Integrand size = 20, antiderivative size = 142 \[ \int \frac {9+6 x+4 x^2}{\left (729-64 x^6\right )^2} \, dx=\frac {1}{157464 (3-2 x)}-\frac {1}{472392 (3+2 x)}-\frac {3-4 x}{236196 \left (9-6 x+4 x^2\right )}-\frac {\arctan \left (\frac {3-4 x}{3 \sqrt {3}}\right )}{52488 \sqrt {3}}+\frac {\arctan \left (\frac {3+4 x}{3 \sqrt {3}}\right )}{472392 \sqrt {3}}-\frac {\log (3-2 x)}{118098}+\frac {\log (3+2 x)}{354294}+\frac {5 \log \left (9-6 x+4 x^2\right )}{2834352}+\frac {\log \left (9+6 x+4 x^2\right )}{944784} \] Output:
1/(472392-314928*x)-1/(1417176+944784*x)-(3-4*x)/(944784*x^2-1417176*x+212 5764)-1/157464*arctan(1/9*(3-4*x)*3^(1/2))*3^(1/2)+1/1417176*arctan(1/9*(3 +4*x)*3^(1/2))*3^(1/2)-1/118098*ln(3-2*x)+1/354294*ln(3+2*x)+5/2834352*ln( 4*x^2-6*x+9)+1/944784*ln(4*x^2+6*x+9)
Time = 0.04 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.78 \[ \int \frac {9+6 x+4 x^2}{\left (729-64 x^6\right )^2} \, dx=\frac {\frac {648 x}{81-54 x+24 x^3-16 x^4}+18 \sqrt {3} \arctan \left (\frac {-3+4 x}{3 \sqrt {3}}\right )+2 \sqrt {3} \arctan \left (\frac {3+4 x}{3 \sqrt {3}}\right )-24 \log (3-2 x)+8 \log (3+2 x)+5 \log \left (9-6 x+4 x^2\right )+3 \log \left (9+6 x+4 x^2\right )}{2834352} \] Input:
Integrate[(9 + 6*x + 4*x^2)/(729 - 64*x^6)^2,x]
Output:
((648*x)/(81 - 54*x + 24*x^3 - 16*x^4) + 18*Sqrt[3]*ArcTan[(-3 + 4*x)/(3*S qrt[3])] + 2*Sqrt[3]*ArcTan[(3 + 4*x)/(3*Sqrt[3])] - 24*Log[3 - 2*x] + 8*L og[3 + 2*x] + 5*Log[9 - 6*x + 4*x^2] + 3*Log[9 + 6*x + 4*x^2])/2834352
Time = 0.58 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, 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.
| \(\displaystyle \int \frac {4 x^2+6 x+9}{\left (729-64 x^6\right )^2} \, dx\) |
| \(\Big \downarrow \) 2019 |
| \(\displaystyle \int \frac {1}{\left (4 x^2+6 x+9\right ) \left (-16 x^4+24 x^3-54 x+81\right )^2}dx\) |
| \(\Big \downarrow \) 2462 |
| \(\displaystyle \int \left (\frac {2 x+3}{236196 \left (4 x^2+6 x+9\right )}+\frac {10 x+21}{708588 \left (4 x^2-6 x+9\right )}+\frac {1}{4374 \left (4 x^2-6 x+9\right )^2}-\frac {1}{59049 (2 x-3)}+\frac {1}{78732 (2 x-3)^2}+\frac {1}{177147 (2 x+3)}+\frac {1}{236196 (2 x+3)^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\arctan \left (\frac {3-4 x}{3 \sqrt {3}}\right )}{52488 \sqrt {3}}+\frac {\arctan \left (\frac {4 x+3}{3 \sqrt {3}}\right )}{472392 \sqrt {3}}-\frac {3-4 x}{236196 \left (4 x^2-6 x+9\right )}+\frac {5 \log \left (4 x^2-6 x+9\right )}{2834352}+\frac {\log \left (4 x^2+6 x+9\right )}{944784}+\frac {1}{157464 (3-2 x)}-\frac {1}{472392 (2 x+3)}-\frac {\log (3-2 x)}{118098}+\frac {\log (2 x+3)}{354294}\) |
Input:
Int[(9 + 6*x + 4*x^2)/(729 - 64*x^6)^2,x]
Output:
1/(157464*(3 - 2*x)) - 1/(472392*(3 + 2*x)) - (3 - 4*x)/(236196*(9 - 6*x + 4*x^2)) - ArcTan[(3 - 4*x)/(3*Sqrt[3])]/(52488*Sqrt[3]) + ArcTan[(3 + 4*x )/(3*Sqrt[3])]/(472392*Sqrt[3]) - Log[3 - 2*x]/118098 + Log[3 + 2*x]/35429 4 + (5*Log[9 - 6*x + 4*x^2])/2834352 + Log[9 + 6*x + 4*x^2]/944784
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 = 0.12 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.66
| method | result | size |
| risch | \(-\frac {x}{69984 \left (x^{4}-\frac {3}{2} x^{3}+\frac {27}{8} x -\frac {81}{16}\right )}+\frac {5 \ln \left (16 x^{2}-24 x +36\right )}{2834352}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (4 x -3\right ) \sqrt {3}}{9}\right )}{157464}+\frac {\arctan \left (\frac {\left (3+4 x \right ) \sqrt {3}}{9}\right ) \sqrt {3}}{1417176}+\frac {\ln \left (16 x^{2}+24 x +36\right )}{944784}+\frac {\ln \left (2 x +3\right )}{354294}-\frac {\ln \left (2 x -3\right )}{118098}\) | \(94\) |
| default | \(\frac {3 x -\frac {9}{4}}{708588 x^{2}-1062882 x +1594323}+\frac {5 \ln \left (4 x^{2}-6 x +9\right )}{2834352}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (8 x -6\right ) \sqrt {3}}{18}\right )}{157464}-\frac {1}{157464 \left (2 x -3\right )}-\frac {\ln \left (2 x -3\right )}{118098}+\frac {\ln \left (4 x^{2}+6 x +9\right )}{944784}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (8 x +6\right ) \sqrt {3}}{18}\right )}{1417176}-\frac {1}{472392 \left (2 x +3\right )}+\frac {\ln \left (2 x +3\right )}{354294}\) | \(111\) |
| meijerg | \(-\frac {\left (-1\right )^{\frac {5}{6}} \left (\frac {4 x \left (-1\right )^{\frac {1}{6}}}{6-\frac {128 x^{6}}{243}}-\frac {5 x \left (-1\right )^{\frac {1}{6}} \left (\ln \left (1-\frac {2 \left (x^{6}\right )^{\frac {1}{6}}}{3}\right )-\ln \left (1+\frac {2 \left (x^{6}\right )^{\frac {1}{6}}}{3}\right )+\frac {\ln \left (1-\frac {2 \left (x^{6}\right )^{\frac {1}{6}}}{3}+\frac {4 \left (x^{6}\right )^{\frac {1}{3}}}{9}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{6}\right )^{\frac {1}{6}}}{3-\left (x^{6}\right )^{\frac {1}{6}}}\right )-\frac {\ln \left (1+\frac {2 \left (x^{6}\right )^{\frac {1}{6}}}{3}+\frac {4 \left (x^{6}\right )^{\frac {1}{3}}}{9}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{6}\right )^{\frac {1}{6}}}{3+\left (x^{6}\right )^{\frac {1}{6}}}\right )\right )}{6 \left (x^{6}\right )^{\frac {1}{6}}}\right )}{236196}-\frac {i \left (\frac {16 i x^{3}}{27 \left (-\frac {128 x^{6}}{729}+2\right )}+i \operatorname {arctanh}\left (\frac {8 x^{3}}{27}\right )\right )}{236196}-\frac {\left (-1\right )^{\frac {2}{3}} \left (\frac {4 x^{2} \left (-1\right )^{\frac {1}{3}}}{3 \left (3-\frac {64 x^{6}}{243}\right )}-\frac {2 x^{2} \left (-1\right )^{\frac {1}{3}} \left (\ln \left (1-\frac {4 \left (x^{6}\right )^{\frac {1}{3}}}{9}\right )-\frac {\ln \left (1+\frac {4 \left (x^{6}\right )^{\frac {1}{3}}}{9}+\frac {16 \left (x^{6}\right )^{\frac {2}{3}}}{81}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {2 \sqrt {3}\, \left (x^{6}\right )^{\frac {1}{3}}}{9 \left (1+\frac {2 \left (x^{6}\right )^{\frac {1}{3}}}{9}\right )}\right )\right )}{3 \left (x^{6}\right )^{\frac {1}{3}}}\right )}{236196}\) | \(270\) |
Input:
int((4*x^2+6*x+9)/(-64*x^6+729)^2,x,method=_RETURNVERBOSE)
Output:
-1/69984*x/(x^4-3/2*x^3+27/8*x-81/16)+5/2834352*ln(16*x^2-24*x+36)+1/15746 4*3^(1/2)*arctan(1/9*(4*x-3)*3^(1/2))+1/1417176*arctan(1/9*(3+4*x)*3^(1/2) )*3^(1/2)+1/944784*ln(16*x^2+24*x+36)+1/354294*ln(2*x+3)-1/118098*ln(2*x-3 )
Time = 0.08 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.32 \[ \int \frac {9+6 x+4 x^2}{\left (729-64 x^6\right )^2} \, dx=\frac {2 \, \sqrt {3} {\left (16 \, x^{4} - 24 \, x^{3} + 54 \, x - 81\right )} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x + 3\right )}\right ) + 18 \, \sqrt {3} {\left (16 \, x^{4} - 24 \, x^{3} + 54 \, x - 81\right )} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x - 3\right )}\right ) + 3 \, {\left (16 \, x^{4} - 24 \, x^{3} + 54 \, x - 81\right )} \log \left (4 \, x^{2} + 6 \, x + 9\right ) + 5 \, {\left (16 \, x^{4} - 24 \, x^{3} + 54 \, x - 81\right )} \log \left (4 \, x^{2} - 6 \, x + 9\right ) + 8 \, {\left (16 \, x^{4} - 24 \, x^{3} + 54 \, x - 81\right )} \log \left (2 \, x + 3\right ) - 24 \, {\left (16 \, x^{4} - 24 \, x^{3} + 54 \, x - 81\right )} \log \left (2 \, x - 3\right ) - 648 \, x}{2834352 \, {\left (16 \, x^{4} - 24 \, x^{3} + 54 \, x - 81\right )}} \] Input:
integrate((4*x^2+6*x+9)/(-64*x^6+729)^2,x, algorithm="fricas")
Output:
1/2834352*(2*sqrt(3)*(16*x^4 - 24*x^3 + 54*x - 81)*arctan(1/9*sqrt(3)*(4*x + 3)) + 18*sqrt(3)*(16*x^4 - 24*x^3 + 54*x - 81)*arctan(1/9*sqrt(3)*(4*x - 3)) + 3*(16*x^4 - 24*x^3 + 54*x - 81)*log(4*x^2 + 6*x + 9) + 5*(16*x^4 - 24*x^3 + 54*x - 81)*log(4*x^2 - 6*x + 9) + 8*(16*x^4 - 24*x^3 + 54*x - 81 )*log(2*x + 3) - 24*(16*x^4 - 24*x^3 + 54*x - 81)*log(2*x - 3) - 648*x)/(1 6*x^4 - 24*x^3 + 54*x - 81)
Time = 0.42 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.82 \[ \int \frac {9+6 x+4 x^2}{\left (729-64 x^6\right )^2} \, dx=- \frac {x}{69984 x^{4} - 104976 x^{3} + 236196 x - 354294} - \frac {\log {\left (x - \frac {3}{2} \right )}}{118098} + \frac {\log {\left (x + \frac {3}{2} \right )}}{354294} + \frac {5 \log {\left (x^{2} - \frac {3 x}{2} + \frac {9}{4} \right )}}{2834352} + \frac {\log {\left (x^{2} + \frac {3 x}{2} + \frac {9}{4} \right )}}{944784} + \frac {\sqrt {3} \operatorname {atan}{\left (\frac {4 \sqrt {3} x}{9} - \frac {\sqrt {3}}{3} \right )}}{157464} + \frac {\sqrt {3} \operatorname {atan}{\left (\frac {4 \sqrt {3} x}{9} + \frac {\sqrt {3}}{3} \right )}}{1417176} \] Input:
integrate((4*x**2+6*x+9)/(-64*x**6+729)**2,x)
Output:
-x/(69984*x**4 - 104976*x**3 + 236196*x - 354294) - log(x - 3/2)/118098 + log(x + 3/2)/354294 + 5*log(x**2 - 3*x/2 + 9/4)/2834352 + log(x**2 + 3*x/2 + 9/4)/944784 + sqrt(3)*atan(4*sqrt(3)*x/9 - sqrt(3)/3)/157464 + sqrt(3)* atan(4*sqrt(3)*x/9 + sqrt(3)/3)/1417176
Time = 0.11 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.67 \[ \int \frac {9+6 x+4 x^2}{\left (729-64 x^6\right )^2} \, dx=\frac {1}{1417176} \, \sqrt {3} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x + 3\right )}\right ) + \frac {1}{157464} \, \sqrt {3} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x - 3\right )}\right ) - \frac {x}{4374 \, {\left (16 \, x^{4} - 24 \, x^{3} + 54 \, x - 81\right )}} + \frac {1}{944784} \, \log \left (4 \, x^{2} + 6 \, x + 9\right ) + \frac {5}{2834352} \, \log \left (4 \, x^{2} - 6 \, x + 9\right ) + \frac {1}{354294} \, \log \left (2 \, x + 3\right ) - \frac {1}{118098} \, \log \left (2 \, x - 3\right ) \] Input:
integrate((4*x^2+6*x+9)/(-64*x^6+729)^2,x, algorithm="maxima")
Output:
1/1417176*sqrt(3)*arctan(1/9*sqrt(3)*(4*x + 3)) + 1/157464*sqrt(3)*arctan( 1/9*sqrt(3)*(4*x - 3)) - 1/4374*x/(16*x^4 - 24*x^3 + 54*x - 81) + 1/944784 *log(4*x^2 + 6*x + 9) + 5/2834352*log(4*x^2 - 6*x + 9) + 1/354294*log(2*x + 3) - 1/118098*log(2*x - 3)
Time = 0.12 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.75 \[ \int \frac {9+6 x+4 x^2}{\left (729-64 x^6\right )^2} \, dx=\frac {1}{1417176} \, \sqrt {3} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x + 3\right )}\right ) + \frac {1}{157464} \, \sqrt {3} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x - 3\right )}\right ) - \frac {x}{4374 \, {\left (4 \, x^{2} - 6 \, x + 9\right )} {\left (2 \, x + 3\right )} {\left (2 \, x - 3\right )}} + \frac {1}{944784} \, \log \left (4 \, x^{2} + 6 \, x + 9\right ) + \frac {5}{2834352} \, \log \left (4 \, x^{2} - 6 \, x + 9\right ) + \frac {1}{354294} \, \log \left ({\left | 2 \, x + 3 \right |}\right ) - \frac {1}{118098} \, \log \left ({\left | 2 \, x - 3 \right |}\right ) \] Input:
integrate((4*x^2+6*x+9)/(-64*x^6+729)^2,x, algorithm="giac")
Output:
1/1417176*sqrt(3)*arctan(1/9*sqrt(3)*(4*x + 3)) + 1/157464*sqrt(3)*arctan( 1/9*sqrt(3)*(4*x - 3)) - 1/4374*x/((4*x^2 - 6*x + 9)*(2*x + 3)*(2*x - 3)) + 1/944784*log(4*x^2 + 6*x + 9) + 5/2834352*log(4*x^2 - 6*x + 9) + 1/35429 4*log(abs(2*x + 3)) - 1/118098*log(abs(2*x - 3))
Time = 0.20 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.78 \[ \int \frac {9+6 x+4 x^2}{\left (729-64 x^6\right )^2} \, dx=\frac {\ln \left (x+\frac {3}{2}\right )}{354294}-\frac {\ln \left (x-\frac {3}{2}\right )}{118098}-\frac {x}{69984\,\left (x^4-\frac {3\,x^3}{2}+\frac {27\,x}{8}-\frac {81}{16}\right )}-\ln \left (x-\frac {3}{4}-\frac {\sqrt {3}\,3{}\mathrm {i}}{4}\right )\,\left (-\frac {5}{2834352}+\frac {\sqrt {3}\,1{}\mathrm {i}}{314928}\right )+\ln \left (x-\frac {3}{4}+\frac {\sqrt {3}\,3{}\mathrm {i}}{4}\right )\,\left (\frac {5}{2834352}+\frac {\sqrt {3}\,1{}\mathrm {i}}{314928}\right )-\ln \left (x+\frac {3}{4}-\frac {\sqrt {3}\,3{}\mathrm {i}}{4}\right )\,\left (-\frac {1}{944784}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2834352}\right )+\ln \left (x+\frac {3}{4}+\frac {\sqrt {3}\,3{}\mathrm {i}}{4}\right )\,\left (\frac {1}{944784}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2834352}\right ) \] Input:
int((6*x + 4*x^2 + 9)/(64*x^6 - 729)^2,x)
Output:
log(x + 3/2)/354294 - log(x - 3/2)/118098 - x/(69984*((27*x)/8 - (3*x^3)/2 + x^4 - 81/16)) - log(x - (3^(1/2)*3i)/4 - 3/4)*((3^(1/2)*1i)/314928 - 5/ 2834352) + log(x + (3^(1/2)*3i)/4 - 3/4)*((3^(1/2)*1i)/314928 + 5/2834352) - log(x - (3^(1/2)*3i)/4 + 3/4)*((3^(1/2)*1i)/2834352 - 1/944784) + log(x + (3^(1/2)*3i)/4 + 3/4)*((3^(1/2)*1i)/2834352 + 1/944784)
Time = 0.15 (sec) , antiderivative size = 360, normalized size of antiderivative = 2.54 \[ \int \frac {9+6 x+4 x^2}{\left (729-64 x^6\right )^2} \, dx=\frac {-648 x -405 \,\mathrm {log}\left (4 x^{2}-6 x +9\right )-162 \sqrt {3}\, \mathit {atan} \left (\frac {4 x +3}{3 \sqrt {3}}\right )-648 \,\mathrm {log}\left (2 x +3\right )+1944 \,\mathrm {log}\left (2 x -3\right )+288 \sqrt {3}\, \mathit {atan} \left (\frac {4 x -3}{3 \sqrt {3}}\right ) x^{4}-432 \sqrt {3}\, \mathit {atan} \left (\frac {4 x -3}{3 \sqrt {3}}\right ) x^{3}+32 \sqrt {3}\, \mathit {atan} \left (\frac {4 x +3}{3 \sqrt {3}}\right ) x^{4}-48 \sqrt {3}\, \mathit {atan} \left (\frac {4 x +3}{3 \sqrt {3}}\right ) x^{3}-1458 \sqrt {3}\, \mathit {atan} \left (\frac {4 x -3}{3 \sqrt {3}}\right )+270 \,\mathrm {log}\left (4 x^{2}-6 x +9\right ) x +162 \,\mathrm {log}\left (4 x^{2}+6 x +9\right ) x -1296 \,\mathrm {log}\left (2 x -3\right ) x +432 \,\mathrm {log}\left (2 x +3\right ) x -243 \,\mathrm {log}\left (4 x^{2}+6 x +9\right )+972 \sqrt {3}\, \mathit {atan} \left (\frac {4 x -3}{3 \sqrt {3}}\right ) x +108 \sqrt {3}\, \mathit {atan} \left (\frac {4 x +3}{3 \sqrt {3}}\right ) x +80 \,\mathrm {log}\left (4 x^{2}-6 x +9\right ) x^{4}-120 \,\mathrm {log}\left (4 x^{2}-6 x +9\right ) x^{3}+48 \,\mathrm {log}\left (4 x^{2}+6 x +9\right ) x^{4}-72 \,\mathrm {log}\left (4 x^{2}+6 x +9\right ) x^{3}-384 \,\mathrm {log}\left (2 x -3\right ) x^{4}+576 \,\mathrm {log}\left (2 x -3\right ) x^{3}+128 \,\mathrm {log}\left (2 x +3\right ) x^{4}-192 \,\mathrm {log}\left (2 x +3\right ) x^{3}}{45349632 x^{4}-68024448 x^{3}+153055008 x -229582512} \] Input:
int((4*x^2+6*x+9)/(-64*x^6+729)^2,x)
Output:
(288*sqrt(3)*atan((4*x - 3)/(3*sqrt(3)))*x**4 - 432*sqrt(3)*atan((4*x - 3) /(3*sqrt(3)))*x**3 + 972*sqrt(3)*atan((4*x - 3)/(3*sqrt(3)))*x - 1458*sqrt (3)*atan((4*x - 3)/(3*sqrt(3))) + 32*sqrt(3)*atan((4*x + 3)/(3*sqrt(3)))*x **4 - 48*sqrt(3)*atan((4*x + 3)/(3*sqrt(3)))*x**3 + 108*sqrt(3)*atan((4*x + 3)/(3*sqrt(3)))*x - 162*sqrt(3)*atan((4*x + 3)/(3*sqrt(3))) + 80*log(4*x **2 - 6*x + 9)*x**4 - 120*log(4*x**2 - 6*x + 9)*x**3 + 270*log(4*x**2 - 6* x + 9)*x - 405*log(4*x**2 - 6*x + 9) + 48*log(4*x**2 + 6*x + 9)*x**4 - 72* log(4*x**2 + 6*x + 9)*x**3 + 162*log(4*x**2 + 6*x + 9)*x - 243*log(4*x**2 + 6*x + 9) - 384*log(2*x - 3)*x**4 + 576*log(2*x - 3)*x**3 - 1296*log(2*x - 3)*x + 1944*log(2*x - 3) + 128*log(2*x + 3)*x**4 - 192*log(2*x + 3)*x**3 + 432*log(2*x + 3)*x - 648*log(2*x + 3) - 648*x)/(2834352*(16*x**4 - 24*x **3 + 54*x - 81))