\(\int \frac {-59049+59049 x^2+118102 x^4-288680 x^6+275562 x^8-153090 x^{10}+54432 x^{12}-12636 x^{14}+1863 x^{16}-159 x^{18}+6 x^{20}}{-59049 x+177147 x^3-236196 x^5+183708 x^7-91854 x^9+30618 x^{11}-6804 x^{13}+972 x^{15}-81 x^{17}+3 x^{19}} \, dx\) [935]

Optimal result

Integrand size = 104, antiderivative size = 25 \[ \int \frac {-59049+59049 x^2+118102 x^4-288680 x^6+275562 x^8-153090 x^{10}+54432 x^{12}-12636 x^{14}+1863 x^{16}-159 x^{18}+6 x^{20}}{-59049 x+177147 x^3-236196 x^5+183708 x^7-91854 x^9+30618 x^{11}-6804 x^{13}+972 x^{15}-81 x^{17}+3 x^{19}} \, dx=x \left (x+\frac {-\frac {x^4}{9 \left (-3+x^2\right )^8}+\log (x)}{x}\right ) \]

[Out]

(x+(ln(x)-1/9*x^4/(x^2-3)^8)/x)*x

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.72, number of steps used = 5, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.019, Rules used = {2099, 267} \[ \int \frac {-59049+59049 x^2+118102 x^4-288680 x^6+275562 x^8-153090 x^{10}+54432 x^{12}-12636 x^{14}+1863 x^{16}-159 x^{18}+6 x^{20}}{-59049 x+177147 x^3-236196 x^5+183708 x^7-91854 x^9+30618 x^{11}-6804 x^{13}+972 x^{15}-81 x^{17}+3 x^{19}} \, dx=x^2-\frac {1}{9 \left (3-x^2\right )^6}+\frac {2}{3 \left (3-x^2\right )^7}-\frac {1}{\left (3-x^2\right )^8}+\log (x) \]

[In]

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^1
3 + 972*x^15 - 81*x^17 + 3*x^19),x]

[Out]

x^2 - (3 - x^2)^(-8) + 2/(3*(3 - x^2)^7) - 1/(9*(3 - x^2)^6) + Log[x]

Rule 267

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rule 2099

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /;  !SumQ[N
onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{x}+2 x+\frac {16 x}{\left (-3+x^2\right )^9}+\frac {28 x}{3 \left (-3+x^2\right )^8}+\frac {4 x}{3 \left (-3+x^2\right )^7}\right ) \, dx \\ & = x^2+\log (x)+\frac {4}{3} \int \frac {x}{\left (-3+x^2\right )^7} \, dx+\frac {28}{3} \int \frac {x}{\left (-3+x^2\right )^8} \, dx+16 \int \frac {x}{\left (-3+x^2\right )^9} \, dx \\ & = x^2-\frac {1}{\left (3-x^2\right )^8}+\frac {2}{3 \left (3-x^2\right )^7}-\frac {1}{9 \left (3-x^2\right )^6}+\log (x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.12 \[ \int \frac {-59049+59049 x^2+118102 x^4-288680 x^6+275562 x^8-153090 x^{10}+54432 x^{12}-12636 x^{14}+1863 x^{16}-159 x^{18}+6 x^{20}}{-59049 x+177147 x^3-236196 x^5+183708 x^7-91854 x^9+30618 x^{11}-6804 x^{13}+972 x^{15}-81 x^{17}+3 x^{19}} \, dx=\frac {1}{3} \left (3 x^2-\frac {x^4}{3 \left (-3+x^2\right )^8}+3 \log (x)\right ) \]

[In]

Integrate[(-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 - 68
04*x^13 + 972*x^15 - 81*x^17 + 3*x^19),x]

[Out]

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

Maple [A] (verified)

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	\(\frac {1417176-216 x^{14} \ln \left (x \right )-13608 x^{10} \ln \left (x \right )-122472 x^{6} \ln \left (x \right )+2268 x^{12} \ln \left (x \right )+51030 x^{8} \ln \left (x \right )+183708 x^{4} \ln \left (x \right )+9 x^{18}+40824 x^{12}-2916 x^{14}-275562 x^{10}+1102248 x^{8}+4251527 x^{4}-3720087 x^{2}-2755620 x^{6}+59049 \ln \left (x \right )-157464 x^{2} \ln \left (x \right )+9 x^{16} \ln \left (x \right )}{9 x^{16}-216 x^{14}+2268 x^{12}-13608 x^{10}+51030 x^{8}-122472 x^{6}+183708 x^{4}-157464 x^{2}+59049}\)	\(147\)

[In]

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^2-59
049)/(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),x,metho
d=_RETURNVERBOSE)

[Out]

x^2+ln(x)-1/9/(x^2-3)^6-1/(x^2-3)^8-2/3/(x^2-3)^7

Fricas [B] (verification not implemented)

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 \[ \int \frac {-59049+59049 x^2+118102 x^4-288680 x^6+275562 x^8-153090 x^{10}+54432 x^{12}-12636 x^{14}+1863 x^{16}-159 x^{18}+6 x^{20}}{-59049 x+177147 x^3-236196 x^5+183708 x^7-91854 x^9+30618 x^{11}-6804 x^{13}+972 x^{15}-81 x^{17}+3 x^{19}} \, dx=\frac {9 \, x^{18} - 216 \, x^{16} + 2268 \, x^{14} - 13608 \, x^{12} + 51030 \, x^{10} - 122472 \, x^{8} + 183708 \, x^{6} - 157465 \, x^{4} + 59049 \, x^{2} + 9 \, {\left (x^{16} - 24 \, x^{14} + 252 \, x^{12} - 1512 \, x^{10} + 5670 \, x^{8} - 13608 \, x^{6} + 20412 \, x^{4} - 17496 \, x^{2} + 6561\right )} \log \left (x\right )}{9 \, {\left (x^{16} - 24 \, x^{14} + 252 \, x^{12} - 1512 \, x^{10} + 5670 \, x^{8} - 13608 \, x^{6} + 20412 \, x^{4} - 17496 \, x^{2} + 6561\right )}} \]

[In]

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+177147*x^3-59049*x),x
, algorithm="fricas")

[Out]

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 - 1512*x^10 + 5670*x^8 - 13608*x^6 + 20412*x^4 - 17496*x^2 + 6561)*log(x))/(x
^16 - 24*x^14 + 252*x^12 - 1512*x^10 + 5670*x^8 - 13608*x^6 + 20412*x^4 - 17496*x^2 + 6561)

Sympy [B] (verification not implemented)

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 \[ \int \frac {-59049+59049 x^2+118102 x^4-288680 x^6+275562 x^8-153090 x^{10}+54432 x^{12}-12636 x^{14}+1863 x^{16}-159 x^{18}+6 x^{20}}{-59049 x+177147 x^3-236196 x^5+183708 x^7-91854 x^9+30618 x^{11}-6804 x^{13}+972 x^{15}-81 x^{17}+3 x^{19}} \, dx=- \frac {x^{4}}{9 x^{16} - 216 x^{14} + 2268 x^{12} - 13608 x^{10} + 51030 x^{8} - 122472 x^{6} + 183708 x^{4} - 157464 x^{2} + 59049} + x^{2} + \log {\left (x \right )} \]

[In]

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+177
147*x**3-59049*x),x)

[Out]

-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(x)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.12 \[ \int \frac {-59049+59049 x^2+118102 x^4-288680 x^6+275562 x^8-153090 x^{10}+54432 x^{12}-12636 x^{14}+1863 x^{16}-159 x^{18}+6 x^{20}}{-59049 x+177147 x^3-236196 x^5+183708 x^7-91854 x^9+30618 x^{11}-6804 x^{13}+972 x^{15}-81 x^{17}+3 x^{19}} \, dx=-\frac {x^{4}}{9 \, {\left (x^{16} - 24 \, x^{14} + 252 \, x^{12} - 1512 \, x^{10} + 5670 \, x^{8} - 13608 \, x^{6} + 20412 \, x^{4} - 17496 \, x^{2} + 6561\right )}} + x^{2} + \log \left (x\right ) \]

[In]

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+177147*x^3-59049*x),x
, algorithm="maxima")

[Out]

-1/9*x^4/(x^16 - 24*x^14 + 252*x^12 - 1512*x^10 + 5670*x^8 - 13608*x^6 + 20412*x^4 - 17496*x^2 + 6561) + x^2 +
log(x)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88 \[ \int \frac {-59049+59049 x^2+118102 x^4-288680 x^6+275562 x^8-153090 x^{10}+54432 x^{12}-12636 x^{14}+1863 x^{16}-159 x^{18}+6 x^{20}}{-59049 x+177147 x^3-236196 x^5+183708 x^7-91854 x^9+30618 x^{11}-6804 x^{13}+972 x^{15}-81 x^{17}+3 x^{19}} \, dx=x^{2} - \frac {x^{4}}{9 \, {\left (x^{2} - 3\right )}^{8}} + \frac {1}{2} \, \log \left (x^{2}\right ) \]

[In]

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+177147*x^3-59049*x),x
, algorithm="giac")

[Out]

x^2 - 1/9*x^4/(x^2 - 3)^8 + 1/2*log(x^2)

Mupad [B] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 55, normalized size of antiderivative = 2.20 \[ \int \frac {-59049+59049 x^2+118102 x^4-288680 x^6+275562 x^8-153090 x^{10}+54432 x^{12}-12636 x^{14}+1863 x^{16}-159 x^{18}+6 x^{20}}{-59049 x+177147 x^3-236196 x^5+183708 x^7-91854 x^9+30618 x^{11}-6804 x^{13}+972 x^{15}-81 x^{17}+3 x^{19}} \, dx=\ln \left (x\right )-\frac {x^4}{9\,\left (x^{16}-24\,x^{14}+252\,x^{12}-1512\,x^{10}+5670\,x^8-13608\,x^6+20412\,x^4-17496\,x^2+6561\right )}+x^2 \]

[In]

int(-(59049*x^2 + 118102*x^4 - 288680*x^6 + 275562*x^8 - 153090*x^10 + 54432*x^12 - 12636*x^14 + 1863*x^16 - 1
59*x^18 + 6*x^20 - 59049)/(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)

[Out]

log(x) - x^4/(9*(20412*x^4 - 17496*x^2 - 13608*x^6 + 5670*x^8 - 1512*x^10 + 252*x^12 - 24*x^14 + x^16 + 6561))
+ x^2