Optimal. Leaf size=49 \[ \frac{1}{2} \tan ^{-1}\left (\frac{x \left (x^2+1\right )}{\sqrt{1-x^4}}\right )+\frac{1}{2} \tanh ^{-1}\left (\frac{x \left (1-x^2\right )}{\sqrt{1-x^4}}\right ) \]
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Rubi [A] time = 0.0072942, 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, Rules used = {405} \[ \frac{1}{2} \tan ^{-1}\left (\frac{x \left (x^2+1\right )}{\sqrt{1-x^4}}\right )+\frac{1}{2} \tanh ^{-1}\left (\frac{x \left (1-x^2\right )}{\sqrt{1-x^4}}\right ) \]
Antiderivative was successfully verified.
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Rule 405
Rubi steps
\begin{align*} \int \frac{\sqrt{1-x^4}}{1+x^4} \, dx &=\frac{1}{2} \tan ^{-1}\left (\frac{x \left (1+x^2\right )}{\sqrt{1-x^4}}\right )+\frac{1}{2} \tanh ^{-1}\left (\frac{x \left (1-x^2\right )}{\sqrt{1-x^4}}\right )\\ \end{align*}
Mathematica [C] time = 0.0968655, 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.
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Maple [B] time = 0.013, size = 100, normalized size = 2. \begin{align*}{\frac{1}{4}\arctan \left ( -{\frac{1}{x}\sqrt{-{x}^{4}+1}}+1 \right ) }-{\frac{1}{8}\ln \left ({ \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 ) }-{\frac{1}{4}\arctan \left ({\frac{1}{x}\sqrt{-{x}^{4}+1}}+1 \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-x^{4} + 1}}{x^{4} + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.51574, size = 138, normalized size = 2.82 \begin{align*} -\frac{1}{2} \, \arctan \left (\frac{\sqrt{-x^{4} + 1} x}{x^{2} - 1}\right ) + \frac{1}{4} \, \log \left (-\frac{x^{4} - 2 \, x^{2} - 2 \, \sqrt{-x^{4} + 1} x - 1}{x^{4} + 1}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{- \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}}{x^{4} + 1}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-x^{4} + 1}}{x^{4} + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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