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

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 = 0.122869, 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 \[ \frac{2 x}{3 \left (3-4 x^2\right )}+\frac{1}{18 (1-2 x)}+\frac{1}{36 (1-x)}-\frac{1}{36 (x+1)}-\frac{1}{18 (2 x+1)}+\frac{67}{54} \tanh ^{-1}(x)-\frac{7}{27} \tanh ^{-1}(2 x)-\frac{5 \tanh ^{-1}\left (\frac{2 x}{\sqrt{3}}\right )}{3 \sqrt{3}} \]

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])

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(-16*x**6+32*x**4-19*x**2+3)**2,x)

[Out]

Timed out

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Mathematica [A]  time = 0.0878126, 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 = 0.031, size = 84, normalized size = 0.9 \[ -{\frac{1}{-36+36\,x}}-{\frac{67\,\ln \left ( -1+x \right ) }{108}}-{\frac{x}{6} \left ({x}^{2}-{\frac{3}{4}} \right ) ^{-1}}-{\frac{5\,\sqrt{3}}{9}{\it Artanh} \left ({\frac{2\,x\sqrt{3}}{3}} \right ) }-{\frac{1}{36+36\,x}}+{\frac{67\,\ln \left ( 1+x \right ) }{108}}-{\frac{1}{36\,x-18}}+{\frac{7\,\ln \left ( 2\,x-1 \right ) }{54}}-{\frac{1}{18+36\,x}}-{\frac{7\,\ln \left ( 1+2\,x \right ) }{54}} \]

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

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Maxima [A]  time = 0.894588, size = 120, normalized size = 1.35 \[ \frac{5}{18} \, \sqrt{3} \log \left (\frac{2 \, x - \sqrt{3}}{2 \, x + \sqrt{3}}\right ) - \frac{80 \, x^{5} - 104 \, x^{3} + 27 \, x}{18 \,{\left (16 \, x^{6} - 32 \, x^{4} + 19 \, x^{2} - 3\right )}} - \frac{7}{54} \, \log \left (2 \, x + 1\right ) + \frac{7}{54} \, \log \left (2 \, x - 1\right ) + \frac{67}{108} \, \log \left (x + 1\right ) - \frac{67}{108} \, \log \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((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) + 67/1
08*log(x + 1) - 67/108*log(x - 1)

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Fricas [A]  time = 0.259509, size = 266, normalized size = 2.99 \[ -\frac{\sqrt{3}{\left (14 \, \sqrt{3}{\left (16 \, x^{6} - 32 \, x^{4} + 19 \, x^{2} - 3\right )} \log \left (2 \, x + 1\right ) - 14 \, \sqrt{3}{\left (16 \, x^{6} - 32 \, x^{4} + 19 \, x^{2} - 3\right )} \log \left (2 \, x - 1\right ) - 67 \, \sqrt{3}{\left (16 \, x^{6} - 32 \, x^{4} + 19 \, x^{2} - 3\right )} \log \left (x + 1\right ) + 67 \, \sqrt{3}{\left (16 \, x^{6} - 32 \, x^{4} + 19 \, x^{2} - 3\right )} \log \left (x - 1\right ) - 90 \,{\left (16 \, x^{6} - 32 \, x^{4} + 19 \, x^{2} - 3\right )} \log \left (\frac{\sqrt{3}{\left (4 \, x^{2} + 3\right )} - 12 \, x}{4 \, x^{2} - 3}\right ) + 6 \, \sqrt{3}{\left (80 \, x^{5} - 104 \, x^{3} + 27 \, x\right )}\right )}}{324 \,{\left (16 \, x^{6} - 32 \, x^{4} + 19 \, x^{2} - 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/324*sqrt(3)*(14*sqrt(3)*(16*x^6 - 32*x^4 + 19*x^2 - 3)*log(2*x + 1) - 14*sqrt
(3)*(16*x^6 - 32*x^4 + 19*x^2 - 3)*log(2*x - 1) - 67*sqrt(3)*(16*x^6 - 32*x^4 +
19*x^2 - 3)*log(x + 1) + 67*sqrt(3)*(16*x^6 - 32*x^4 + 19*x^2 - 3)*log(x - 1) -
90*(16*x^6 - 32*x^4 + 19*x^2 - 3)*log((sqrt(3)*(4*x^2 + 3) - 12*x)/(4*x^2 - 3))
+ 6*sqrt(3)*(80*x^5 - 104*x^3 + 27*x))/(16*x^6 - 32*x^4 + 19*x^2 - 3)

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Sympy [A]  time = 3.85285, size = 104, normalized size = 1.17 \[ - \frac{80 x^{5} - 104 x^{3} + 27 x}{288 x^{6} - 576 x^{4} + 342 x^{2} - 54} - \frac{67 \log{\left (x - 1 \right )}}{108} + \frac{7 \log{\left (x - \frac{1}{2} \right )}}{54} - \frac{7 \log{\left (x + \frac{1}{2} \right )}}{54} + \frac{67 \log{\left (x + 1 \right )}}{108} + \frac{5 \sqrt{3} \log{\left (x - \frac{\sqrt{3}}{2} \right )}}{18} - \frac{5 \sqrt{3} \log{\left (x + \frac{\sqrt{3}}{2} \right )}}{18} \]

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)*l
og(x - sqrt(3)/2)/18 - 5*sqrt(3)*log(x + sqrt(3)/2)/18

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GIAC/XCAS [A]  time = 0.263432, size = 131, normalized size = 1.47 \[ \frac{5}{18} \, \sqrt{3}{\rm ln}\left (\frac{{\left | 8 \, x - 4 \, \sqrt{3} \right |}}{{\left | 8 \, x + 4 \, \sqrt{3} \right |}}\right ) - \frac{80 \, x^{5} - 104 \, x^{3} + 27 \, x}{18 \,{\left (16 \, x^{6} - 32 \, x^{4} + 19 \, x^{2} - 3\right )}} - \frac{7}{54} \,{\rm ln}\left ({\left | 2 \, x + 1 \right |}\right ) + \frac{7}{54} \,{\rm ln}\left ({\left | 2 \, x - 1 \right |}\right ) + \frac{67}{108} \,{\rm ln}\left ({\left | x + 1 \right |}\right ) - \frac{67}{108} \,{\rm ln}\left ({\left | x - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5/18*sqrt(3)*ln(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*ln(abs(2*x + 1)) + 7/54*ln(abs
(2*x - 1)) + 67/108*ln(abs(x + 1)) - 67/108*ln(abs(x - 1))