Optimal. Leaf size=22 \[ \tan ^{-1}\left (\frac{x}{\sqrt{\sqrt{x^4+1}-x^2}}\right ) \]
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Rubi [A] time = 0.0623883, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {2128, 203} \[ \tan ^{-1}\left (\frac{x}{\sqrt{\sqrt{x^4+1}-x^2}}\right ) \]
Antiderivative was successfully verified.
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Rule 2128
Rule 203
Rubi steps
\begin{align*} \int \frac{1}{\left (1+x^4\right ) \sqrt{-x^2+\sqrt{1+x^4}}} \, dx &=\operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\frac{x}{\sqrt{-x^2+\sqrt{1+x^4}}}\right )\\ &=\tan ^{-1}\left (\frac{x}{\sqrt{-x^2+\sqrt{1+x^4}}}\right )\\ \end{align*}
Mathematica [A] time = 1.00432, size = 24, normalized size = 1.09 \[ \cot ^{-1}\left (\frac{\sqrt{\sqrt{x^4+1}-x^2}}{x}\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.025, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{4}+1}{\frac{1}{\sqrt{-{x}^{2}+\sqrt{{x}^{4}+1}}}}}\, dx \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{1}{{\left (x^{4} + 1\right )} \sqrt{-x^{2} + \sqrt{x^{4} + 1}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 6.05857, size = 149, normalized size = 6.77 \begin{align*} -\frac{1}{4} \, \arctan \left (\frac{4 \,{\left (10 \, x^{7} - 6 \, x^{3} +{\left (7 \, x^{5} - x\right )} \sqrt{x^{4} + 1}\right )} \sqrt{-x^{2} + \sqrt{x^{4} + 1}}}{17 \, x^{8} - 46 \, 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{1}{\sqrt{- x^{2} + \sqrt{x^{4} + 1}} \left (x^{4} + 1\right )}\, 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{1}{{\left (x^{4} + 1\right )} \sqrt{-x^{2} + \sqrt{x^{4} + 1}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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