Optimal. Leaf size=49 \[ \frac{1}{2} \tan ^{-1}\left (\frac{x \left (x^2+1\right )}{\sqrt{1-x^4}}\right )+\frac{1}{2} \tanh ^{-1}\left (\frac{x \left (1-x^2\right )}{\sqrt{1-x^4}}\right ) \]
[Out]
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Rubi [A] time = 0.0261983, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053 \[ \frac{1}{2} \tan ^{-1}\left (\frac{x \left (x^2+1\right )}{\sqrt{1-x^4}}\right )+\frac{1}{2} \tanh ^{-1}\left (\frac{x \left (1-x^2\right )}{\sqrt{1-x^4}}\right ) \]
Antiderivative was successfully verified.
[In] Int[Sqrt[1 - x^4]/(1 + x^4),x]
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Rubi in Sympy [A] time = 2.40816, size = 36, normalized size = 0.73 \[ \frac{\operatorname{atan}{\left (\frac{x \left (x^{2} + 1\right )}{\sqrt{- x^{4} + 1}} \right )}}{2} + \frac{\operatorname{atanh}{\left (\frac{x \left (- x^{2} + 1\right )}{\sqrt{- x^{4} + 1}} \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((-x**4+1)**(1/2)/(x**4+1),x)
[Out]
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Mathematica [C] time = 0.144058, size = 110, normalized size = 2.24 \[ -\frac{5 x \sqrt{1-x^4} F_1\left (\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};x^4,-x^4\right )}{\left (x^4+1\right ) \left (2 x^4 \left (2 F_1\left (\frac{5}{4};-\frac{1}{2},2;\frac{9}{4};x^4,-x^4\right )+F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};x^4,-x^4\right )\right )-5 F_1\left (\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};x^4,-x^4\right )\right )} \]
Warning: Unable to verify antiderivative.
[In] Integrate[Sqrt[1 - x^4]/(1 + x^4),x]
[Out]
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Maple [B] time = 0.022, size = 100, normalized size = 2. \[ -{\frac{1}{4}\arctan \left ({\frac{1}{x}\sqrt{-{x}^{4}+1}}+1 \right ) }+{\frac{1}{4}\arctan \left ( -{\frac{1}{x}\sqrt{-{x}^{4}+1}}+1 \right ) }-{\frac{1}{8}\ln \left ({1 \left ({\frac{-{x}^{4}+1}{2\,{x}^{2}}}-{\frac{1}{x}\sqrt{-{x}^{4}+1}}+1 \right ) \left ({\frac{-{x}^{4}+1}{2\,{x}^{2}}}+{\frac{1}{x}\sqrt{-{x}^{4}+1}}+1 \right ) ^{-1}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((-x^4+1)^(1/2)/(x^4+1),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{-x^{4} + 1}}{x^{4} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-x^4 + 1)/(x^4 + 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.257111, size = 105, normalized size = 2.14 \[ \frac{1}{2} \, \arctan \left (-\frac{x^{3} + \sqrt{-x^{4} + 1} x^{2} + x}{x^{3} - x - \sqrt{-x^{4} + 1}}\right ) + \frac{1}{4} \, \log \left (-\frac{x^{4} - 2 \, x^{2} - 2 \, \sqrt{-x^{4} + 1} x - 1}{x^{4} + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-x^4 + 1)/(x^4 + 1),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{- \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}}{x^{4} + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-x**4+1)**(1/2)/(x**4+1),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{-x^{4} + 1}}{x^{4} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-x^4 + 1)/(x^4 + 1),x, algorithm="giac")
[Out]