3.571 \(\int \frac {9-6 x+4 x^2}{(729-64 x^6)^2} \, dx\)

Optimal. Leaf size=142 \[ \frac {4 x+3}{236196 \left (4 x^2+6 x+9\right )}-\frac {\log \left (4 x^2-6 x+9\right )}{944784}-\frac {5 \log \left (4 x^2+6 x+9\right )}{2834352}+\frac {1}{472392 (3-2 x)}-\frac {1}{157464 (2 x+3)}-\frac {\log (3-2 x)}{354294}+\frac {\log (2 x+3)}{118098}-\frac {\tan ^{-1}\left (\frac {3-4 x}{3 \sqrt {3}}\right )}{472392 \sqrt {3}}+\frac {\tan ^{-1}\left (\frac {4 x+3}{3 \sqrt {3}}\right )}{52488 \sqrt {3}} \]

[Out]

1/472392/(3-2*x)-1/157464/(3+2*x)+1/236196*(3+4*x)/(4*x^2+6*x+9)-1/354294*ln(3-2*x)+1/118098*ln(3+2*x)-1/94478
4*ln(4*x^2-6*x+9)-5/2834352*ln(4*x^2+6*x+9)-1/1417176*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)

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Rubi [A]  time = 0.15, antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {1586, 2074, 634, 618, 204, 628, 614} \[ \frac {4 x+3}{236196 \left (4 x^2+6 x+9\right )}-\frac {\log \left (4 x^2-6 x+9\right )}{944784}-\frac {5 \log \left (4 x^2+6 x+9\right )}{2834352}+\frac {1}{472392 (3-2 x)}-\frac {1}{157464 (2 x+3)}-\frac {\log (3-2 x)}{354294}+\frac {\log (2 x+3)}{118098}-\frac {\tan ^{-1}\left (\frac {3-4 x}{3 \sqrt {3}}\right )}{472392 \sqrt {3}}+\frac {\tan ^{-1}\left (\frac {4 x+3}{3 \sqrt {3}}\right )}{52488 \sqrt {3}} \]

Antiderivative was successfully verified.

[In]

Int[(9 - 6*x + 4*x^2)/(729 - 64*x^6)^2,x]

[Out]

1/(472392*(3 - 2*x)) - 1/(157464*(3 + 2*x)) + (3 + 4*x)/(236196*(9 + 6*x + 4*x^2)) - ArcTan[(3 - 4*x)/(3*Sqrt[
3])]/(472392*Sqrt[3]) + ArcTan[(3 + 4*x)/(3*Sqrt[3])]/(52488*Sqrt[3]) - Log[3 - 2*x]/354294 + Log[3 + 2*x]/118
098 - Log[9 - 6*x + 4*x^2]/944784 - (5*Log[9 + 6*x + 4*x^2])/2834352

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 614

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

Rule 618

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 628

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 634

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 1586

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 2074

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 {9-6 x+4 x^2}{\left (729-64 x^6\right )^2} \, dx &=\int \frac {1}{\left (9-6 x+4 x^2\right ) \left (81+54 x-24 x^3-16 x^4\right )^2} \, dx\\ &=\int \left (\frac {1}{236196 (-3+2 x)^2}-\frac {1}{177147 (-3+2 x)}+\frac {1}{78732 (3+2 x)^2}+\frac {1}{59049 (3+2 x)}+\frac {3-2 x}{236196 \left (9-6 x+4 x^2\right )}+\frac {1}{4374 \left (9+6 x+4 x^2\right )^2}+\frac {21-10 x}{708588 \left (9+6 x+4 x^2\right )}\right ) \, dx\\ &=\frac {1}{472392 (3-2 x)}-\frac {1}{157464 (3+2 x)}-\frac {\log (3-2 x)}{354294}+\frac {\log (3+2 x)}{118098}+\frac {\int \frac {21-10 x}{9+6 x+4 x^2} \, dx}{708588}+\frac {\int \frac {3-2 x}{9-6 x+4 x^2} \, dx}{236196}+\frac {\int \frac {1}{\left (9+6 x+4 x^2\right )^2} \, dx}{4374}\\ &=\frac {1}{472392 (3-2 x)}-\frac {1}{157464 (3+2 x)}+\frac {3+4 x}{236196 \left (9+6 x+4 x^2\right )}-\frac {\log (3-2 x)}{354294}+\frac {\log (3+2 x)}{118098}-\frac {\int \frac {-6+8 x}{9-6 x+4 x^2} \, dx}{944784}-\frac {5 \int \frac {6+8 x}{9+6 x+4 x^2} \, dx}{2834352}+\frac {\int \frac {1}{9-6 x+4 x^2} \, dx}{157464}+\frac {\int \frac {1}{9+6 x+4 x^2} \, dx}{59049}+\frac {19 \int \frac {1}{9+6 x+4 x^2} \, dx}{472392}\\ &=\frac {1}{472392 (3-2 x)}-\frac {1}{157464 (3+2 x)}+\frac {3+4 x}{236196 \left (9+6 x+4 x^2\right )}-\frac {\log (3-2 x)}{354294}+\frac {\log (3+2 x)}{118098}-\frac {\log \left (9-6 x+4 x^2\right )}{944784}-\frac {5 \log \left (9+6 x+4 x^2\right )}{2834352}-\frac {\operatorname {Subst}\left (\int \frac {1}{-108-x^2} \, dx,x,-6+8 x\right )}{78732}-\frac {2 \operatorname {Subst}\left (\int \frac {1}{-108-x^2} \, dx,x,6+8 x\right )}{59049}-\frac {19 \operatorname {Subst}\left (\int \frac {1}{-108-x^2} \, dx,x,6+8 x\right )}{236196}\\ &=\frac {1}{472392 (3-2 x)}-\frac {1}{157464 (3+2 x)}+\frac {3+4 x}{236196 \left (9+6 x+4 x^2\right )}-\frac {\tan ^{-1}\left (\frac {3-4 x}{3 \sqrt {3}}\right )}{472392 \sqrt {3}}+\frac {\tan ^{-1}\left (\frac {3+4 x}{3 \sqrt {3}}\right )}{52488 \sqrt {3}}-\frac {\log (3-2 x)}{354294}+\frac {\log (3+2 x)}{118098}-\frac {\log \left (9-6 x+4 x^2\right )}{944784}-\frac {5 \log \left (9+6 x+4 x^2\right )}{2834352}\\ \end {align*}

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Mathematica [A]  time = 0.07, size = 111, normalized size = 0.78 \[ \frac {-3 \log \left (4 x^2-6 x+9\right )-5 \log \left (4 x^2+6 x+9\right )+\frac {648 x}{-16 x^4-24 x^3+54 x+81}-8 \log (3-2 x)+24 \log (2 x+3)+2 \sqrt {3} \tan ^{-1}\left (\frac {4 x-3}{3 \sqrt {3}}\right )+18 \sqrt {3} \tan ^{-1}\left (\frac {4 x+3}{3 \sqrt {3}}\right )}{2834352} \]

Antiderivative was successfully verified.

[In]

Integrate[(9 - 6*x + 4*x^2)/(729 - 64*x^6)^2,x]

[Out]

((648*x)/(81 + 54*x - 24*x^3 - 16*x^4) + 2*Sqrt[3]*ArcTan[(-3 + 4*x)/(3*Sqrt[3])] + 18*Sqrt[3]*ArcTan[(3 + 4*x
)/(3*Sqrt[3])] - 8*Log[3 - 2*x] + 24*Log[3 + 2*x] - 3*Log[9 - 6*x + 4*x^2] - 5*Log[9 + 6*x + 4*x^2])/2834352

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fricas [A]  time = 0.43, size = 187, normalized size = 1.32 \[ \frac {18 \, \sqrt {3} {\left (16 \, x^{4} + 24 \, x^{3} - 54 \, x - 81\right )} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x + 3\right )}\right ) + 2 \, \sqrt {3} {\left (16 \, x^{4} + 24 \, x^{3} - 54 \, x - 81\right )} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x - 3\right )}\right ) - 5 \, {\left (16 \, x^{4} + 24 \, x^{3} - 54 \, x - 81\right )} \log \left (4 \, x^{2} + 6 \, x + 9\right ) - 3 \, {\left (16 \, x^{4} + 24 \, x^{3} - 54 \, x - 81\right )} \log \left (4 \, x^{2} - 6 \, x + 9\right ) + 24 \, {\left (16 \, x^{4} + 24 \, x^{3} - 54 \, x - 81\right )} \log \left (2 \, x + 3\right ) - 8 \, {\left (16 \, x^{4} + 24 \, x^{3} - 54 \, x - 81\right )} \log \left (2 \, x - 3\right ) - 648 \, x}{2834352 \, {\left (16 \, x^{4} + 24 \, x^{3} - 54 \, x - 81\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^2-6*x+9)/(-64*x^6+729)^2,x, algorithm="fricas")

[Out]

1/2834352*(18*sqrt(3)*(16*x^4 + 24*x^3 - 54*x - 81)*arctan(1/9*sqrt(3)*(4*x + 3)) + 2*sqrt(3)*(16*x^4 + 24*x^3
- 54*x - 81)*arctan(1/9*sqrt(3)*(4*x - 3)) - 5*(16*x^4 + 24*x^3 - 54*x - 81)*log(4*x^2 + 6*x + 9) - 3*(16*x^4
+ 24*x^3 - 54*x - 81)*log(4*x^2 - 6*x + 9) + 24*(16*x^4 + 24*x^3 - 54*x - 81)*log(2*x + 3) - 8*(16*x^4 + 24*x
^3 - 54*x - 81)*log(2*x - 3) - 648*x)/(16*x^4 + 24*x^3 - 54*x - 81)

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giac [A]  time = 0.18, size = 106, normalized size = 0.75 \[ \frac {1}{157464} \, \sqrt {3} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x + 3\right )}\right ) + \frac {1}{1417176} \, \sqrt {3} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x - 3\right )}\right ) - \frac {x}{4374 \, {\left (4 \, x^{2} + 6 \, x + 9\right )} {\left (2 \, x + 3\right )} {\left (2 \, x - 3\right )}} - \frac {5}{2834352} \, \log \left (4 \, x^{2} + 6 \, x + 9\right ) - \frac {1}{944784} \, \log \left (4 \, x^{2} - 6 \, x + 9\right ) + \frac {1}{118098} \, \log \left ({\left | 2 \, x + 3 \right |}\right ) - \frac {1}{354294} \, \log \left ({\left | 2 \, x - 3 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^2-6*x+9)/(-64*x^6+729)^2,x, algorithm="giac")

[Out]

1/157464*sqrt(3)*arctan(1/9*sqrt(3)*(4*x + 3)) + 1/1417176*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)) - 5/2834352*log(4*x^2 + 6*x + 9) - 1/944784*log(4*x^2 - 6*x + 9) + 1/118
098*log(abs(2*x + 3)) - 1/354294*log(abs(2*x - 3))

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maple [A]  time = 0.06, size = 111, normalized size = 0.78 \[ \frac {\sqrt {3}\, \arctan \left (\frac {\left (8 x -6\right ) \sqrt {3}}{18}\right )}{1417176}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (8 x +6\right ) \sqrt {3}}{18}\right )}{157464}-\frac {\ln \left (2 x -3\right )}{354294}+\frac {\ln \left (2 x +3\right )}{118098}-\frac {\ln \left (4 x^{2}-6 x +9\right )}{944784}-\frac {5 \ln \left (4 x^{2}+6 x +9\right )}{2834352}-\frac {1}{157464 \left (2 x +3\right )}-\frac {-3 x -\frac {9}{4}}{708588 \left (x^{2}+\frac {3}{2} x +\frac {9}{4}\right )}-\frac {1}{472392 \left (2 x -3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x^2-6*x+9)/(-64*x^6+729)^2,x)

[Out]

-1/944784*ln(4*x^2-6*x+9)+1/1417176*3^(1/2)*arctan(1/18*(8*x-6)*3^(1/2))-1/157464/(2*x+3)+1/118098*ln(2*x+3)-1
/708588*(-3*x-9/4)/(x^2+3/2*x+9/4)-5/2834352*ln(4*x^2+6*x+9)+1/157464*3^(1/2)*arctan(1/18*(8*x+6)*3^(1/2))-1/4
72392/(2*x-3)-1/354294*ln(2*x-3)

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maxima [A]  time = 2.98, size = 95, normalized size = 0.67 \[ \frac {1}{157464} \, \sqrt {3} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x + 3\right )}\right ) + \frac {1}{1417176} \, \sqrt {3} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x - 3\right )}\right ) - \frac {x}{4374 \, {\left (16 \, x^{4} + 24 \, x^{3} - 54 \, x - 81\right )}} - \frac {5}{2834352} \, \log \left (4 \, x^{2} + 6 \, x + 9\right ) - \frac {1}{944784} \, \log \left (4 \, x^{2} - 6 \, x + 9\right ) + \frac {1}{118098} \, \log \left (2 \, x + 3\right ) - \frac {1}{354294} \, \log \left (2 \, x - 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^2-6*x+9)/(-64*x^6+729)^2,x, algorithm="maxima")

[Out]

1/157464*sqrt(3)*arctan(1/9*sqrt(3)*(4*x + 3)) + 1/1417176*sqrt(3)*arctan(1/9*sqrt(3)*(4*x - 3)) - 1/4374*x/(1
6*x^4 + 24*x^3 - 54*x - 81) - 5/2834352*log(4*x^2 + 6*x + 9) - 1/944784*log(4*x^2 - 6*x + 9) + 1/118098*log(2*
x + 3) - 1/354294*log(2*x - 3)

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mupad [B]  time = 5.08, size = 110, normalized size = 0.77 \[ \frac {\ln \left (x+\frac {3}{2}\right )}{118098}-\frac {\ln \left (x-\frac {3}{2}\right )}{354294}-\ln \left (x+\frac {3}{4}-\frac {\sqrt {3}\,3{}\mathrm {i}}{4}\right )\,\left (\frac {5}{2834352}+\frac {\sqrt {3}\,1{}\mathrm {i}}{314928}\right )+\ln \left (x+\frac {3}{4}+\frac {\sqrt {3}\,3{}\mathrm {i}}{4}\right )\,\left (-\frac {5}{2834352}+\frac {\sqrt {3}\,1{}\mathrm {i}}{314928}\right )-\ln \left (x-\frac {3}{4}-\frac {\sqrt {3}\,3{}\mathrm {i}}{4}\right )\,\left (\frac {1}{944784}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2834352}\right )+\ln \left (x-\frac {3}{4}+\frac {\sqrt {3}\,3{}\mathrm {i}}{4}\right )\,\left (-\frac {1}{944784}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2834352}\right )+\frac {x}{69984\,\left (-x^4-\frac {3\,x^3}{2}+\frac {27\,x}{8}+\frac {81}{16}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x^2 - 6*x + 9)/(64*x^6 - 729)^2,x)

[Out]

log(x + 3/2)/118098 - log(x - 3/2)/354294 - 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) + x/(69984*((27*x)/8 - (
3*x^3)/2 - x^4 + 81/16))

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sympy [A]  time = 0.64, size = 116, normalized size = 0.82 \[ - \frac {x}{69984 x^{4} + 104976 x^{3} - 236196 x - 354294} - \frac {\log {\left (x - \frac {3}{2} \right )}}{354294} + \frac {\log {\left (x + \frac {3}{2} \right )}}{118098} - \frac {\log {\left (x^{2} - \frac {3 x}{2} + \frac {9}{4} \right )}}{944784} - \frac {5 \log {\left (x^{2} + \frac {3 x}{2} + \frac {9}{4} \right )}}{2834352} + \frac {\sqrt {3} \operatorname {atan}{\left (\frac {4 \sqrt {3} x}{9} - \frac {\sqrt {3}}{3} \right )}}{1417176} + \frac {\sqrt {3} \operatorname {atan}{\left (\frac {4 \sqrt {3} x}{9} + \frac {\sqrt {3}}{3} \right )}}{157464} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x**2-6*x+9)/(-64*x**6+729)**2,x)

[Out]

-x/(69984*x**4 + 104976*x**3 - 236196*x - 354294) - log(x - 3/2)/354294 + log(x + 3/2)/118098 - log(x**2 - 3*x
/2 + 9/4)/944784 - 5*log(x**2 + 3*x/2 + 9/4)/2834352 + sqrt(3)*atan(4*sqrt(3)*x/9 - sqrt(3)/3)/1417176 + sqrt(
3)*atan(4*sqrt(3)*x/9 + sqrt(3)/3)/157464

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