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

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Rubi [A]  time = 0.15, antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 7, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, 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 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 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 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 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 {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*}

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Mathematica [A]  time = 0.08, size = 111, normalized size = 0.85 \[ \frac {17 \log \left (4 x^2-6 x+9\right )+3 \log \left (4 x^2+6 x+9\right )+\frac {216 x}{-8 x^3+24 x^2-36 x+27}-42 \log (3-2 x)+2 \log (2 x+3)+22 \sqrt {3} \tan ^{-1}\left (\frac {4 x-3}{3 \sqrt {3}}\right )-2 \sqrt {3} \tan ^{-1}\left (\frac {4 x+3}{3 \sqrt {3}}\right )}{944784} \]

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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fricas [A]  time = 0.43, size = 187, normalized size = 1.43 \[ -\frac {2 \, \sqrt {3} {\left (8 \, x^{3} - 24 \, x^{2} + 36 \, x - 27\right )} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x + 3\right )}\right ) - 22 \, \sqrt {3} {\left (8 \, x^{3} - 24 \, x^{2} + 36 \, x - 27\right )} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x - 3\right )}\right ) - 3 \, {\left (8 \, x^{3} - 24 \, x^{2} + 36 \, x - 27\right )} \log \left (4 \, x^{2} + 6 \, x + 9\right ) - 17 \, {\left (8 \, x^{3} - 24 \, x^{2} + 36 \, x - 27\right )} \log \left (4 \, x^{2} - 6 \, x + 9\right ) - 2 \, {\left (8 \, x^{3} - 24 \, x^{2} + 36 \, x - 27\right )} \log \left (2 \, x + 3\right ) + 42 \, {\left (8 \, x^{3} - 24 \, x^{2} + 36 \, x - 27\right )} \log \left (2 \, x - 3\right ) + 216 \, x}{944784 \, {\left (8 \, x^{3} - 24 \, x^{2} + 36 \, x - 27\right )}} \]

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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giac [A]  time = 0.18, size = 99, normalized size = 0.76 \[ -\frac {1}{472392} \, \sqrt {3} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x + 3\right )}\right ) + \frac {11}{472392} \, \sqrt {3} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x - 3\right )}\right ) - \frac {x}{4374 \, {\left (4 \, x^{2} - 6 \, x + 9\right )} {\left (2 \, x - 3\right )}} + \frac {1}{314928} \, \log \left (4 \, x^{2} + 6 \, x + 9\right ) + \frac {17}{944784} \, \log \left (4 \, x^{2} - 6 \, x + 9\right ) + \frac {1}{472392} \, \log \left ({\left | 2 \, x + 3 \right |}\right ) - \frac {7}{157464} \, \log \left ({\left | 2 \, x - 3 \right |}\right ) \]

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))

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maple [A]  time = 0.06, size = 102, normalized size = 0.78 \[ \frac {11 \sqrt {3}\, \arctan \left (\frac {\left (8 x -6\right ) \sqrt {3}}{18}\right )}{472392}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (8 x +6\right ) \sqrt {3}}{18}\right )}{472392}-\frac {7 \ln \left (2 x -3\right )}{157464}+\frac {\ln \left (2 x +3\right )}{472392}+\frac {17 \ln \left (4 x^{2}-6 x +9\right )}{944784}+\frac {\ln \left (4 x^{2}+6 x +9\right )}{314928}+\frac {\frac {9 x}{4}-\frac {27}{8}}{118098 x^{2}-177147 x +\frac {531441}{2}}-\frac {1}{26244 \left (2 x -3\right )} \]

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]

1/118098*(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))
+1/472392*ln(2*x+3)+1/314928*ln(4*x^2+6*x+9)-1/472392*3^(1/2)*arctan(1/18*(8*x+6)*3^(1/2))-1/26244/(2*x-3)-7/1
57464*ln(2*x-3)

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maxima [A]  time = 2.99, size = 95, normalized size = 0.73 \[ -\frac {1}{472392} \, \sqrt {3} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x + 3\right )}\right ) + \frac {11}{472392} \, \sqrt {3} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x - 3\right )}\right ) - \frac {x}{4374 \, {\left (8 \, x^{3} - 24 \, x^{2} + 36 \, x - 27\right )}} + \frac {1}{314928} \, \log \left (4 \, x^{2} + 6 \, x + 9\right ) + \frac {17}{944784} \, \log \left (4 \, x^{2} - 6 \, x + 9\right ) + \frac {1}{472392} \, \log \left (2 \, x + 3\right ) - \frac {7}{157464} \, \log \left (2 \, x - 3\right ) \]

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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mupad [B]  time = 0.19, size = 111, normalized size = 0.85 \[ \frac {\ln \left (x+\frac {3}{2}\right )}{472392}-\frac {7\,\ln \left (x-\frac {3}{2}\right )}{157464}-\frac {x}{34992\,\left (x^3-3\,x^2+\frac {9\,x}{2}-\frac {27}{8}\right )}+\ln \left (x+\frac {3}{4}-\frac {\sqrt {3}\,3{}\mathrm {i}}{4}\right )\,\left (\frac {1}{314928}+\frac {\sqrt {3}\,1{}\mathrm {i}}{944784}\right )-\ln \left (x+\frac {3}{4}+\frac {\sqrt {3}\,3{}\mathrm {i}}{4}\right )\,\left (-\frac {1}{314928}+\frac {\sqrt {3}\,1{}\mathrm {i}}{944784}\right )-\ln \left (x-\frac {3}{4}-\frac {\sqrt {3}\,3{}\mathrm {i}}{4}\right )\,\left (-\frac {17}{944784}+\frac {\sqrt {3}\,11{}\mathrm {i}}{944784}\right )+\ln \left (x-\frac {3}{4}+\frac {\sqrt {3}\,3{}\mathrm {i}}{4}\right )\,\left (\frac {17}{944784}+\frac {\sqrt {3}\,11{}\mathrm {i}}{944784}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x + 3/2)/472392 - (7*log(x - 3/2))/157464 - x/(34992*((9*x)/2 - 3*x^2 + x^3 - 27/8)) + log(x - (3^(1/2)*3i
)/4 + 3/4)*((3^(1/2)*1i)/944784 + 1/314928) - log(x + (3^(1/2)*3i)/4 + 3/4)*((3^(1/2)*1i)/944784 - 1/314928) -
log(x - (3^(1/2)*3i)/4 - 3/4)*((3^(1/2)*11i)/944784 - 17/944784) + log(x + (3^(1/2)*3i)/4 - 3/4)*((3^(1/2)*11
i)/944784 + 17/944784)

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sympy [A]  time = 0.70, size = 119, normalized size = 0.91 \[ - \frac {x}{34992 x^{3} - 104976 x^{2} + 157464 x - 118098} - \frac {7 \log {\left (x - \frac {3}{2} \right )}}{157464} + \frac {\log {\left (x + \frac {3}{2} \right )}}{472392} + \frac {17 \log {\left (x^{2} - \frac {3 x}{2} + \frac {9}{4} \right )}}{944784} + \frac {\log {\left (x^{2} + \frac {3 x}{2} + \frac {9}{4} \right )}}{314928} + \frac {11 \sqrt {3} \operatorname {atan}{\left (\frac {4 \sqrt {3} x}{9} - \frac {\sqrt {3}}{3} \right )}}{472392} - \frac {\sqrt {3} \operatorname {atan}{\left (\frac {4 \sqrt {3} x}{9} + \frac {\sqrt {3}}{3} \right )}}{472392} \]

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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