3.6.63 \(\int \frac {27-8 x^3}{729-64 x^6} \, dx\) [563]

Optimal. Leaf size=50 \[ -\frac {\tan ^{-1}\left (\frac {3-4 x}{3 \sqrt {3}}\right )}{18 \sqrt {3}}+\frac {1}{54} \log (3+2 x)-\frac {1}{108} \log \left (9-6 x+4 x^2\right ) \]

[Out]

1/54*ln(3+2*x)-1/108*ln(4*x^2-6*x+9)-1/54*arctan(1/9*(3-4*x)*3^(1/2))*3^(1/2)

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Rubi [A]
time = 0.02, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {26, 206, 31, 648, 632, 210, 642} \begin {gather*} -\frac {\text {ArcTan}\left (\frac {3-4 x}{3 \sqrt {3}}\right )}{18 \sqrt {3}}-\frac {1}{108} \log \left (4 x^2-6 x+9\right )+\frac {1}{54} \log (2 x+3) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(27 - 8*x^3)/(729 - 64*x^6),x]

[Out]

-1/18*ArcTan[(3 - 4*x)/(3*Sqrt[3])]/Sqrt[3] + Log[3 + 2*x]/54 - Log[9 - 6*x + 4*x^2]/108

Rule 26

Int[(u_.)*((a_) + (b_.)*(x_)^(n_.))^(m_.)*((c_) + (d_.)*(x_)^(j_))^(p_.), x_Symbol] :> Dist[(-b^2/d)^m, Int[u/
(a - b*x^n)^m, x], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && EqQ[j, 2*n] && EqQ[p, -m] && EqQ[b^2*c + a^2*d, 0]
&& GtQ[a, 0] && LtQ[d, 0]

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 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]

Rubi steps

\begin {align*} \int \frac {27-8 x^3}{729-64 x^6} \, dx &=\int \frac {1}{27+8 x^3} \, dx\\ &=\frac {1}{27} \int \frac {1}{3+2 x} \, dx+\frac {1}{27} \int \frac {6-2 x}{9-6 x+4 x^2} \, dx\\ &=\frac {1}{54} \log (3+2 x)-\frac {1}{108} \int \frac {-6+8 x}{9-6 x+4 x^2} \, dx+\frac {1}{6} \int \frac {1}{9-6 x+4 x^2} \, dx\\ &=\frac {1}{54} \log (3+2 x)-\frac {1}{108} \log \left (9-6 x+4 x^2\right )-\frac {1}{3} \text {Subst}\left (\int \frac {1}{-108-x^2} \, dx,x,-6+8 x\right )\\ &=-\frac {\tan ^{-1}\left (\frac {3-4 x}{3 \sqrt {3}}\right )}{18 \sqrt {3}}+\frac {1}{54} \log (3+2 x)-\frac {1}{108} \log \left (9-6 x+4 x^2\right )\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 50, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\frac {-3+4 x}{3 \sqrt {3}}\right )}{18 \sqrt {3}}+\frac {1}{54} \log (3+2 x)-\frac {1}{108} \log \left (9-6 x+4 x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTan[(-3 + 4*x)/(3*Sqrt[3])]/(18*Sqrt[3]) + Log[3 + 2*x]/54 - Log[9 - 6*x + 4*x^2]/108

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Maple [A]
time = 0.36, size = 39, normalized size = 0.78

method	result	size
default	\(-\frac {\ln \left (4 x^{2}-6 x +9\right )}{108}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (8 x -6\right ) \sqrt {3}}{18}\right )}{54}+\frac {\ln \left (2 x +3\right )}{54}\)	\(39\)
risch	\(-\frac {\ln \left (16 x^{2}-24 x +36\right )}{108}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (-3+4 x \right ) \sqrt {3}}{9}\right )}{54}+\frac {\ln \left (2 x +3\right )}{54}\)	\(39\)
meijerg	\(-\frac {x \left (\ln \left (1-\frac {2 \left (x^{6}\right )^{\frac {1}{6}}}{3}\right )-\ln \left (1+\frac {2 \left (x^{6}\right )^{\frac {1}{6}}}{3}\right )+\frac {\ln \left (1-\frac {2 \left (x^{6}\right )^{\frac {1}{6}}}{3}+\frac {4 \left (x^{6}\right )^{\frac {1}{3}}}{9}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{6}\right )^{\frac {1}{6}}}{3-\left (x^{6}\right )^{\frac {1}{6}}}\right )-\frac {\ln \left (1+\frac {2 \left (x^{6}\right )^{\frac {1}{6}}}{3}+\frac {4 \left (x^{6}\right )^{\frac {1}{3}}}{9}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{6}\right )^{\frac {1}{6}}}{3+\left (x^{6}\right )^{\frac {1}{6}}}\right )\right )}{108 \left (x^{6}\right )^{\frac {1}{6}}}+\frac {x^{4} \left (\ln \left (1-\frac {4 \left (x^{6}\right )^{\frac {1}{3}}}{9}\right )-\frac {\ln \left (1+\frac {4 \left (x^{6}\right )^{\frac {1}{3}}}{9}+\frac {16 \left (x^{6}\right )^{\frac {2}{3}}}{81}\right )}{2}+\sqrt {3}\, \arctan \left (\frac {2 \sqrt {3}\, \left (x^{6}\right )^{\frac {1}{3}}}{9 \left (1+\frac {2 \left (x^{6}\right )^{\frac {1}{3}}}{9}\right )}\right )\right )}{108 \left (x^{6}\right )^{\frac {2}{3}}}\)	\(191\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/108*ln(4*x^2-6*x+9)+1/54*3^(1/2)*arctan(1/18*(8*x-6)*3^(1/2))+1/54*ln(2*x+3)

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Maxima [A]
time = 0.50, size = 38, normalized size = 0.76 \begin {gather*} \frac {1}{54} \, \sqrt {3} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x - 3\right )}\right ) - \frac {1}{108} \, \log \left (4 \, x^{2} - 6 \, x + 9\right ) + \frac {1}{54} \, \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),x, algorithm="maxima")

[Out]

1/54*sqrt(3)*arctan(1/9*sqrt(3)*(4*x - 3)) - 1/108*log(4*x^2 - 6*x + 9) + 1/54*log(2*x + 3)

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Fricas [A]
time = 0.39, size = 38, normalized size = 0.76 \begin {gather*} \frac {1}{54} \, \sqrt {3} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x - 3\right )}\right ) - \frac {1}{108} \, \log \left (4 \, x^{2} - 6 \, x + 9\right ) + \frac {1}{54} \, \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),x, algorithm="fricas")

[Out]

1/54*sqrt(3)*arctan(1/9*sqrt(3)*(4*x - 3)) - 1/108*log(4*x^2 - 6*x + 9) + 1/54*log(2*x + 3)

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Sympy [A]
time = 0.05, size = 48, normalized size = 0.96 \begin {gather*} \frac {\log {\left (x + \frac {3}{2} \right )}}{54} - \frac {\log {\left (x^{2} - \frac {3 x}{2} + \frac {9}{4} \right )}}{108} + \frac {\sqrt {3} \operatorname {atan}{\left (\frac {4 \sqrt {3} x}{9} - \frac {\sqrt {3}}{3} \right )}}{54} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x + 3/2)/54 - log(x**2 - 3*x/2 + 9/4)/108 + sqrt(3)*atan(4*sqrt(3)*x/9 - sqrt(3)/3)/54

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Giac [A]
time = 0.68, size = 35, normalized size = 0.70 \begin {gather*} \frac {1}{54} \, \sqrt {3} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x - 3\right )}\right ) - \frac {1}{108} \, \log \left (x^{2} - \frac {3}{2} \, x + \frac {9}{4}\right ) + \frac {1}{54} \, \log \left ({\left | x + \frac {3}{2} \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/54*sqrt(3)*arctan(1/9*sqrt(3)*(4*x - 3)) - 1/108*log(x^2 - 3/2*x + 9/4) + 1/54*log(abs(x + 3/2))

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Mupad [B]
time = 0.09, size = 46, normalized size = 0.92 \begin {gather*} \frac {\ln \left (x+\frac {3}{2}\right )}{54}-\ln \left (x-\frac {3}{4}-\frac {\sqrt {3}\,3{}\mathrm {i}}{4}\right )\,\left (\frac {1}{108}+\frac {\sqrt {3}\,1{}\mathrm {i}}{108}\right )+\ln \left (x-\frac {3}{4}+\frac {\sqrt {3}\,3{}\mathrm {i}}{4}\right )\,\left (-\frac {1}{108}+\frac {\sqrt {3}\,1{}\mathrm {i}}{108}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((8*x^3 - 27)/(64*x^6 - 729),x)

[Out]

log(x + 3/2)/54 - log(x - (3^(1/2)*3i)/4 - 3/4)*((3^(1/2)*1i)/108 + 1/108) + log(x + (3^(1/2)*3i)/4 - 3/4)*((3
^(1/2)*1i)/108 - 1/108)

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