3.561 \(\int \frac {9-6 x+4 x^2}{729-64 x^6} \, dx\)

Optimal. Leaf size=60 \[ -\frac {1}{324} \log \left (4 x^2+6 x+9\right )-\frac {1}{324} \log (3-2 x)+\frac {1}{108} \log (2 x+3)+\frac {\tan ^{-1}\left (\frac {4 x+3}{3 \sqrt {3}}\right )}{54 \sqrt {3}} \]

[Out]

-1/324*ln(3-2*x)+1/108*ln(3+2*x)-1/324*ln(4*x^2+6*x+9)+1/162*arctan(1/9*(3+4*x)*3^(1/2))*3^(1/2)

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Rubi [A]  time = 0.05, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {1586, 2058, 634, 618, 204, 628} \[ -\frac {1}{324} \log \left (4 x^2+6 x+9\right )-\frac {1}{324} \log (3-2 x)+\frac {1}{108} \log (2 x+3)+\frac {\tan ^{-1}\left (\frac {4 x+3}{3 \sqrt {3}}\right )}{54 \sqrt {3}} \]

Antiderivative was successfully verified.

[In]

Int[(9 - 6*x + 4*x^2)/(729 - 64*x^6),x]

[Out]

ArcTan[(3 + 4*x)/(3*Sqrt[3])]/(54*Sqrt[3]) - Log[3 - 2*x]/324 + Log[3 + 2*x]/108 - Log[9 + 6*x + 4*x^2]/324

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 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 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]

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 1586

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]

Rule 2058

Int[(P_)^(p_), x_Symbol] :> With[{u = Factor[P]}, Int[ExpandIntegrand[u^p, x], x] /;  !SumQ[NonfreeFactors[u,
x]]] /; PolyQ[P, x] && ILtQ[p, 0]

Rubi steps

\begin {align*} \int \frac {9-6 x+4 x^2}{729-64 x^6} \, dx &=\int \frac {1}{81+54 x-24 x^3-16 x^4} \, dx\\ &=\int \left (-\frac {1}{162 (-3+2 x)}+\frac {1}{54 (3+2 x)}+\frac {3-2 x}{81 \left (9+6 x+4 x^2\right )}\right ) \, dx\\ &=-\frac {1}{324} \log (3-2 x)+\frac {1}{108} \log (3+2 x)+\frac {1}{81} \int \frac {3-2 x}{9+6 x+4 x^2} \, dx\\ &=-\frac {1}{324} \log (3-2 x)+\frac {1}{108} \log (3+2 x)-\frac {1}{324} \int \frac {6+8 x}{9+6 x+4 x^2} \, dx+\frac {1}{18} \int \frac {1}{9+6 x+4 x^2} \, dx\\ &=-\frac {1}{324} \log (3-2 x)+\frac {1}{108} \log (3+2 x)-\frac {1}{324} \log \left (9+6 x+4 x^2\right )-\frac {1}{9} \operatorname {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 )}{54 \sqrt {3}}-\frac {1}{324} \log (3-2 x)+\frac {1}{108} \log (3+2 x)-\frac {1}{324} \log \left (9+6 x+4 x^2\right )\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 56, normalized size = 0.93 \[ \frac {1}{324} \left (-\log \left (4 x^2+6 x+9\right )-\log (3-2 x)+3 \log (2 x+3)+2 \sqrt {3} \tan ^{-1}\left (\frac {4 x+3}{3 \sqrt {3}}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(9 - 6*x + 4*x^2)/(729 - 64*x^6),x]

[Out]

(2*Sqrt[3]*ArcTan[(3 + 4*x)/(3*Sqrt[3])] - Log[3 - 2*x] + 3*Log[3 + 2*x] - Log[9 + 6*x + 4*x^2])/324

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fricas [A]  time = 0.44, size = 46, normalized size = 0.77 \[ \frac {1}{162} \, \sqrt {3} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x + 3\right )}\right ) - \frac {1}{324} \, \log \left (4 \, x^{2} + 6 \, x + 9\right ) + \frac {1}{108} \, \log \left (2 \, x + 3\right ) - \frac {1}{324} \, \log \left (2 \, x - 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^2-6*x+9)/(-64*x^6+729),x, algorithm="fricas")

[Out]

1/162*sqrt(3)*arctan(1/9*sqrt(3)*(4*x + 3)) - 1/324*log(4*x^2 + 6*x + 9) + 1/108*log(2*x + 3) - 1/324*log(2*x
- 3)

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giac [A]  time = 0.19, size = 48, normalized size = 0.80 \[ \frac {1}{162} \, \sqrt {3} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x + 3\right )}\right ) - \frac {1}{324} \, \log \left (4 \, x^{2} + 6 \, x + 9\right ) + \frac {1}{108} \, \log \left ({\left | 2 \, x + 3 \right |}\right ) - \frac {1}{324} \, \log \left ({\left | 2 \, x - 3 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^2-6*x+9)/(-64*x^6+729),x, algorithm="giac")

[Out]

1/162*sqrt(3)*arctan(1/9*sqrt(3)*(4*x + 3)) - 1/324*log(4*x^2 + 6*x + 9) + 1/108*log(abs(2*x + 3)) - 1/324*log
(abs(2*x - 3))

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maple [A]  time = 0.05, size = 47, normalized size = 0.78 \[ \frac {\sqrt {3}\, \arctan \left (\frac {\left (8 x +6\right ) \sqrt {3}}{18}\right )}{162}-\frac {\ln \left (2 x -3\right )}{324}+\frac {\ln \left (2 x +3\right )}{108}-\frac {\ln \left (4 x^{2}+6 x +9\right )}{324} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x^2-6*x+9)/(-64*x^6+729),x)

[Out]

1/108*ln(2*x+3)-1/324*ln(4*x^2+6*x+9)+1/162*3^(1/2)*arctan(1/18*(8*x+6)*3^(1/2))-1/324*ln(2*x-3)

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maxima [A]  time = 2.94, size = 46, normalized size = 0.77 \[ \frac {1}{162} \, \sqrt {3} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x + 3\right )}\right ) - \frac {1}{324} \, \log \left (4 \, x^{2} + 6 \, x + 9\right ) + \frac {1}{108} \, \log \left (2 \, x + 3\right ) - \frac {1}{324} \, \log \left (2 \, x - 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^2-6*x+9)/(-64*x^6+729),x, algorithm="maxima")

[Out]

1/162*sqrt(3)*arctan(1/9*sqrt(3)*(4*x + 3)) - 1/324*log(4*x^2 + 6*x + 9) + 1/108*log(2*x + 3) - 1/324*log(2*x
- 3)

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mupad [B]  time = 5.01, size = 52, normalized size = 0.87 \[ \frac {\ln \left (x+\frac {3}{2}\right )}{108}-\frac {\ln \left (x-\frac {3}{2}\right )}{324}-\ln \left (x+\frac {3}{4}-\frac {\sqrt {3}\,3{}\mathrm {i}}{4}\right )\,\left (\frac {1}{324}+\frac {\sqrt {3}\,1{}\mathrm {i}}{324}\right )+\ln \left (x+\frac {3}{4}+\frac {\sqrt {3}\,3{}\mathrm {i}}{4}\right )\,\left (-\frac {1}{324}+\frac {\sqrt {3}\,1{}\mathrm {i}}{324}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(4*x^2 - 6*x + 9)/(64*x^6 - 729),x)

[Out]

log(x + 3/2)/108 - log(x - 3/2)/324 - log(x - (3^(1/2)*3i)/4 + 3/4)*((3^(1/2)*1i)/324 + 1/324) + log(x + (3^(1
/2)*3i)/4 + 3/4)*((3^(1/2)*1i)/324 - 1/324)

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sympy [A]  time = 0.23, size = 56, normalized size = 0.93 \[ - \frac {\log {\left (x - \frac {3}{2} \right )}}{324} + \frac {\log {\left (x + \frac {3}{2} \right )}}{108} - \frac {\log {\left (x^{2} + \frac {3 x}{2} + \frac {9}{4} \right )}}{324} + \frac {\sqrt {3} \operatorname {atan}{\left (\frac {4 \sqrt {3} x}{9} + \frac {\sqrt {3}}{3} \right )}}{162} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x**2-6*x+9)/(-64*x**6+729),x)

[Out]

-log(x - 3/2)/324 + log(x + 3/2)/108 - log(x**2 + 3*x/2 + 9/4)/324 + sqrt(3)*atan(4*sqrt(3)*x/9 + sqrt(3)/3)/1
62

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