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 = 0.06, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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 63
Rule 206
Rule 266
Rule 807
Rule 1807
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*}
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Mathematica [A] time = 0.02, 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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fricas [A] time = 0.84, size = 43, normalized size = 0.84 \[ \frac {3 \, x^{2} \log \left (\frac {\sqrt {-x^{2} + 1} - 1}{x}\right ) - \sqrt {-x^{2} + 1} {\left (4 \, x + 1\right )}}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.18, size = 91, normalized size = 1.78 \[ \frac {x^{2} {\left (\frac {8 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}}{x} - 1\right )}}{8 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}^{2}} - \frac {\sqrt {-x^{2} + 1} - 1}{x} + \frac {{\left (\sqrt {-x^{2} + 1} - 1\right )}^{2}}{8 \, x^{2}} + \frac {3}{2} \, \log \left (-\frac {\sqrt {-x^{2} + 1} - 1}{{\left | x \right |}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 42, normalized size = 0.82 \[ -\frac {3 \arctanh \left (\frac {1}{\sqrt {-x^{2}+1}}\right )}{2}-\frac {2 \sqrt {-x^{2}+1}}{x}-\frac {\sqrt {-x^{2}+1}}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.97, size = 54, normalized size = 1.06 \[ -\frac {2 \, \sqrt {-x^{2} + 1}}{x} - \frac {\sqrt {-x^{2} + 1}}{2 \, x^{2}} - \frac {3}{2} \, \log \left (\frac {2 \, \sqrt {-x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.49, size = 47, normalized size = 0.92 \[ \frac {3\,\ln \left (\sqrt {\frac {1}{x^2}-1}-\sqrt {\frac {1}{x^2}}\right )}{2}-\frac {2\,\sqrt {1-x^2}}{x}-\frac {\sqrt {1-x^2}}{2\,x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 7.03, size = 116, normalized size = 2.27 \[ 2 \left (\begin {cases} - \frac {i \sqrt {x^{2} - 1}}{x} & \text {for}\: \left |{x^{2}}\right | > 1 \\- \frac {\sqrt {1 - x^{2}}}{x} & \text {otherwise} \end {cases}\right ) + \begin {cases} - \frac {\operatorname {acosh}{\left (\frac {1}{x} \right )}}{2} - \frac {\sqrt {-1 + \frac {1}{x^{2}}}}{2 x} & \text {for}\: \frac {1}{\left |{x^{2}}\right |} > 1 \\\frac {i \operatorname {asin}{\left (\frac {1}{x} \right )}}{2} - \frac {i}{2 x \sqrt {1 - \frac {1}{x^{2}}}} + \frac {i}{2 x^{3} \sqrt {1 - \frac {1}{x^{2}}}} & \text {otherwise} \end {cases} + \begin {cases} - \operatorname {acosh}{\left (\frac {1}{x} \right )} & \text {for}\: \frac {1}{\left |{x^{2}}\right |} > 1 \\i \operatorname {asin}{\left (\frac {1}{x} \right )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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