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

Optimal. Leaf size=13 \[ \frac{1}{2} \tan ^{-1}(x)+\frac{1}{2} \tanh ^{-1}(x) \]

[Out]

ArcTan[x]/2 + ArcTanh[x]/2

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Rubi [A]  time = 0.0109466, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{1}{2} \tan ^{-1}(x)+\frac{1}{2} \tanh ^{-1}(x) \]

Antiderivative was successfully verified.

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

[Out]

ArcTan[x]/2 + ArcTanh[x]/2

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Rubi in Sympy [A]  time = 3.97275, size = 8, normalized size = 0.62 \[ \frac{\operatorname{atan}{\left (x \right )}}{2} + \frac{\operatorname{atanh}{\left (x \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

atan(x)/2 + atanh(x)/2

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Mathematica [A]  time = 0.00645278, size = 25, normalized size = 1.92 \[ -\frac{1}{4} \log (1-x)+\frac{1}{4} \log (x+1)+\frac{1}{2} \tan ^{-1}(x) \]

Antiderivative was successfully verified.

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

[Out]

ArcTan[x]/2 - Log[1 - x]/4 + Log[1 + x]/4

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Maple [A]  time = 0.002, size = 10, normalized size = 0.8 \[{\frac{\arctan \left ( x \right ) }{2}}+{\frac{{\it Artanh} \left ( x \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-x^4+1)/(x^8-2*x^4+1),x)

[Out]

1/2*arctan(x)+1/2*arctanh(x)

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Maxima [A]  time = 0.817391, size = 23, normalized size = 1.77 \[ \frac{1}{2} \, \arctan \left (x\right ) + \frac{1}{4} \, \log \left (x + 1\right ) - \frac{1}{4} \, \log \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*arctan(x) + 1/4*log(x + 1) - 1/4*log(x - 1)

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Fricas [A]  time = 0.267531, size = 23, normalized size = 1.77 \[ \frac{1}{2} \, \arctan \left (x\right ) + \frac{1}{4} \, \log \left (x + 1\right ) - \frac{1}{4} \, \log \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*arctan(x) + 1/4*log(x + 1) - 1/4*log(x - 1)

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Sympy [A]  time = 0.353536, size = 17, normalized size = 1.31 \[ - \frac{\log{\left (x - 1 \right )}}{4} + \frac{\log{\left (x + 1 \right )}}{4} + \frac{\operatorname{atan}{\left (x \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-log(x - 1)/4 + log(x + 1)/4 + atan(x)/2

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GIAC/XCAS [A]  time = 0.266856, size = 26, normalized size = 2. \[ \frac{1}{2} \, \arctan \left (x\right ) + \frac{1}{4} \,{\rm ln}\left ({\left | x + 1 \right |}\right ) - \frac{1}{4} \,{\rm ln}\left ({\left | x - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*arctan(x) + 1/4*ln(abs(x + 1)) - 1/4*ln(abs(x - 1))