\(\int \frac {81+36 x^2+16 x^4}{(729-64 x^6)^2} \, dx\) [567]

Optimal result

Integrand size = 22, antiderivative size = 81 \[ \int \frac {81+36 x^2+16 x^4}{\left (729-64 x^6\right )^2} \, dx=\frac {1}{17496 (3-2 x)}-\frac {1}{17496 (3+2 x)}-\frac {\arctan \left (\frac {3-4 x}{3 \sqrt {3}}\right )}{13122 \sqrt {3}}+\frac {\arctan \left (\frac {3+4 x}{3 \sqrt {3}}\right )}{13122 \sqrt {3}}+\frac {\text {arctanh}\left (\frac {2 x}{3}\right )}{8748} \]

[Out]

1/17496/(3-2*x)-1/17496/(3+2*x)+1/8748*arctanh(2/3*x)-1/39366*arctan(1/9*(3-4*x)*3^(1/2))*3^(1/2)+1/39366*arct
an(1/9*(3+4*x)*3^(1/2))*3^(1/2)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {1600, 1184, 213, 632, 210} \[ \int \frac {81+36 x^2+16 x^4}{\left (729-64 x^6\right )^2} \, dx=-\frac {\arctan \left (\frac {3-4 x}{3 \sqrt {3}}\right )}{13122 \sqrt {3}}+\frac {\arctan \left (\frac {4 x+3}{3 \sqrt {3}}\right )}{13122 \sqrt {3}}+\frac {\text {arctanh}\left (\frac {2 x}{3}\right )}{8748}+\frac {1}{17496 (3-2 x)}-\frac {1}{17496 (2 x+3)} \]

[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 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 213

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

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 1184

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 1600

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]

Rubi steps \begin{align*} \text {integral}& = \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 {\text {Subst}\left (\int \frac {1}{-108-x^2} \, dx,x,-6+8 x\right )}{2187}-\frac {\text {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] (verified)

Result contains complex when optimal does not.

Time = 0.53 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.51 \[ \int \frac {81+36 x^2+16 x^4}{\left (729-64 x^6\right )^2} \, dx=\frac {\frac {36 x}{9-4 x^2}+3 \sqrt {3} \arctan \left (\frac {1}{3} \left (-i+\sqrt {3}\right ) x\right )+4 i \sqrt {3} \text {arctanh}\left (\frac {1}{3} \left (1-i \sqrt {3}\right ) x\right )+\left (-3+\frac {2}{\sqrt {\frac {1}{6} \left (1+i \sqrt {3}\right )}}\right ) \text {arctanh}\left (\frac {1}{3} \left (x+i \sqrt {3} x\right )\right )-9 \log (3-2 x)+9 \log (3+2 x)}{157464} \]

[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

Maple [A] (verified)

Time = 1.59 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.75

method	result	size
risch	\(-\frac {x}{17496 \left (x^{2}-\frac {9}{4}\right )}-\frac {\ln \left (-3+2 x \right )}{17496}+\frac {\ln \left (2 x +3\right )}{17496}+\frac {\sqrt {3}\, \arctan \left (\frac {2 \sqrt {3}\, x}{9}\right )}{39366}+\frac {\sqrt {3}\, \arctan \left (\frac {8 x^{3} \sqrt {3}}{81}+\frac {4 \sqrt {3}\, x}{9}\right )}{39366}\)	\(61\)
default	\(-\frac {1}{17496 \left (-3+2 x \right )}-\frac {\ln \left (-3+2 x \right )}{17496}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (8 x -6\right ) \sqrt {3}}{18}\right )}{39366}-\frac {1}{17496 \left (2 x +3\right )}+\frac {\ln \left (2 x +3\right )}{17496}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (8 x +6\right ) \sqrt {3}}{18}\right )}{39366}\)	\(68\)
meijerg	\(-\frac {\left (-1\right )^{\frac {5}{6}} \left (\frac {4 x \left (-1\right )^{\frac {1}{6}}}{6-\frac {128 x^{6}}{243}}-\frac {5 x \left (-1\right )^{\frac {1}{6}} \left (\ln \left (1-\frac {2 \left (x^{6}\right )^{\frac {1}{6}}}{3}\right )-\ln \left (1+\frac {2 \left (x^{6}\right )^{\frac {1}{6}}}{3}\right )+\frac {\ln \left (1-\frac {2 \left (x^{6}\right )^{\frac {1}{6}}}{3}+\frac {4 \left (x^{6}\right )^{\frac {1}{3}}}{9}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{6}\right )^{\frac {1}{6}}}{3-\left (x^{6}\right )^{\frac {1}{6}}}\right )-\frac {\ln \left (1+\frac {2 \left (x^{6}\right )^{\frac {1}{6}}}{3}+\frac {4 \left (x^{6}\right )^{\frac {1}{3}}}{9}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{6}\right )^{\frac {1}{6}}}{3+\left (x^{6}\right )^{\frac {1}{6}}}\right )\right )}{6 \left (x^{6}\right )^{\frac {1}{6}}}\right )}{26244}-\frac {\left (-1\right )^{\frac {1}{6}} \left (\frac {64 x^{5} \left (-1\right )^{\frac {5}{6}}}{81 \left (6-\frac {128 x^{6}}{243}\right )}-\frac {x^{5} \left (-1\right )^{\frac {5}{6}} \left (\ln \left (1-\frac {2 \left (x^{6}\right )^{\frac {1}{6}}}{3}\right )-\ln \left (1+\frac {2 \left (x^{6}\right )^{\frac {1}{6}}}{3}\right )+\frac {\ln \left (1-\frac {2 \left (x^{6}\right )^{\frac {1}{6}}}{3}+\frac {4 \left (x^{6}\right )^{\frac {1}{3}}}{9}\right )}{2}+\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{6}\right )^{\frac {1}{6}}}{3-\left (x^{6}\right )^{\frac {1}{6}}}\right )-\frac {\ln \left (1+\frac {2 \left (x^{6}\right )^{\frac {1}{6}}}{3}+\frac {4 \left (x^{6}\right )^{\frac {1}{3}}}{9}\right )}{2}+\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{6}\right )^{\frac {1}{6}}}{3+\left (x^{6}\right )^{\frac {1}{6}}}\right )\right )}{6 \left (x^{6}\right )^{\frac {5}{6}}}\right )}{26244}-\frac {i \left (\frac {16 i x^{3}}{27 \left (-\frac {128 x^{6}}{729}+2\right )}+i \operatorname {arctanh}\left (\frac {8 x^{3}}{27}\right )\right )}{26244}\)	\(326\)

[In]

int((16*x^4+36*x^2+81)/(-64*x^6+729)^2,x,method=_RETURNVERBOSE)

[Out]

-1/17496*x/(x^2-9/4)-1/17496*ln(-3+2*x)+1/17496*ln(2*x+3)+1/39366*3^(1/2)*arctan(2/9*3^(1/2)*x)+1/39366*3^(1/2
)*arctan(8/81*x^3*3^(1/2)+4/9*3^(1/2)*x)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.12 \[ \int \frac {81+36 x^2+16 x^4}{\left (729-64 x^6\right )^2} \, dx=\frac {4 \, \sqrt {3} {\left (4 \, x^{2} - 9\right )} \arctan \left (\frac {4}{81} \, \sqrt {3} {\left (2 \, x^{3} + 9 \, x\right )}\right ) + 4 \, \sqrt {3} {\left (4 \, x^{2} - 9\right )} \arctan \left (\frac {2}{9} \, \sqrt {3} x\right ) + 9 \, {\left (4 \, x^{2} - 9\right )} \log \left (2 \, x + 3\right ) - 9 \, {\left (4 \, x^{2} - 9\right )} \log \left (2 \, x - 3\right ) - 36 \, x}{157464 \, {\left (4 \, x^{2} - 9\right )}} \]

[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)

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.86 \[ \int \frac {81+36 x^2+16 x^4}{\left (729-64 x^6\right )^2} \, dx=- \frac {x}{17496 x^{2} - 39366} + \frac {\sqrt {3} \cdot \left (2 \operatorname {atan}{\left (\frac {2 \sqrt {3} x}{9} \right )} + 2 \operatorname {atan}{\left (\frac {8 \sqrt {3} x^{3}}{81} + \frac {4 \sqrt {3} x}{9} \right )}\right )}{78732} - \frac {\log {\left (x - \frac {3}{2} \right )}}{17496} + \frac {\log {\left (x + \frac {3}{2} \right )}}{17496} \]

[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

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.75 \[ \int \frac {81+36 x^2+16 x^4}{\left (729-64 x^6\right )^2} \, dx=\frac {1}{39366} \, \sqrt {3} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x + 3\right )}\right ) + \frac {1}{39366} \, \sqrt {3} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x - 3\right )}\right ) - \frac {x}{4374 \, {\left (4 \, x^{2} - 9\right )}} + \frac {1}{17496} \, \log \left (2 \, x + 3\right ) - \frac {1}{17496} \, \log \left (2 \, x - 3\right ) \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.78 \[ \int \frac {81+36 x^2+16 x^4}{\left (729-64 x^6\right )^2} \, dx=\frac {1}{39366} \, \sqrt {3} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x + 3\right )}\right ) + \frac {1}{39366} \, \sqrt {3} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x - 3\right )}\right ) - \frac {x}{4374 \, {\left (4 \, x^{2} - 9\right )}} + \frac {1}{17496} \, \log \left ({\left | 2 \, x + 3 \right |}\right ) - \frac {1}{17496} \, \log \left ({\left | 2 \, x - 3 \right |}\right ) \]

[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))

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.64 \[ \int \frac {81+36 x^2+16 x^4}{\left (729-64 x^6\right )^2} \, dx=\frac {\mathrm {atanh}\left (\frac {2\,x}{3}\right )}{8748}+\frac {\sqrt {3}\,\left (2\,\mathrm {atan}\left (\frac {8\,\sqrt {3}\,x^3}{81}+\frac {4\,\sqrt {3}\,x}{9}\right )+2\,\mathrm {atan}\left (\frac {2\,\sqrt {3}\,x}{9}\right )\right )}{78732}-\frac {x}{17496\,\left (x^2-\frac {9}{4}\right )} \]

[In]

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

[Out]

atanh((2*x)/3)/8748 + (3^(1/2)*(2*atan((4*3^(1/2)*x)/9 + (8*3^(1/2)*x^3)/81) + 2*atan((2*3^(1/2)*x)/9)))/78732
- x/(17496*(x^2 - 9/4))