3.75 \(\int \frac{1}{(3-19 x^2+32 x^4-16 x^6)^2} \, dx\)

Optimal. Leaf size=89 \[ \frac{2 x}{3 \left (3-4 x^2\right )}+\frac{1}{18 (1-2 x)}+\frac{1}{36 (1-x)}-\frac{1}{36 (x+1)}-\frac{1}{18 (2 x+1)}+\frac{67}{54} \tanh ^{-1}(x)-\frac{7}{27} \tanh ^{-1}(2 x)-\frac{5 \tanh ^{-1}\left (\frac{2 x}{\sqrt{3}}\right )}{3 \sqrt{3}} \]

[Out]

1/(18*(1 - 2*x)) + 1/(36*(1 - x)) - 1/(36*(1 + x)) - 1/(18*(1 + 2*x)) + (2*x)/(3*(3 - 4*x^2)) + (67*ArcTanh[x]
)/54 - (7*ArcTanh[2*x])/27 - (5*ArcTanh[(2*x)/Sqrt[3]])/(3*Sqrt[3])

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Rubi [A]  time = 0.0614828, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {2057, 207, 199} \[ \frac{2 x}{3 \left (3-4 x^2\right )}+\frac{1}{18 (1-2 x)}+\frac{1}{36 (1-x)}-\frac{1}{36 (x+1)}-\frac{1}{18 (2 x+1)}+\frac{67}{54} \tanh ^{-1}(x)-\frac{7}{27} \tanh ^{-1}(2 x)-\frac{5 \tanh ^{-1}\left (\frac{2 x}{\sqrt{3}}\right )}{3 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Int[(3 - 19*x^2 + 32*x^4 - 16*x^6)^(-2),x]

[Out]

1/(18*(1 - 2*x)) + 1/(36*(1 - x)) - 1/(36*(1 + x)) - 1/(18*(1 + 2*x)) + (2*x)/(3*(3 - 4*x^2)) + (67*ArcTanh[x]
)/54 - (7*ArcTanh[2*x])/27 - (5*ArcTanh[(2*x)/Sqrt[3]])/(3*Sqrt[3])

Rule 2057

Int[(P_)^(p_), x_Symbol] :> With[{u = Factor[P /. x -> Sqrt[x]]}, Int[ExpandIntegrand[(u /. x -> x^2)^p, x], x
] /;  !SumQ[NonfreeFactors[u, x]]] /; PolyQ[P, x^2] && ILtQ[p, 0]

Rule 207

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

Rule 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

Rubi steps

\begin{align*} \int \frac{1}{\left (3-19 x^2+32 x^4-16 x^6\right )^2} \, dx &=\int \left (\frac{1}{36 (-1+x)^2}+\frac{1}{36 (1+x)^2}+\frac{1}{9 (-1+2 x)^2}+\frac{1}{9 (1+2 x)^2}-\frac{67}{54 \left (-1+x^2\right )}+\frac{4}{\left (-3+4 x^2\right )^2}+\frac{4}{-3+4 x^2}+\frac{14}{27 \left (-1+4 x^2\right )}\right ) \, dx\\ &=\frac{1}{18 (1-2 x)}+\frac{1}{36 (1-x)}-\frac{1}{36 (1+x)}-\frac{1}{18 (1+2 x)}+\frac{14}{27} \int \frac{1}{-1+4 x^2} \, dx-\frac{67}{54} \int \frac{1}{-1+x^2} \, dx+4 \int \frac{1}{\left (-3+4 x^2\right )^2} \, dx+4 \int \frac{1}{-3+4 x^2} \, dx\\ &=\frac{1}{18 (1-2 x)}+\frac{1}{36 (1-x)}-\frac{1}{36 (1+x)}-\frac{1}{18 (1+2 x)}+\frac{2 x}{3 \left (3-4 x^2\right )}+\frac{67}{54} \tanh ^{-1}(x)-\frac{7}{27} \tanh ^{-1}(2 x)-\frac{2 \tanh ^{-1}\left (\frac{2 x}{\sqrt{3}}\right )}{\sqrt{3}}-\frac{2}{3} \int \frac{1}{-3+4 x^2} \, dx\\ &=\frac{1}{18 (1-2 x)}+\frac{1}{36 (1-x)}-\frac{1}{36 (1+x)}-\frac{1}{18 (1+2 x)}+\frac{2 x}{3 \left (3-4 x^2\right )}+\frac{67}{54} \tanh ^{-1}(x)-\frac{7}{27} \tanh ^{-1}(2 x)-\frac{5 \tanh ^{-1}\left (\frac{2 x}{\sqrt{3}}\right )}{3 \sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.0510452, size = 103, normalized size = 1.16 \[ \frac{1}{108} \left (-\frac{6 x \left (80 x^4-104 x^2+27\right )}{16 x^6-32 x^4+19 x^2-3}+14 \log (1-2 x)+30 \sqrt{3} \log \left (\sqrt{3}-2 x\right )-67 \log (1-x)+67 \log (x+1)-14 \log (2 x+1)-30 \sqrt{3} \log \left (2 x+\sqrt{3}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(3 - 19*x^2 + 32*x^4 - 16*x^6)^(-2),x]

[Out]

((-6*x*(27 - 104*x^2 + 80*x^4))/(-3 + 19*x^2 - 32*x^4 + 16*x^6) + 14*Log[1 - 2*x] + 30*Sqrt[3]*Log[Sqrt[3] - 2
*x] - 67*Log[1 - x] + 67*Log[1 + x] - 14*Log[1 + 2*x] - 30*Sqrt[3]*Log[Sqrt[3] + 2*x])/108

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Maple [A]  time = 0.02, size = 84, normalized size = 0.9 \begin{align*} -{\frac{1}{36\,x-36}}-{\frac{67\,\ln \left ( x-1 \right ) }{108}}-{\frac{x}{6} \left ({x}^{2}-{\frac{3}{4}} \right ) ^{-1}}-{\frac{5\,\sqrt{3}}{9}{\it Artanh} \left ({\frac{2\,x\sqrt{3}}{3}} \right ) }-{\frac{1}{36\,x-18}}+{\frac{7\,\ln \left ( 2\,x-1 \right ) }{54}}-{\frac{1}{18+36\,x}}-{\frac{7\,\ln \left ( 1+2\,x \right ) }{54}}-{\frac{1}{36+36\,x}}+{\frac{67\,\ln \left ( 1+x \right ) }{108}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-16*x^6+32*x^4-19*x^2+3)^2,x)

[Out]

-1/36/(x-1)-67/108*ln(x-1)-1/6*x/(x^2-3/4)-5/9*arctanh(2/3*x*3^(1/2))*3^(1/2)-1/18/(2*x-1)+7/54*ln(2*x-1)-1/18
/(1+2*x)-7/54*ln(1+2*x)-1/36/(1+x)+67/108*ln(1+x)

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Maxima [A]  time = 1.74873, size = 120, normalized size = 1.35 \begin{align*} \frac{5}{18} \, \sqrt{3} \log \left (\frac{2 \, x - \sqrt{3}}{2 \, x + \sqrt{3}}\right ) - \frac{80 \, x^{5} - 104 \, x^{3} + 27 \, x}{18 \,{\left (16 \, x^{6} - 32 \, x^{4} + 19 \, x^{2} - 3\right )}} - \frac{7}{54} \, \log \left (2 \, x + 1\right ) + \frac{7}{54} \, \log \left (2 \, x - 1\right ) + \frac{67}{108} \, \log \left (x + 1\right ) - \frac{67}{108} \, \log \left (x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-16*x^6+32*x^4-19*x^2+3)^2,x, algorithm="maxima")

[Out]

5/18*sqrt(3)*log((2*x - sqrt(3))/(2*x + sqrt(3))) - 1/18*(80*x^5 - 104*x^3 + 27*x)/(16*x^6 - 32*x^4 + 19*x^2 -
3) - 7/54*log(2*x + 1) + 7/54*log(2*x - 1) + 67/108*log(x + 1) - 67/108*log(x - 1)

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Fricas [B]  time = 1.7795, size = 467, normalized size = 5.25 \begin{align*} -\frac{480 \, x^{5} - 624 \, x^{3} - 30 \, \sqrt{3}{\left (16 \, x^{6} - 32 \, x^{4} + 19 \, x^{2} - 3\right )} \log \left (\frac{4 \, x^{2} - 4 \, \sqrt{3} x + 3}{4 \, x^{2} - 3}\right ) + 14 \,{\left (16 \, x^{6} - 32 \, x^{4} + 19 \, x^{2} - 3\right )} \log \left (2 \, x + 1\right ) - 14 \,{\left (16 \, x^{6} - 32 \, x^{4} + 19 \, x^{2} - 3\right )} \log \left (2 \, x - 1\right ) - 67 \,{\left (16 \, x^{6} - 32 \, x^{4} + 19 \, x^{2} - 3\right )} \log \left (x + 1\right ) + 67 \,{\left (16 \, x^{6} - 32 \, x^{4} + 19 \, x^{2} - 3\right )} \log \left (x - 1\right ) + 162 \, x}{108 \,{\left (16 \, x^{6} - 32 \, x^{4} + 19 \, x^{2} - 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-16*x^6+32*x^4-19*x^2+3)^2,x, algorithm="fricas")

[Out]

-1/108*(480*x^5 - 624*x^3 - 30*sqrt(3)*(16*x^6 - 32*x^4 + 19*x^2 - 3)*log((4*x^2 - 4*sqrt(3)*x + 3)/(4*x^2 - 3
)) + 14*(16*x^6 - 32*x^4 + 19*x^2 - 3)*log(2*x + 1) - 14*(16*x^6 - 32*x^4 + 19*x^2 - 3)*log(2*x - 1) - 67*(16*
x^6 - 32*x^4 + 19*x^2 - 3)*log(x + 1) + 67*(16*x^6 - 32*x^4 + 19*x^2 - 3)*log(x - 1) + 162*x)/(16*x^6 - 32*x^4
+ 19*x^2 - 3)

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Sympy [A]  time = 1.10252, size = 104, normalized size = 1.17 \begin{align*} - \frac{80 x^{5} - 104 x^{3} + 27 x}{288 x^{6} - 576 x^{4} + 342 x^{2} - 54} - \frac{67 \log{\left (x - 1 \right )}}{108} + \frac{7 \log{\left (x - \frac{1}{2} \right )}}{54} - \frac{7 \log{\left (x + \frac{1}{2} \right )}}{54} + \frac{67 \log{\left (x + 1 \right )}}{108} + \frac{5 \sqrt{3} \log{\left (x - \frac{\sqrt{3}}{2} \right )}}{18} - \frac{5 \sqrt{3} \log{\left (x + \frac{\sqrt{3}}{2} \right )}}{18} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-16*x**6+32*x**4-19*x**2+3)**2,x)

[Out]

-(80*x**5 - 104*x**3 + 27*x)/(288*x**6 - 576*x**4 + 342*x**2 - 54) - 67*log(x - 1)/108 + 7*log(x - 1/2)/54 - 7
*log(x + 1/2)/54 + 67*log(x + 1)/108 + 5*sqrt(3)*log(x - sqrt(3)/2)/18 - 5*sqrt(3)*log(x + sqrt(3)/2)/18

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Giac [A]  time = 1.1205, size = 131, normalized size = 1.47 \begin{align*} \frac{5}{18} \, \sqrt{3} \log \left (\frac{{\left | 8 \, x - 4 \, \sqrt{3} \right |}}{{\left | 8 \, x + 4 \, \sqrt{3} \right |}}\right ) - \frac{80 \, x^{5} - 104 \, x^{3} + 27 \, x}{18 \,{\left (16 \, x^{6} - 32 \, x^{4} + 19 \, x^{2} - 3\right )}} - \frac{7}{54} \, \log \left ({\left | 2 \, x + 1 \right |}\right ) + \frac{7}{54} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) + \frac{67}{108} \, \log \left ({\left | x + 1 \right |}\right ) - \frac{67}{108} \, \log \left ({\left | x - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-16*x^6+32*x^4-19*x^2+3)^2,x, algorithm="giac")

[Out]

5/18*sqrt(3)*log(abs(8*x - 4*sqrt(3))/abs(8*x + 4*sqrt(3))) - 1/18*(80*x^5 - 104*x^3 + 27*x)/(16*x^6 - 32*x^4
+ 19*x^2 - 3) - 7/54*log(abs(2*x + 1)) + 7/54*log(abs(2*x - 1)) + 67/108*log(abs(x + 1)) - 67/108*log(abs(x -
1))