3.573 \(\int \frac{27-8 x^3}{(729-64 x^6)^2} \, dx\)

Optimal. Leaf size=113 \[ \frac{x}{4374 \left (8 x^3+27\right )}-\frac{7 \log \left (4 x^2-6 x+9\right )}{944784}+\frac{\log \left (4 x^2+6 x+9\right )}{314928}-\frac{\log (3-2 x)}{157464}+\frac{7 \log (2 x+3)}{472392}-\frac{7 \tan ^{-1}\left (\frac{3-4 x}{3 \sqrt{3}}\right )}{157464 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{4 x+3}{3 \sqrt{3}}\right )}{52488 \sqrt{3}} \]

[Out]

x/(4374*(27 + 8*x^3)) - (7*ArcTan[(3 - 4*x)/(3*Sqrt[3])])/(157464*Sqrt[3]) + ArcTan[(3 + 4*x)/(3*Sqrt[3])]/(52
488*Sqrt[3]) - Log[3 - 2*x]/157464 + (7*Log[3 + 2*x])/472392 - (7*Log[9 - 6*x + 4*x^2])/944784 + Log[9 + 6*x +
4*x^2]/314928

________________________________________________________________________________________

Rubi [A]  time = 0.0818849, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 9, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.529, Rules used = {1404, 414, 522, 200, 31, 634, 618, 204, 628} \[ \frac{x}{4374 \left (8 x^3+27\right )}-\frac{7 \log \left (4 x^2-6 x+9\right )}{944784}+\frac{\log \left (4 x^2+6 x+9\right )}{314928}-\frac{\log (3-2 x)}{157464}+\frac{7 \log (2 x+3)}{472392}-\frac{7 \tan ^{-1}\left (\frac{3-4 x}{3 \sqrt{3}}\right )}{157464 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{4 x+3}{3 \sqrt{3}}\right )}{52488 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Int[(27 - 8*x^3)/(729 - 64*x^6)^2,x]

[Out]

x/(4374*(27 + 8*x^3)) - (7*ArcTan[(3 - 4*x)/(3*Sqrt[3])])/(157464*Sqrt[3]) + ArcTan[(3 + 4*x)/(3*Sqrt[3])]/(52
488*Sqrt[3]) - Log[3 - 2*x]/157464 + (7*Log[3 + 2*x])/472392 - (7*Log[9 - 6*x + 4*x^2])/944784 + Log[9 + 6*x +
4*x^2]/314928

Rule 1404

Int[((d_) + (e_.)*(x_)^(n_))^(q_.)*((a_) + (c_.)*(x_)^(n2_))^(p_.), x_Symbol] :> Int[(d + e*x^n)^(p + q)*(a/d
+ (c*x^n)/e)^p, x] /; FreeQ[{a, c, d, e, n, q}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 + a*e^2, 0] && IntegerQ[p]

Rule 414

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(b*x*(a + b*x^n)^(p + 1)*(
c + d*x^n)^(q + 1))/(a*n*(p + 1)*(b*c - a*d)), x] + Dist[1/(a*n*(p + 1)*(b*c - a*d)), Int[(a + b*x^n)^(p + 1)*
(c + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d,
n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &&  !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomial
Q[a, b, c, d, n, p, q, x]

Rule 522

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[(b*e - a*f
)/(b*c - a*d), Int[1/(a + b*x^n), x], x] - Dist[(d*e - c*f)/(b*c - a*d), Int[1/(c + d*x^n), x], x] /; FreeQ[{a
, b, c, d, e, f, n}, x]

Rule 200

Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Dist[1/(3*Rt[a, 3]^2), Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Di
st[1/(3*Rt[a, 3]^2), Int[(2*Rt[a, 3] - Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], x]
/; FreeQ[{a, b}, x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

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 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 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{27-8 x^3}{\left (729-64 x^6\right )^2} \, dx &=\int \frac{1}{\left (27-8 x^3\right ) \left (27+8 x^3\right )^2} \, dx\\ &=\frac{x}{4374 \left (27+8 x^3\right )}-\frac{\int \frac{-1080+128 x^3}{\left (27-8 x^3\right ) \left (27+8 x^3\right )} \, dx}{34992}\\ &=\frac{x}{4374 \left (27+8 x^3\right )}+\frac{\int \frac{1}{27-8 x^3} \, dx}{2916}+\frac{7 \int \frac{1}{27+8 x^3} \, dx}{8748}\\ &=\frac{x}{4374 \left (27+8 x^3\right )}+\frac{\int \frac{1}{3-2 x} \, dx}{78732}+\frac{\int \frac{6+2 x}{9+6 x+4 x^2} \, dx}{78732}+\frac{7 \int \frac{1}{3+2 x} \, dx}{236196}+\frac{7 \int \frac{6-2 x}{9-6 x+4 x^2} \, dx}{236196}\\ &=\frac{x}{4374 \left (27+8 x^3\right )}-\frac{\log (3-2 x)}{157464}+\frac{7 \log (3+2 x)}{472392}+\frac{\int \frac{6+8 x}{9+6 x+4 x^2} \, dx}{314928}-\frac{7 \int \frac{-6+8 x}{9-6 x+4 x^2} \, dx}{944784}+\frac{\int \frac{1}{9+6 x+4 x^2} \, dx}{17496}+\frac{7 \int \frac{1}{9-6 x+4 x^2} \, dx}{52488}\\ &=\frac{x}{4374 \left (27+8 x^3\right )}-\frac{\log (3-2 x)}{157464}+\frac{7 \log (3+2 x)}{472392}-\frac{7 \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 )}{8748}-\frac{7 \operatorname{Subst}\left (\int \frac{1}{-108-x^2} \, dx,x,-6+8 x\right )}{26244}\\ &=\frac{x}{4374 \left (27+8 x^3\right )}-\frac{7 \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 )}{52488 \sqrt{3}}-\frac{\log (3-2 x)}{157464}+\frac{7 \log (3+2 x)}{472392}-\frac{7 \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 = 0.0501018, size = 103, normalized size = 0.91 \[ \frac{\frac{216 x}{8 x^3+27}-7 \log \left (4 x^2-6 x+9\right )+3 \log \left (4 x^2+6 x+9\right )-6 \log (3-2 x)+14 \log (2 x+3)+14 \sqrt{3} \tan ^{-1}\left (\frac{4 x-3}{3 \sqrt{3}}\right )+6 \sqrt{3} \tan ^{-1}\left (\frac{4 x+3}{3 \sqrt{3}}\right )}{944784} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((216*x)/(27 + 8*x^3) + 14*Sqrt[3]*ArcTan[(-3 + 4*x)/(3*Sqrt[3])] + 6*Sqrt[3]*ArcTan[(3 + 4*x)/(3*Sqrt[3])] -
6*Log[3 - 2*x] + 14*Log[3 + 2*x] - 7*Log[9 - 6*x + 4*x^2] + 3*Log[9 + 6*x + 4*x^2])/944784

________________________________________________________________________________________

Maple [A]  time = 0.013, size = 102, normalized size = 0.9 \begin{align*} -{\frac{1}{236196+157464\,x}}+{\frac{7\,\ln \left ( 3+2\,x \right ) }{472392}}-{\frac{\ln \left ( -3+2\,x \right ) }{157464}}+{\frac{\ln \left ( 4\,{x}^{2}+6\,x+9 \right ) }{314928}}+{\frac{\sqrt{3}}{157464}\arctan \left ({\frac{ \left ( 8\,x+6 \right ) \sqrt{3}}{18}} \right ) }-{\frac{1}{118098} \left ( -{\frac{3\,x}{4}}-{\frac{9}{8}} \right ) \left ({x}^{2}-{\frac{3\,x}{2}}+{\frac{9}{4}} \right ) ^{-1}}-{\frac{7\,\ln \left ( 4\,{x}^{2}-6\,x+9 \right ) }{944784}}+{\frac{7\,\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+27)/(-64*x^6+729)^2,x)

[Out]

-1/78732/(3+2*x)+7/472392*ln(3+2*x)-1/157464*ln(-3+2*x)+1/314928*ln(4*x^2+6*x+9)+1/157464*3^(1/2)*arctan(1/18*
(8*x+6)*3^(1/2))-1/118098*(-3/4*x-9/8)/(x^2-3/2*x+9/4)-7/944784*ln(4*x^2-6*x+9)+7/472392*3^(1/2)*arctan(1/18*(
8*x-6)*3^(1/2))

________________________________________________________________________________________

Maxima [A]  time = 1.38086, size = 117, normalized size = 1.04 \begin{align*} \frac{1}{157464} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x + 3\right )}\right ) + \frac{7}{472392} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x - 3\right )}\right ) + \frac{x}{4374 \,{\left (8 \, x^{3} + 27\right )}} + \frac{1}{314928} \, \log \left (4 \, x^{2} + 6 \, x + 9\right ) - \frac{7}{944784} \, \log \left (4 \, x^{2} - 6 \, x + 9\right ) + \frac{7}{472392} \, \log \left (2 \, x + 3\right ) - \frac{1}{157464} \, \log \left (2 \, x - 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/157464*sqrt(3)*arctan(1/9*sqrt(3)*(4*x + 3)) + 7/472392*sqrt(3)*arctan(1/9*sqrt(3)*(4*x - 3)) + 1/4374*x/(8*
x^3 + 27) + 1/314928*log(4*x^2 + 6*x + 9) - 7/944784*log(4*x^2 - 6*x + 9) + 7/472392*log(2*x + 3) - 1/157464*l
og(2*x - 3)

________________________________________________________________________________________

Fricas [A]  time = 1.42664, size = 377, normalized size = 3.34 \begin{align*} \frac{6 \, \sqrt{3}{\left (8 \, x^{3} + 27\right )} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x + 3\right )}\right ) + 14 \, \sqrt{3}{\left (8 \, x^{3} + 27\right )} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x - 3\right )}\right ) + 3 \,{\left (8 \, x^{3} + 27\right )} \log \left (4 \, x^{2} + 6 \, x + 9\right ) - 7 \,{\left (8 \, x^{3} + 27\right )} \log \left (4 \, x^{2} - 6 \, x + 9\right ) + 14 \,{\left (8 \, x^{3} + 27\right )} \log \left (2 \, x + 3\right ) - 6 \,{\left (8 \, x^{3} + 27\right )} \log \left (2 \, x - 3\right ) + 216 \, x}{944784 \,{\left (8 \, x^{3} + 27\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/944784*(6*sqrt(3)*(8*x^3 + 27)*arctan(1/9*sqrt(3)*(4*x + 3)) + 14*sqrt(3)*(8*x^3 + 27)*arctan(1/9*sqrt(3)*(4
*x - 3)) + 3*(8*x^3 + 27)*log(4*x^2 + 6*x + 9) - 7*(8*x^3 + 27)*log(4*x^2 - 6*x + 9) + 14*(8*x^3 + 27)*log(2*x
+ 3) - 6*(8*x^3 + 27)*log(2*x - 3) + 216*x)/(8*x^3 + 27)

________________________________________________________________________________________

Sympy [A]  time = 0.3835, size = 110, normalized size = 0.97 \begin{align*} \frac{x}{34992 x^{3} + 118098} - \frac{\log{\left (x - \frac{3}{2} \right )}}{157464} + \frac{7 \log{\left (x + \frac{3}{2} \right )}}{472392} - \frac{7 \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{7 \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 )}}{157464} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/(34992*x**3 + 118098) - log(x - 3/2)/157464 + 7*log(x + 3/2)/472392 - 7*log(x**2 - 3*x/2 + 9/4)/944784 + log
(x**2 + 3*x/2 + 9/4)/314928 + 7*sqrt(3)*atan(4*sqrt(3)*x/9 - sqrt(3)/3)/472392 + sqrt(3)*atan(4*sqrt(3)*x/9 +
sqrt(3)/3)/157464

________________________________________________________________________________________

Giac [A]  time = 1.05941, size = 120, normalized size = 1.06 \begin{align*} \frac{1}{157464} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x + 3\right )}\right ) + \frac{7}{472392} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x - 3\right )}\right ) + \frac{x}{4374 \,{\left (8 \, x^{3} + 27\right )}} + \frac{1}{314928} \, \log \left (4 \, x^{2} + 6 \, x + 9\right ) - \frac{7}{944784} \, \log \left (4 \, x^{2} - 6 \, x + 9\right ) + \frac{7}{472392} \, \log \left ({\left | 2 \, x + 3 \right |}\right ) - \frac{1}{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+27)/(-64*x^6+729)^2,x, algorithm="giac")

[Out]

1/157464*sqrt(3)*arctan(1/9*sqrt(3)*(4*x + 3)) + 7/472392*sqrt(3)*arctan(1/9*sqrt(3)*(4*x - 3)) + 1/4374*x/(8*
x^3 + 27) + 1/314928*log(4*x^2 + 6*x + 9) - 7/944784*log(4*x^2 - 6*x + 9) + 7/472392*log(abs(2*x + 3)) - 1/157
464*log(abs(2*x - 3))