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

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

[Out]

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]/354
294 + (5*Log[9 - 6*x + 4*x^2])/2834352 + Log[9 + 6*x + 4*x^2]/944784

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Rubi [A]  time = 0.144815, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35, Rules used = {1586, 2074, 614, 618, 204, 634, 628} \[ -\frac{3-4 x}{236196 \left (4 x^2-6 x+9\right )}+\frac{5 \log \left (4 x^2-6 x+9\right )}{2834352}+\frac{\log \left (4 x^2+6 x+9\right )}{944784}+\frac{1}{157464 (3-2 x)}-\frac{1}{472392 (2 x+3)}-\frac{\log (3-2 x)}{118098}+\frac{\log (2 x+3)}{354294}-\frac{\tan ^{-1}\left (\frac{3-4 x}{3 \sqrt{3}}\right )}{52488 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{4 x+3}{3 \sqrt{3}}\right )}{472392 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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]/354
294 + (5*Log[9 - 6*x + 4*x^2])/2834352 + Log[9 + 6*x + 4*x^2]/944784

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]

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 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 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 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]

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}{78732 (-3+2 x)^2}-\frac{1}{59049 (-3+2 x)}+\frac{1}{236196 (3+2 x)^2}+\frac{1}{177147 (3+2 x)}+\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 )}+\frac{3+2 x}{236196 \left (9+6 x+4 x^2\right )}\right ) \, dx\\ &=\frac{1}{157464 (3-2 x)}-\frac{1}{472392 (3+2 x)}-\frac{\log (3-2 x)}{118098}+\frac{\log (3+2 x)}{354294}+\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}{157464 (3-2 x)}-\frac{1}{472392 (3+2 x)}-\frac{3-4 x}{236196 \left (9-6 x+4 x^2\right )}-\frac{\log (3-2 x)}{118098}+\frac{\log (3+2 x)}{354294}+\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}{157464 (3-2 x)}-\frac{1}{472392 (3+2 x)}-\frac{3-4 x}{236196 \left (9-6 x+4 x^2\right )}-\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}-\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}{157464 (3-2 x)}-\frac{1}{472392 (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 )}{52488 \sqrt{3}}+\frac{\tan ^{-1}\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}\\ \end{align*}

Mathematica [A]  time = 0.0619316, size = 111, normalized size = 0.78 \[ \frac{\frac{648 x}{-16 x^4+24 x^3-54 x+81}+5 \log \left (4 x^2-6 x+9\right )+3 \log \left (4 x^2+6 x+9\right )-24 \log (3-2 x)+8 \log (2 x+3)+18 \sqrt{3} \tan ^{-1}\left (\frac{4 x-3}{3 \sqrt{3}}\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{4 x+3}{3 \sqrt{3}}\right )}{2834352} \]

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

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Maple [A]  time = 0.013, size = 111, normalized size = 0.8 \begin{align*} -{\frac{1}{1417176+944784\,x}}+{\frac{\ln \left ( 3+2\,x \right ) }{354294}}-{\frac{1}{-472392+314928\,x}}-{\frac{\ln \left ( -3+2\,x \right ) }{118098}}+{\frac{\ln \left ( 4\,{x}^{2}+6\,x+9 \right ) }{944784}}+{\frac{\sqrt{3}}{1417176}\arctan \left ({\frac{ \left ( 8\,x+6 \right ) \sqrt{3}}{18}} \right ) }+{\frac{1}{708588} \left ( 3\,x-{\frac{9}{4}} \right ) \left ({x}^{2}-{\frac{3\,x}{2}}+{\frac{9}{4}} \right ) ^{-1}}+{\frac{5\,\ln \left ( 4\,{x}^{2}-6\,x+9 \right ) }{2834352}}+{\frac{\sqrt{3}}{157464}\arctan \left ({\frac{ \left ( 8\,x-6 \right ) \sqrt{3}}{18}} \right ) } \end{align*}

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]

-1/472392/(3+2*x)+1/354294*ln(3+2*x)-1/157464/(-3+2*x)-1/118098*ln(-3+2*x)+1/944784*ln(4*x^2+6*x+9)+1/1417176*
3^(1/2)*arctan(1/18*(8*x+6)*3^(1/2))+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))

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Maxima [A]  time = 1.37685, size = 128, normalized size = 0.9 \begin{align*} \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 ) \end{align*}

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]

1/1417176*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/(1
6*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)

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Fricas [A]  time = 1.4611, size = 539, normalized size = 3.8 \begin{align*} \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 )}} \end{align*}

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*(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)/(16*x^4 - 24*x^3 + 54*x - 81)

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Sympy [A]  time = 0.46865, size = 116, normalized size = 0.82 \begin{align*} - \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} \end{align*}

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)/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

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Giac [A]  time = 1.06027, size = 143, normalized size = 1.01 \begin{align*} \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 ) \end{align*}

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]

1/1417176*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) + 1/354
294*log(abs(2*x + 3)) - 1/118098*log(abs(2*x - 3))