Optimal. Leaf size=14 \[ \frac{1}{2} \tan ^{-1}\left (\sqrt{x^4-1}\right ) \]
[Out]
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Rubi [A] time = 0.0236733, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{1}{2} \tan ^{-1}\left (\sqrt{x^4-1}\right ) \]
Antiderivative was successfully verified.
[In] Int[1/(x*Sqrt[-1 + x^4]),x]
[Out]
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Rubi in Sympy [A] time = 3.20581, size = 10, normalized size = 0.71 \[ \frac{\operatorname{atan}{\left (\sqrt{x^{4} - 1} \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x/(x**4-1)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0320399, size = 14, normalized size = 1. \[ \frac{1}{2} \tan ^{-1}\left (\sqrt{x^4-1}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[1/(x*Sqrt[-1 + x^4]),x]
[Out]
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Maple [A] time = 0.016, size = 11, normalized size = 0.8 \[ -{\frac{1}{2}\arctan \left ({\frac{1}{\sqrt{{x}^{4}-1}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x/(x^4-1)^(1/2),x)
[Out]
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Maxima [A] time = 1.61973, size = 14, normalized size = 1. \[ \frac{1}{2} \, \arctan \left (\sqrt{x^{4} - 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(x^4 - 1)*x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.275404, size = 14, normalized size = 1. \[ \frac{1}{2} \, \arctan \left (\sqrt{x^{4} - 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(x^4 - 1)*x),x, algorithm="fricas")
[Out]
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Sympy [A] time = 3.33344, size = 24, normalized size = 1.71 \[ \begin{cases} \frac{i \operatorname{acosh}{\left (\frac{1}{x^{2}} \right )}}{2} & \text{for}\: \left |{\frac{1}{x^{4}}}\right | > 1 \\- \frac{\operatorname{asin}{\left (\frac{1}{x^{2}} \right )}}{2} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x/(x**4-1)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.213125, size = 14, normalized size = 1. \[ \frac{1}{2} \, \arctan \left (\sqrt{x^{4} - 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(x^4 - 1)*x),x, algorithm="giac")
[Out]