Optimal. Leaf size=32 \[ -\sqrt{1-x^2}-\tanh ^{-1}\left (\sqrt{1-x^2}\right )+2 \sin ^{-1}(x) \]
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Rubi [A] time = 0.0589199, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {1809, 844, 216, 266, 63, 206} \[ -\sqrt{1-x^2}-\tanh ^{-1}\left (\sqrt{1-x^2}\right )+2 \sin ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 1809
Rule 844
Rule 216
Rule 266
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{(1+x)^2}{x \sqrt{1-x^2}} \, dx &=-\sqrt{1-x^2}-\int \frac{-1-2 x}{x \sqrt{1-x^2}} \, dx\\ &=-\sqrt{1-x^2}+2 \int \frac{1}{\sqrt{1-x^2}} \, dx+\int \frac{1}{x \sqrt{1-x^2}} \, dx\\ &=-\sqrt{1-x^2}+2 \sin ^{-1}(x)+\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x} x} \, dx,x,x^2\right )\\ &=-\sqrt{1-x^2}+2 \sin ^{-1}(x)-\operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\sqrt{1-x^2}\right )\\ &=-\sqrt{1-x^2}+2 \sin ^{-1}(x)-\tanh ^{-1}\left (\sqrt{1-x^2}\right )\\ \end{align*}
Mathematica [A] time = 0.0105869, size = 32, normalized size = 1. \[ -\sqrt{1-x^2}-\tanh ^{-1}\left (\sqrt{1-x^2}\right )+2 \sin ^{-1}(x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.048, size = 29, normalized size = 0.9 \begin{align*} -\sqrt{-{x}^{2}+1}+2\,\arcsin \left ( x \right ) -{\it Artanh} \left ({\frac{1}{\sqrt{-{x}^{2}+1}}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.48763, size = 55, normalized size = 1.72 \begin{align*} -\sqrt{-x^{2} + 1} + 2 \, \arcsin \left (x\right ) - \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 = 2.13574, size = 111, normalized size = 3.47 \begin{align*} -\sqrt{-x^{2} + 1} - 4 \, \arctan \left (\frac{\sqrt{-x^{2} + 1} - 1}{x}\right ) + \log \left (\frac{\sqrt{-x^{2} + 1} - 1}{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.91425, size = 31, normalized size = 0.97 \begin{align*} - \sqrt{1 - x^{2}} + \begin{cases} - \operatorname{acosh}{\left (\frac{1}{x} \right )} & \text{for}\: \frac{1}{\left |{x^{2}}\right |} > 1 \\i \operatorname{asin}{\left (\frac{1}{x} \right )} & \text{otherwise} \end{cases} + 2 \operatorname{asin}{\left (x \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14875, size = 46, normalized size = 1.44 \begin{align*} -\sqrt{-x^{2} + 1} + 2 \, \arcsin \left (x\right ) + \log \left (-\frac{\sqrt{-x^{2} + 1} - 1}{{\left | x \right |}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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