Optimal. Leaf size=140 \[ \frac{\sqrt{x^4-1}}{x}-\frac{x \left (x^2+1\right )}{\sqrt{x^4-1}}-\frac{\sqrt{x^2-1} \sqrt{x^2+1} F\left (\sin ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{x^2-1}}\right )|\frac{1}{2}\right )}{\sqrt{2} \sqrt{x^4-1}}+\frac{\sqrt{2} \sqrt{x^2-1} \sqrt{x^2+1} E\left (\sin ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{x^2-1}}\right )|\frac{1}{2}\right )}{\sqrt{x^4-1}} \]
[Out]
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Rubi [A] time = 0.0613276, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ \frac{\sqrt{x^4-1}}{x}-\frac{x \left (x^2+1\right )}{\sqrt{x^4-1}}-\frac{\sqrt{x^2-1} \sqrt{x^2+1} F\left (\sin ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{x^2-1}}\right )|\frac{1}{2}\right )}{\sqrt{2} \sqrt{x^4-1}}+\frac{\sqrt{2} \sqrt{x^2-1} \sqrt{x^2+1} E\left (\sin ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{x^2-1}}\right )|\frac{1}{2}\right )}{\sqrt{x^4-1}} \]
Antiderivative was successfully verified.
[In] Int[1/(x^2*Sqrt[-1 + x^4]),x]
[Out]
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Rubi in Sympy [A] time = 6.84739, size = 99, normalized size = 0.71 \[ - \frac{x \left (x^{2} + 1\right )}{\sqrt{x^{4} - 1}} + \frac{\sqrt{2} \sqrt{x^{2} - 1} \sqrt{x^{2} + 1} E\left (\operatorname{asin}{\left (\frac{\sqrt{2} x}{\sqrt{x^{2} - 1}} \right )}\middle | \frac{1}{2}\right )}{\sqrt{x^{4} - 1}} - \frac{\sqrt{- x^{4} + 1} F\left (\operatorname{asin}{\left (x \right )}\middle | -1\right )}{\sqrt{x^{4} - 1}} + \frac{\sqrt{x^{4} - 1}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**2/(x**4-1)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0460273, size = 55, normalized size = 0.39 \[ \frac{\sqrt{x^4-1}}{x}+\frac{\sqrt{1-x^2} \sqrt{x^2+1} \left (F\left (\left .\sin ^{-1}(x)\right |-1\right )-E\left (\left .\sin ^{-1}(x)\right |-1\right )\right )}{\sqrt{x^4-1}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^2*Sqrt[-1 + x^4]),x]
[Out]
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Maple [C] time = 0.013, size = 56, normalized size = 0.4 \[{\frac{1}{x}\sqrt{{x}^{4}-1}}+{i \left ({\it EllipticF} \left ( ix,i \right ) -{\it EllipticE} \left ( ix,i \right ) \right ) \sqrt{{x}^{2}+1}\sqrt{-{x}^{2}+1}{\frac{1}{\sqrt{{x}^{4}-1}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^2/(x^4-1)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{x^{4} - 1} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(x^4 - 1)*x^2),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{x^{4} - 1} x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(x^4 - 1)*x^2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.00875, size = 29, normalized size = 0.21 \[ - \frac{i \Gamma \left (- \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{1}{2} \\ \frac{3}{4} \end{matrix}\middle |{x^{4}} \right )}}{4 x \Gamma \left (\frac{3}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**2/(x**4-1)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{x^{4} - 1} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(x^4 - 1)*x^2),x, algorithm="giac")
[Out]