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

Optimal. Leaf size=35 \[ -\frac{\tan ^{-1}\left (\frac{x}{\sqrt [4]{2}}\right )}{2\ 2^{3/4}}-\frac{\tanh ^{-1}\left (\frac{x}{\sqrt [4]{2}}\right )}{2\ 2^{3/4}} \]

[Out]

-ArcTan[x/2^(1/4)]/(2*2^(3/4)) - ArcTanh[x/2^(1/4)]/(2*2^(3/4))

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Rubi [A]  time = 0.0265055, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429 \[ -\frac{\tan ^{-1}\left (\frac{x}{\sqrt [4]{2}}\right )}{2\ 2^{3/4}}-\frac{\tanh ^{-1}\left (\frac{x}{\sqrt [4]{2}}\right )}{2\ 2^{3/4}} \]

Antiderivative was successfully verified.

[In]  Int[(-2 + x^4)^(-1),x]

[Out]

-ArcTan[x/2^(1/4)]/(2*2^(3/4)) - ArcTanh[x/2^(1/4)]/(2*2^(3/4))

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Rubi in Sympy [A]  time = 0.933167, size = 34, normalized size = 0.97 \[ - \frac{\sqrt [4]{2} \operatorname{atan}{\left (\frac{2^{\frac{3}{4}} x}{2} \right )}}{4} - \frac{\sqrt [4]{2} \operatorname{atanh}{\left (\frac{2^{\frac{3}{4}} x}{2} \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(x**4-2),x)

[Out]

-2**(1/4)*atan(2**(3/4)*x/2)/4 - 2**(1/4)*atanh(2**(3/4)*x/2)/4

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Mathematica [A]  time = 0.025382, size = 43, normalized size = 1.23 \[ -\frac{-\log \left (2-2^{3/4} x\right )+\log \left (2^{3/4} x+2\right )+2 \tan ^{-1}\left (\frac{x}{\sqrt [4]{2}}\right )}{4\ 2^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(-2 + x^4)^(-1),x]

[Out]

-(2*ArcTan[x/2^(1/4)] - Log[2 - 2^(3/4)*x] + Log[2 + 2^(3/4)*x])/(4*2^(3/4))

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Maple [A]  time = 0.002, size = 35, normalized size = 1. \[ -{\frac{\sqrt [4]{2}}{4}\arctan \left ({\frac{x{2}^{{\frac{3}{4}}}}{2}} \right ) }-{\frac{\sqrt [4]{2}}{8}\ln \left ({\frac{x+\sqrt [4]{2}}{x-\sqrt [4]{2}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(x^4-2),x)

[Out]

-1/4*arctan(1/2*x*2^(3/4))*2^(1/4)-1/8*2^(1/4)*ln((x+2^(1/4))/(x-2^(1/4)))

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Maxima [A]  time = 1.51745, size = 50, normalized size = 1.43 \[ -\frac{1}{4} \cdot 2^{\frac{1}{4}} \arctan \left (\frac{1}{2} \cdot 2^{\frac{3}{4}} x\right ) + \frac{1}{8} \cdot 2^{\frac{1}{4}} \log \left (\frac{2 \,{\left (x - 2^{\frac{1}{4}}\right )}}{2 \, x + 2 \cdot 2^{\frac{1}{4}}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*2^(1/4)*arctan(1/2*2^(3/4)*x) + 1/8*2^(1/4)*log(2*(x - 2^(1/4))/((2*2^(1/4)
) + 2*x))

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Fricas [A]  time = 0.219564, size = 80, normalized size = 2.29 \[ \frac{1}{32} \cdot 8^{\frac{3}{4}}{\left (4 \, \arctan \left (\frac{2}{8^{\frac{1}{4}} \sqrt{\frac{1}{2}} \sqrt{\sqrt{2}{\left (\sqrt{2} x^{2} + 2\right )}} + 8^{\frac{1}{4}} x}\right ) - \log \left (8^{\frac{1}{4}} x + 2\right ) + \log \left (8^{\frac{1}{4}} x - 2\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/32*8^(3/4)*(4*arctan(2/(8^(1/4)*sqrt(1/2)*sqrt(sqrt(2)*(sqrt(2)*x^2 + 2)) + 8^
(1/4)*x)) - log(8^(1/4)*x + 2) + log(8^(1/4)*x - 2))

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Sympy [A]  time = 0.623137, size = 46, normalized size = 1.31 \[ \frac{\sqrt [4]{2} \log{\left (x - \sqrt [4]{2} \right )}}{8} - \frac{\sqrt [4]{2} \log{\left (x + \sqrt [4]{2} \right )}}{8} - \frac{\sqrt [4]{2} \operatorname{atan}{\left (\frac{2^{\frac{3}{4}} x}{2} \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x**4-2),x)

[Out]

2**(1/4)*log(x - 2**(1/4))/8 - 2**(1/4)*log(x + 2**(1/4))/8 - 2**(1/4)*atan(2**(
3/4)*x/2)/4

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GIAC/XCAS [A]  time = 0.213099, size = 53, normalized size = 1.51 \[ -\frac{1}{4} \cdot 2^{\frac{1}{4}} \arctan \left (\frac{1}{2} \cdot 2^{\frac{3}{4}} x\right ) - \frac{1}{8} \cdot 2^{\frac{1}{4}}{\rm ln}\left ({\left | x + 2^{\frac{1}{4}} \right |}\right ) + \frac{1}{8} \cdot 2^{\frac{1}{4}}{\rm ln}\left ({\left | x - 2^{\frac{1}{4}} \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*2^(1/4)*arctan(1/2*2^(3/4)*x) - 1/8*2^(1/4)*ln(abs(x + 2^(1/4))) + 1/8*2^(1
/4)*ln(abs(x - 2^(1/4)))