3.574 \(\int \frac{27+36 x+24 x^2+8 x^3}{(729-64 x^6)^2} \, dx\)

Optimal. Leaf size=131 \[ -\frac{3-2 x}{26244 \left (4 x^2-6 x+9\right )}+\frac{17 \log \left (4 x^2-6 x+9\right )}{944784}+\frac{\log \left (4 x^2+6 x+9\right )}{314928}+\frac{1}{26244 (3-2 x)}-\frac{7 \log (3-2 x)}{157464}+\frac{\log (2 x+3)}{472392}-\frac{11 \tan ^{-1}\left (\frac{3-4 x}{3 \sqrt{3}}\right )}{157464 \sqrt{3}}-\frac{\tan ^{-1}\left (\frac{4 x+3}{3 \sqrt{3}}\right )}{157464 \sqrt{3}} \]

[Out]

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])]/(157464*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

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Rubi [A]  time = 0.147723, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28, Rules used = {1586, 2074, 638, 618, 204, 634, 628} \[ -\frac{3-2 x}{26244 \left (4 x^2-6 x+9\right )}+\frac{17 \log \left (4 x^2-6 x+9\right )}{944784}+\frac{\log \left (4 x^2+6 x+9\right )}{314928}+\frac{1}{26244 (3-2 x)}-\frac{7 \log (3-2 x)}{157464}+\frac{\log (2 x+3)}{472392}-\frac{11 \tan ^{-1}\left (\frac{3-4 x}{3 \sqrt{3}}\right )}{157464 \sqrt{3}}-\frac{\tan ^{-1}\left (\frac{4 x+3}{3 \sqrt{3}}\right )}{157464 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Int[(27 + 36*x + 24*x^2 + 8*x^3)/(729 - 64*x^6)^2,x]

[Out]

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])]/(157464*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

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 638

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)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), 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 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{27+36 x+24 x^2+8 x^3}{\left (729-64 x^6\right )^2} \, dx &=\int \frac{1}{\left (27-36 x+24 x^2-8 x^3\right )^2 \left (27+36 x+24 x^2+8 x^3\right )} \, dx\\ &=\int \left (\frac{1}{13122 (-3+2 x)^2}-\frac{7}{78732 (-3+2 x)}+\frac{1}{236196 (3+2 x)}+\frac{3+2 x}{4374 \left (9-6 x+4 x^2\right )^2}+\frac{3+17 x}{118098 \left (9-6 x+4 x^2\right )}+\frac{x}{39366 \left (9+6 x+4 x^2\right )}\right ) \, dx\\ &=\frac{1}{26244 (3-2 x)}-\frac{7 \log (3-2 x)}{157464}+\frac{\log (3+2 x)}{472392}+\frac{\int \frac{3+17 x}{9-6 x+4 x^2} \, dx}{118098}+\frac{\int \frac{x}{9+6 x+4 x^2} \, dx}{39366}+\frac{\int \frac{3+2 x}{\left (9-6 x+4 x^2\right )^2} \, dx}{4374}\\ &=\frac{1}{26244 (3-2 x)}-\frac{3-2 x}{26244 \left (9-6 x+4 x^2\right )}-\frac{7 \log (3-2 x)}{157464}+\frac{\log (3+2 x)}{472392}+\frac{\int \frac{6+8 x}{9+6 x+4 x^2} \, dx}{314928}+\frac{17 \int \frac{-6+8 x}{9-6 x+4 x^2} \, dx}{944784}-\frac{\int \frac{1}{9+6 x+4 x^2} \, dx}{52488}+\frac{\int \frac{1}{9-6 x+4 x^2} \, dx}{13122}+\frac{7 \int \frac{1}{9-6 x+4 x^2} \, dx}{52488}\\ &=\frac{1}{26244 (3-2 x)}-\frac{3-2 x}{26244 \left (9-6 x+4 x^2\right )}-\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}+\frac{\operatorname{Subst}\left (\int \frac{1}{-108-x^2} \, dx,x,6+8 x\right )}{26244}-\frac{\operatorname{Subst}\left (\int \frac{1}{-108-x^2} \, dx,x,-6+8 x\right )}{6561}-\frac{7 \operatorname{Subst}\left (\int \frac{1}{-108-x^2} \, dx,x,-6+8 x\right )}{26244}\\ &=\frac{1}{26244 (3-2 x)}-\frac{3-2 x}{26244 \left (9-6 x+4 x^2\right )}-\frac{11 \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 )}{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}\\ \end{align*}

Mathematica [A]  time = 0.0637757, size = 111, normalized size = 0.85 \[ \frac{\frac{216 x}{-8 x^3+24 x^2-36 x+27}+17 \log \left (4 x^2-6 x+9\right )+3 \log \left (4 x^2+6 x+9\right )-42 \log (3-2 x)+2 \log (2 x+3)+22 \sqrt{3} \tan ^{-1}\left (\frac{4 x-3}{3 \sqrt{3}}\right )-2 \sqrt{3} \tan ^{-1}\left (\frac{4 x+3}{3 \sqrt{3}}\right )}{944784} \]

Antiderivative was successfully verified.

[In]

Integrate[(27 + 36*x + 24*x^2 + 8*x^3)/(729 - 64*x^6)^2,x]

[Out]

((216*x)/(27 - 36*x + 24*x^2 - 8*x^3) + 22*Sqrt[3]*ArcTan[(-3 + 4*x)/(3*Sqrt[3])] - 2*Sqrt[3]*ArcTan[(3 + 4*x)
/(3*Sqrt[3])] - 42*Log[3 - 2*x] + 2*Log[3 + 2*x] + 17*Log[9 - 6*x + 4*x^2] + 3*Log[9 + 6*x + 4*x^2])/944784

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Maple [A]  time = 0.013, size = 102, normalized size = 0.8 \begin{align*}{\frac{\ln \left ( 3+2\,x \right ) }{472392}}-{\frac{1}{-78732+52488\,x}}-{\frac{7\,\ln \left ( -3+2\,x \right ) }{157464}}+{\frac{\ln \left ( 4\,{x}^{2}+6\,x+9 \right ) }{314928}}-{\frac{\sqrt{3}}{472392}\arctan \left ({\frac{ \left ( 8\,x+6 \right ) \sqrt{3}}{18}} \right ) }+{\frac{1}{118098} \left ({\frac{9\,x}{4}}-{\frac{27}{8}} \right ) \left ({x}^{2}-{\frac{3\,x}{2}}+{\frac{9}{4}} \right ) ^{-1}}+{\frac{17\,\ln \left ( 4\,{x}^{2}-6\,x+9 \right ) }{944784}}+{\frac{11\,\sqrt{3}}{472392}\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((8*x^3+24*x^2+36*x+27)/(-64*x^6+729)^2,x)

[Out]

1/472392*ln(3+2*x)-1/26244/(-3+2*x)-7/157464*ln(-3+2*x)+1/314928*ln(4*x^2+6*x+9)-1/472392*3^(1/2)*arctan(1/18*
(8*x+6)*3^(1/2))+1/118098*(9/4*x-27/8)/(x^2-3/2*x+9/4)+17/944784*ln(4*x^2-6*x+9)+11/472392*3^(1/2)*arctan(1/18
*(8*x-6)*3^(1/2))

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Maxima [A]  time = 1.3948, size = 128, normalized size = 0.98 \begin{align*} -\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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*x^3+24*x^2+36*x+27)/(-64*x^6+729)^2,x, algorithm="maxima")

[Out]

-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)

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Fricas [A]  time = 1.44256, size = 531, normalized size = 4.05 \begin{align*} -\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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*x^3+24*x^2+36*x+27)/(-64*x^6+729)^2,x, algorithm="fricas")

[Out]

-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)*log(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)

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Sympy [A]  time = 0.46846, size = 119, normalized size = 0.91 \begin{align*} - \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*x**3+24*x**2+36*x+27)/(-64*x**6+729)**2,x)

[Out]

-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 - s
qrt(3)*atan(4*sqrt(3)*x/9 + sqrt(3)/3)/472392

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Giac [A]  time = 1.04995, size = 134, normalized size = 1.02 \begin{align*} -\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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*x^3+24*x^2+36*x+27)/(-64*x^6+729)^2,x, algorithm="giac")

[Out]

-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)) + 1/314928*log(4*x^2 + 6*x + 9) + 17/944784*log(4*x^2 - 6*x + 9) + 1/472392*log(a
bs(2*x + 3)) - 7/157464*log(abs(2*x - 3))