Optimal. Leaf size=51 \[ -\frac{2 \sqrt{1-x^2}}{x}-\frac{\sqrt{1-x^2}}{2 x^2}-\frac{3}{2} \tanh ^{-1}\left (\sqrt{1-x^2}\right ) \]
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Rubi [A] time = 0.0614837, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {1807, 807, 266, 63, 206} \[ -\frac{2 \sqrt{1-x^2}}{x}-\frac{\sqrt{1-x^2}}{2 x^2}-\frac{3}{2} \tanh ^{-1}\left (\sqrt{1-x^2}\right ) \]
Antiderivative was successfully verified.
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Rule 1807
Rule 807
Rule 266
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{(1+x)^2}{x^3 \sqrt{1-x^2}} \, dx &=-\frac{\sqrt{1-x^2}}{2 x^2}-\frac{1}{2} \int \frac{-4-3 x}{x^2 \sqrt{1-x^2}} \, dx\\ &=-\frac{\sqrt{1-x^2}}{2 x^2}-\frac{2 \sqrt{1-x^2}}{x}+\frac{3}{2} \int \frac{1}{x \sqrt{1-x^2}} \, dx\\ &=-\frac{\sqrt{1-x^2}}{2 x^2}-\frac{2 \sqrt{1-x^2}}{x}+\frac{3}{4} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x} x} \, dx,x,x^2\right )\\ &=-\frac{\sqrt{1-x^2}}{2 x^2}-\frac{2 \sqrt{1-x^2}}{x}-\frac{3}{2} \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\sqrt{1-x^2}\right )\\ &=-\frac{\sqrt{1-x^2}}{2 x^2}-\frac{2 \sqrt{1-x^2}}{x}-\frac{3}{2} \tanh ^{-1}\left (\sqrt{1-x^2}\right )\\ \end{align*}
Mathematica [A] time = 0.0195676, size = 40, normalized size = 0.78 \[ -\frac{\sqrt{1-x^2} (4 x+1)}{2 x^2}-\frac{3}{2} \tanh ^{-1}\left (\sqrt{1-x^2}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.052, size = 42, normalized size = 0.8 \begin{align*} -{\frac{3}{2}{\it Artanh} \left ({\frac{1}{\sqrt{-{x}^{2}+1}}} \right ) }-{\frac{1}{2\,{x}^{2}}\sqrt{-{x}^{2}+1}}-2\,{\frac{\sqrt{-{x}^{2}+1}}{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.48397, size = 73, normalized size = 1.43 \begin{align*} -\frac{2 \, \sqrt{-x^{2} + 1}}{x} - \frac{\sqrt{-x^{2} + 1}}{2 \, x^{2}} - \frac{3}{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 = 1.98484, size = 97, normalized size = 1.9 \begin{align*} \frac{3 \, x^{2} \log \left (\frac{\sqrt{-x^{2} + 1} - 1}{x}\right ) - \sqrt{-x^{2} + 1}{\left (4 \, x + 1\right )}}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 6.60171, size = 116, normalized size = 2.27 \begin{align*} 2 \left (\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}\right ) + \begin{cases} - \frac{\operatorname{acosh}{\left (\frac{1}{x} \right )}}{2} - \frac{\sqrt{-1 + \frac{1}{x^{2}}}}{2 x} & \text{for}\: \frac{1}{\left |{x^{2}}\right |} > 1 \\\frac{i \operatorname{asin}{\left (\frac{1}{x} \right )}}{2} - \frac{i}{2 x \sqrt{1 - \frac{1}{x^{2}}}} + \frac{i}{2 x^{3} \sqrt{1 - \frac{1}{x^{2}}}} & \text{otherwise} \end{cases} + \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.1394, size = 123, normalized size = 2.41 \begin{align*} \frac{x^{2}{\left (\frac{8 \,{\left (\sqrt{-x^{2} + 1} - 1\right )}}{x} - 1\right )}}{8 \,{\left (\sqrt{-x^{2} + 1} - 1\right )}^{2}} - \frac{\sqrt{-x^{2} + 1} - 1}{x} + \frac{{\left (\sqrt{-x^{2} + 1} - 1\right )}^{2}}{8 \, x^{2}} + \frac{3}{2} \, \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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