Optimal. Leaf size=67 \[ -\frac{5 \sqrt{1-x^2}}{3 x}-\frac{\sqrt{1-x^2}}{x^2}-\frac{\sqrt{1-x^2}}{3 x^3}-\tanh ^{-1}\left (\sqrt{1-x^2}\right ) \]
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Rubi [A] time = 0.0737916, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 6, 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} \[ -\frac{5 \sqrt{1-x^2}}{3 x}-\frac{\sqrt{1-x^2}}{x^2}-\frac{\sqrt{1-x^2}}{3 x^3}-\tanh ^{-1}\left (\sqrt{1-x^2}\right ) \]
Antiderivative was successfully verified.
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Rule 1807
Rule 835
Rule 807
Rule 266
Rule 63
Rule 206
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*}
Mathematica [A] time = 0.0231655, size = 43, normalized size = 0.64 \[ -\frac{\sqrt{1-x^2} \left (5 x^2+3 x+1\right )}{3 x^3}-\tanh ^{-1}\left (\sqrt{1-x^2}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.051, size = 56, normalized size = 0.8 \begin{align*} -{\frac{1}{3\,{x}^{3}}\sqrt{-{x}^{2}+1}}-{\frac{5}{3\,x}\sqrt{-{x}^{2}+1}}-{\frac{1}{{x}^{2}}\sqrt{-{x}^{2}+1}}-{\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 = 1.48349, size = 92, normalized size = 1.37 \begin{align*} -\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.84516, size = 108, normalized size = 1.61 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 7.73114, size = 128, normalized size = 1.91 \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{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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.11811, size = 169, normalized size = 2.52 \begin{align*} -\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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