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

Optimal. Leaf size=34 \[ \frac{x^5}{4 \left (1-x^4\right )}+\frac{5 x}{4}-\frac{5}{8} \tan ^{-1}(x)-\frac{5}{8} \tanh ^{-1}(x) \]

[Out]

(5*x)/4 + x^5/(4*(1 - x^4)) - (5*ArcTan[x])/8 - (5*ArcTanh[x])/8

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Rubi [A]  time = 0.0264783, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375 \[ \frac{x^5}{4 \left (1-x^4\right )}+\frac{5 x}{4}-\frac{5}{8} \tan ^{-1}(x)-\frac{5}{8} \tanh ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[x^8/(1 - 2*x^4 + x^8),x]

[Out]

(5*x)/4 + x^5/(4*(1 - x^4)) - (5*ArcTan[x])/8 - (5*ArcTanh[x])/8

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Rubi in Sympy [A]  time = 6.27079, size = 27, normalized size = 0.79 \[ \frac{x^{5}}{4 \left (- x^{4} + 1\right )} + \frac{5 x}{4} - \frac{5 \operatorname{atan}{\left (x \right )}}{8} - \frac{5 \operatorname{atanh}{\left (x \right )}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**5/(4*(-x**4 + 1)) + 5*x/4 - 5*atan(x)/8 - 5*atanh(x)/8

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Mathematica [A]  time = 0.0253772, size = 38, normalized size = 1.12 \[ -\frac{x}{4 \left (x^4-1\right )}+x+\frac{5}{16} \log (1-x)-\frac{5}{16} \log (x+1)-\frac{5}{8} \tan ^{-1}(x) \]

Antiderivative was successfully verified.

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

[Out]

x - x/(4*(-1 + x^4)) - (5*ArcTan[x])/8 + (5*Log[1 - x])/16 - (5*Log[1 + x])/16

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Maple [A]  time = 0.02, size = 43, normalized size = 1.3 \[ x-{\frac{1}{-16+16\,x}}+{\frac{5\,\ln \left ( -1+x \right ) }{16}}-{\frac{1}{16+16\,x}}-{\frac{5\,\ln \left ( 1+x \right ) }{16}}+{\frac{x}{8\,{x}^{2}+8}}-{\frac{5\,\arctan \left ( x \right ) }{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^8/(x^8-2*x^4+1),x)

[Out]

x-1/16/(-1+x)+5/16*ln(-1+x)-1/16/(1+x)-5/16*ln(1+x)+1/8*x/(x^2+1)-5/8*arctan(x)

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Maxima [A]  time = 0.845837, size = 38, normalized size = 1.12 \[ x - \frac{x}{4 \,{\left (x^{4} - 1\right )}} - \frac{5}{8} \, \arctan \left (x\right ) - \frac{5}{16} \, \log \left (x + 1\right ) + \frac{5}{16} \, \log \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x - 1/4*x/(x^4 - 1) - 5/8*arctan(x) - 5/16*log(x + 1) + 5/16*log(x - 1)

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Fricas [A]  time = 0.264062, size = 66, normalized size = 1.94 \[ \frac{16 \, x^{5} - 10 \,{\left (x^{4} - 1\right )} \arctan \left (x\right ) - 5 \,{\left (x^{4} - 1\right )} \log \left (x + 1\right ) + 5 \,{\left (x^{4} - 1\right )} \log \left (x - 1\right ) - 20 \, x}{16 \,{\left (x^{4} - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/16*(16*x^5 - 10*(x^4 - 1)*arctan(x) - 5*(x^4 - 1)*log(x + 1) + 5*(x^4 - 1)*log
(x - 1) - 20*x)/(x^4 - 1)

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Sympy [A]  time = 0.45463, size = 32, normalized size = 0.94 \[ x - \frac{x}{4 x^{4} - 4} + \frac{5 \log{\left (x - 1 \right )}}{16} - \frac{5 \log{\left (x + 1 \right )}}{16} - \frac{5 \operatorname{atan}{\left (x \right )}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x - x/(4*x**4 - 4) + 5*log(x - 1)/16 - 5*log(x + 1)/16 - 5*atan(x)/8

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GIAC/XCAS [A]  time = 0.304627, size = 41, normalized size = 1.21 \[ x - \frac{x}{4 \,{\left (x^{4} - 1\right )}} - \frac{5}{8} \, \arctan \left (x\right ) - \frac{5}{16} \,{\rm ln}\left ({\left | x + 1 \right |}\right ) + \frac{5}{16} \,{\rm ln}\left ({\left | x - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x - 1/4*x/(x^4 - 1) - 5/8*arctan(x) - 5/16*ln(abs(x + 1)) + 5/16*ln(abs(x - 1))