Optimal. Leaf size=26 \[ \frac{2 \sqrt{1-x^2}}{x}+\frac{2}{x}+2 \sin ^{-1}(x) \]
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Rubi [A] time = 0.205232, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 42, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {6688, 6742, 277, 216} \[ \frac{2 \sqrt{1-x^2}}{x}+\frac{2}{x}+2 \sin ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 6688
Rule 6742
Rule 277
Rule 216
Rubi steps
\begin{align*} \int \frac{\left (-\sqrt{1-x}-\sqrt{1+x}\right ) \left (\sqrt{1-x}+\sqrt{1+x}\right )}{x^2} \, dx &=-\int \frac{\left (\sqrt{1-x}+\sqrt{1+x}\right )^2}{x^2} \, dx\\ &=-\int \left (\frac{2}{x^2}+\frac{2 \sqrt{1-x^2}}{x^2}\right ) \, dx\\ &=\frac{2}{x}-2 \int \frac{\sqrt{1-x^2}}{x^2} \, dx\\ &=\frac{2}{x}+\frac{2 \sqrt{1-x^2}}{x}+2 \int \frac{1}{\sqrt{1-x^2}} \, dx\\ &=\frac{2}{x}+\frac{2 \sqrt{1-x^2}}{x}+2 \sin ^{-1}(x)\\ \end{align*}
Mathematica [A] time = 0.0253906, size = 22, normalized size = 0.85 \[ \frac{2 \left (\sqrt{1-x^2}+x \sin ^{-1}(x)+1\right )}{x} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.002, size = 50, normalized size = 1.9 \begin{align*} 2\,{x}^{-1}-2\,{\frac{ \left ( -\arcsin \left ( x \right ) x-\sqrt{-{x}^{2}+1} \right ) \sqrt{1+x}\sqrt{1-x}}{x\sqrt{-{x}^{2}+1}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.65774, size = 32, normalized size = 1.23 \begin{align*} \frac{2 \, \sqrt{-x^{2} + 1}}{x} + \frac{2}{x} + 2 \, \arcsin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.968857, size = 113, normalized size = 4.35 \begin{align*} -\frac{2 \,{\left (2 \, x \arctan \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) - \sqrt{x + 1} \sqrt{-x + 1} - 1\right )}}{x} \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{2}{x^{2}}\, dx - \int \frac{2 \sqrt{1 - x} \sqrt{x + 1}}{x^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.19291, size = 201, normalized size = 7.73 \begin{align*} 2 \, \pi + \frac{8 \,{\left (\frac{\sqrt{2} - \sqrt{-x + 1}}{\sqrt{x + 1}} - \frac{\sqrt{x + 1}}{\sqrt{2} - \sqrt{-x + 1}}\right )}}{{\left (\frac{\sqrt{2} - \sqrt{-x + 1}}{\sqrt{x + 1}} - \frac{\sqrt{x + 1}}{\sqrt{2} - \sqrt{-x + 1}}\right )}^{2} - 4} + \frac{2}{x} + 4 \, \arctan \left (\frac{\sqrt{x + 1}{\left (\frac{{\left (\sqrt{2} - \sqrt{-x + 1}\right )}^{2}}{x + 1} - 1\right )}}{2 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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