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 = 0.06, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, 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 63
Rule 206
Rule 216
Rule 266
Rule 844
Rule 1809
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*}
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Mathematica [A] time = 0.01, size = 32, normalized size = 1.00 \[ -\sqrt {1-x^2}-\tanh ^{-1}\left (\sqrt {1-x^2}\right )+2 \sin ^{-1}(x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.86, size = 46, normalized size = 1.44 \[ -\sqrt {-x^{2} + 1} - 4 \, \arctan \left (\frac {\sqrt {-x^{2} + 1} - 1}{x}\right ) + \log \left (\frac {\sqrt {-x^{2} + 1} - 1}{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 34, normalized size = 1.06 \[ -\sqrt {-x^{2} + 1} + 2 \, \arcsin \relax (x) + \log \left (-\frac {\sqrt {-x^{2} + 1} - 1}{{\left | x \right |}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 29, normalized size = 0.91 \[ -\arctanh \left (\frac {1}{\sqrt {-x^{2}+1}}\right )+2 \arcsin \relax (x )-\sqrt {-x^{2}+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.98, size = 41, normalized size = 1.28 \[ -\sqrt {-x^{2} + 1} + 2 \, \arcsin \relax (x) - \log \left (\frac {2 \, \sqrt {-x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 32, normalized size = 1.00 \[ 2\,\mathrm {asin}\relax (x)+\ln \left (\sqrt {\frac {1}{x^2}-1}-\sqrt {\frac {1}{x^2}}\right )-\sqrt {1-x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 6.30, size = 31, normalized size = 0.97 \[ - \sqrt {1 - x^{2}} + \begin {cases} - \operatorname {acosh}{\left (\frac {1}{x} \right )} & \text {for}\: \frac {1}{\left |{x^{2}}\right |} > 1 \\i \operatorname {asin}{\left (\frac {1}{x} \right )} & \text {otherwise} \end {cases} + 2 \operatorname {asin}{\relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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