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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]