3.6.73 \(\int \frac {27-8 x^3}{(729-64 x^6)^2} \, dx\) [573]

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

[Out]

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

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Rubi [A]
time = 0.05, antiderivative size = 113, normalized size of antiderivative = 1.00, 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 = {1418, 425, 536, 206, 31, 648, 632, 210, 642} \begin {gather*} -\frac {7 \text {ArcTan}\left (\frac {3-4 x}{3 \sqrt {3}}\right )}{157464 \sqrt {3}}+\frac {\text {ArcTan}\left (\frac {4 x+3}{3 \sqrt {3}}\right )}{52488 \sqrt {3}}+\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} \end {gather*}

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 31

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

Rule 206

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 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 425

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]) && IntBinomi
alQ[a, b, c, d, n, p, q, x]

Rule 536

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 632

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 642

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 648

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 1418

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

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 {\text {Subst}\left (\int \frac {1}{-108-x^2} \, dx,x,6+8 x\right )}{8748}-\frac {7 \text {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*}

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Mathematica [A]
time = 0.04, size = 103, normalized size = 0.91 \begin {gather*} \frac {\frac {216 x}{27+8 x^3}+14 \sqrt {3} \tan ^{-1}\left (\frac {-3+4 x}{3 \sqrt {3}}\right )+6 \sqrt {3} \tan ^{-1}\left (\frac {3+4 x}{3 \sqrt {3}}\right )-6 \log (3-2 x)+14 \log (3+2 x)-7 \log \left (9-6 x+4 x^2\right )+3 \log \left (9+6 x+4 x^2\right )}{944784} \end {gather*}

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

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Maple [A]
time = 0.39, size = 102, normalized size = 0.90

method	result	size
risch	\(\frac {x}{34992 x^{3}+118098}-\frac {\ln \left (-3+2 x \right )}{157464}-\frac {7 \ln \left (4 x^{2}-6 x +9\right )}{944784}+\frac {7 \sqrt {3}\, \arctan \left (\frac {2 \left (2 x -\frac {3}{2}\right ) \sqrt {3}}{9}\right )}{472392}+\frac {7 \ln \left (2 x +3\right )}{472392}+\frac {\ln \left (4 x^{2}+6 x +9\right )}{314928}+\frac {\sqrt {3}\, \arctan \left (\frac {2 \left (2 x +\frac {3}{2}\right ) \sqrt {3}}{9}\right )}{157464}\)	\(86\)
default	\(-\frac {-\frac {3 x}{4}-\frac {9}{8}}{118098 \left (x^{2}-\frac {3}{2} x +\frac {9}{4}\right )}-\frac {7 \ln \left (4 x^{2}-6 x +9\right )}{944784}+\frac {7 \sqrt {3}\, \arctan \left (\frac {\left (8 x -6\right ) \sqrt {3}}{18}\right )}{472392}+\frac {\ln \left (4 x^{2}+6 x +9\right )}{314928}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (8 x +6\right ) \sqrt {3}}{18}\right )}{157464}-\frac {1}{78732 \left (2 x +3\right )}+\frac {7 \ln \left (2 x +3\right )}{472392}-\frac {\ln \left (-3+2 x \right )}{157464}\)	\(102\)
meijerg	\(-\frac {\left (-1\right )^{\frac {5}{6}} \left (\frac {4 x \left (-1\right )^{\frac {1}{6}}}{6-\frac {128 x^{6}}{243}}-\frac {5 x \left (-1\right )^{\frac {1}{6}} \left (\ln \left (1-\frac {2 \left (x^{6}\right )^{\frac {1}{6}}}{3}\right )-\ln \left (1+\frac {2 \left (x^{6}\right )^{\frac {1}{6}}}{3}\right )+\frac {\ln \left (1-\frac {2 \left (x^{6}\right )^{\frac {1}{6}}}{3}+\frac {4 \left (x^{6}\right )^{\frac {1}{3}}}{9}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{6}\right )^{\frac {1}{6}}}{3-\left (x^{6}\right )^{\frac {1}{6}}}\right )-\frac {\ln \left (1+\frac {2 \left (x^{6}\right )^{\frac {1}{6}}}{3}+\frac {4 \left (x^{6}\right )^{\frac {1}{3}}}{9}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{6}\right )^{\frac {1}{6}}}{3+\left (x^{6}\right )^{\frac {1}{6}}}\right )\right )}{6 \left (x^{6}\right )^{\frac {1}{6}}}\right )}{78732}+\frac {\left (-1\right )^{\frac {1}{3}} \left (\frac {16 x^{4} \left (-1\right )^{\frac {2}{3}}}{27 \left (3-\frac {64 x^{6}}{243}\right )}-\frac {x^{4} \left (-1\right )^{\frac {2}{3}} \left (\ln \left (1-\frac {4 \left (x^{6}\right )^{\frac {1}{3}}}{9}\right )-\frac {\ln \left (1+\frac {4 \left (x^{6}\right )^{\frac {1}{3}}}{9}+\frac {16 \left (x^{6}\right )^{\frac {2}{3}}}{81}\right )}{2}+\sqrt {3}\, \arctan \left (\frac {2 \sqrt {3}\, \left (x^{6}\right )^{\frac {1}{3}}}{9 \left (1+\frac {2 \left (x^{6}\right )^{\frac {1}{3}}}{9}\right )}\right )\right )}{3 \left (x^{6}\right )^{\frac {2}{3}}}\right )}{78732}\)	\(241\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-8*x^3+27)/(-64*x^6+729)^2,x,method=_RETURNVERBOSE)

[Out]

-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))+
1/314928*ln(4*x^2+6*x+9)+1/157464*3^(1/2)*arctan(1/18*(8*x+6)*3^(1/2))-1/78732/(2*x+3)+7/472392*ln(2*x+3)-1/15
7464*ln(-3+2*x)

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Maxima [A]
time = 0.52, size = 87, normalized size = 0.77 \begin {gather*} \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 {gather*}

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)

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Fricas [A]
time = 0.37, size = 131, normalized size = 1.16 \begin {gather*} \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 {gather*}

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)

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Sympy [A]
time = 0.22, size = 110, normalized size = 0.97 \begin {gather*} \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 {gather*}

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

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Giac [A]
time = 1.53, size = 89, normalized size = 0.79 \begin {gather*} \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 {gather*}

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))

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Mupad [B]
time = 0.17, size = 102, normalized size = 0.90 \begin {gather*} \frac {7\,\ln \left (x+\frac {3}{2}\right )}{472392}-\frac {\ln \left (x-\frac {3}{2}\right )}{157464}+\frac {x}{34992\,\left (x^3+\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}}{314928}\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}}{314928}\right )-\ln \left (x-\frac {3}{4}-\frac {\sqrt {3}\,3{}\mathrm {i}}{4}\right )\,\left (\frac {7}{944784}+\frac {\sqrt {3}\,7{}\mathrm {i}}{944784}\right )+\ln \left (x-\frac {3}{4}+\frac {\sqrt {3}\,3{}\mathrm {i}}{4}\right )\,\left (-\frac {7}{944784}+\frac {\sqrt {3}\,7{}\mathrm {i}}{944784}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(8*x^3 - 27)/(64*x^6 - 729)^2,x)

[Out]

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

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