Optimal. Leaf size=44 \[ -\frac{\sqrt{1-x} \sqrt{x+1}}{x}+\sin ^{-1}(x)-2 \tanh ^{-1}\left (\sqrt{1-x} \sqrt{x+1}\right ) \]
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Rubi [A] time = 0.0165006, antiderivative size = 44, 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 = {98, 157, 41, 216, 92, 206} \[ -\frac{\sqrt{1-x} \sqrt{x+1}}{x}+\sin ^{-1}(x)-2 \tanh ^{-1}\left (\sqrt{1-x} \sqrt{x+1}\right ) \]
Antiderivative was successfully verified.
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Rule 98
Rule 157
Rule 41
Rule 216
Rule 92
Rule 206
Rubi steps
\begin{align*} \int \frac{(1+x)^{3/2}}{\sqrt{1-x} x^2} \, dx &=-\frac{\sqrt{1-x} \sqrt{1+x}}{x}-\int \frac{-2-x}{\sqrt{1-x} x \sqrt{1+x}} \, dx\\ &=-\frac{\sqrt{1-x} \sqrt{1+x}}{x}+2 \int \frac{1}{\sqrt{1-x} x \sqrt{1+x}} \, dx+\int \frac{1}{\sqrt{1-x} \sqrt{1+x}} \, dx\\ &=-\frac{\sqrt{1-x} \sqrt{1+x}}{x}-2 \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\sqrt{1-x} \sqrt{1+x}\right )+\int \frac{1}{\sqrt{1-x^2}} \, dx\\ &=-\frac{\sqrt{1-x} \sqrt{1+x}}{x}+\sin ^{-1}(x)-2 \tanh ^{-1}\left (\sqrt{1-x} \sqrt{1+x}\right )\\ \end{align*}
Mathematica [A] time = 0.0474442, size = 66, normalized size = 1.5 \[ \frac{x}{\sqrt{1-x^2}}-\frac{1}{\sqrt{1-x^2} x}-2 \tanh ^{-1}\left (\sqrt{1-x^2}\right )+2 \sin ^{-1}\left (\frac{\sqrt{1-x}}{\sqrt{2}}\right )+2 \sin ^{-1}(x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 55, normalized size = 1.3 \begin{align*}{\frac{1}{x} \left ( -2\,{\it Artanh} \left ({\frac{1}{\sqrt{-{x}^{2}+1}}} \right ) x+\arcsin \left ( x \right ) x-\sqrt{-{x}^{2}+1} \right ) \sqrt{1+x}\sqrt{1-x}{\frac{1}{\sqrt{-{x}^{2}+1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.75508, size = 57, normalized size = 1.3 \begin{align*} -\frac{\sqrt{-x^{2} + 1}}{x} + \arcsin \left (x\right ) - 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 = 1.75544, size = 165, normalized size = 3.75 \begin{align*} -\frac{2 \, x \arctan \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) - 2 \, x \log \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) + \sqrt{x + 1} \sqrt{-x + 1}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (x + 1\right )^{\frac{3}{2}}}{x^{2} \sqrt{1 - x}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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