dx=2 \, \log \left (-2 \, {\left (\log \left (x^{4} + 2 \, x^{2} + 6\right ) + \log \left (x\right )\right )} \log \left (x^{6} + 2 \, x^{4} + 6 \, x^{2} + 9\right ) + \log \left (x^{6} + 2 \, x^{4} + 6 \, x^{2} + 9\right )^{2} + \log \left (x^{4} + 2 \, x^{2} + 6\right )^{2} + 2 \, \log \left (x^{4} + 2 \, x^{2} + 6\right ) \log \left (x\right ) + \log \left (x\right )^{2} + \frac {1}{4}\right ) \]
integrate((16*x^10+64*x^8+256*x^6-336*x^4-288*x^2-864)*log((x^6+2*x^4+6*x^ 2+9)/(x^5+2*x^3+6*x))/((4*x^11+16*x^9+64*x^7+132*x^5+216*x^3+216*x)*log((x ^6+2*x^4+6*x^2+9)/(x^5+2*x^3+6*x))^2+x^11+4*x^9+16*x^7+33*x^5+54*x^3+54*x) ,x, algorithm=\
2*log(-2*(log(x^4 + 2*x^2 + 6) + log(x))*log(x^6 + 2*x^4 + 6*x^2 + 9) + lo g(x^6 + 2*x^4 + 6*x^2 + 9)^2 + log(x^4 + 2*x^2 + 6)^2 + 2*log(x^4 + 2*x^2 + 6)*log(x) + log(x)^2 + 1/4)
Time = 0.58 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.43 \[ \int \frac {\left (-864-288 x^2-336 x^4+256 x^6+64 x^8+16 x^{10}\right ) \log \left (\frac {9+6 x^2+2 x^4+x^6}{6 x+2 x^3+x^5}\right )}{54 x+54 x^3+33 x^5+16 x^7+4 x^9+x^{11}+\left (216 x+216 x^3+132 x^5+64 x^7+16 x^9+4 x^{11}\right ) \log ^2\left (\frac {9+6 x^2+2 x^4+x^6}{6 x+2 x^3+x^5}\right )} \, dx=2 \, \log \left (4 \, \log \left (\frac {x^{6} + 2 \, x^{4} + 6 \, x^{2} + 9}{x^{5} + 2 \, x^{3} + 6 \, x}\right )^{2} + 1\right ) \]
integrate((16*x^10+64*x^8+256*x^6-336*x^4-288*x^2-864)*log((x^6+2*x^4+6*x^ 2+9)/(x^5+2*x^3+6*x))/((4*x^11+16*x^9+64*x^7+132*x^5+216*x^3+216*x)*log((x ^6+2*x^4+6*x^2+9)/(x^5+2*x^3+6*x))^2+x^11+4*x^9+16*x^7+33*x^5+54*x^3+54*x) ,x, algorithm=\
Time = 19.57 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.36 \[ \int \frac {\left (-864-288 x^2-336 x^4+256 x^6+64 x^8+16 x^{10}\right ) \log \left (\frac {9+6 x^2+2 x^4+x^6}{6 x+2 x^3+x^5}\right )}{54 x+54 x^3+33 x^5+16 x^7+4 x^9+x^{11}+\left (216 x+216 x^3+132 x^5+64 x^7+16 x^9+4 x^{11}\right ) \log ^2\left (\frac {9+6 x^2+2 x^4+x^6}{6 x+2 x^3+x^5}\right )} \, dx=2\,\ln \left ({\ln \left (\frac {x^6+2\,x^4+6\,x^2+9}{x^5+2\,x^3+6\,x}\right )}^2+\frac {1}{4}\right ) \]