Optimal. Leaf size=33 \[ -\frac {\sqrt {1-x^2}}{x}-2 \tanh ^{-1}\left (\sqrt {1-x^2}\right )+\sin ^{-1}(x) \]
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Rubi [A] time = 0.06, antiderivative size = 33, 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 = {1807, 844, 216, 266, 63, 206} \[ -\frac {\sqrt {1-x^2}}{x}-2 \tanh ^{-1}\left (\sqrt {1-x^2}\right )+\sin ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 63
Rule 206
Rule 216
Rule 266
Rule 844
Rule 1807
Rubi steps
\begin {align*} \int \frac {(1+x)^2}{x^2 \sqrt {1-x^2}} \, dx &=-\frac {\sqrt {1-x^2}}{x}-\int \frac {-2-x}{x \sqrt {1-x^2}} \, dx\\ &=-\frac {\sqrt {1-x^2}}{x}+2 \int \frac {1}{x \sqrt {1-x^2}} \, dx+\int \frac {1}{\sqrt {1-x^2}} \, dx\\ &=-\frac {\sqrt {1-x^2}}{x}+\sin ^{-1}(x)+\operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x} x} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {1-x^2}}{x}+\sin ^{-1}(x)-2 \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {1-x^2}\right )\\ &=-\frac {\sqrt {1-x^2}}{x}+\sin ^{-1}(x)-2 \tanh ^{-1}\left (\sqrt {1-x^2}\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 33, normalized size = 1.00 \[ -\frac {\sqrt {1-x^2}}{x}-2 \tanh ^{-1}\left (\sqrt {1-x^2}\right )+\sin ^{-1}(x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.83, size = 53, normalized size = 1.61 \[ -\frac {2 \, x \arctan \left (\frac {\sqrt {-x^{2} + 1} - 1}{x}\right ) - 2 \, x \log \left (\frac {\sqrt {-x^{2} + 1} - 1}{x}\right ) + \sqrt {-x^{2} + 1}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 55, normalized size = 1.67 \[ \frac {x}{2 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}} - \frac {\sqrt {-x^{2} + 1} - 1}{2 \, x} + \arcsin \relax (x) + 2 \, \log \left (-\frac {\sqrt {-x^{2} + 1} - 1}{{\left | x \right |}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 30, normalized size = 0.91 \[ -2 \arctanh \left (\frac {1}{\sqrt {-x^{2}+1}}\right )+\arcsin \relax (x )-\frac {\sqrt {-x^{2}+1}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.97, size = 42, normalized size = 1.27 \[ -\frac {\sqrt {-x^{2} + 1}}{x} + \arcsin \relax (x) - 2 \, \log \left (\frac {2 \, \sqrt {-x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 35, normalized size = 1.06 \[ \mathrm {asin}\relax (x)+2\,\ln \left (\sqrt {\frac {1}{x^2}-1}-\sqrt {\frac {1}{x^2}}\right )-\frac {\sqrt {1-x^2}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 4.68, size = 51, normalized size = 1.55 \[ \begin {cases} - \frac {i \sqrt {x^{2} - 1}}{x} & \text {for}\: \left |{x^{2}}\right | > 1 \\- \frac {\sqrt {1 - x^{2}}}{x} & \text {otherwise} \end {cases} + 2 \left (\begin {cases} - \operatorname {acosh}{\left (\frac {1}{x} \right )} & \text {for}\: \frac {1}{\left |{x^{2}}\right |} > 1 \\i \operatorname {asin}{\left (\frac {1}{x} \right )} & \text {otherwise} \end {cases}\right ) + \operatorname {asin}{\relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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