Integrand size = 104, antiderivative size = 25 \[ \int \frac {-59049+59049 x^2+118102 x^4-288680 x^6+275562 x^8-153090 x^{10}+54432 x^{12}-12636 x^{14}+1863 x^{16}-159 x^{18}+6 x^{20}}{-59049 x+177147 x^3-236196 x^5+183708 x^7-91854 x^9+30618 x^{11}-6804 x^{13}+972 x^{15}-81 x^{17}+3 x^{19}} \, dx=x \left (x+\frac {-\frac {x^4}{9 \left (-3+x^2\right )^8}+\log (x)}{x}\right ) \]
Time = 0.03 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.12 \[ \int \frac {-59049+59049 x^2+118102 x^4-288680 x^6+275562 x^8-153090 x^{10}+54432 x^{12}-12636 x^{14}+1863 x^{16}-159 x^{18}+6 x^{20}}{-59049 x+177147 x^3-236196 x^5+183708 x^7-91854 x^9+30618 x^{11}-6804 x^{13}+972 x^{15}-81 x^{17}+3 x^{19}} \, dx=\frac {1}{3} \left (3 x^2-\frac {x^4}{3 \left (-3+x^2\right )^8}+3 \log (x)\right ) \]
Integrate[(-59049 + 59049*x^2 + 118102*x^4 - 288680*x^6 + 275562*x^8 - 153 090*x^10 + 54432*x^12 - 12636*x^14 + 1863*x^16 - 159*x^18 + 6*x^20)/(-5904 9*x + 177147*x^3 - 236196*x^5 + 183708*x^7 - 91854*x^9 + 30618*x^11 - 6804 *x^13 + 972*x^15 - 81*x^17 + 3*x^19),x]
Leaf count is larger than twice the leaf count of optimal. \(51\) vs. \(2(25)=50\).
Time = 0.76 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.04, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.058, Rules used = {2026, 2070, 2331, 27, 2123, 2009}
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.
\(\displaystyle \int \frac {6 x^{20}-159 x^{18}+1863 x^{16}-12636 x^{14}+54432 x^{12}-153090 x^{10}+275562 x^8-288680 x^6+118102 x^4+59049 x^2-59049}{3 x^{19}-81 x^{17}+972 x^{15}-6804 x^{13}+30618 x^{11}-91854 x^9+183708 x^7-236196 x^5+177147 x^3-59049 x} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {6 x^{20}-159 x^{18}+1863 x^{16}-12636 x^{14}+54432 x^{12}-153090 x^{10}+275562 x^8-288680 x^6+118102 x^4+59049 x^2-59049}{x \left (3 x^{18}-81 x^{16}+972 x^{14}-6804 x^{12}+30618 x^{10}-91854 x^8+183708 x^6-236196 x^4+177147 x^2-59049\right )}dx\) |
\(\Big \downarrow \) 2070 |
\(\displaystyle \int \frac {6 x^{20}-159 x^{18}+1863 x^{16}-12636 x^{14}+54432 x^{12}-153090 x^{10}+275562 x^8-288680 x^6+118102 x^4+59049 x^2-59049}{x \left (\sqrt [9]{3} x^2-3 \sqrt [9]{3}\right )^9}dx\) |
\(\Big \downarrow \) 2331 |
\(\displaystyle \frac {1}{2} \int \frac {-6 x^{20}+159 x^{18}-1863 x^{16}+12636 x^{14}-54432 x^{12}+153090 x^{10}-275562 x^8+288680 x^6-118102 x^4-59049 x^2+59049}{3 x^2 \left (3-x^2\right )^9}dx^2\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{6} \int \frac {-6 x^{20}+159 x^{18}-1863 x^{16}+12636 x^{14}-54432 x^{12}+153090 x^{10}-275562 x^8+288680 x^6-118102 x^4-59049 x^2+59049}{x^2 \left (3-x^2\right )^9}dx^2\) |
\(\Big \downarrow \) 2123 |
\(\displaystyle \frac {1}{6} \int \left (\frac {4}{\left (x^2-3\right )^7}+\frac {28}{\left (x^2-3\right )^8}+\frac {48}{\left (x^2-3\right )^9}+6+\frac {3}{x^2}\right )dx^2\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{6} \left (6 x^2-\frac {2}{3 \left (3-x^2\right )^6}+\frac {4}{\left (3-x^2\right )^7}-\frac {6}{\left (3-x^2\right )^8}+3 \log \left (x^2\right )\right )\) |
Int[(-59049 + 59049*x^2 + 118102*x^4 - 288680*x^6 + 275562*x^8 - 153090*x^ 10 + 54432*x^12 - 12636*x^14 + 1863*x^16 - 159*x^18 + 6*x^20)/(-59049*x + 177147*x^3 - 236196*x^5 + 183708*x^7 - 91854*x^9 + 30618*x^11 - 6804*x^13 + 972*x^15 - 81*x^17 + 3*x^19),x]
3.10.35.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p *r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ erQ[p] && !MonomialQ[Px, x] && (ILtQ[p, 0] || !PolyQ[u, x])
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{a = Rt[Coeff[Px, x^2, 0], Expon[Px , x^2]], b = Rt[Coeff[Px, x^2, Expon[Px, x^2]], Expon[Px, x^2]]}, Int[u*(a + b*x^2)^(Expon[Px, x^2]*p), x] /; EqQ[Px, (a + b*x^2)^Expon[Px, x^2]]] /; IntegerQ[p] && PolyQ[Px, x^2] && GtQ[Expon[Px, x^2], 1] && NeQ[Coeff[Px, x^ 2, 0], 0]
Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c , d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2])
Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/2 S ubst[Int[x^((m - 1)/2)*SubstFor[x^2, Pq, x]*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x^2] && IntegerQ[(m - 1)/2]
Time = 0.06 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.36
method | result | size |
default | \(x^{2}+\ln \left (x \right )-\frac {1}{9 \left (x^{2}-3\right )^{6}}-\frac {1}{\left (x^{2}-3\right )^{8}}-\frac {2}{3 \left (x^{2}-3\right )^{7}}\) | \(34\) |
norman | \(\frac {x^{18}-413343 x^{2}-306180 x^{6}-30618 x^{10}-324 x^{14}+4536 x^{12}+122472 x^{8}+\frac {4251527}{9} x^{4}+157464}{\left (x^{2}-3\right )^{8}}+\ln \left (x \right )\) | \(52\) |
risch | \(x^{2}-\frac {x^{4}}{9 \left (x^{16}-24 x^{14}+252 x^{12}-1512 x^{10}+5670 x^{8}-13608 x^{6}+20412 x^{4}-17496 x^{2}+6561\right )}+\ln \left (x \right )\) | \(54\) |
parallelrisch | \(\frac {1417176-216 x^{14} \ln \left (x \right )-13608 x^{10} \ln \left (x \right )-122472 x^{6} \ln \left (x \right )+2268 x^{12} \ln \left (x \right )+51030 x^{8} \ln \left (x \right )+183708 x^{4} \ln \left (x \right )+9 x^{18}+40824 x^{12}-2916 x^{14}-275562 x^{10}+1102248 x^{8}+4251527 x^{4}-3720087 x^{2}-2755620 x^{6}+59049 \ln \left (x \right )-157464 x^{2} \ln \left (x \right )+9 x^{16} \ln \left (x \right )}{9 x^{16}-216 x^{14}+2268 x^{12}-13608 x^{10}+51030 x^{8}-122472 x^{6}+183708 x^{4}-157464 x^{2}+59049}\) | \(147\) |
int((6*x^20-159*x^18+1863*x^16-12636*x^14+54432*x^12-153090*x^10+275562*x^ 8-288680*x^6+118102*x^4+59049*x^2-59049)/(3*x^19-81*x^17+972*x^15-6804*x^1 3+30618*x^11-91854*x^9+183708*x^7-236196*x^5+177147*x^3-59049*x),x,method= _RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 134 vs. \(2 (28) = 56\).
Time = 0.26 (sec) , antiderivative size = 134, normalized size of antiderivative = 5.36 \[ \int \frac {-59049+59049 x^2+118102 x^4-288680 x^6+275562 x^8-153090 x^{10}+54432 x^{12}-12636 x^{14}+1863 x^{16}-159 x^{18}+6 x^{20}}{-59049 x+177147 x^3-236196 x^5+183708 x^7-91854 x^9+30618 x^{11}-6804 x^{13}+972 x^{15}-81 x^{17}+3 x^{19}} \, dx=\frac {9 \, x^{18} - 216 \, x^{16} + 2268 \, x^{14} - 13608 \, x^{12} + 51030 \, x^{10} - 122472 \, x^{8} + 183708 \, x^{6} - 157465 \, x^{4} + 59049 \, x^{2} + 9 \, {\left (x^{16} - 24 \, x^{14} + 252 \, x^{12} - 1512 \, x^{10} + 5670 \, x^{8} - 13608 \, x^{6} + 20412 \, x^{4} - 17496 \, x^{2} + 6561\right )} \log \left (x\right )}{9 \, {\left (x^{16} - 24 \, x^{14} + 252 \, x^{12} - 1512 \, x^{10} + 5670 \, x^{8} - 13608 \, x^{6} + 20412 \, x^{4} - 17496 \, x^{2} + 6561\right )}} \]
integrate((6*x^20-159*x^18+1863*x^16-12636*x^14+54432*x^12-153090*x^10+275 562*x^8-288680*x^6+118102*x^4+59049*x^2-59049)/(3*x^19-81*x^17+972*x^15-68 04*x^13+30618*x^11-91854*x^9+183708*x^7-236196*x^5+177147*x^3-59049*x),x, algorithm=\
1/9*(9*x^18 - 216*x^16 + 2268*x^14 - 13608*x^12 + 51030*x^10 - 122472*x^8 + 183708*x^6 - 157465*x^4 + 59049*x^2 + 9*(x^16 - 24*x^14 + 252*x^12 - 151 2*x^10 + 5670*x^8 - 13608*x^6 + 20412*x^4 - 17496*x^2 + 6561)*log(x))/(x^1 6 - 24*x^14 + 252*x^12 - 1512*x^10 + 5670*x^8 - 13608*x^6 + 20412*x^4 - 17 496*x^2 + 6561)
Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (19) = 38\).
Time = 0.09 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.04 \[ \int \frac {-59049+59049 x^2+118102 x^4-288680 x^6+275562 x^8-153090 x^{10}+54432 x^{12}-12636 x^{14}+1863 x^{16}-159 x^{18}+6 x^{20}}{-59049 x+177147 x^3-236196 x^5+183708 x^7-91854 x^9+30618 x^{11}-6804 x^{13}+972 x^{15}-81 x^{17}+3 x^{19}} \, dx=- \frac {x^{4}}{9 x^{16} - 216 x^{14} + 2268 x^{12} - 13608 x^{10} + 51030 x^{8} - 122472 x^{6} + 183708 x^{4} - 157464 x^{2} + 59049} + x^{2} + \log {\left (x \right )} \]
integrate((6*x**20-159*x**18+1863*x**16-12636*x**14+54432*x**12-153090*x** 10+275562*x**8-288680*x**6+118102*x**4+59049*x**2-59049)/(3*x**19-81*x**17 +972*x**15-6804*x**13+30618*x**11-91854*x**9+183708*x**7-236196*x**5+17714 7*x**3-59049*x),x)
-x**4/(9*x**16 - 216*x**14 + 2268*x**12 - 13608*x**10 + 51030*x**8 - 12247 2*x**6 + 183708*x**4 - 157464*x**2 + 59049) + x**2 + log(x)
Time = 0.19 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.12 \[ \int \frac {-59049+59049 x^2+118102 x^4-288680 x^6+275562 x^8-153090 x^{10}+54432 x^{12}-12636 x^{14}+1863 x^{16}-159 x^{18}+6 x^{20}}{-59049 x+177147 x^3-236196 x^5+183708 x^7-91854 x^9+30618 x^{11}-6804 x^{13}+972 x^{15}-81 x^{17}+3 x^{19}} \, dx=-\frac {x^{4}}{9 \, {\left (x^{16} - 24 \, x^{14} + 252 \, x^{12} - 1512 \, x^{10} + 5670 \, x^{8} - 13608 \, x^{6} + 20412 \, x^{4} - 17496 \, x^{2} + 6561\right )}} + x^{2} + \log \left (x\right ) \]
integrate((6*x^20-159*x^18+1863*x^16-12636*x^14+54432*x^12-153090*x^10+275 562*x^8-288680*x^6+118102*x^4+59049*x^2-59049)/(3*x^19-81*x^17+972*x^15-68 04*x^13+30618*x^11-91854*x^9+183708*x^7-236196*x^5+177147*x^3-59049*x),x, algorithm=\
-1/9*x^4/(x^16 - 24*x^14 + 252*x^12 - 1512*x^10 + 5670*x^8 - 13608*x^6 + 2 0412*x^4 - 17496*x^2 + 6561) + x^2 + log(x)
Time = 0.29 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88 \[ \int \frac {-59049+59049 x^2+118102 x^4-288680 x^6+275562 x^8-153090 x^{10}+54432 x^{12}-12636 x^{14}+1863 x^{16}-159 x^{18}+6 x^{20}}{-59049 x+177147 x^3-236196 x^5+183708 x^7-91854 x^9+30618 x^{11}-6804 x^{13}+972 x^{15}-81 x^{17}+3 x^{19}} \, dx=x^{2} - \frac {x^{4}}{9 \, {\left (x^{2} - 3\right )}^{8}} + \frac {1}{2} \, \log \left (x^{2}\right ) \]
integrate((6*x^20-159*x^18+1863*x^16-12636*x^14+54432*x^12-153090*x^10+275 562*x^8-288680*x^6+118102*x^4+59049*x^2-59049)/(3*x^19-81*x^17+972*x^15-68 04*x^13+30618*x^11-91854*x^9+183708*x^7-236196*x^5+177147*x^3-59049*x),x, algorithm=\
Time = 0.11 (sec) , antiderivative size = 55, normalized size of antiderivative = 2.20 \[ \int \frac {-59049+59049 x^2+118102 x^4-288680 x^6+275562 x^8-153090 x^{10}+54432 x^{12}-12636 x^{14}+1863 x^{16}-159 x^{18}+6 x^{20}}{-59049 x+177147 x^3-236196 x^5+183708 x^7-91854 x^9+30618 x^{11}-6804 x^{13}+972 x^{15}-81 x^{17}+3 x^{19}} \, dx=\ln \left (x\right )-\frac {x^4}{9\,\left (x^{16}-24\,x^{14}+252\,x^{12}-1512\,x^{10}+5670\,x^8-13608\,x^6+20412\,x^4-17496\,x^2+6561\right )}+x^2 \]
int(-(59049*x^2 + 118102*x^4 - 288680*x^6 + 275562*x^8 - 153090*x^10 + 544 32*x^12 - 12636*x^14 + 1863*x^16 - 159*x^18 + 6*x^20 - 59049)/(59049*x - 1 77147*x^3 + 236196*x^5 - 183708*x^7 + 91854*x^9 - 30618*x^11 + 6804*x^13 - 972*x^15 + 81*x^17 - 3*x^19),x)