Integrand size = 17, antiderivative size = 113 \[ \int \frac {27-8 x^3}{\left (729-64 x^6\right )^2} \, dx=\frac {x}{4374 \left (27+8 x^3\right )}-\frac {7 \arctan \left (\frac {3-4 x}{3 \sqrt {3}}\right )}{157464 \sqrt {3}}+\frac {\arctan \left (\frac {3+4 x}{3 \sqrt {3}}\right )}{52488 \sqrt {3}}-\frac {\log (3-2 x)}{157464}+\frac {7 \log (3+2 x)}{472392}-\frac {7 \log \left (9-6 x+4 x^2\right )}{944784}+\frac {\log \left (9+6 x+4 x^2\right )}{314928} \]
1/4374*x/(8*x^3+27)-1/157464*ln(3-2*x)+7/472392*ln(3+2*x)-7/944784*ln(4*x^ 2-6*x+9)+1/314928*ln(4*x^2+6*x+9)-7/472392*arctan(1/9*(3-4*x)*3^(1/2))*3^( 1/2)+1/157464*arctan(1/9*(3+4*x)*3^(1/2))*3^(1/2)
Time = 0.06 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.91 \[ \int \frac {27-8 x^3}{\left (729-64 x^6\right )^2} \, dx=\frac {\frac {216 x}{27+8 x^3}+14 \sqrt {3} \arctan \left (\frac {-3+4 x}{3 \sqrt {3}}\right )+6 \sqrt {3} \arctan \left (\frac {3+4 x}{3 \sqrt {3}}\right )-6 \log (3-2 x)+14 \log (3+2 x)-7 \log \left (9-6 x+4 x^2\right )+3 \log \left (9+6 x+4 x^2\right )}{944784} \]
((216*x)/(27 + 8*x^3) + 14*Sqrt[3]*ArcTan[(-3 + 4*x)/(3*Sqrt[3])] + 6*Sqrt [3]*ArcTan[(3 + 4*x)/(3*Sqrt[3])] - 6*Log[3 - 2*x] + 14*Log[3 + 2*x] - 7*L og[9 - 6*x + 4*x^2] + 3*Log[9 + 6*x + 4*x^2])/944784
Time = 0.33 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.22, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.706, Rules used = {1388, 931, 27, 1020, 750, 16, 27, 1142, 27, 1083, 217, 1103}
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.
\(\displaystyle \int \frac {27-8 x^3}{\left (729-64 x^6\right )^2} \, dx\) |
\(\Big \downarrow \) 1388 |
\(\displaystyle \int \frac {1}{\left (27-8 x^3\right ) \left (8 x^3+27\right )^2}dx\) |
\(\Big \downarrow \) 931 |
\(\displaystyle \frac {x}{4374 \left (8 x^3+27\right )}-\frac {\int -\frac {8 \left (135-16 x^3\right )}{\left (27-8 x^3\right ) \left (8 x^3+27\right )}dx}{34992}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {135-16 x^3}{\left (27-8 x^3\right ) \left (8 x^3+27\right )}dx}{4374}+\frac {x}{4374 \left (8 x^3+27\right )}\) |
\(\Big \downarrow \) 1020 |
\(\displaystyle \frac {\frac {3}{2} \int \frac {1}{27-8 x^3}dx+\frac {7}{2} \int \frac {1}{8 x^3+27}dx}{4374}+\frac {x}{4374 \left (8 x^3+27\right )}\) |
\(\Big \downarrow \) 750 |
\(\displaystyle \frac {\frac {7}{2} \left (\frac {1}{27} \int \frac {2 (3-x)}{4 x^2-6 x+9}dx+\frac {1}{27} \int \frac {1}{2 x+3}dx\right )+\frac {3}{2} \left (\frac {1}{27} \int \frac {2 (x+3)}{4 x^2+6 x+9}dx+\frac {1}{27} \int \frac {1}{3-2 x}dx\right )}{4374}+\frac {x}{4374 \left (8 x^3+27\right )}\) |
\(\Big \downarrow \) 16 |
\(\displaystyle \frac {\frac {7}{2} \left (\frac {1}{27} \int \frac {2 (3-x)}{4 x^2-6 x+9}dx+\frac {1}{54} \log (2 x+3)\right )+\frac {3}{2} \left (\frac {1}{27} \int \frac {2 (x+3)}{4 x^2+6 x+9}dx-\frac {1}{54} \log (3-2 x)\right )}{4374}+\frac {x}{4374 \left (8 x^3+27\right )}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {7}{2} \left (\frac {2}{27} \int \frac {3-x}{4 x^2-6 x+9}dx+\frac {1}{54} \log (2 x+3)\right )+\frac {3}{2} \left (\frac {2}{27} \int \frac {x+3}{4 x^2+6 x+9}dx-\frac {1}{54} \log (3-2 x)\right )}{4374}+\frac {x}{4374 \left (8 x^3+27\right )}\) |
\(\Big \downarrow \) 1142 |
\(\displaystyle \frac {\frac {7}{2} \left (\frac {2}{27} \left (\frac {9}{4} \int \frac {1}{4 x^2-6 x+9}dx-\frac {1}{8} \int -\frac {2 (3-4 x)}{4 x^2-6 x+9}dx\right )+\frac {1}{54} \log (2 x+3)\right )+\frac {3}{2} \left (\frac {2}{27} \left (\frac {9}{4} \int \frac {1}{4 x^2+6 x+9}dx+\frac {1}{8} \int \frac {2 (4 x+3)}{4 x^2+6 x+9}dx\right )-\frac {1}{54} \log (3-2 x)\right )}{4374}+\frac {x}{4374 \left (8 x^3+27\right )}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {7}{2} \left (\frac {2}{27} \left (\frac {9}{4} \int \frac {1}{4 x^2-6 x+9}dx+\frac {1}{4} \int \frac {3-4 x}{4 x^2-6 x+9}dx\right )+\frac {1}{54} \log (2 x+3)\right )+\frac {3}{2} \left (\frac {2}{27} \left (\frac {9}{4} \int \frac {1}{4 x^2+6 x+9}dx+\frac {1}{4} \int \frac {4 x+3}{4 x^2+6 x+9}dx\right )-\frac {1}{54} \log (3-2 x)\right )}{4374}+\frac {x}{4374 \left (8 x^3+27\right )}\) |
\(\Big \downarrow \) 1083 |
\(\displaystyle \frac {\frac {7}{2} \left (\frac {2}{27} \left (\frac {1}{4} \int \frac {3-4 x}{4 x^2-6 x+9}dx-\frac {9}{2} \int \frac {1}{-(8 x-6)^2-108}d(8 x-6)\right )+\frac {1}{54} \log (2 x+3)\right )+\frac {3}{2} \left (\frac {2}{27} \left (\frac {1}{4} \int \frac {4 x+3}{4 x^2+6 x+9}dx-\frac {9}{2} \int \frac {1}{-(8 x+6)^2-108}d(8 x+6)\right )-\frac {1}{54} \log (3-2 x)\right )}{4374}+\frac {x}{4374 \left (8 x^3+27\right )}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {\frac {7}{2} \left (\frac {2}{27} \left (\frac {1}{4} \int \frac {3-4 x}{4 x^2-6 x+9}dx+\frac {1}{4} \sqrt {3} \arctan \left (\frac {8 x-6}{6 \sqrt {3}}\right )\right )+\frac {1}{54} \log (2 x+3)\right )+\frac {3}{2} \left (\frac {2}{27} \left (\frac {1}{4} \int \frac {4 x+3}{4 x^2+6 x+9}dx+\frac {1}{4} \sqrt {3} \arctan \left (\frac {8 x+6}{6 \sqrt {3}}\right )\right )-\frac {1}{54} \log (3-2 x)\right )}{4374}+\frac {x}{4374 \left (8 x^3+27\right )}\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle \frac {\frac {7}{2} \left (\frac {2}{27} \left (\frac {1}{4} \sqrt {3} \arctan \left (\frac {8 x-6}{6 \sqrt {3}}\right )-\frac {1}{8} \log \left (4 x^2-6 x+9\right )\right )+\frac {1}{54} \log (2 x+3)\right )+\frac {3}{2} \left (\frac {2}{27} \left (\frac {1}{4} \sqrt {3} \arctan \left (\frac {8 x+6}{6 \sqrt {3}}\right )+\frac {1}{8} \log \left (4 x^2+6 x+9\right )\right )-\frac {1}{54} \log (3-2 x)\right )}{4374}+\frac {x}{4374 \left (8 x^3+27\right )}\) |
x/(4374*(27 + 8*x^3)) + ((7*(Log[3 + 2*x]/54 + (2*((Sqrt[3]*ArcTan[(-6 + 8 *x)/(6*Sqrt[3])])/4 - Log[9 - 6*x + 4*x^2]/8))/27))/2 + (3*(-1/54*Log[3 - 2*x] + (2*((Sqrt[3]*ArcTan[(6 + 8*x)/(6*Sqrt[3])])/4 + Log[9 + 6*x + 4*x^2 ]/8))/27))/2)/4374
3.6.73.3.1 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[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Simp[1/(3*Rt[a, 3]^2) Int[1/ (Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[1/(3*Rt[a, 3]^2) Int[(2*Rt[a, 3] - Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], x] /; FreeQ[{a, b}, x]
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-b)*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*n*(p + 1)*(b*c - a*d))), x] + Simp[1/(a*n*(p + 1)*(b*c - a*d)) Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomialQ[a, b, c, d, n, p, q, x]
Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^( n_))), x_Symbol] :> Simp[(b*e - a*f)/(b*c - a*d) Int[1/(a + b*x^n), x], x ] - Simp[(d*e - c*f)/(b*c - a*d) Int[1/(c + d*x^n), x], x] /; FreeQ[{a, b , c, d, e, f, n}, x]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[u*(d + e*x^n)^(p + q)*(a/d + (c/e)*x^n)^p, x] /; FreeQ[{a, c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 + a*e^2, 0] && (Integer Q[p] || (GtQ[a, 0] && GtQ[d, 0]))
Time = 1.58 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.76
method | result | size |
risch | \(\frac {x}{34992 x^{3}+118098}-\frac {\ln \left (-3+2 x \right )}{157464}+\frac {7 \ln \left (2 x +3\right )}{472392}-\frac {7 \ln \left (4 x^{2}-6 x +9\right )}{944784}+\frac {7 \sqrt {3}\, \arctan \left (\frac {2 \left (-\frac {3}{2}+2 x \right ) \sqrt {3}}{9}\right )}{472392}+\frac {\ln \left (4 x^{2}+6 x +9\right )}{314928}+\frac {\sqrt {3}\, \arctan \left (\frac {2 \left (2 x +\frac {3}{2}\right ) \sqrt {3}}{9}\right )}{157464}\) | \(86\) |
default | \(-\frac {\ln \left (-3+2 x \right )}{157464}-\frac {-\frac {3 x}{4}-\frac {9}{8}}{118098 \left (x^{2}-\frac {3}{2} x +\frac {9}{4}\right )}-\frac {7 \ln \left (4 x^{2}-6 x +9\right )}{944784}+\frac {7 \sqrt {3}\, \arctan \left (\frac {\left (8 x -6\right ) \sqrt {3}}{18}\right )}{472392}-\frac {1}{78732 \left (2 x +3\right )}+\frac {7 \ln \left (2 x +3\right )}{472392}+\frac {\ln \left (4 x^{2}+6 x +9\right )}{314928}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (8 x +6\right ) \sqrt {3}}{18}\right )}{157464}\) | \(102\) |
meijerg | \(-\frac {\left (-1\right )^{\frac {5}{6}} \left (\frac {4 x \left (-1\right )^{\frac {1}{6}}}{6-\frac {128 x^{6}}{243}}-\frac {5 x \left (-1\right )^{\frac {1}{6}} \left (\ln \left (1-\frac {2 \left (x^{6}\right )^{\frac {1}{6}}}{3}\right )-\ln \left (1+\frac {2 \left (x^{6}\right )^{\frac {1}{6}}}{3}\right )+\frac {\ln \left (1-\frac {2 \left (x^{6}\right )^{\frac {1}{6}}}{3}+\frac {4 \left (x^{6}\right )^{\frac {1}{3}}}{9}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{6}\right )^{\frac {1}{6}}}{3-\left (x^{6}\right )^{\frac {1}{6}}}\right )-\frac {\ln \left (1+\frac {2 \left (x^{6}\right )^{\frac {1}{6}}}{3}+\frac {4 \left (x^{6}\right )^{\frac {1}{3}}}{9}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{6}\right )^{\frac {1}{6}}}{3+\left (x^{6}\right )^{\frac {1}{6}}}\right )\right )}{6 \left (x^{6}\right )^{\frac {1}{6}}}\right )}{78732}+\frac {\left (-1\right )^{\frac {1}{3}} \left (\frac {16 x^{4} \left (-1\right )^{\frac {2}{3}}}{27 \left (3-\frac {64 x^{6}}{243}\right )}-\frac {x^{4} \left (-1\right )^{\frac {2}{3}} \left (\ln \left (1-\frac {4 \left (x^{6}\right )^{\frac {1}{3}}}{9}\right )-\frac {\ln \left (1+\frac {4 \left (x^{6}\right )^{\frac {1}{3}}}{9}+\frac {16 \left (x^{6}\right )^{\frac {2}{3}}}{81}\right )}{2}+\sqrt {3}\, \arctan \left (\frac {2 \sqrt {3}\, \left (x^{6}\right )^{\frac {1}{3}}}{9 \left (1+\frac {2 \left (x^{6}\right )^{\frac {1}{3}}}{9}\right )}\right )\right )}{3 \left (x^{6}\right )^{\frac {2}{3}}}\right )}{78732}\) | \(241\) |
1/34992*x/(x^3+27/8)-1/157464*ln(-3+2*x)+7/472392*ln(2*x+3)-7/944784*ln(4* x^2-6*x+9)+7/472392*3^(1/2)*arctan(2/9*(-3/2+2*x)*3^(1/2))+1/314928*ln(4*x ^2+6*x+9)+1/157464*3^(1/2)*arctan(2/9*(2*x+3/2)*3^(1/2))
Time = 0.48 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.16 \[ \int \frac {27-8 x^3}{\left (729-64 x^6\right )^2} \, dx=\frac {6 \, \sqrt {3} {\left (8 \, x^{3} + 27\right )} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x + 3\right )}\right ) + 14 \, \sqrt {3} {\left (8 \, x^{3} + 27\right )} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x - 3\right )}\right ) + 3 \, {\left (8 \, x^{3} + 27\right )} \log \left (4 \, x^{2} + 6 \, x + 9\right ) - 7 \, {\left (8 \, x^{3} + 27\right )} \log \left (4 \, x^{2} - 6 \, x + 9\right ) + 14 \, {\left (8 \, x^{3} + 27\right )} \log \left (2 \, x + 3\right ) - 6 \, {\left (8 \, x^{3} + 27\right )} \log \left (2 \, x - 3\right ) + 216 \, x}{944784 \, {\left (8 \, x^{3} + 27\right )}} \]
1/944784*(6*sqrt(3)*(8*x^3 + 27)*arctan(1/9*sqrt(3)*(4*x + 3)) + 14*sqrt(3 )*(8*x^3 + 27)*arctan(1/9*sqrt(3)*(4*x - 3)) + 3*(8*x^3 + 27)*log(4*x^2 + 6*x + 9) - 7*(8*x^3 + 27)*log(4*x^2 - 6*x + 9) + 14*(8*x^3 + 27)*log(2*x + 3) - 6*(8*x^3 + 27)*log(2*x - 3) + 216*x)/(8*x^3 + 27)
Time = 0.24 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.97 \[ \int \frac {27-8 x^3}{\left (729-64 x^6\right )^2} \, dx=\frac {x}{34992 x^{3} + 118098} - \frac {\log {\left (x - \frac {3}{2} \right )}}{157464} + \frac {7 \log {\left (x + \frac {3}{2} \right )}}{472392} - \frac {7 \log {\left (x^{2} - \frac {3 x}{2} + \frac {9}{4} \right )}}{944784} + \frac {\log {\left (x^{2} + \frac {3 x}{2} + \frac {9}{4} \right )}}{314928} + \frac {7 \sqrt {3} \operatorname {atan}{\left (\frac {4 \sqrt {3} x}{9} - \frac {\sqrt {3}}{3} \right )}}{472392} + \frac {\sqrt {3} \operatorname {atan}{\left (\frac {4 \sqrt {3} x}{9} + \frac {\sqrt {3}}{3} \right )}}{157464} \]
x/(34992*x**3 + 118098) - log(x - 3/2)/157464 + 7*log(x + 3/2)/472392 - 7* log(x**2 - 3*x/2 + 9/4)/944784 + log(x**2 + 3*x/2 + 9/4)/314928 + 7*sqrt(3 )*atan(4*sqrt(3)*x/9 - sqrt(3)/3)/472392 + sqrt(3)*atan(4*sqrt(3)*x/9 + sq rt(3)/3)/157464
Time = 0.29 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.77 \[ \int \frac {27-8 x^3}{\left (729-64 x^6\right )^2} \, dx=\frac {1}{157464} \, \sqrt {3} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x + 3\right )}\right ) + \frac {7}{472392} \, \sqrt {3} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x - 3\right )}\right ) + \frac {x}{4374 \, {\left (8 \, x^{3} + 27\right )}} + \frac {1}{314928} \, \log \left (4 \, x^{2} + 6 \, x + 9\right ) - \frac {7}{944784} \, \log \left (4 \, x^{2} - 6 \, x + 9\right ) + \frac {7}{472392} \, \log \left (2 \, x + 3\right ) - \frac {1}{157464} \, \log \left (2 \, x - 3\right ) \]
1/157464*sqrt(3)*arctan(1/9*sqrt(3)*(4*x + 3)) + 7/472392*sqrt(3)*arctan(1 /9*sqrt(3)*(4*x - 3)) + 1/4374*x/(8*x^3 + 27) + 1/314928*log(4*x^2 + 6*x + 9) - 7/944784*log(4*x^2 - 6*x + 9) + 7/472392*log(2*x + 3) - 1/157464*log (2*x - 3)
Time = 0.27 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.79 \[ \int \frac {27-8 x^3}{\left (729-64 x^6\right )^2} \, dx=\frac {1}{157464} \, \sqrt {3} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x + 3\right )}\right ) + \frac {7}{472392} \, \sqrt {3} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x - 3\right )}\right ) + \frac {x}{4374 \, {\left (8 \, x^{3} + 27\right )}} + \frac {1}{314928} \, \log \left (4 \, x^{2} + 6 \, x + 9\right ) - \frac {7}{944784} \, \log \left (4 \, x^{2} - 6 \, x + 9\right ) + \frac {7}{472392} \, \log \left ({\left | 2 \, x + 3 \right |}\right ) - \frac {1}{157464} \, \log \left ({\left | 2 \, x - 3 \right |}\right ) \]
1/157464*sqrt(3)*arctan(1/9*sqrt(3)*(4*x + 3)) + 7/472392*sqrt(3)*arctan(1 /9*sqrt(3)*(4*x - 3)) + 1/4374*x/(8*x^3 + 27) + 1/314928*log(4*x^2 + 6*x + 9) - 7/944784*log(4*x^2 - 6*x + 9) + 7/472392*log(abs(2*x + 3)) - 1/15746 4*log(abs(2*x - 3))
Time = 9.75 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.90 \[ \int \frac {27-8 x^3}{\left (729-64 x^6\right )^2} \, dx=\frac {7\,\ln \left (x+\frac {3}{2}\right )}{472392}-\frac {\ln \left (x-\frac {3}{2}\right )}{157464}+\frac {x}{34992\,\left (x^3+\frac {27}{8}\right )}-\ln \left (x+\frac {3}{4}-\frac {\sqrt {3}\,3{}\mathrm {i}}{4}\right )\,\left (-\frac {1}{314928}+\frac {\sqrt {3}\,1{}\mathrm {i}}{314928}\right )+\ln \left (x+\frac {3}{4}+\frac {\sqrt {3}\,3{}\mathrm {i}}{4}\right )\,\left (\frac {1}{314928}+\frac {\sqrt {3}\,1{}\mathrm {i}}{314928}\right )-\ln \left (x-\frac {3}{4}-\frac {\sqrt {3}\,3{}\mathrm {i}}{4}\right )\,\left (\frac {7}{944784}+\frac {\sqrt {3}\,7{}\mathrm {i}}{944784}\right )+\ln \left (x-\frac {3}{4}+\frac {\sqrt {3}\,3{}\mathrm {i}}{4}\right )\,\left (-\frac {7}{944784}+\frac {\sqrt {3}\,7{}\mathrm {i}}{944784}\right ) \]
(7*log(x + 3/2))/472392 - log(x - 3/2)/157464 + x/(34992*(x^3 + 27/8)) - l og(x - (3^(1/2)*3i)/4 + 3/4)*((3^(1/2)*1i)/314928 - 1/314928) + log(x + (3 ^(1/2)*3i)/4 + 3/4)*((3^(1/2)*1i)/314928 + 1/314928) - log(x - (3^(1/2)*3i )/4 - 3/4)*((3^(1/2)*7i)/944784 + 7/944784) + log(x + (3^(1/2)*3i)/4 - 3/4 )*((3^(1/2)*7i)/944784 - 7/944784)