Optimal. Leaf size=54 \[ \frac{\sqrt{x^2-1} \sqrt{x^2+1} F\left (\sin ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{x^2-1}}\right )|\frac{1}{2}\right )}{\sqrt{2} \sqrt{x^4-1}} \]
[Out]
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Rubi [A] time = 0.0146213, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{\sqrt{x^2-1} \sqrt{x^2+1} F\left (\sin ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{x^2-1}}\right )|\frac{1}{2}\right )}{\sqrt{2} \sqrt{x^4-1}} \]
Antiderivative was successfully verified.
[In] Int[1/Sqrt[-1 + x^4],x]
[Out]
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Rubi in Sympy [A] time = 1.2738, size = 22, normalized size = 0.41 \[ \frac{\sqrt{- x^{4} + 1} F\left (\operatorname{asin}{\left (x \right )}\middle | -1\right )}{\sqrt{x^{4} - 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(x**4-1)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0178122, size = 25, normalized size = 0.46 \[ \frac{\sqrt{1-x^4} F\left (\left .\sin ^{-1}(x)\right |-1\right )}{\sqrt{x^4-1}} \]
Antiderivative was successfully verified.
[In] Integrate[1/Sqrt[-1 + x^4],x]
[Out]
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Maple [C] time = 0.007, size = 34, normalized size = 0.6 \[{-i{\it EllipticF} \left ( ix,i \right ) \sqrt{{x}^{2}+1}\sqrt{-{x}^{2}+1}{\frac{1}{\sqrt{{x}^{4}-1}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(x^4-1)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{x^{4} - 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(x^4 - 1),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{x^{4} - 1}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(x^4 - 1),x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.66939, size = 26, normalized size = 0.48 \[ - \frac{i x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{1}{2} \\ \frac{5}{4} \end{matrix}\middle |{x^{4}} \right )}}{4 \Gamma \left (\frac{5}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(x**4-1)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{x^{4} - 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(x^4 - 1),x, algorithm="giac")
[Out]