Optimal. Leaf size=23 \[ \frac{1}{2} \sqrt{1-x^2} x+\frac{1}{2} \sin ^{-1}(x) \]
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Rubi [A] time = 0.0030979, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {26, 195, 216} \[ \frac{1}{2} \sqrt{1-x^2} x+\frac{1}{2} \sin ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 26
Rule 195
Rule 216
Rubi steps
\begin{align*} \int \frac{\sqrt{1-x^4}}{\sqrt{1+x^2}} \, dx &=\int \sqrt{1-x^2} \, dx\\ &=\frac{1}{2} x \sqrt{1-x^2}+\frac{1}{2} \int \frac{1}{\sqrt{1-x^2}} \, dx\\ &=\frac{1}{2} x \sqrt{1-x^2}+\frac{1}{2} \sin ^{-1}(x)\\ \end{align*}
Mathematica [B] time = 0.0411808, size = 50, normalized size = 2.17 \[ \frac{1}{2} \left (\frac{\sqrt{1-x^4} x}{\sqrt{x^2+1}}+\tan ^{-1}\left (\frac{x \sqrt{x^2+1}}{\sqrt{1-x^4}}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.008, size = 42, normalized size = 1.8 \begin{align*}{\frac{1}{2}\sqrt{-{x}^{4}+1} \left ( x\sqrt{-{x}^{2}+1}+\arcsin \left ( x \right ) \right ){\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^{4} + 1}}{\sqrt{x^{2} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.45123, size = 144, normalized size = 6.26 \begin{align*} \frac{\sqrt{-x^{4} + 1} \sqrt{x^{2} + 1} x -{\left (x^{2} + 1\right )} \arctan \left (\frac{\sqrt{-x^{4} + 1} \sqrt{x^{2} + 1}}{x^{3} + x}\right )}{2 \,{\left (x^{2} + 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 )}}{\sqrt{x^{2} + 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}}{\sqrt{x^{2} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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