Optimal. Leaf size=107 \[ -\frac{6 \sqrt{1-x^2}}{5 x}-\frac{3 \sqrt{1-x^2}}{4 x^2}-\frac{3 \sqrt{1-x^2}}{5 x^3}-\frac{\sqrt{1-x^2}}{2 x^4}-\frac{\sqrt{1-x^2}}{5 x^5}-\frac{3}{4} \tanh ^{-1}\left (\sqrt{1-x^2}\right ) \]
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Rubi [A] time = 0.103015, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {1807, 835, 807, 266, 63, 206} \[ -\frac{6 \sqrt{1-x^2}}{5 x}-\frac{3 \sqrt{1-x^2}}{4 x^2}-\frac{3 \sqrt{1-x^2}}{5 x^3}-\frac{\sqrt{1-x^2}}{2 x^4}-\frac{\sqrt{1-x^2}}{5 x^5}-\frac{3}{4} \tanh ^{-1}\left (\sqrt{1-x^2}\right ) \]
Antiderivative was successfully verified.
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Rule 1807
Rule 835
Rule 807
Rule 266
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{(1+x)^2}{x^6 \sqrt{1-x^2}} \, dx &=-\frac{\sqrt{1-x^2}}{5 x^5}-\frac{1}{5} \int \frac{-10-9 x}{x^5 \sqrt{1-x^2}} \, dx\\ &=-\frac{\sqrt{1-x^2}}{5 x^5}-\frac{\sqrt{1-x^2}}{2 x^4}+\frac{1}{20} \int \frac{36+30 x}{x^4 \sqrt{1-x^2}} \, dx\\ &=-\frac{\sqrt{1-x^2}}{5 x^5}-\frac{\sqrt{1-x^2}}{2 x^4}-\frac{3 \sqrt{1-x^2}}{5 x^3}-\frac{1}{60} \int \frac{-90-72 x}{x^3 \sqrt{1-x^2}} \, dx\\ &=-\frac{\sqrt{1-x^2}}{5 x^5}-\frac{\sqrt{1-x^2}}{2 x^4}-\frac{3 \sqrt{1-x^2}}{5 x^3}-\frac{3 \sqrt{1-x^2}}{4 x^2}+\frac{1}{120} \int \frac{144+90 x}{x^2 \sqrt{1-x^2}} \, dx\\ &=-\frac{\sqrt{1-x^2}}{5 x^5}-\frac{\sqrt{1-x^2}}{2 x^4}-\frac{3 \sqrt{1-x^2}}{5 x^3}-\frac{3 \sqrt{1-x^2}}{4 x^2}-\frac{6 \sqrt{1-x^2}}{5 x}+\frac{3}{4} \int \frac{1}{x \sqrt{1-x^2}} \, dx\\ &=-\frac{\sqrt{1-x^2}}{5 x^5}-\frac{\sqrt{1-x^2}}{2 x^4}-\frac{3 \sqrt{1-x^2}}{5 x^3}-\frac{3 \sqrt{1-x^2}}{4 x^2}-\frac{6 \sqrt{1-x^2}}{5 x}+\frac{3}{8} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x} x} \, dx,x,x^2\right )\\ &=-\frac{\sqrt{1-x^2}}{5 x^5}-\frac{\sqrt{1-x^2}}{2 x^4}-\frac{3 \sqrt{1-x^2}}{5 x^3}-\frac{3 \sqrt{1-x^2}}{4 x^2}-\frac{6 \sqrt{1-x^2}}{5 x}-\frac{3}{4} \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\sqrt{1-x^2}\right )\\ &=-\frac{\sqrt{1-x^2}}{5 x^5}-\frac{\sqrt{1-x^2}}{2 x^4}-\frac{3 \sqrt{1-x^2}}{5 x^3}-\frac{3 \sqrt{1-x^2}}{4 x^2}-\frac{6 \sqrt{1-x^2}}{5 x}-\frac{3}{4} \tanh ^{-1}\left (\sqrt{1-x^2}\right )\\ \end{align*}
Mathematica [C] time = 0.0177882, size = 50, normalized size = 0.47 \[ -\frac{\sqrt{1-x^2} \left (10 x^5 \, _2F_1\left (\frac{1}{2},3;\frac{3}{2};1-x^2\right )+6 x^4+3 x^2+1\right )}{5 x^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.05, size = 84, normalized size = 0.8 \begin{align*} -{\frac{3}{5\,{x}^{3}}\sqrt{-{x}^{2}+1}}-{\frac{6}{5\,x}\sqrt{-{x}^{2}+1}}-{\frac{1}{5\,{x}^{5}}\sqrt{-{x}^{2}+1}}-{\frac{1}{2\,{x}^{4}}\sqrt{-{x}^{2}+1}}-{\frac{3}{4\,{x}^{2}}\sqrt{-{x}^{2}+1}}-{\frac{3}{4}{\it Artanh} \left ({\frac{1}{\sqrt{-{x}^{2}+1}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.70761, size = 130, normalized size = 1.21 \begin{align*} -\frac{6 \, \sqrt{-x^{2} + 1}}{5 \, x} - \frac{3 \, \sqrt{-x^{2} + 1}}{4 \, x^{2}} - \frac{3 \, \sqrt{-x^{2} + 1}}{5 \, x^{3}} - \frac{\sqrt{-x^{2} + 1}}{2 \, x^{4}} - \frac{\sqrt{-x^{2} + 1}}{5 \, x^{5}} - \frac{3}{4} \, \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 = 1.85401, size = 138, normalized size = 1.29 \begin{align*} \frac{15 \, x^{5} \log \left (\frac{\sqrt{-x^{2} + 1} - 1}{x}\right ) -{\left (24 \, x^{4} + 15 \, x^{3} + 12 \, x^{2} + 10 \, x + 4\right )} \sqrt{-x^{2} + 1}}{20 \, x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 14.0711, size = 201, normalized size = 1.88 \begin{align*} \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{\sqrt{1 - x^{2}}}{x} - \frac{2 \left (1 - x^{2}\right )^{\frac{3}{2}}}{3 x^{3}} - \frac{\left (1 - x^{2}\right )^{\frac{5}{2}}}{5 x^{5}} & \text{for}\: x > -1 \wedge x < 1 \end{cases} + 2 \left (\begin{cases} - \frac{3 \operatorname{acosh}{\left (\frac{1}{x} \right )}}{8} + \frac{3}{8 x \sqrt{-1 + \frac{1}{x^{2}}}} - \frac{1}{8 x^{3} \sqrt{-1 + \frac{1}{x^{2}}}} - \frac{1}{4 x^{5} \sqrt{-1 + \frac{1}{x^{2}}}} & \text{for}\: \frac{1}{\left |{x^{2}}\right |} > 1 \\\frac{3 i \operatorname{asin}{\left (\frac{1}{x} \right )}}{8} - \frac{3 i}{8 x \sqrt{1 - \frac{1}{x^{2}}}} + \frac{i}{8 x^{3} \sqrt{1 - \frac{1}{x^{2}}}} + \frac{i}{4 x^{5} \sqrt{1 - \frac{1}{x^{2}}}} & \text{otherwise} \end{cases}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.13081, size = 269, normalized size = 2.51 \begin{align*} -\frac{x^{5}{\left (\frac{5 \,{\left (\sqrt{-x^{2} + 1} - 1\right )}}{x} - \frac{15 \,{\left (\sqrt{-x^{2} + 1} - 1\right )}^{2}}{x^{2}} + \frac{40 \,{\left (\sqrt{-x^{2} + 1} - 1\right )}^{3}}{x^{3}} - \frac{110 \,{\left (\sqrt{-x^{2} + 1} - 1\right )}^{4}}{x^{4}} - 1\right )}}{160 \,{\left (\sqrt{-x^{2} + 1} - 1\right )}^{5}} - \frac{11 \,{\left (\sqrt{-x^{2} + 1} - 1\right )}}{16 \, x} + \frac{{\left (\sqrt{-x^{2} + 1} - 1\right )}^{2}}{4 \, x^{2}} - \frac{3 \,{\left (\sqrt{-x^{2} + 1} - 1\right )}^{3}}{32 \, x^{3}} + \frac{{\left (\sqrt{-x^{2} + 1} - 1\right )}^{4}}{32 \, x^{4}} - \frac{{\left (\sqrt{-x^{2} + 1} - 1\right )}^{5}}{160 \, x^{5}} + \frac{3}{4} \, \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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