\(\int \frac {27-8 x^3}{(729-64 x^6)^2} \, dx\) [48]

Optimal result

Integrand size = 17, antiderivative size = 113 \[ \int \frac {27-8 x^3}{\left (729-64 x^6\right )^2} \, dx=\frac {x}{4374 \left (27+8 x^3\right )}-\frac {7 \arctan \left (\frac {3-4 x}{3 \sqrt {3}}\right )}{157464 \sqrt {3}}+\frac {\arctan \left (\frac {3+4 x}{3 \sqrt {3}}\right )}{52488 \sqrt {3}}-\frac {\log (3-2 x)}{157464}+\frac {7 \log (3+2 x)}{472392}-\frac {7 \log \left (9-6 x+4 x^2\right )}{944784}+\frac {\log \left (9+6 x+4 x^2\right )}{314928} \] Output:

x/(34992*x^3+118098)-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)-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)
 

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.91 \[ \int \frac {27-8 x^3}{\left (729-64 x^6\right )^2} \, dx=\frac {\frac {216 x}{27+8 x^3}+14 \sqrt {3} \arctan \left (\frac {-3+4 x}{3 \sqrt {3}}\right )+6 \sqrt {3} \arctan \left (\frac {3+4 x}{3 \sqrt {3}}\right )-6 \log (3-2 x)+14 \log (3+2 x)-7 \log \left (9-6 x+4 x^2\right )+3 \log \left (9+6 x+4 x^2\right )}{944784} \] Input:

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

Output:

((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*L 
og[9 - 6*x + 4*x^2] + 3*Log[9 + 6*x + 4*x^2])/944784
 

Rubi [A] (verified)

Time = 0.55 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.22, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.706, Rules used = {1388, 931, 27, 1020, 750, 16, 27, 1142, 27, 1083, 217, 1103}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 1388

\(\displaystyle \int \frac {1}{\left (27-8 x^3\right ) \left (8 x^3+27\right )^2}dx\)

\(\Big \downarrow \) 931

\(\displaystyle \frac {x}{4374 \left (8 x^3+27\right )}-\frac {\int -\frac {8 \left (135-16 x^3\right )}{\left (27-8 x^3\right ) \left (8 x^3+27\right )}dx}{34992}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {135-16 x^3}{\left (27-8 x^3\right ) \left (8 x^3+27\right )}dx}{4374}+\frac {x}{4374 \left (8 x^3+27\right )}\)

\(\Big \downarrow \) 1020

\(\displaystyle \frac {\frac {3}{2} \int \frac {1}{27-8 x^3}dx+\frac {7}{2} \int \frac {1}{8 x^3+27}dx}{4374}+\frac {x}{4374 \left (8 x^3+27\right )}\)

\(\Big \downarrow \) 750

\(\displaystyle \frac {\frac {7}{2} \left (\frac {1}{27} \int \frac {2 (3-x)}{4 x^2-6 x+9}dx+\frac {1}{27} \int \frac {1}{2 x+3}dx\right )+\frac {3}{2} \left (\frac {1}{27} \int \frac {2 (x+3)}{4 x^2+6 x+9}dx+\frac {1}{27} \int \frac {1}{3-2 x}dx\right )}{4374}+\frac {x}{4374 \left (8 x^3+27\right )}\)

\(\Big \downarrow \) 16

\(\displaystyle \frac {\frac {7}{2} \left (\frac {1}{27} \int \frac {2 (3-x)}{4 x^2-6 x+9}dx+\frac {1}{54} \log (2 x+3)\right )+\frac {3}{2} \left (\frac {1}{27} \int \frac {2 (x+3)}{4 x^2+6 x+9}dx-\frac {1}{54} \log (3-2 x)\right )}{4374}+\frac {x}{4374 \left (8 x^3+27\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {7}{2} \left (\frac {2}{27} \int \frac {3-x}{4 x^2-6 x+9}dx+\frac {1}{54} \log (2 x+3)\right )+\frac {3}{2} \left (\frac {2}{27} \int \frac {x+3}{4 x^2+6 x+9}dx-\frac {1}{54} \log (3-2 x)\right )}{4374}+\frac {x}{4374 \left (8 x^3+27\right )}\)

\(\Big \downarrow \) 1142

\(\displaystyle \frac {\frac {7}{2} \left (\frac {2}{27} \left (\frac {9}{4} \int \frac {1}{4 x^2-6 x+9}dx-\frac {1}{8} \int -\frac {2 (3-4 x)}{4 x^2-6 x+9}dx\right )+\frac {1}{54} \log (2 x+3)\right )+\frac {3}{2} \left (\frac {2}{27} \left (\frac {9}{4} \int \frac {1}{4 x^2+6 x+9}dx+\frac {1}{8} \int \frac {2 (4 x+3)}{4 x^2+6 x+9}dx\right )-\frac {1}{54} \log (3-2 x)\right )}{4374}+\frac {x}{4374 \left (8 x^3+27\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {7}{2} \left (\frac {2}{27} \left (\frac {9}{4} \int \frac {1}{4 x^2-6 x+9}dx+\frac {1}{4} \int \frac {3-4 x}{4 x^2-6 x+9}dx\right )+\frac {1}{54} \log (2 x+3)\right )+\frac {3}{2} \left (\frac {2}{27} \left (\frac {9}{4} \int \frac {1}{4 x^2+6 x+9}dx+\frac {1}{4} \int \frac {4 x+3}{4 x^2+6 x+9}dx\right )-\frac {1}{54} \log (3-2 x)\right )}{4374}+\frac {x}{4374 \left (8 x^3+27\right )}\)

\(\Big \downarrow \) 1083

\(\displaystyle \frac {\frac {7}{2} \left (\frac {2}{27} \left (\frac {1}{4} \int \frac {3-4 x}{4 x^2-6 x+9}dx-\frac {9}{2} \int \frac {1}{-(8 x-6)^2-108}d(8 x-6)\right )+\frac {1}{54} \log (2 x+3)\right )+\frac {3}{2} \left (\frac {2}{27} \left (\frac {1}{4} \int \frac {4 x+3}{4 x^2+6 x+9}dx-\frac {9}{2} \int \frac {1}{-(8 x+6)^2-108}d(8 x+6)\right )-\frac {1}{54} \log (3-2 x)\right )}{4374}+\frac {x}{4374 \left (8 x^3+27\right )}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {\frac {7}{2} \left (\frac {2}{27} \left (\frac {1}{4} \int \frac {3-4 x}{4 x^2-6 x+9}dx+\frac {1}{4} \sqrt {3} \arctan \left (\frac {8 x-6}{6 \sqrt {3}}\right )\right )+\frac {1}{54} \log (2 x+3)\right )+\frac {3}{2} \left (\frac {2}{27} \left (\frac {1}{4} \int \frac {4 x+3}{4 x^2+6 x+9}dx+\frac {1}{4} \sqrt {3} \arctan \left (\frac {8 x+6}{6 \sqrt {3}}\right )\right )-\frac {1}{54} \log (3-2 x)\right )}{4374}+\frac {x}{4374 \left (8 x^3+27\right )}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {\frac {7}{2} \left (\frac {2}{27} \left (\frac {1}{4} \sqrt {3} \arctan \left (\frac {8 x-6}{6 \sqrt {3}}\right )-\frac {1}{8} \log \left (4 x^2-6 x+9\right )\right )+\frac {1}{54} \log (2 x+3)\right )+\frac {3}{2} \left (\frac {2}{27} \left (\frac {1}{4} \sqrt {3} \arctan \left (\frac {8 x+6}{6 \sqrt {3}}\right )+\frac {1}{8} \log \left (4 x^2+6 x+9\right )\right )-\frac {1}{54} \log (3-2 x)\right )}{4374}+\frac {x}{4374 \left (8 x^3+27\right )}\)

Input:

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

Output:

x/(4374*(27 + 8*x^3)) + ((7*(Log[3 + 2*x]/54 + (2*((Sqrt[3]*ArcTan[(-6 + 8 
*x)/(6*Sqrt[3])])/4 - Log[9 - 6*x + 4*x^2]/8))/27))/2 + (3*(-1/54*Log[3 - 
2*x] + (2*((Sqrt[3]*ArcTan[(6 + 8*x)/(6*Sqrt[3])])/4 + Log[9 + 6*x + 4*x^2 
]/8))/27))/2)/4374
 

Defintions of rubi rules used

rule 16

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

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 217

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 750

Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Simp[1/(3*Rt[a, 3]^2)   Int[1/ 
(Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[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 931

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] + Simp[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]) && IntBinomialQ[a, b, 
 c, d, n, p, q, x]
 

rule 1020

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^( 
n_))), x_Symbol] :> Simp[(b*e - a*f)/(b*c - a*d)   Int[1/(a + b*x^n), x], x 
] - Simp[(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 1083

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2   Subst[I 
nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, 
x]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[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 1142

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[(2*c*d - b*e)/(2*c)   Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) 
Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
 

rule 1388

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

Maple [A] (verified)

Time = 0.14 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.76

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

Input:

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

Output:

1/34992*x/(x^3+27/8)-1/157464*ln(2*x-3)+7/472392*ln(2*x+3)+1/314928*ln(4*x 
^2+6*x+9)+1/157464*3^(1/2)*arctan(2/9*(2*x+3/2)*3^(1/2))-7/944784*ln(4*x^2 
-6*x+9)+7/472392*3^(1/2)*arctan(2/9*(2*x-3/2)*3^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.16 \[ \int \frac {27-8 x^3}{\left (729-64 x^6\right )^2} \, dx=\frac {6 \, \sqrt {3} {\left (8 \, x^{3} + 27\right )} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x + 3\right )}\right ) + 14 \, \sqrt {3} {\left (8 \, x^{3} + 27\right )} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x - 3\right )}\right ) + 3 \, {\left (8 \, x^{3} + 27\right )} \log \left (4 \, x^{2} + 6 \, x + 9\right ) - 7 \, {\left (8 \, x^{3} + 27\right )} \log \left (4 \, x^{2} - 6 \, x + 9\right ) + 14 \, {\left (8 \, x^{3} + 27\right )} \log \left (2 \, x + 3\right ) - 6 \, {\left (8 \, x^{3} + 27\right )} \log \left (2 \, x - 3\right ) + 216 \, x}{944784 \, {\left (8 \, x^{3} + 27\right )}} \] Input:

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

Output:

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] (verification not implemented)

Time = 0.32 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.97 \[ \int \frac {27-8 x^3}{\left (729-64 x^6\right )^2} \, dx=\frac {x}{34992 x^{3} + 118098} - \frac {\log {\left (x - \frac {3}{2} \right )}}{157464} + \frac {7 \log {\left (x + \frac {3}{2} \right )}}{472392} - \frac {7 \log {\left (x^{2} - \frac {3 x}{2} + \frac {9}{4} \right )}}{944784} + \frac {\log {\left (x^{2} + \frac {3 x}{2} + \frac {9}{4} \right )}}{314928} + \frac {7 \sqrt {3} \operatorname {atan}{\left (\frac {4 \sqrt {3} x}{9} - \frac {\sqrt {3}}{3} \right )}}{472392} + \frac {\sqrt {3} \operatorname {atan}{\left (\frac {4 \sqrt {3} x}{9} + \frac {\sqrt {3}}{3} \right )}}{157464} \] Input:

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

Output:

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 + sq 
rt(3)/3)/157464
 

Maxima [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.77 \[ \int \frac {27-8 x^3}{\left (729-64 x^6\right )^2} \, dx=\frac {1}{157464} \, \sqrt {3} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x + 3\right )}\right ) + \frac {7}{472392} \, \sqrt {3} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x - 3\right )}\right ) + \frac {x}{4374 \, {\left (8 \, x^{3} + 27\right )}} + \frac {1}{314928} \, \log \left (4 \, x^{2} + 6 \, x + 9\right ) - \frac {7}{944784} \, \log \left (4 \, x^{2} - 6 \, x + 9\right ) + \frac {7}{472392} \, \log \left (2 \, x + 3\right ) - \frac {1}{157464} \, \log \left (2 \, x - 3\right ) \] Input:

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

Output:

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*log 
(2*x - 3)
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.79 \[ \int \frac {27-8 x^3}{\left (729-64 x^6\right )^2} \, dx=\frac {1}{157464} \, \sqrt {3} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x + 3\right )}\right ) + \frac {7}{472392} \, \sqrt {3} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x - 3\right )}\right ) + \frac {x}{4374 \, {\left (8 \, x^{3} + 27\right )}} + \frac {1}{314928} \, \log \left (4 \, x^{2} + 6 \, x + 9\right ) - \frac {7}{944784} \, \log \left (4 \, x^{2} - 6 \, x + 9\right ) + \frac {7}{472392} \, \log \left ({\left | 2 \, x + 3 \right |}\right ) - \frac {1}{157464} \, \log \left ({\left | 2 \, x - 3 \right |}\right ) \] Input:

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

Output:

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)) - 1/15746 
4*log(abs(2*x - 3))
 

Mupad [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.90 \[ \int \frac {27-8 x^3}{\left (729-64 x^6\right )^2} \, dx=\frac {7\,\ln \left (x+\frac {3}{2}\right )}{472392}-\frac {\ln \left (x-\frac {3}{2}\right )}{157464}+\frac {x}{34992\,\left (x^3+\frac {27}{8}\right )}-\ln \left (x+\frac {3}{4}-\frac {\sqrt {3}\,3{}\mathrm {i}}{4}\right )\,\left (-\frac {1}{314928}+\frac {\sqrt {3}\,1{}\mathrm {i}}{314928}\right )+\ln \left (x+\frac {3}{4}+\frac {\sqrt {3}\,3{}\mathrm {i}}{4}\right )\,\left (\frac {1}{314928}+\frac {\sqrt {3}\,1{}\mathrm {i}}{314928}\right )-\ln \left (x-\frac {3}{4}-\frac {\sqrt {3}\,3{}\mathrm {i}}{4}\right )\,\left (\frac {7}{944784}+\frac {\sqrt {3}\,7{}\mathrm {i}}{944784}\right )+\ln \left (x-\frac {3}{4}+\frac {\sqrt {3}\,3{}\mathrm {i}}{4}\right )\,\left (-\frac {7}{944784}+\frac {\sqrt {3}\,7{}\mathrm {i}}{944784}\right ) \] Input:

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

Output:

(7*log(x + 3/2))/472392 - log(x - 3/2)/157464 + x/(34992*(x^3 + 27/8)) - l 
og(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)
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.59 \[ \int \frac {27-8 x^3}{\left (729-64 x^6\right )^2} \, dx=\frac {112 \sqrt {3}\, \mathit {atan} \left (\frac {4 x -3}{3 \sqrt {3}}\right ) x^{3}+378 \sqrt {3}\, \mathit {atan} \left (\frac {4 x -3}{3 \sqrt {3}}\right )+48 \sqrt {3}\, \mathit {atan} \left (\frac {4 x +3}{3 \sqrt {3}}\right ) x^{3}+162 \sqrt {3}\, \mathit {atan} \left (\frac {4 x +3}{3 \sqrt {3}}\right )-56 \,\mathrm {log}\left (4 x^{2}-6 x +9\right ) x^{3}-189 \,\mathrm {log}\left (4 x^{2}-6 x +9\right )+24 \,\mathrm {log}\left (4 x^{2}+6 x +9\right ) x^{3}+81 \,\mathrm {log}\left (4 x^{2}+6 x +9\right )-48 \,\mathrm {log}\left (2 x -3\right ) x^{3}-162 \,\mathrm {log}\left (2 x -3\right )+112 \,\mathrm {log}\left (2 x +3\right ) x^{3}+378 \,\mathrm {log}\left (2 x +3\right )+216 x}{7558272 x^{3}[phone]} \] Input:

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

Output:

(112*sqrt(3)*atan((4*x - 3)/(3*sqrt(3)))*x**3 + 378*sqrt(3)*atan((4*x - 3) 
/(3*sqrt(3))) + 48*sqrt(3)*atan((4*x + 3)/(3*sqrt(3)))*x**3 + 162*sqrt(3)* 
atan((4*x + 3)/(3*sqrt(3))) - 56*log(4*x**2 - 6*x + 9)*x**3 - 189*log(4*x* 
*2 - 6*x + 9) + 24*log(4*x**2 + 6*x + 9)*x**3 + 81*log(4*x**2 + 6*x + 9) - 
 48*log(2*x - 3)*x**3 - 162*log(2*x - 3) + 112*log(2*x + 3)*x**3 + 378*log 
(2*x + 3) + 216*x)/(944784*(8*x**3 + 27))