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

Optimal. Leaf size=161 \[ \frac{5 x}{3 \left (3-4 x^2\right )}-\frac{2 x}{3 \left (3-4 x^2\right )^2}-\frac{7}{108 (1-2 x)}+\frac{67}{432 (1-x)}-\frac{67}{432 (x+1)}+\frac{7}{108 (2 x+1)}+\frac{1}{108 (1-2 x)^2}+\frac{1}{432 (1-x)^2}-\frac{1}{432 (x+1)^2}-\frac{1}{108 (2 x+1)^2}+\frac{3913}{648} \tanh ^{-1}(x)+\frac{67}{162} \tanh ^{-1}(2 x)-4 \sqrt{3} \tanh ^{-1}\left (\frac{2 x}{\sqrt{3}}\right )+\frac{5 \tanh ^{-1}\left (\frac{2 x}{\sqrt{3}}\right )}{6 \sqrt{3}} \]

[Out]

1/(108*(1 - 2*x)^2) - 7/(108*(1 - 2*x)) + 1/(432*(1 - x)^2) + 67/(432*(1 - x)) -
1/(432*(1 + x)^2) - 67/(432*(1 + x)) - 1/(108*(1 + 2*x)^2) + 7/(108*(1 + 2*x))
- (2*x)/(3*(3 - 4*x^2)^2) + (5*x)/(3*(3 - 4*x^2)) + (3913*ArcTanh[x])/648 + (67*
ArcTanh[2*x])/162 + (5*ArcTanh[(2*x)/Sqrt[3]])/(6*Sqrt[3]) - 4*Sqrt[3]*ArcTanh[(
2*x)/Sqrt[3]]

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Rubi [A]  time = 0.243963, antiderivative size = 161, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \frac{5 x}{3 \left (3-4 x^2\right )}-\frac{2 x}{3 \left (3-4 x^2\right )^2}-\frac{7}{108 (1-2 x)}+\frac{67}{432 (1-x)}-\frac{67}{432 (x+1)}+\frac{7}{108 (2 x+1)}+\frac{1}{108 (1-2 x)^2}+\frac{1}{432 (1-x)^2}-\frac{1}{432 (x+1)^2}-\frac{1}{108 (2 x+1)^2}+\frac{3913}{648} \tanh ^{-1}(x)+\frac{67}{162} \tanh ^{-1}(2 x)-4 \sqrt{3} \tanh ^{-1}\left (\frac{2 x}{\sqrt{3}}\right )+\frac{5 \tanh ^{-1}\left (\frac{2 x}{\sqrt{3}}\right )}{6 \sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

1/(108*(1 - 2*x)^2) - 7/(108*(1 - 2*x)) + 1/(432*(1 - x)^2) + 67/(432*(1 - x)) -
1/(432*(1 + x)^2) - 67/(432*(1 + x)) - 1/(108*(1 + 2*x)^2) + 7/(108*(1 + 2*x))
- (2*x)/(3*(3 - 4*x^2)^2) + (5*x)/(3*(3 - 4*x^2)) + (3913*ArcTanh[x])/648 + (67*
ArcTanh[2*x])/162 + (5*ArcTanh[(2*x)/Sqrt[3]])/(6*Sqrt[3]) - 4*Sqrt[3]*ArcTanh[(
2*x)/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)**3,x)

[Out]

Timed out

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Mathematica [A]  time = 0.155048, size = 137, normalized size = 0.85 \[ \frac{\frac{36 x \left (80 x^4-104 x^2+27\right )}{\left (-16 x^6+32 x^4-19 x^2+3\right )^2}-\frac{6 x \left (2288 x^4-2384 x^2+345\right )}{16 x^6-32 x^4+19 x^2-3}-268 \log (1-2 x)+2412 \sqrt{3} \log \left (\sqrt{3}-2 x\right )-3913 \log (1-x)+3913 \log (x+1)+268 \log (2 x+1)-2412 \sqrt{3} \log \left (2 x+\sqrt{3}\right )}{1296} \]

Antiderivative was successfully verified.

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

[Out]

((36*x*(27 - 104*x^2 + 80*x^4))/(3 - 19*x^2 + 32*x^4 - 16*x^6)^2 - (6*x*(345 - 2
384*x^2 + 2288*x^4))/(-3 + 19*x^2 - 32*x^4 + 16*x^6) - 268*Log[1 - 2*x] + 2412*S
qrt[3]*Log[Sqrt[3] - 2*x] - 3913*Log[1 - x] + 3913*Log[1 + x] + 268*Log[1 + 2*x]
- 2412*Sqrt[3]*Log[Sqrt[3] + 2*x])/1296

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Maple [A]  time = 0.035, size = 126, normalized size = 0.8 \[{\frac{1}{432\, \left ( -1+x \right ) ^{2}}}-{\frac{67}{-432+432\,x}}-{\frac{3913\,\ln \left ( -1+x \right ) }{1296}}+64\,{\frac{1}{ \left ( 4\,{x}^{2}-3 \right ) ^{2}} \left ( -{\frac{5\,{x}^{3}}{48}}+{\frac{13\,x}{192}} \right ) }-{\frac{67\,\sqrt{3}}{18}{\it Artanh} \left ({\frac{2\,x\sqrt{3}}{3}} \right ) }-{\frac{1}{432\, \left ( 1+x \right ) ^{2}}}-{\frac{67}{432+432\,x}}+{\frac{3913\,\ln \left ( 1+x \right ) }{1296}}+{\frac{1}{108\, \left ( 2\,x-1 \right ) ^{2}}}+{\frac{7}{216\,x-108}}-{\frac{67\,\ln \left ( 2\,x-1 \right ) }{324}}-{\frac{1}{108\, \left ( 1+2\,x \right ) ^{2}}}+{\frac{7}{108+216\,x}}+{\frac{67\,\ln \left ( 1+2\,x \right ) }{324}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/432/(-1+x)^2-67/432/(-1+x)-3913/1296*ln(-1+x)+64*(-5/48*x^3+13/192*x)/(4*x^2-3
)^2-67/18*arctanh(2/3*x*3^(1/2))*3^(1/2)-1/432/(1+x)^2-67/432/(1+x)+3913/1296*ln
(1+x)+1/108/(2*x-1)^2+7/108/(2*x-1)-67/324*ln(2*x-1)-1/108/(1+2*x)^2+7/108/(1+2*
x)+67/324*ln(1+2*x)

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Maxima [A]  time = 0.917111, size = 161, normalized size = 1. \[ \frac{67}{36} \, \sqrt{3} \log \left (\frac{2 \, x - \sqrt{3}}{2 \, x + \sqrt{3}}\right ) - \frac{36608 \, x^{11} - 111360 \, x^{9} + 125280 \, x^{7} - 63680 \, x^{5} + 14331 \, x^{3} - 1197 \, x}{216 \,{\left (256 \, x^{12} - 1024 \, x^{10} + 1632 \, x^{8} - 1312 \, x^{6} + 553 \, x^{4} - 114 \, x^{2} + 9\right )}} + \frac{67}{324} \, \log \left (2 \, x + 1\right ) - \frac{67}{324} \, \log \left (2 \, x - 1\right ) + \frac{3913}{1296} \, \log \left (x + 1\right ) - \frac{3913}{1296} \, \log \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

67/36*sqrt(3)*log((2*x - sqrt(3))/(2*x + sqrt(3))) - 1/216*(36608*x^11 - 111360*
x^9 + 125280*x^7 - 63680*x^5 + 14331*x^3 - 1197*x)/(256*x^12 - 1024*x^10 + 1632*
x^8 - 1312*x^6 + 553*x^4 - 114*x^2 + 9) + 67/324*log(2*x + 1) - 67/324*log(2*x -
1) + 3913/1296*log(x + 1) - 3913/1296*log(x - 1)

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Fricas [A]  time = 0.274147, size = 408, normalized size = 2.53 \[ \frac{\sqrt{3}{\left (268 \, \sqrt{3}{\left (256 \, x^{12} - 1024 \, x^{10} + 1632 \, x^{8} - 1312 \, x^{6} + 553 \, x^{4} - 114 \, x^{2} + 9\right )} \log \left (2 \, x + 1\right ) - 268 \, \sqrt{3}{\left (256 \, x^{12} - 1024 \, x^{10} + 1632 \, x^{8} - 1312 \, x^{6} + 553 \, x^{4} - 114 \, x^{2} + 9\right )} \log \left (2 \, x - 1\right ) + 3913 \, \sqrt{3}{\left (256 \, x^{12} - 1024 \, x^{10} + 1632 \, x^{8} - 1312 \, x^{6} + 553 \, x^{4} - 114 \, x^{2} + 9\right )} \log \left (x + 1\right ) - 3913 \, \sqrt{3}{\left (256 \, x^{12} - 1024 \, x^{10} + 1632 \, x^{8} - 1312 \, x^{6} + 553 \, x^{4} - 114 \, x^{2} + 9\right )} \log \left (x - 1\right ) + 7236 \,{\left (256 \, x^{12} - 1024 \, x^{10} + 1632 \, x^{8} - 1312 \, x^{6} + 553 \, x^{4} - 114 \, x^{2} + 9\right )} \log \left (\frac{\sqrt{3}{\left (4 \, x^{2} + 3\right )} - 12 \, x}{4 \, x^{2} - 3}\right ) - 6 \, \sqrt{3}{\left (36608 \, x^{11} - 111360 \, x^{9} + 125280 \, x^{7} - 63680 \, x^{5} + 14331 \, x^{3} - 1197 \, x\right )}\right )}}{3888 \,{\left (256 \, x^{12} - 1024 \, x^{10} + 1632 \, x^{8} - 1312 \, x^{6} + 553 \, x^{4} - 114 \, x^{2} + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3888*sqrt(3)*(268*sqrt(3)*(256*x^12 - 1024*x^10 + 1632*x^8 - 1312*x^6 + 553*x^
4 - 114*x^2 + 9)*log(2*x + 1) - 268*sqrt(3)*(256*x^12 - 1024*x^10 + 1632*x^8 - 1
312*x^6 + 553*x^4 - 114*x^2 + 9)*log(2*x - 1) + 3913*sqrt(3)*(256*x^12 - 1024*x^
10 + 1632*x^8 - 1312*x^6 + 553*x^4 - 114*x^2 + 9)*log(x + 1) - 3913*sqrt(3)*(256
*x^12 - 1024*x^10 + 1632*x^8 - 1312*x^6 + 553*x^4 - 114*x^2 + 9)*log(x - 1) + 72
36*(256*x^12 - 1024*x^10 + 1632*x^8 - 1312*x^6 + 553*x^4 - 114*x^2 + 9)*log((sqr
t(3)*(4*x^2 + 3) - 12*x)/(4*x^2 - 3)) - 6*sqrt(3)*(36608*x^11 - 111360*x^9 + 125
280*x^7 - 63680*x^5 + 14331*x^3 - 1197*x))/(256*x^12 - 1024*x^10 + 1632*x^8 - 13
12*x^6 + 553*x^4 - 114*x^2 + 9)

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Sympy [A]  time = 4.31092, size = 134, normalized size = 0.83 \[ - \frac{36608 x^{11} - 111360 x^{9} + 125280 x^{7} - 63680 x^{5} + 14331 x^{3} - 1197 x}{55296 x^{12} - 221184 x^{10} + 352512 x^{8} - 283392 x^{6} + 119448 x^{4} - 24624 x^{2} + 1944} - \frac{3913 \log{\left (x - 1 \right )}}{1296} - \frac{67 \log{\left (x - \frac{1}{2} \right )}}{324} + \frac{67 \log{\left (x + \frac{1}{2} \right )}}{324} + \frac{3913 \log{\left (x + 1 \right )}}{1296} + \frac{67 \sqrt{3} \log{\left (x - \frac{\sqrt{3}}{2} \right )}}{36} - \frac{67 \sqrt{3} \log{\left (x + \frac{\sqrt{3}}{2} \right )}}{36} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(36608*x**11 - 111360*x**9 + 125280*x**7 - 63680*x**5 + 14331*x**3 - 1197*x)/(5
5296*x**12 - 221184*x**10 + 352512*x**8 - 283392*x**6 + 119448*x**4 - 24624*x**2
+ 1944) - 3913*log(x - 1)/1296 - 67*log(x - 1/2)/324 + 67*log(x + 1/2)/324 + 39
13*log(x + 1)/1296 + 67*sqrt(3)*log(x - sqrt(3)/2)/36 - 67*sqrt(3)*log(x + sqrt(
3)/2)/36

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GIAC/XCAS [A]  time = 0.26227, size = 151, normalized size = 0.94 \[ \frac{67}{36} \, \sqrt{3}{\rm ln}\left (\frac{{\left | 8 \, x - 4 \, \sqrt{3} \right |}}{{\left | 8 \, x + 4 \, \sqrt{3} \right |}}\right ) - \frac{36608 \, x^{11} - 111360 \, x^{9} + 125280 \, x^{7} - 63680 \, x^{5} + 14331 \, x^{3} - 1197 \, x}{216 \,{\left (16 \, x^{6} - 32 \, x^{4} + 19 \, x^{2} - 3\right )}^{2}} + \frac{67}{324} \,{\rm ln}\left ({\left | 2 \, x + 1 \right |}\right ) - \frac{67}{324} \,{\rm ln}\left ({\left | 2 \, x - 1 \right |}\right ) + \frac{3913}{1296} \,{\rm ln}\left ({\left | x + 1 \right |}\right ) - \frac{3913}{1296} \,{\rm ln}\left ({\left | x - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

67/36*sqrt(3)*ln(abs(8*x - 4*sqrt(3))/abs(8*x + 4*sqrt(3))) - 1/216*(36608*x^11
- 111360*x^9 + 125280*x^7 - 63680*x^5 + 14331*x^3 - 1197*x)/(16*x^6 - 32*x^4 + 1
9*x^2 - 3)^2 + 67/324*ln(abs(2*x + 1)) - 67/324*ln(abs(2*x - 1)) + 3913/1296*ln(
abs(x + 1)) - 3913/1296*ln(abs(x - 1))