Optimal. Leaf size=32 \[ 2 \sqrt{1-x^2}-2 \tanh ^{-1}\left (\sqrt{1-x^2}\right )+2 \log (x) \]
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Rubi [A] time = 0.0887197, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {6742, 266, 50, 63, 206} \[ 2 \sqrt{1-x^2}-2 \tanh ^{-1}\left (\sqrt{1-x^2}\right )+2 \log (x) \]
Antiderivative was successfully verified.
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Rule 6742
Rule 266
Rule 50
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{\left (\sqrt{1-x}+\sqrt{1+x}\right )^2}{x} \, dx &=\int \left (\frac{2}{x}+\frac{2 \sqrt{1-x^2}}{x}\right ) \, dx\\ &=2 \log (x)+2 \int \frac{\sqrt{1-x^2}}{x} \, dx\\ &=2 \log (x)+\operatorname{Subst}\left (\int \frac{\sqrt{1-x}}{x} \, dx,x,x^2\right )\\ &=2 \sqrt{1-x^2}+2 \log (x)+\operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x} x} \, dx,x,x^2\right )\\ &=2 \sqrt{1-x^2}+2 \log (x)-2 \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\sqrt{1-x^2}\right )\\ &=2 \sqrt{1-x^2}-2 \tanh ^{-1}\left (\sqrt{1-x^2}\right )+2 \log (x)\\ \end{align*}
Mathematica [A] time = 0.0284104, size = 32, normalized size = 1. \[ 2 \sqrt{1-x^2}-2 \tanh ^{-1}\left (\sqrt{1-x^2}\right )+2 \log (x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 51, normalized size = 1.6 \begin{align*} 2\,\ln \left ( x \right ) +2\,{\frac{\sqrt{1-x}\sqrt{1+x} \left ( \sqrt{-{x}^{2}+1}-{\it Artanh} \left ({\frac{1}{\sqrt{-{x}^{2}+1}}} \right ) \right ) }{\sqrt{-{x}^{2}+1}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.7694, size = 55, normalized size = 1.72 \begin{align*} 2 \, \sqrt{-x^{2} + 1} + 2 \, \log \left (x\right ) - 2 \, \log \left (\frac{2 \, \sqrt{-x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.27879, size = 109, normalized size = 3.41 \begin{align*} 2 \, \sqrt{x + 1} \sqrt{-x + 1} + 2 \, \log \left (x\right ) + 2 \, \log \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{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{\left (\sqrt{1 - x} + \sqrt{x + 1}\right )^{2}}{x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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