3.304 \(\int \frac{1}{1-2 x^4+x^8} \, dx\)

Optimal. Leaf size=27 \[ \frac{x}{4 \left (1-x^4\right )}+\frac{3}{8} \tan ^{-1}(x)+\frac{3}{8} \tanh ^{-1}(x) \]

[Out]

x/(4*(1 - x^4)) + (3*ArcTan[x])/8 + (3*ArcTanh[x])/8

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Rubi [A]  time = 0.0139036, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417 \[ \frac{x}{4 \left (1-x^4\right )}+\frac{3}{8} \tan ^{-1}(x)+\frac{3}{8} \tanh ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[(1 - 2*x^4 + x^8)^(-1),x]

[Out]

x/(4*(1 - x^4)) + (3*ArcTan[x])/8 + (3*ArcTanh[x])/8

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Rubi in Sympy [A]  time = 2.00162, size = 20, normalized size = 0.74 \[ \frac{x}{4 \left (- x^{4} + 1\right )} + \frac{3 \operatorname{atan}{\left (x \right )}}{8} + \frac{3 \operatorname{atanh}{\left (x \right )}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(x**8-2*x**4+1),x)

[Out]

x/(4*(-x**4 + 1)) + 3*atan(x)/8 + 3*atanh(x)/8

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Mathematica [A]  time = 0.0149598, size = 33, normalized size = 1.22 \[ \frac{1}{16} \left (-\frac{4 x}{x^4-1}-3 \log (1-x)+3 \log (x+1)+6 \tan ^{-1}(x)\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(1 - 2*x^4 + x^8)^(-1),x]

[Out]

((-4*x)/(-1 + x^4) + 6*ArcTan[x] - 3*Log[1 - x] + 3*Log[1 + x])/16

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Maple [A]  time = 0.019, size = 42, normalized size = 1.6 \[ -{\frac{1}{-16+16\,x}}-{\frac{3\,\ln \left ( -1+x \right ) }{16}}-{\frac{1}{16+16\,x}}+{\frac{3\,\ln \left ( 1+x \right ) }{16}}+{\frac{x}{8\,{x}^{2}+8}}+{\frac{3\,\arctan \left ( x \right ) }{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(x^8-2*x^4+1),x)

[Out]

-1/16/(-1+x)-3/16*ln(-1+x)-1/16/(1+x)+3/16*ln(1+x)+1/8*x/(x^2+1)+3/8*arctan(x)

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Maxima [A]  time = 0.85635, size = 36, normalized size = 1.33 \[ -\frac{x}{4 \,{\left (x^{4} - 1\right )}} + \frac{3}{8} \, \arctan \left (x\right ) + \frac{3}{16} \, \log \left (x + 1\right ) - \frac{3}{16} \, \log \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x^8 - 2*x^4 + 1),x, algorithm="maxima")

[Out]

-1/4*x/(x^4 - 1) + 3/8*arctan(x) + 3/16*log(x + 1) - 3/16*log(x - 1)

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Fricas [A]  time = 0.258219, size = 59, normalized size = 2.19 \[ \frac{6 \,{\left (x^{4} - 1\right )} \arctan \left (x\right ) + 3 \,{\left (x^{4} - 1\right )} \log \left (x + 1\right ) - 3 \,{\left (x^{4} - 1\right )} \log \left (x - 1\right ) - 4 \, x}{16 \,{\left (x^{4} - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x^8 - 2*x^4 + 1),x, algorithm="fricas")

[Out]

1/16*(6*(x^4 - 1)*arctan(x) + 3*(x^4 - 1)*log(x + 1) - 3*(x^4 - 1)*log(x - 1) -
4*x)/(x^4 - 1)

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Sympy [A]  time = 0.456182, size = 31, normalized size = 1.15 \[ - \frac{x}{4 x^{4} - 4} - \frac{3 \log{\left (x - 1 \right )}}{16} + \frac{3 \log{\left (x + 1 \right )}}{16} + \frac{3 \operatorname{atan}{\left (x \right )}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x**8-2*x**4+1),x)

[Out]

-x/(4*x**4 - 4) - 3*log(x - 1)/16 + 3*log(x + 1)/16 + 3*atan(x)/8

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GIAC/XCAS [A]  time = 0.28432, size = 39, normalized size = 1.44 \[ -\frac{x}{4 \,{\left (x^{4} - 1\right )}} + \frac{3}{8} \, \arctan \left (x\right ) + \frac{3}{16} \,{\rm ln}\left ({\left | x + 1 \right |}\right ) - \frac{3}{16} \,{\rm ln}\left ({\left | x - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x^8 - 2*x^4 + 1),x, algorithm="giac")

[Out]

-1/4*x/(x^4 - 1) + 3/8*arctan(x) + 3/16*ln(abs(x + 1)) - 3/16*ln(abs(x - 1))