Optimal. Leaf size=67 \[ -\frac {5 \sqrt {1-x^2}}{3 x}-\frac {\sqrt {1-x^2}}{x^2}-\tanh ^{-1}\left (\sqrt {1-x^2}\right )-\frac {\sqrt {1-x^2}}{3 x^3} \]
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Rubi [A] time = 0.07, antiderivative size = 67, 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, 835, 807, 266, 63, 206} \[ -\frac {5 \sqrt {1-x^2}}{3 x}-\frac {\sqrt {1-x^2}}{x^2}-\frac {\sqrt {1-x^2}}{3 x^3}-\tanh ^{-1}\left (\sqrt {1-x^2}\right ) \]
Antiderivative was successfully verified.
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Rule 63
Rule 206
Rule 266
Rule 807
Rule 835
Rule 1807
Rubi steps
\begin {align*} \int \frac {(1+x)^2}{x^4 \sqrt {1-x^2}} \, dx &=-\frac {\sqrt {1-x^2}}{3 x^3}-\frac {1}{3} \int \frac {-6-5 x}{x^3 \sqrt {1-x^2}} \, dx\\ &=-\frac {\sqrt {1-x^2}}{3 x^3}-\frac {\sqrt {1-x^2}}{x^2}+\frac {1}{6} \int \frac {10+6 x}{x^2 \sqrt {1-x^2}} \, dx\\ &=-\frac {\sqrt {1-x^2}}{3 x^3}-\frac {\sqrt {1-x^2}}{x^2}-\frac {5 \sqrt {1-x^2}}{3 x}+\int \frac {1}{x \sqrt {1-x^2}} \, dx\\ &=-\frac {\sqrt {1-x^2}}{3 x^3}-\frac {\sqrt {1-x^2}}{x^2}-\frac {5 \sqrt {1-x^2}}{3 x}+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x} x} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {1-x^2}}{3 x^3}-\frac {\sqrt {1-x^2}}{x^2}-\frac {5 \sqrt {1-x^2}}{3 x}-\operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {1-x^2}\right )\\ &=-\frac {\sqrt {1-x^2}}{3 x^3}-\frac {\sqrt {1-x^2}}{x^2}-\frac {5 \sqrt {1-x^2}}{3 x}-\tanh ^{-1}\left (\sqrt {1-x^2}\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 43, normalized size = 0.64 \[ -\tanh ^{-1}\left (\sqrt {1-x^2}\right )-\frac {\sqrt {1-x^2} \left (5 x^2+3 x+1\right )}{3 x^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 48, normalized size = 0.72 \[ \frac {3 \, x^{3} \log \left (\frac {\sqrt {-x^{2} + 1} - 1}{x}\right ) - {\left (5 \, x^{2} + 3 \, x + 1\right )} \sqrt {-x^{2} + 1}}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.19, size = 125, normalized size = 1.87 \[ -\frac {x^{3} {\left (\frac {6 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}}{x} - \frac {21 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}^{2}}{x^{2}} - 1\right )}}{24 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}^{3}} - \frac {7 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}}{8 \, x} + \frac {{\left (\sqrt {-x^{2} + 1} - 1\right )}^{2}}{4 \, x^{2}} - \frac {{\left (\sqrt {-x^{2} + 1} - 1\right )}^{3}}{24 \, x^{3}} + \log \left (-\frac {\sqrt {-x^{2} + 1} - 1}{{\left | x \right |}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 56, normalized size = 0.84 \[ -\arctanh \left (\frac {1}{\sqrt {-x^{2}+1}}\right )-\frac {5 \sqrt {-x^{2}+1}}{3 x}-\frac {\sqrt {-x^{2}+1}}{x^{2}}-\frac {\sqrt {-x^{2}+1}}{3 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.96, size = 68, normalized size = 1.01 \[ -\frac {5 \, \sqrt {-x^{2} + 1}}{3 \, x} - \frac {\sqrt {-x^{2} + 1}}{x^{2}} - \frac {\sqrt {-x^{2} + 1}}{3 \, x^{3}} - \log \left (\frac {2 \, \sqrt {-x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 67, normalized size = 1.00 \[ \ln \left (\sqrt {\frac {1}{x^2}-1}-\sqrt {\frac {1}{x^2}}\right )-\sqrt {1-x^2}\,\left (\frac {2}{3\,x}+\frac {1}{3\,x^3}\right )-\frac {\sqrt {1-x^2}}{x}-\frac {\sqrt {1-x^2}}{x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 8.35, size = 128, normalized size = 1.91 \[ \begin {cases} - \frac {\sqrt {1 - x^{2}}}{x} - \frac {\left (1 - x^{2}\right )^{\frac {3}{2}}}{3 x^{3}} & \text {for}\: x > -1 \wedge x < 1 \end {cases} + \begin {cases} - \frac {i \sqrt {x^{2} - 1}}{x} & \text {for}\: \left |{x^{2}}\right | > 1 \\- \frac {\sqrt {1 - x^{2}}}{x} & \text {otherwise} \end {cases} + 2 \left (\begin {cases} - \frac {\operatorname {acosh}{\left (\frac {1}{x} \right )}}{2} - \frac {\sqrt {-1 + \frac {1}{x^{2}}}}{2 x} & \text {for}\: \frac {1}{\left |{x^{2}}\right |} > 1 \\\frac {i \operatorname {asin}{\left (\frac {1}{x} \right )}}{2} - \frac {i}{2 x \sqrt {1 - \frac {1}{x^{2}}}} + \frac {i}{2 x^{3} \sqrt {1 - \frac {1}{x^{2}}}} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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