Integrand size = 25, antiderivative size = 131 \[ \int \frac {27+36 x+24 x^2+8 x^3}{\left (729-64 x^6\right )^2} \, dx=\frac {1}{26244 (3-2 x)}-\frac {3-2 x}{26244 \left (9-6 x+4 x^2\right )}-\frac {11 \arctan \left (\frac {3-4 x}{3 \sqrt {3}}\right )}{157464 \sqrt {3}}-\frac {\arctan \left (\frac {3+4 x}{3 \sqrt {3}}\right )}{157464 \sqrt {3}}-\frac {7 \log (3-2 x)}{157464}+\frac {\log (3+2 x)}{472392}+\frac {17 \log \left (9-6 x+4 x^2\right )}{944784}+\frac {\log \left (9+6 x+4 x^2\right )}{314928} \]
1/26244/(3-2*x)+1/26244*(-3+2*x)/(4*x^2-6*x+9)-7/157464*ln(3-2*x)+1/472392 *ln(3+2*x)+17/944784*ln(4*x^2-6*x+9)+1/314928*ln(4*x^2+6*x+9)-11/472392*ar ctan(1/9*(3-4*x)*3^(1/2))*3^(1/2)-1/472392*arctan(1/9*(3+4*x)*3^(1/2))*3^( 1/2)
Time = 0.07 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.85 \[ \int \frac {27+36 x+24 x^2+8 x^3}{\left (729-64 x^6\right )^2} \, dx=\frac {\frac {216 x}{27-36 x+24 x^2-8 x^3}+22 \sqrt {3} \arctan \left (\frac {-3+4 x}{3 \sqrt {3}}\right )-2 \sqrt {3} \arctan \left (\frac {3+4 x}{3 \sqrt {3}}\right )-42 \log (3-2 x)+2 \log (3+2 x)+17 \log \left (9-6 x+4 x^2\right )+3 \log \left (9+6 x+4 x^2\right )}{944784} \]
((216*x)/(27 - 36*x + 24*x^2 - 8*x^3) + 22*Sqrt[3]*ArcTan[(-3 + 4*x)/(3*Sq rt[3])] - 2*Sqrt[3]*ArcTan[(3 + 4*x)/(3*Sqrt[3])] - 42*Log[3 - 2*x] + 2*Lo g[3 + 2*x] + 17*Log[9 - 6*x + 4*x^2] + 3*Log[9 + 6*x + 4*x^2])/944784
Time = 0.35 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2019, 2462, 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.
| \(\displaystyle \int \frac {8 x^3+24 x^2+36 x+27}{\left (729-64 x^6\right )^2} \, dx\) |
| \(\Big \downarrow \) 2019 |
| \(\displaystyle \int \frac {1}{\left (-8 x^3+24 x^2-36 x+27\right )^2 \left (8 x^3+24 x^2+36 x+27\right )}dx\) |
| \(\Big \downarrow \) 2462 |
| \(\displaystyle \int \left (\frac {x}{39366 \left (4 x^2+6 x+9\right )}+\frac {17 x+3}{118098 \left (4 x^2-6 x+9\right )}+\frac {2 x+3}{4374 \left (4 x^2-6 x+9\right )^2}-\frac {7}{78732 (2 x-3)}+\frac {1}{236196 (2 x+3)}+\frac {1}{13122 (2 x-3)^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {11 \arctan \left (\frac {3-4 x}{3 \sqrt {3}}\right )}{157464 \sqrt {3}}-\frac {\arctan \left (\frac {4 x+3}{3 \sqrt {3}}\right )}{157464 \sqrt {3}}-\frac {3-2 x}{26244 \left (4 x^2-6 x+9\right )}+\frac {17 \log \left (4 x^2-6 x+9\right )}{944784}+\frac {\log \left (4 x^2+6 x+9\right )}{314928}+\frac {1}{26244 (3-2 x)}-\frac {7 \log (3-2 x)}{157464}+\frac {\log (2 x+3)}{472392}\) |
1/(26244*(3 - 2*x)) - (3 - 2*x)/(26244*(9 - 6*x + 4*x^2)) - (11*ArcTan[(3 - 4*x)/(3*Sqrt[3])])/(157464*Sqrt[3]) - ArcTan[(3 + 4*x)/(3*Sqrt[3])]/(157 464*Sqrt[3]) - (7*Log[3 - 2*x])/157464 + Log[3 + 2*x]/472392 + (17*Log[9 - 6*x + 4*x^2])/944784 + Log[9 + 6*x + 4*x^2]/314928
3.6.74.3.1 Defintions of rubi rules used
Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px , Qx, x]^p*Qx^(p + q), x] /; FreeQ[q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u*Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && GtQ [Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0 ] && RationalFunctionQ[u, x]
Time = 1.54 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.72
| method | result | size |
| risch | \(-\frac {x}{34992 \left (x^{3}-3 x^{2}+\frac {9}{2} x -\frac {27}{8}\right )}-\frac {7 \ln \left (-3+2 x \right )}{157464}+\frac {\ln \left (4 x^{2}+6 x +9\right )}{314928}-\frac {\sqrt {3}\, \arctan \left (\frac {2 \left (2 x +\frac {3}{2}\right ) \sqrt {3}}{9}\right )}{472392}+\frac {17 \ln \left (484 x^{2}-726 x +1089\right )}{944784}+\frac {11 \sqrt {3}\, \arctan \left (\frac {2 \left (22 x -\frac {33}{2}\right ) \sqrt {3}}{99}\right )}{472392}+\frac {\ln \left (2 x +3\right )}{472392}\) | \(94\) |
| default | \(-\frac {1}{26244 \left (-3+2 x \right )}-\frac {7 \ln \left (-3+2 x \right )}{157464}+\frac {\frac {9 x}{4}-\frac {27}{8}}{118098 x^{2}-177147 x +\frac {531441}{2}}+\frac {17 \ln \left (4 x^{2}-6 x +9\right )}{944784}+\frac {11 \sqrt {3}\, \arctan \left (\frac {\left (8 x -6\right ) \sqrt {3}}{18}\right )}{472392}+\frac {\ln \left (2 x +3\right )}{472392}+\frac {\ln \left (4 x^{2}+6 x +9\right )}{314928}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (8 x +6\right ) \sqrt {3}}{18}\right )}{472392}\) | \(102\) |
| 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}-\frac {i \left (\frac {16 i x^{3}}{27 \left (-\frac {128 x^{6}}{729}+2\right )}+i \operatorname {arctanh}\left (\frac {8 x^{3}}{27}\right )\right )}{39366}-\frac {\left (-1\right )^{\frac {2}{3}} \left (\frac {4 x^{2} \left (-1\right )^{\frac {1}{3}}}{3 \left (3-\frac {64 x^{6}}{243}\right )}-\frac {2 x^{2} \left (-1\right )^{\frac {1}{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 {1}{3}}}\right )}{39366}\) | \(362\) |
-1/34992*x/(x^3-3*x^2+9/2*x-27/8)-7/157464*ln(-3+2*x)+1/314928*ln(4*x^2+6* x+9)-1/472392*3^(1/2)*arctan(2/9*(2*x+3/2)*3^(1/2))+17/944784*ln(484*x^2-7 26*x+1089)+11/472392*3^(1/2)*arctan(2/99*(22*x-33/2)*3^(1/2))+1/472392*ln( 2*x+3)
Time = 0.46 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.43 \[ \int \frac {27+36 x+24 x^2+8 x^3}{\left (729-64 x^6\right )^2} \, dx=-\frac {2 \, \sqrt {3} {\left (8 \, x^{3} - 24 \, x^{2} + 36 \, x - 27\right )} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x + 3\right )}\right ) - 22 \, \sqrt {3} {\left (8 \, x^{3} - 24 \, x^{2} + 36 \, x - 27\right )} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x - 3\right )}\right ) - 3 \, {\left (8 \, x^{3} - 24 \, x^{2} + 36 \, x - 27\right )} \log \left (4 \, x^{2} + 6 \, x + 9\right ) - 17 \, {\left (8 \, x^{3} - 24 \, x^{2} + 36 \, x - 27\right )} \log \left (4 \, x^{2} - 6 \, x + 9\right ) - 2 \, {\left (8 \, x^{3} - 24 \, x^{2} + 36 \, x - 27\right )} \log \left (2 \, x + 3\right ) + 42 \, {\left (8 \, x^{3} - 24 \, x^{2} + 36 \, x - 27\right )} \log \left (2 \, x - 3\right ) + 216 \, x}{944784 \, {\left (8 \, x^{3} - 24 \, x^{2} + 36 \, x - 27\right )}} \]
-1/944784*(2*sqrt(3)*(8*x^3 - 24*x^2 + 36*x - 27)*arctan(1/9*sqrt(3)*(4*x + 3)) - 22*sqrt(3)*(8*x^3 - 24*x^2 + 36*x - 27)*arctan(1/9*sqrt(3)*(4*x - 3)) - 3*(8*x^3 - 24*x^2 + 36*x - 27)*log(4*x^2 + 6*x + 9) - 17*(8*x^3 - 24 *x^2 + 36*x - 27)*log(4*x^2 - 6*x + 9) - 2*(8*x^3 - 24*x^2 + 36*x - 27)*lo g(2*x + 3) + 42*(8*x^3 - 24*x^2 + 36*x - 27)*log(2*x - 3) + 216*x)/(8*x^3 - 24*x^2 + 36*x - 27)
Time = 0.29 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.91 \[ \int \frac {27+36 x+24 x^2+8 x^3}{\left (729-64 x^6\right )^2} \, dx=- \frac {x}{34992 x^{3} - 104976 x^{2} + 157464 x - 118098} - \frac {7 \log {\left (x - \frac {3}{2} \right )}}{157464} + \frac {\log {\left (x + \frac {3}{2} \right )}}{472392} + \frac {17 \log {\left (x^{2} - \frac {3 x}{2} + \frac {9}{4} \right )}}{944784} + \frac {\log {\left (x^{2} + \frac {3 x}{2} + \frac {9}{4} \right )}}{314928} + \frac {11 \sqrt {3} \operatorname {atan}{\left (\frac {4 \sqrt {3} x}{9} - \frac {\sqrt {3}}{3} \right )}}{472392} - \frac {\sqrt {3} \operatorname {atan}{\left (\frac {4 \sqrt {3} x}{9} + \frac {\sqrt {3}}{3} \right )}}{472392} \]
-x/(34992*x**3 - 104976*x**2 + 157464*x - 118098) - 7*log(x - 3/2)/157464 + log(x + 3/2)/472392 + 17*log(x**2 - 3*x/2 + 9/4)/944784 + log(x**2 + 3*x /2 + 9/4)/314928 + 11*sqrt(3)*atan(4*sqrt(3)*x/9 - sqrt(3)/3)/472392 - sqr t(3)*atan(4*sqrt(3)*x/9 + sqrt(3)/3)/472392
Time = 0.29 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.73 \[ \int \frac {27+36 x+24 x^2+8 x^3}{\left (729-64 x^6\right )^2} \, dx=-\frac {1}{472392} \, \sqrt {3} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x + 3\right )}\right ) + \frac {11}{472392} \, \sqrt {3} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x - 3\right )}\right ) - \frac {x}{4374 \, {\left (8 \, x^{3} - 24 \, x^{2} + 36 \, x - 27\right )}} + \frac {1}{314928} \, \log \left (4 \, x^{2} + 6 \, x + 9\right ) + \frac {17}{944784} \, \log \left (4 \, x^{2} - 6 \, x + 9\right ) + \frac {1}{472392} \, \log \left (2 \, x + 3\right ) - \frac {7}{157464} \, \log \left (2 \, x - 3\right ) \]
-1/472392*sqrt(3)*arctan(1/9*sqrt(3)*(4*x + 3)) + 11/472392*sqrt(3)*arctan (1/9*sqrt(3)*(4*x - 3)) - 1/4374*x/(8*x^3 - 24*x^2 + 36*x - 27) + 1/314928 *log(4*x^2 + 6*x + 9) + 17/944784*log(4*x^2 - 6*x + 9) + 1/472392*log(2*x + 3) - 7/157464*log(2*x - 3)
Time = 0.28 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.76 \[ \int \frac {27+36 x+24 x^2+8 x^3}{\left (729-64 x^6\right )^2} \, dx=-\frac {1}{472392} \, \sqrt {3} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x + 3\right )}\right ) + \frac {11}{472392} \, \sqrt {3} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x - 3\right )}\right ) - \frac {x}{4374 \, {\left (4 \, x^{2} - 6 \, x + 9\right )} {\left (2 \, x - 3\right )}} + \frac {1}{314928} \, \log \left (4 \, x^{2} + 6 \, x + 9\right ) + \frac {17}{944784} \, \log \left (4 \, x^{2} - 6 \, x + 9\right ) + \frac {1}{472392} \, \log \left ({\left | 2 \, x + 3 \right |}\right ) - \frac {7}{157464} \, \log \left ({\left | 2 \, x - 3 \right |}\right ) \]
-1/472392*sqrt(3)*arctan(1/9*sqrt(3)*(4*x + 3)) + 11/472392*sqrt(3)*arctan (1/9*sqrt(3)*(4*x - 3)) - 1/4374*x/((4*x^2 - 6*x + 9)*(2*x - 3)) + 1/31492 8*log(4*x^2 + 6*x + 9) + 17/944784*log(4*x^2 - 6*x + 9) + 1/472392*log(abs (2*x + 3)) - 7/157464*log(abs(2*x - 3))
Time = 0.19 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.85 \[ \int \frac {27+36 x+24 x^2+8 x^3}{\left (729-64 x^6\right )^2} \, dx=\frac {\ln \left (x+\frac {3}{2}\right )}{472392}-\frac {7\,\ln \left (x-\frac {3}{2}\right )}{157464}-\frac {x}{34992\,\left (x^3-3\,x^2+\frac {9\,x}{2}-\frac {27}{8}\right )}+\ln \left (x+\frac {3}{4}-\frac {\sqrt {3}\,3{}\mathrm {i}}{4}\right )\,\left (\frac {1}{314928}+\frac {\sqrt {3}\,1{}\mathrm {i}}{944784}\right )-\ln \left (x+\frac {3}{4}+\frac {\sqrt {3}\,3{}\mathrm {i}}{4}\right )\,\left (-\frac {1}{314928}+\frac {\sqrt {3}\,1{}\mathrm {i}}{944784}\right )-\ln \left (x-\frac {3}{4}-\frac {\sqrt {3}\,3{}\mathrm {i}}{4}\right )\,\left (-\frac {17}{944784}+\frac {\sqrt {3}\,11{}\mathrm {i}}{944784}\right )+\ln \left (x-\frac {3}{4}+\frac {\sqrt {3}\,3{}\mathrm {i}}{4}\right )\,\left (\frac {17}{944784}+\frac {\sqrt {3}\,11{}\mathrm {i}}{944784}\right ) \]
log(x + 3/2)/472392 - (7*log(x - 3/2))/157464 - x/(34992*((9*x)/2 - 3*x^2 + x^3 - 27/8)) + log(x - (3^(1/2)*3i)/4 + 3/4)*((3^(1/2)*1i)/944784 + 1/31 4928) - log(x + (3^(1/2)*3i)/4 + 3/4)*((3^(1/2)*1i)/944784 - 1/314928) - l og(x - (3^(1/2)*3i)/4 - 3/4)*((3^(1/2)*11i)/944784 - 17/944784) + log(x + (3^(1/2)*3i)/4 - 3/4)*((3^(1/2)*11i)/944784 + 17/944784)