Optimal. Leaf size=2 \[ \sin ^{-1}(x) \]
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Rubi [A] time = 0.0013687, antiderivative size = 2, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {26, 216} \[ \sin ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 26
Rule 216
Rubi steps
\begin{align*} \int \frac{\sqrt{1+x^2}}{\sqrt{1-x^4}} \, dx &=\int \frac{1}{\sqrt{1-x^2}} \, dx\\ &=\sin ^{-1}(x)\\ \end{align*}
Mathematica [B] time = 0.0254332, size = 32, normalized size = 16. \[ -\tan ^{-1}\left (\frac{x \sqrt{x^2+1} \sqrt{1-x^4}}{x^4-1}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.013, size = 29, normalized size = 14.5 \begin{align*}{\arcsin \left ( x \right ) \sqrt{-{x}^{4}+1}{\frac{1}{\sqrt{{x}^{2}+1}}}{\frac{1}{\sqrt{-{x}^{2}+1}}}} \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^{2} + 1}}{\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 = 1.47706, size = 66, normalized size = 33. \begin{align*} -\arctan \left (\frac{\sqrt{-x^{4} + 1} \sqrt{x^{2} + 1}}{x^{3} + x}\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{x^{2} + 1}}{\sqrt{- \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 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{\sqrt{x^{2} + 1}}{\sqrt{-x^{4} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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