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

Optimal. Leaf size=24 \[ -\frac{\tan ^{-1}\left (\frac{3-4 x}{3 \sqrt{3}}\right )}{3 \sqrt{3}} \]

[Out]

-ArcTan[(3 - 4*x)/(3*Sqrt[3])]/(3*Sqrt[3])

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Rubi [A]  time = 0.0237435, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {1586, 618, 204} \[ -\frac{\tan ^{-1}\left (\frac{3-4 x}{3 \sqrt{3}}\right )}{3 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-ArcTan[(3 - 4*x)/(3*Sqrt[3])]/(3*Sqrt[3])

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 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])

Rubi steps

\begin{align*} \int \frac{81+54 x-24 x^3-16 x^4}{729-64 x^6} \, dx &=\int \frac{1}{9-6 x+4 x^2} \, dx\\ &=-\left (2 \operatorname{Subst}\left (\int \frac{1}{-108-x^2} \, dx,x,-6+8 x\right )\right )\\ &=-\frac{\tan ^{-1}\left (\frac{3-4 x}{3 \sqrt{3}}\right )}{3 \sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.0063597, size = 24, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{4 x-3}{3 \sqrt{3}}\right )}{3 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTan[(-3 + 4*x)/(3*Sqrt[3])]/(3*Sqrt[3])

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Maple [A]  time = 0.003, size = 17, normalized size = 0.7 \begin{align*}{\frac{\sqrt{3}}{9}\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),x)

[Out]

1/9*3^(1/2)*arctan(1/18*(8*x-6)*3^(1/2))

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Maxima [A]  time = 1.37219, size = 22, normalized size = 0.92 \begin{align*} \frac{1}{9} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, 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),x, algorithm="maxima")

[Out]

1/9*sqrt(3)*arctan(1/9*sqrt(3)*(4*x - 3))

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Fricas [A]  time = 1.35482, size = 58, normalized size = 2.42 \begin{align*} \frac{1}{9} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, 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),x, algorithm="fricas")

[Out]

1/9*sqrt(3)*arctan(1/9*sqrt(3)*(4*x - 3))

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Sympy [A]  time = 0.114066, size = 24, normalized size = 1. \begin{align*} \frac{\sqrt{3} \operatorname{atan}{\left (\frac{4 \sqrt{3} x}{9} - \frac{\sqrt{3}}{3} \right )}}{9} \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),x)

[Out]

sqrt(3)*atan(4*sqrt(3)*x/9 - sqrt(3)/3)/9

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Giac [A]  time = 1.05202, size = 22, normalized size = 0.92 \begin{align*} \frac{1}{9} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, 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),x, algorithm="giac")

[Out]

1/9*sqrt(3)*arctan(1/9*sqrt(3)*(4*x - 3))