3.568 \(\int \frac{81+54 x-24 x^3-16 x^4}{(729-64 x^6)^2} \, dx\)

Optimal. Leaf size=92 \[ \frac{x}{4374 \left (4 x^2-6 x+9\right )}-\frac{\log \left (4 x^2-6 x+9\right )}{157464}+\frac{\log \left (4 x^2+6 x+9\right )}{52488}-\frac{\log (3-2 x)}{26244}+\frac{\log (2 x+3)}{78732}-\frac{\tan ^{-1}\left (\frac{3-4 x}{3 \sqrt{3}}\right )}{4374 \sqrt{3}} \]

[Out]

x/(4374*(9 - 6*x + 4*x^2)) - ArcTan[(3 - 4*x)/(3*Sqrt[3])]/(4374*Sqrt[3]) - Log[3 - 2*x]/26244 + Log[3 + 2*x]/
78732 - Log[9 - 6*x + 4*x^2]/157464 + Log[9 + 6*x + 4*x^2]/52488

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Rubi [A]  time = 0.116453, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 11, 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} \[ \frac{x}{4374 \left (4 x^2-6 x+9\right )}-\frac{\log \left (4 x^2-6 x+9\right )}{157464}+\frac{\log \left (4 x^2+6 x+9\right )}{52488}-\frac{\log (3-2 x)}{26244}+\frac{\log (2 x+3)}{78732}-\frac{\tan ^{-1}\left (\frac{3-4 x}{3 \sqrt{3}}\right )}{4374 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Int[(81 + 54*x - 24*x^3 - 16*x^4)/(729 - 64*x^6)^2,x]

[Out]

x/(4374*(9 - 6*x + 4*x^2)) - ArcTan[(3 - 4*x)/(3*Sqrt[3])]/(4374*Sqrt[3]) - Log[3 - 2*x]/26244 + Log[3 + 2*x]/
78732 - Log[9 - 6*x + 4*x^2]/157464 + Log[9 + 6*x + 4*x^2]/52488

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{81+54 x-24 x^3-16 x^4}{\left (729-64 x^6\right )^2} \, dx &=\int \frac{1}{\left (9-6 x+4 x^2\right )^2 \left (81+54 x-24 x^3-16 x^4\right )} \, dx\\ &=\int \left (-\frac{1}{13122 (-3+2 x)}+\frac{1}{39366 (3+2 x)}+\frac{3-x}{729 \left (9-6 x+4 x^2\right )^2}+\frac{39-4 x}{78732 \left (9-6 x+4 x^2\right )}+\frac{3+4 x}{26244 \left (9+6 x+4 x^2\right )}\right ) \, dx\\ &=-\frac{\log (3-2 x)}{26244}+\frac{\log (3+2 x)}{78732}+\frac{\int \frac{39-4 x}{9-6 x+4 x^2} \, dx}{78732}+\frac{\int \frac{3+4 x}{9+6 x+4 x^2} \, dx}{26244}+\frac{1}{729} \int \frac{3-x}{\left (9-6 x+4 x^2\right )^2} \, dx\\ &=\frac{x}{4374 \left (9-6 x+4 x^2\right )}-\frac{\log (3-2 x)}{26244}+\frac{\log (3+2 x)}{78732}+\frac{\log \left (9+6 x+4 x^2\right )}{52488}-\frac{\int \frac{-6+8 x}{9-6 x+4 x^2} \, dx}{157464}+\frac{\int \frac{1}{9-6 x+4 x^2} \, dx}{4374}+\frac{\int \frac{1}{9-6 x+4 x^2} \, dx}{2187}\\ &=\frac{x}{4374 \left (9-6 x+4 x^2\right )}-\frac{\log (3-2 x)}{26244}+\frac{\log (3+2 x)}{78732}-\frac{\log \left (9-6 x+4 x^2\right )}{157464}+\frac{\log \left (9+6 x+4 x^2\right )}{52488}-\frac{\operatorname{Subst}\left (\int \frac{1}{-108-x^2} \, dx,x,-6+8 x\right )}{2187}-\frac{2 \operatorname{Subst}\left (\int \frac{1}{-108-x^2} \, dx,x,-6+8 x\right )}{2187}\\ &=\frac{x}{4374 \left (9-6 x+4 x^2\right )}-\frac{\tan ^{-1}\left (\frac{3-4 x}{3 \sqrt{3}}\right )}{4374 \sqrt{3}}-\frac{\log (3-2 x)}{26244}+\frac{\log (3+2 x)}{78732}-\frac{\log \left (9-6 x+4 x^2\right )}{157464}+\frac{\log \left (9+6 x+4 x^2\right )}{52488}\\ \end{align*}

Mathematica [A]  time = 0.029823, size = 84, normalized size = 0.91 \[ \frac{\frac{36 x}{4 x^2-6 x+9}-\log \left (4 x^2-6 x+9\right )+3 \log \left (4 x^2+6 x+9\right )-6 \log (3-2 x)+2 \log (2 x+3)+12 \sqrt{3} \tan ^{-1}\left (\frac{4 x-3}{3 \sqrt{3}}\right )}{157464} \]

Antiderivative was successfully verified.

[In]

Integrate[(81 + 54*x - 24*x^3 - 16*x^4)/(729 - 64*x^6)^2,x]

[Out]

((36*x)/(9 - 6*x + 4*x^2) + 12*Sqrt[3]*ArcTan[(-3 + 4*x)/(3*Sqrt[3])] - 6*Log[3 - 2*x] + 2*Log[3 + 2*x] - Log[
9 - 6*x + 4*x^2] + 3*Log[9 + 6*x + 4*x^2])/157464

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Maple [A]  time = 0.01, size = 73, normalized size = 0.8 \begin{align*}{\frac{\ln \left ( 3+2\,x \right ) }{78732}}-{\frac{\ln \left ( -3+2\,x \right ) }{26244}}+{\frac{\ln \left ( 4\,{x}^{2}+6\,x+9 \right ) }{52488}}+{\frac{x}{17496} \left ({x}^{2}-{\frac{3\,x}{2}}+{\frac{9}{4}} \right ) ^{-1}}-{\frac{\ln \left ( 4\,{x}^{2}-6\,x+9 \right ) }{157464}}+{\frac{\sqrt{3}}{13122}\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((-16*x^4-24*x^3+54*x+81)/(-64*x^6+729)^2,x)

[Out]

1/78732*ln(3+2*x)-1/26244*ln(-3+2*x)+1/52488*ln(4*x^2+6*x+9)+1/17496*x/(x^2-3/2*x+9/4)-1/157464*ln(4*x^2-6*x+9
)+1/13122*3^(1/2)*arctan(1/18*(8*x-6)*3^(1/2))

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Maxima [A]  time = 1.38606, size = 100, normalized size = 1.09 \begin{align*} \frac{1}{13122} \, \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 )}} + \frac{1}{52488} \, \log \left (4 \, x^{2} + 6 \, x + 9\right ) - \frac{1}{157464} \, \log \left (4 \, x^{2} - 6 \, x + 9\right ) + \frac{1}{78732} \, \log \left (2 \, x + 3\right ) - \frac{1}{26244} \, \log \left (2 \, x - 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-16*x^4-24*x^3+54*x+81)/(-64*x^6+729)^2,x, algorithm="maxima")

[Out]

1/13122*sqrt(3)*arctan(1/9*sqrt(3)*(4*x - 3)) + 1/4374*x/(4*x^2 - 6*x + 9) + 1/52488*log(4*x^2 + 6*x + 9) - 1/
157464*log(4*x^2 - 6*x + 9) + 1/78732*log(2*x + 3) - 1/26244*log(2*x - 3)

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Fricas [A]  time = 1.65431, size = 338, normalized size = 3.67 \begin{align*} \frac{12 \, \sqrt{3}{\left (4 \, x^{2} - 6 \, x + 9\right )} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x - 3\right )}\right ) + 3 \,{\left (4 \, x^{2} - 6 \, x + 9\right )} \log \left (4 \, x^{2} + 6 \, x + 9\right ) -{\left (4 \, x^{2} - 6 \, x + 9\right )} \log \left (4 \, x^{2} - 6 \, x + 9\right ) + 2 \,{\left (4 \, x^{2} - 6 \, x + 9\right )} \log \left (2 \, x + 3\right ) - 6 \,{\left (4 \, x^{2} - 6 \, x + 9\right )} \log \left (2 \, x - 3\right ) + 36 \, x}{157464 \,{\left (4 \, x^{2} - 6 \, x + 9\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-16*x^4-24*x^3+54*x+81)/(-64*x^6+729)^2,x, algorithm="fricas")

[Out]

1/157464*(12*sqrt(3)*(4*x^2 - 6*x + 9)*arctan(1/9*sqrt(3)*(4*x - 3)) + 3*(4*x^2 - 6*x + 9)*log(4*x^2 + 6*x + 9
) - (4*x^2 - 6*x + 9)*log(4*x^2 - 6*x + 9) + 2*(4*x^2 - 6*x + 9)*log(2*x + 3) - 6*(4*x^2 - 6*x + 9)*log(2*x -
3) + 36*x)/(4*x^2 - 6*x + 9)

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Sympy [A]  time = 0.284686, size = 82, normalized size = 0.89 \begin{align*} \frac{x}{17496 x^{2} - 26244 x + 39366} - \frac{\log{\left (x - \frac{3}{2} \right )}}{26244} + \frac{\log{\left (x + \frac{3}{2} \right )}}{78732} - \frac{\log{\left (x^{2} - \frac{3 x}{2} + \frac{9}{4} \right )}}{157464} + \frac{\log{\left (4 x^{2} + 6 x + 9 \right )}}{52488} + \frac{\sqrt{3} \operatorname{atan}{\left (\frac{4 \sqrt{3} x}{9} - \frac{\sqrt{3}}{3} \right )}}{13122} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-16*x**4-24*x**3+54*x+81)/(-64*x**6+729)**2,x)

[Out]

x/(17496*x**2 - 26244*x + 39366) - log(x - 3/2)/26244 + log(x + 3/2)/78732 - log(x**2 - 3*x/2 + 9/4)/157464 +
log(4*x**2 + 6*x + 9)/52488 + sqrt(3)*atan(4*sqrt(3)*x/9 - sqrt(3)/3)/13122

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Giac [A]  time = 1.05603, size = 103, normalized size = 1.12 \begin{align*} \frac{1}{13122} \, \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 )}} + \frac{1}{52488} \, \log \left (4 \, x^{2} + 6 \, x + 9\right ) - \frac{1}{157464} \, \log \left (4 \, x^{2} - 6 \, x + 9\right ) + \frac{1}{78732} \, \log \left ({\left | 2 \, x + 3 \right |}\right ) - \frac{1}{26244} \, \log \left ({\left | 2 \, x - 3 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-16*x^4-24*x^3+54*x+81)/(-64*x^6+729)^2,x, algorithm="giac")

[Out]

1/13122*sqrt(3)*arctan(1/9*sqrt(3)*(4*x - 3)) + 1/4374*x/(4*x^2 - 6*x + 9) + 1/52488*log(4*x^2 + 6*x + 9) - 1/
157464*log(4*x^2 - 6*x + 9) + 1/78732*log(abs(2*x + 3)) - 1/26244*log(abs(2*x - 3))