3.6.70 \(\int \frac {3+2 x}{(729-64 x^6)^2} \, dx\) [570]

Optimal. Leaf size=146 \[ \frac {1}{708588 (3-2 x)}+\frac {x}{236196 \left (9-6 x+4 x^2\right )}-\frac {3+x}{708588 \left (9+6 x+4 x^2\right )}-\frac {\tan ^{-1}\left (\frac {3-4 x}{3 \sqrt {3}}\right )}{157464 \sqrt {3}}+\frac {\tan ^{-1}\left (\frac {3+4 x}{3 \sqrt {3}}\right )}{1417176 \sqrt {3}}-\frac {\log (3-2 x)}{472392}+\frac {\log (3+2 x)}{4251528}-\frac {\log \left (9-6 x+4 x^2\right )}{8503056}+\frac {\log \left (9+6 x+4 x^2\right )}{944784} \]

[Out]

1/708588/(3-2*x)+1/236196*x/(4*x^2-6*x+9)+1/708588*(-3-x)/(4*x^2+6*x+9)-1/472392*ln(3-2*x)+1/4251528*ln(3+2*x)
-1/8503056*ln(4*x^2-6*x+9)+1/944784*ln(4*x^2+6*x+9)-1/472392*arctan(1/9*(3-4*x)*3^(1/2))*3^(1/2)+1/4251528*arc
tan(1/9*(3+4*x)*3^(1/2))*3^(1/2)

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Rubi [A]
time = 0.12, antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 7, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {1600, 2099, 652, 632, 210, 648, 642} \begin {gather*} -\frac {\text {ArcTan}\left (\frac {3-4 x}{3 \sqrt {3}}\right )}{157464 \sqrt {3}}+\frac {\text {ArcTan}\left (\frac {4 x+3}{3 \sqrt {3}}\right )}{1417176 \sqrt {3}}+\frac {x}{236196 \left (4 x^2-6 x+9\right )}-\frac {x+3}{708588 \left (4 x^2+6 x+9\right )}-\frac {\log \left (4 x^2-6 x+9\right )}{8503056}+\frac {\log \left (4 x^2+6 x+9\right )}{944784}+\frac {1}{708588 (3-2 x)}-\frac {\log (3-2 x)}{472392}+\frac {\log (2 x+3)}{4251528} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(3 + 2*x)/(729 - 64*x^6)^2,x]

[Out]

1/(708588*(3 - 2*x)) + x/(236196*(9 - 6*x + 4*x^2)) - (3 + x)/(708588*(9 + 6*x + 4*x^2)) - ArcTan[(3 - 4*x)/(3
*Sqrt[3])]/(157464*Sqrt[3]) + ArcTan[(3 + 4*x)/(3*Sqrt[3])]/(1417176*Sqrt[3]) - Log[3 - 2*x]/472392 + Log[3 +
2*x]/4251528 - Log[9 - 6*x + 4*x^2]/8503056 + Log[9 + 6*x + 4*x^2]/944784

Rule 210

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])

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[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]

Rule 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 652

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b*d - 2*a*e + (2*c*d -
b*e)*x)/((p + 1)*(b^2 - 4*a*c)))*(a + b*x + c*x^2)^(p + 1), x] - Dist[(2*p + 3)*((2*c*d - b*e)/((p + 1)*(b^2 -
4*a*c))), Int[(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^
2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2]

Rule 1600

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]

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*} \int \frac {3+2 x}{\left (729-64 x^6\right )^2} \, dx &=\int \frac {1}{(3+2 x) \left (243-162 x+108 x^2-72 x^3+48 x^4-32 x^5\right )^2} \, dx\\ &=\int \left (\frac {1}{354294 (-3+2 x)^2}-\frac {1}{236196 (-3+2 x)}+\frac {1}{2125764 (3+2 x)}+\frac {3-x}{39366 \left (9-6 x+4 x^2\right )^2}+\frac {33-2 x}{2125764 \left (9-6 x+4 x^2\right )}+\frac {x}{39366 \left (9+6 x+4 x^2\right )^2}+\frac {7+6 x}{708588 \left (9+6 x+4 x^2\right )}\right ) \, dx\\ &=\frac {1}{708588 (3-2 x)}-\frac {\log (3-2 x)}{472392}+\frac {\log (3+2 x)}{4251528}+\frac {\int \frac {33-2 x}{9-6 x+4 x^2} \, dx}{2125764}+\frac {\int \frac {7+6 x}{9+6 x+4 x^2} \, dx}{708588}+\frac {\int \frac {3-x}{\left (9-6 x+4 x^2\right )^2} \, dx}{39366}+\frac {\int \frac {x}{\left (9+6 x+4 x^2\right )^2} \, dx}{39366}\\ &=\frac {1}{708588 (3-2 x)}+\frac {x}{236196 \left (9-6 x+4 x^2\right )}-\frac {3+x}{708588 \left (9+6 x+4 x^2\right )}-\frac {\log (3-2 x)}{472392}+\frac {\log (3+2 x)}{4251528}-\frac {\int \frac {-6+8 x}{9-6 x+4 x^2} \, dx}{8503056}+\frac {\int \frac {6+8 x}{9+6 x+4 x^2} \, dx}{944784}-\frac {\int \frac {1}{9+6 x+4 x^2} \, dx}{708588}+\frac {5 \int \frac {1}{9+6 x+4 x^2} \, dx}{1417176}+\frac {\int \frac {1}{9-6 x+4 x^2} \, dx}{236196}+\frac {7 \int \frac {1}{9-6 x+4 x^2} \, dx}{472392}\\ &=\frac {1}{708588 (3-2 x)}+\frac {x}{236196 \left (9-6 x+4 x^2\right )}-\frac {3+x}{708588 \left (9+6 x+4 x^2\right )}-\frac {\log (3-2 x)}{472392}+\frac {\log (3+2 x)}{4251528}-\frac {\log \left (9-6 x+4 x^2\right )}{8503056}+\frac {\log \left (9+6 x+4 x^2\right )}{944784}+\frac {\text {Subst}\left (\int \frac {1}{-108-x^2} \, dx,x,6+8 x\right )}{354294}-\frac {5 \text {Subst}\left (\int \frac {1}{-108-x^2} \, dx,x,6+8 x\right )}{708588}-\frac {\text {Subst}\left (\int \frac {1}{-108-x^2} \, dx,x,-6+8 x\right )}{118098}-\frac {7 \text {Subst}\left (\int \frac {1}{-108-x^2} \, dx,x,-6+8 x\right )}{236196}\\ &=\frac {1}{708588 (3-2 x)}+\frac {x}{236196 \left (9-6 x+4 x^2\right )}-\frac {3+x}{708588 \left (9+6 x+4 x^2\right )}-\frac {\tan ^{-1}\left (\frac {3-4 x}{3 \sqrt {3}}\right )}{157464 \sqrt {3}}+\frac {\tan ^{-1}\left (\frac {3+4 x}{3 \sqrt {3}}\right )}{1417176 \sqrt {3}}-\frac {\log (3-2 x)}{472392}+\frac {\log (3+2 x)}{4251528}-\frac {\log \left (9-6 x+4 x^2\right )}{8503056}+\frac {\log \left (9+6 x+4 x^2\right )}{944784}\\ \end {align*}

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Mathematica [A]
time = 0.05, size = 121, normalized size = 0.83 \begin {gather*} \frac {\frac {1944 x}{243-162 x+108 x^2-72 x^3+48 x^4-32 x^5}+18 \sqrt {3} \tan ^{-1}\left (\frac {-3+4 x}{3 \sqrt {3}}\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {3+4 x}{3 \sqrt {3}}\right )-18 \log (3-2 x)+2 \log (3+2 x)-\log \left (9-6 x+4 x^2\right )+9 \log \left (9+6 x+4 x^2\right )}{8503056} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3 + 2*x)/(729 - 64*x^6)^2,x]

[Out]

((1944*x)/(243 - 162*x + 108*x^2 - 72*x^3 + 48*x^4 - 32*x^5) + 18*Sqrt[3]*ArcTan[(-3 + 4*x)/(3*Sqrt[3])] + 2*S
qrt[3]*ArcTan[(3 + 4*x)/(3*Sqrt[3])] - 18*Log[3 - 2*x] + 2*Log[3 + 2*x] - Log[9 - 6*x + 4*x^2] + 9*Log[9 + 6*x
+ 4*x^2])/8503056

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Maple [A]
time = 0.40, size = 115, normalized size = 0.79

method	result	size
risch	\(-\frac {x}{139968 \left (x^{5}-\frac {3}{2} x^{4}+\frac {9}{4} x^{3}-\frac {27}{8} x^{2}+\frac {81}{16} x -\frac {243}{32}\right )}+\frac {\ln \left (16 x^{2}+24 x +36\right )}{944784}+\frac {\arctan \left (\frac {\left (3+4 x \right ) \sqrt {3}}{9}\right ) \sqrt {3}}{4251528}+\frac {\ln \left (2 x +3\right )}{4251528}-\frac {\ln \left (36 x^{2}-54 x +81\right )}{8503056}+\frac {\sqrt {3}\, \arctan \left (\frac {2 \left (6 x -\frac {9}{2}\right ) \sqrt {3}}{27}\right )}{472392}-\frac {\ln \left (-3+2 x \right )}{472392}\)	\(104\)
default	\(\frac {x}{944784 x^{2}-1417176 x +2125764}-\frac {\ln \left (4 x^{2}-6 x +9\right )}{8503056}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (8 x -6\right ) \sqrt {3}}{18}\right )}{472392}+\frac {-\frac {x}{4}-\frac {3}{4}}{708588 x^{2}+1062882 x +1594323}+\frac {\ln \left (4 x^{2}+6 x +9\right )}{944784}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (8 x +6\right ) \sqrt {3}}{18}\right )}{4251528}+\frac {\ln \left (2 x +3\right )}{4251528}-\frac {1}{708588 \left (-3+2 x \right )}-\frac {\ln \left (-3+2 x \right )}{472392}\)	\(115\)
meijerg	\(-\frac {\left (-1\right )^{\frac {5}{6}} \left (\frac {4 x \left (-1\right )^{\frac {1}{6}}}{6-\frac {128 x^{6}}{243}}-\frac {5 x \left (-1\right )^{\frac {1}{6}} \left (\ln \left (1-\frac {2 \left (x^{6}\right )^{\frac {1}{6}}}{3}\right )-\ln \left (1+\frac {2 \left (x^{6}\right )^{\frac {1}{6}}}{3}\right )+\frac {\ln \left (1-\frac {2 \left (x^{6}\right )^{\frac {1}{6}}}{3}+\frac {4 \left (x^{6}\right )^{\frac {1}{3}}}{9}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{6}\right )^{\frac {1}{6}}}{3-\left (x^{6}\right )^{\frac {1}{6}}}\right )-\frac {\ln \left (1+\frac {2 \left (x^{6}\right )^{\frac {1}{6}}}{3}+\frac {4 \left (x^{6}\right )^{\frac {1}{3}}}{9}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{6}\right )^{\frac {1}{6}}}{3+\left (x^{6}\right )^{\frac {1}{6}}}\right )\right )}{6 \left (x^{6}\right )^{\frac {1}{6}}}\right )}{708588}-\frac {\left (-1\right )^{\frac {2}{3}} \left (\frac {4 x^{2} \left (-1\right )^{\frac {1}{3}}}{3 \left (3-\frac {64 x^{6}}{243}\right )}-\frac {2 x^{2} \left (-1\right )^{\frac {1}{3}} \left (\ln \left (1-\frac {4 \left (x^{6}\right )^{\frac {1}{3}}}{9}\right )-\frac {\ln \left (1+\frac {4 \left (x^{6}\right )^{\frac {1}{3}}}{9}+\frac {16 \left (x^{6}\right )^{\frac {2}{3}}}{81}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {2 \sqrt {3}\, \left (x^{6}\right )^{\frac {1}{3}}}{9 \left (1+\frac {2 \left (x^{6}\right )^{\frac {1}{3}}}{9}\right )}\right )\right )}{3 \left (x^{6}\right )^{\frac {1}{3}}}\right )}{708588}\)	\(242\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x+3)/(-64*x^6+729)^2,x,method=_RETURNVERBOSE)

[Out]

1/944784*x/(x^2-3/2*x+9/4)-1/8503056*ln(4*x^2-6*x+9)+1/472392*3^(1/2)*arctan(1/18*(8*x-6)*3^(1/2))+1/708588*(-
1/4*x-3/4)/(x^2+3/2*x+9/4)+1/944784*ln(4*x^2+6*x+9)+1/4251528*3^(1/2)*arctan(1/18*(8*x+6)*3^(1/2))+1/4251528*l
n(2*x+3)-1/708588/(-3+2*x)-1/472392*ln(-3+2*x)

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Maxima [A]
time = 0.52, size = 105, normalized size = 0.72 \begin {gather*} \frac {1}{4251528} \, \sqrt {3} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x + 3\right )}\right ) + \frac {1}{472392} \, \sqrt {3} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x - 3\right )}\right ) - \frac {x}{4374 \, {\left (32 \, x^{5} - 48 \, x^{4} + 72 \, x^{3} - 108 \, x^{2} + 162 \, x - 243\right )}} + \frac {1}{944784} \, \log \left (4 \, x^{2} + 6 \, x + 9\right ) - \frac {1}{8503056} \, \log \left (4 \, x^{2} - 6 \, x + 9\right ) + \frac {1}{4251528} \, \log \left (2 \, x + 3\right ) - \frac {1}{472392} \, \log \left (2 \, x - 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+2*x)/(-64*x^6+729)^2,x, algorithm="maxima")

[Out]

1/4251528*sqrt(3)*arctan(1/9*sqrt(3)*(4*x + 3)) + 1/472392*sqrt(3)*arctan(1/9*sqrt(3)*(4*x - 3)) - 1/4374*x/(3
2*x^5 - 48*x^4 + 72*x^3 - 108*x^2 + 162*x - 243) + 1/944784*log(4*x^2 + 6*x + 9) - 1/8503056*log(4*x^2 - 6*x +
9) + 1/4251528*log(2*x + 3) - 1/472392*log(2*x - 3)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 257 vs. \(2 (116) = 232\).
time = 0.42, size = 257, normalized size = 1.76 \begin {gather*} \frac {2 \, \sqrt {3} {\left (32 \, x^{5} - 48 \, x^{4} + 72 \, x^{3} - 108 \, x^{2} + 162 \, x - 243\right )} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x + 3\right )}\right ) + 18 \, \sqrt {3} {\left (32 \, x^{5} - 48 \, x^{4} + 72 \, x^{3} - 108 \, x^{2} + 162 \, x - 243\right )} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x - 3\right )}\right ) + 9 \, {\left (32 \, x^{5} - 48 \, x^{4} + 72 \, x^{3} - 108 \, x^{2} + 162 \, x - 243\right )} \log \left (4 \, x^{2} + 6 \, x + 9\right ) - {\left (32 \, x^{5} - 48 \, x^{4} + 72 \, x^{3} - 108 \, x^{2} + 162 \, x - 243\right )} \log \left (4 \, x^{2} - 6 \, x + 9\right ) + 2 \, {\left (32 \, x^{5} - 48 \, x^{4} + 72 \, x^{3} - 108 \, x^{2} + 162 \, x - 243\right )} \log \left (2 \, x + 3\right ) - 18 \, {\left (32 \, x^{5} - 48 \, x^{4} + 72 \, x^{3} - 108 \, x^{2} + 162 \, x - 243\right )} \log \left (2 \, x - 3\right ) - 1944 \, x}{8503056 \, {\left (32 \, x^{5} - 48 \, x^{4} + 72 \, x^{3} - 108 \, x^{2} + 162 \, x - 243\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+2*x)/(-64*x^6+729)^2,x, algorithm="fricas")

[Out]

1/8503056*(2*sqrt(3)*(32*x^5 - 48*x^4 + 72*x^3 - 108*x^2 + 162*x - 243)*arctan(1/9*sqrt(3)*(4*x + 3)) + 18*sqr
t(3)*(32*x^5 - 48*x^4 + 72*x^3 - 108*x^2 + 162*x - 243)*arctan(1/9*sqrt(3)*(4*x - 3)) + 9*(32*x^5 - 48*x^4 + 7
2*x^3 - 108*x^2 + 162*x - 243)*log(4*x^2 + 6*x + 9) - (32*x^5 - 48*x^4 + 72*x^3 - 108*x^2 + 162*x - 243)*log(4
*x^2 - 6*x + 9) + 2*(32*x^5 - 48*x^4 + 72*x^3 - 108*x^2 + 162*x - 243)*log(2*x + 3) - 18*(32*x^5 - 48*x^4 + 72
*x^3 - 108*x^2 + 162*x - 243)*log(2*x - 3) - 1944*x)/(32*x^5 - 48*x^4 + 72*x^3 - 108*x^2 + 162*x - 243)

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Sympy [A]
time = 0.31, size = 124, normalized size = 0.85 \begin {gather*} - \frac {x}{139968 x^{5} - 209952 x^{4} + 314928 x^{3} - 472392 x^{2} + 708588 x - 1062882} - \frac {\log {\left (x - \frac {3}{2} \right )}}{472392} + \frac {\log {\left (x + \frac {3}{2} \right )}}{4251528} - \frac {\log {\left (x^{2} - \frac {3 x}{2} + \frac {9}{4} \right )}}{8503056} + \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 )}}{472392} + \frac {\sqrt {3} \operatorname {atan}{\left (\frac {4 \sqrt {3} x}{9} + \frac {\sqrt {3}}{3} \right )}}{4251528} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+2*x)/(-64*x**6+729)**2,x)

[Out]

-x/(139968*x**5 - 209952*x**4 + 314928*x**3 - 472392*x**2 + 708588*x - 1062882) - log(x - 3/2)/472392 + log(x
+ 3/2)/4251528 - log(x**2 - 3*x/2 + 9/4)/8503056 + log(x**2 + 3*x/2 + 9/4)/944784 + sqrt(3)*atan(4*sqrt(3)*x/9
- sqrt(3)/3)/472392 + sqrt(3)*atan(4*sqrt(3)*x/9 + sqrt(3)/3)/4251528

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Giac [A]
time = 2.11, size = 111, normalized size = 0.76 \begin {gather*} \frac {1}{4251528} \, \sqrt {3} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x + 3\right )}\right ) + \frac {1}{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 (4 \, x^{2} - 6 \, x + 9\right )} {\left (2 \, x - 3\right )}} + \frac {1}{944784} \, \log \left (4 \, x^{2} + 6 \, x + 9\right ) - \frac {1}{8503056} \, \log \left (4 \, x^{2} - 6 \, x + 9\right ) + \frac {1}{4251528} \, \log \left ({\left | 2 \, x + 3 \right |}\right ) - \frac {1}{472392} \, \log \left ({\left | 2 \, x - 3 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+2*x)/(-64*x^6+729)^2,x, algorithm="giac")

[Out]

1/4251528*sqrt(3)*arctan(1/9*sqrt(3)*(4*x + 3)) + 1/472392*sqrt(3)*arctan(1/9*sqrt(3)*(4*x - 3)) - 1/4374*x/((
4*x^2 + 6*x + 9)*(4*x^2 - 6*x + 9)*(2*x - 3)) + 1/944784*log(4*x^2 + 6*x + 9) - 1/8503056*log(4*x^2 - 6*x + 9)
+ 1/4251528*log(abs(2*x + 3)) - 1/472392*log(abs(2*x - 3))

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Mupad [B]
time = 5.09, size = 121, normalized size = 0.83 \begin {gather*} \frac {\ln \left (x+\frac {3}{2}\right )}{4251528}-\frac {\ln \left (x-\frac {3}{2}\right )}{472392}-\ln \left (x-\frac {3}{4}-\frac {\sqrt {3}\,3{}\mathrm {i}}{4}\right )\,\left (\frac {1}{8503056}+\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}{944784}+\frac {\sqrt {3}\,1{}\mathrm {i}}{8503056}\right )+\ln \left (x-\frac {3}{4}+\frac {\sqrt {3}\,3{}\mathrm {i}}{4}\right )\,\left (-\frac {1}{8503056}+\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}{944784}+\frac {\sqrt {3}\,1{}\mathrm {i}}{8503056}\right )-\frac {x}{139968\,\left (x^5-\frac {3\,x^4}{2}+\frac {9\,x^3}{4}-\frac {27\,x^2}{8}+\frac {81\,x}{16}-\frac {243}{32}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x + 3)/(64*x^6 - 729)^2,x)

[Out]

log(x + 3/2)/4251528 - log(x - 3/2)/472392 - log(x - (3^(1/2)*3i)/4 - 3/4)*((3^(1/2)*1i)/944784 + 1/8503056) -
log(x - (3^(1/2)*3i)/4 + 3/4)*((3^(1/2)*1i)/8503056 - 1/944784) + log(x + (3^(1/2)*3i)/4 - 3/4)*((3^(1/2)*1i)
/944784 - 1/8503056) + log(x + (3^(1/2)*3i)/4 + 3/4)*((3^(1/2)*1i)/8503056 + 1/944784) - x/(139968*((81*x)/16
- (27*x^2)/8 + (9*x^3)/4 - (3*x^4)/2 + x^5 - 243/32))

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