Optimal. Leaf size=33 \[ \sqrt{\frac{1}{x^2-1}} \sqrt{x^2-1} \tanh ^{-1}\left (\frac{x}{\sqrt{x^2-1}}\right ) \]
[Out]
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Rubi [A] time = 0.0396453, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ \sqrt{\frac{1}{x^2-1}} \sqrt{x^2-1} \tanh ^{-1}\left (\frac{x}{\sqrt{x^2-1}}\right ) \]
Antiderivative was successfully verified.
[In] Int[Sqrt[(1 + x^2)/(-1 + x^4)],x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{\frac{x^{2} + 1}{x^{4} - 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(((x**2+1)/(x**4-1))**(1/2),x)
[Out]
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Mathematica [A] time = 0.00501797, size = 56, normalized size = 1.7 \[ \frac{1}{2} \sqrt{\frac{1}{x^2-1}} \sqrt{x^2-1} \left (\log \left (\frac{x}{\sqrt{x^2-1}}+1\right )-\log \left (1-\frac{x}{\sqrt{x^2-1}}\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[(1 + x^2)/(-1 + x^4)],x]
[Out]
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Maple [A] time = 0.005, size = 28, normalized size = 0.9 \[ \sqrt{ \left ({x}^{2}-1 \right ) ^{-1}}\sqrt{{x}^{2}-1}\ln \left ( x+\sqrt{{x}^{2}-1} \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(((x^2+1)/(x^4-1))^(1/2),x)
[Out]
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Maxima [A] time = 0.71637, size = 19, normalized size = 0.58 \[ \log \left (2 \, x + 2 \, \sqrt{x^{2} - 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((x^2 + 1)/(x^4 - 1)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.266031, size = 31, normalized size = 0.94 \[ -\log \left (-\sqrt{x^{2} - 1}{\left (\frac{x}{\sqrt{x^{2} - 1}} - 1\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((x^2 + 1)/(x^4 - 1)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{\frac{x^{2} + 1}{x^{4} - 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((x**2+1)/(x**4-1))**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.264541, size = 28, normalized size = 0.85 \[ -{\rm ln}\left ({\left | -x + \sqrt{x^{2} - 1} \right |}\right ){\rm sign}\left (x^{2} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((x^2 + 1)/(x^4 - 1)),x, algorithm="giac")
[Out]