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 = 0.01, antiderivative size = 49, normalized size of antiderivative = 1.00, 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*}
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Mathematica [C] time = 0.10, 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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fricas [A] time = 0.68, size = 56, normalized size = 1.14 \[ -\frac {1}{2} \, \arctan \left (\frac {\sqrt {-x^{4} + 1} x}{x^{2} - 1}\right ) + \frac {1}{4} \, \log \left (-\frac {x^{4} - 2 \, x^{2} - 2 \, \sqrt {-x^{4} + 1} x - 1}{x^{4} + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-x^{4} + 1}}{x^{4} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 100, normalized size = 2.04 \[ \frac {\arctan \left (-\frac {\sqrt {-x^{4}+1}}{x}+1\right )}{4}-\frac {\arctan \left (\frac {\sqrt {-x^{4}+1}}{x}+1\right )}{4}-\frac {\ln \left (\frac {-\frac {\sqrt {-x^{4}+1}}{x}+\frac {-x^{4}+1}{2 x^{2}}+1}{\frac {\sqrt {-x^{4}+1}}{x}+\frac {-x^{4}+1}{2 x^{2}}+1}\right )}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-x^{4} + 1}}{x^{4} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\sqrt {1-x^4}}{x^4+1} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {- \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}}{x^{4} + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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