3.567 \(\int \frac{81+36 x^2+16 x^4}{(729-64 x^6)^2} \, dx\)

Optimal. Leaf size=81 \[ \frac{1}{17496 (3-2 x)}-\frac{1}{17496 (2 x+3)}-\frac{\tan ^{-1}\left (\frac{3-4 x}{3 \sqrt{3}}\right )}{13122 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{4 x+3}{3 \sqrt{3}}\right )}{13122 \sqrt{3}}+\frac{\tanh ^{-1}\left (\frac{2 x}{3}\right )}{8748} \]

[Out]

1/(17496*(3 - 2*x)) - 1/(17496*(3 + 2*x)) - ArcTan[(3 - 4*x)/(3*Sqrt[3])]/(13122*Sqrt[3]) + ArcTan[(3 + 4*x)/(
3*Sqrt[3])]/(13122*Sqrt[3]) + ArcTanh[(2*x)/3]/8748

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Rubi [A]  time = 0.0692554, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {1586, 1170, 207, 618, 204} \[ \frac{1}{17496 (3-2 x)}-\frac{1}{17496 (2 x+3)}-\frac{\tan ^{-1}\left (\frac{3-4 x}{3 \sqrt{3}}\right )}{13122 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{4 x+3}{3 \sqrt{3}}\right )}{13122 \sqrt{3}}+\frac{\tanh ^{-1}\left (\frac{2 x}{3}\right )}{8748} \]

Antiderivative was successfully verified.

[In]

Int[(81 + 36*x^2 + 16*x^4)/(729 - 64*x^6)^2,x]

[Out]

1/(17496*(3 - 2*x)) - 1/(17496*(3 + 2*x)) - ArcTan[(3 - 4*x)/(3*Sqrt[3])]/(13122*Sqrt[3]) + ArcTan[(3 + 4*x)/(
3*Sqrt[3])]/(13122*Sqrt[3]) + ArcTanh[(2*x)/3]/8748

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 1170

Int[((d_) + (e_.)*(x_)^2)^(q_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Int[ExpandIntegrand[(d + e*x
^2)^q/(a + b*x^2 + c*x^4), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a
*e^2, 0] && IntegerQ[q]

Rule 207

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[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 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+36 x^2+16 x^4}{\left (729-64 x^6\right )^2} \, dx &=\int \frac{1}{\left (9-4 x^2\right )^2 \left (81+36 x^2+16 x^4\right )} \, dx\\ &=\int \left (\frac{1}{8748 (-3+2 x)^2}+\frac{1}{8748 (3+2 x)^2}-\frac{1}{1458 \left (-9+4 x^2\right )}+\frac{1}{4374 \left (9-6 x+4 x^2\right )}+\frac{1}{4374 \left (9+6 x+4 x^2\right )}\right ) \, dx\\ &=\frac{1}{17496 (3-2 x)}-\frac{1}{17496 (3+2 x)}+\frac{\int \frac{1}{9-6 x+4 x^2} \, dx}{4374}+\frac{\int \frac{1}{9+6 x+4 x^2} \, dx}{4374}-\frac{\int \frac{1}{-9+4 x^2} \, dx}{1458}\\ &=\frac{1}{17496 (3-2 x)}-\frac{1}{17496 (3+2 x)}+\frac{\tanh ^{-1}\left (\frac{2 x}{3}\right )}{8748}-\frac{\operatorname{Subst}\left (\int \frac{1}{-108-x^2} \, dx,x,-6+8 x\right )}{2187}-\frac{\operatorname{Subst}\left (\int \frac{1}{-108-x^2} \, dx,x,6+8 x\right )}{2187}\\ &=\frac{1}{17496 (3-2 x)}-\frac{1}{17496 (3+2 x)}-\frac{\tan ^{-1}\left (\frac{3-4 x}{3 \sqrt{3}}\right )}{13122 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{3+4 x}{3 \sqrt{3}}\right )}{13122 \sqrt{3}}+\frac{\tanh ^{-1}\left (\frac{2 x}{3}\right )}{8748}\\ \end{align*}

Mathematica [C]  time = 0.430905, size = 122, normalized size = 1.51 \[ \frac{\frac{36 x}{9-4 x^2}-9 \log (3-2 x)+9 \log (2 x+3)+3 \sqrt{3} \tan ^{-1}\left (\frac{1}{3} \left (\sqrt{3}-i\right ) x\right )+4 i \sqrt{3} \tanh ^{-1}\left (\frac{1}{3} \left (1-i \sqrt{3}\right ) x\right )+\left (-3+\frac{2}{\sqrt{\frac{1}{6} \left (1+i \sqrt{3}\right )}}\right ) \tanh ^{-1}\left (\frac{1}{3} \left (x+i \sqrt{3} x\right )\right )}{157464} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(81 + 36*x^2 + 16*x^4)/(729 - 64*x^6)^2,x]

[Out]

((36*x)/(9 - 4*x^2) + 3*Sqrt[3]*ArcTan[((-I + Sqrt[3])*x)/3] + (4*I)*Sqrt[3]*ArcTanh[((1 - I*Sqrt[3])*x)/3] +
(-3 + 2/Sqrt[(1 + I*Sqrt[3])/6])*ArcTanh[(x + I*Sqrt[3]*x)/3] - 9*Log[3 - 2*x] + 9*Log[3 + 2*x])/157464

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Maple [A]  time = 0.01, size = 68, normalized size = 0.8 \begin{align*} -{\frac{1}{52488+34992\,x}}+{\frac{\ln \left ( 3+2\,x \right ) }{17496}}-{\frac{1}{-52488+34992\,x}}-{\frac{\ln \left ( -3+2\,x \right ) }{17496}}+{\frac{\sqrt{3}}{39366}\arctan \left ({\frac{ \left ( 8\,x+6 \right ) \sqrt{3}}{18}} \right ) }+{\frac{\sqrt{3}}{39366}\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+36*x^2+81)/(-64*x^6+729)^2,x)

[Out]

-1/17496/(3+2*x)+1/17496*ln(3+2*x)-1/17496/(-3+2*x)-1/17496*ln(-3+2*x)+1/39366*3^(1/2)*arctan(1/18*(8*x+6)*3^(
1/2))+1/39366*3^(1/2)*arctan(1/18*(8*x-6)*3^(1/2))

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Maxima [A]  time = 1.37462, size = 82, normalized size = 1.01 \begin{align*} \frac{1}{39366} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x + 3\right )}\right ) + \frac{1}{39366} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x - 3\right )}\right ) - \frac{x}{4374 \,{\left (4 \, x^{2} - 9\right )}} + \frac{1}{17496} \, \log \left (2 \, x + 3\right ) - \frac{1}{17496} \, \log \left (2 \, x - 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((16*x^4+36*x^2+81)/(-64*x^6+729)^2,x, algorithm="maxima")

[Out]

1/39366*sqrt(3)*arctan(1/9*sqrt(3)*(4*x + 3)) + 1/39366*sqrt(3)*arctan(1/9*sqrt(3)*(4*x - 3)) - 1/4374*x/(4*x^
2 - 9) + 1/17496*log(2*x + 3) - 1/17496*log(2*x - 3)

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Fricas [A]  time = 1.52705, size = 259, normalized size = 3.2 \begin{align*} \frac{4 \, \sqrt{3}{\left (4 \, x^{2} - 9\right )} \arctan \left (\frac{4}{81} \, \sqrt{3}{\left (2 \, x^{3} + 9 \, x\right )}\right ) + 4 \, \sqrt{3}{\left (4 \, x^{2} - 9\right )} \arctan \left (\frac{2}{9} \, \sqrt{3} x\right ) + 9 \,{\left (4 \, x^{2} - 9\right )} \log \left (2 \, x + 3\right ) - 9 \,{\left (4 \, x^{2} - 9\right )} \log \left (2 \, x - 3\right ) - 36 \, x}{157464 \,{\left (4 \, x^{2} - 9\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((16*x^4+36*x^2+81)/(-64*x^6+729)^2,x, algorithm="fricas")

[Out]

1/157464*(4*sqrt(3)*(4*x^2 - 9)*arctan(4/81*sqrt(3)*(2*x^3 + 9*x)) + 4*sqrt(3)*(4*x^2 - 9)*arctan(2/9*sqrt(3)*
x) + 9*(4*x^2 - 9)*log(2*x + 3) - 9*(4*x^2 - 9)*log(2*x - 3) - 36*x)/(4*x^2 - 9)

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Sympy [A]  time = 0.201229, size = 70, normalized size = 0.86 \begin{align*} - \frac{x}{17496 x^{2} - 39366} + \frac{\sqrt{3} \left (2 \operatorname{atan}{\left (\frac{2 \sqrt{3} x}{9} \right )} + 2 \operatorname{atan}{\left (\frac{8 \sqrt{3} x^{3}}{81} + \frac{4 \sqrt{3} x}{9} \right )}\right )}{78732} - \frac{\log{\left (x - \frac{3}{2} \right )}}{17496} + \frac{\log{\left (x + \frac{3}{2} \right )}}{17496} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((16*x**4+36*x**2+81)/(-64*x**6+729)**2,x)

[Out]

-x/(17496*x**2 - 39366) + sqrt(3)*(2*atan(2*sqrt(3)*x/9) + 2*atan(8*sqrt(3)*x**3/81 + 4*sqrt(3)*x/9))/78732 -
log(x - 3/2)/17496 + log(x + 3/2)/17496

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Giac [A]  time = 1.06105, size = 85, normalized size = 1.05 \begin{align*} \frac{1}{39366} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x + 3\right )}\right ) + \frac{1}{39366} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x - 3\right )}\right ) - \frac{x}{4374 \,{\left (4 \, x^{2} - 9\right )}} + \frac{1}{17496} \, \log \left ({\left | 2 \, x + 3 \right |}\right ) - \frac{1}{17496} \, \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+36*x^2+81)/(-64*x^6+729)^2,x, algorithm="giac")

[Out]

1/39366*sqrt(3)*arctan(1/9*sqrt(3)*(4*x + 3)) + 1/39366*sqrt(3)*arctan(1/9*sqrt(3)*(4*x - 3)) - 1/4374*x/(4*x^
2 - 9) + 1/17496*log(abs(2*x + 3)) - 1/17496*log(abs(2*x - 3))