Optimal. Leaf size=43 \[ \frac{4 \sqrt{x+1}}{\sqrt{1-x}}-\sin ^{-1}(x)-\tanh ^{-1}\left (\sqrt{1-x} \sqrt{x+1}\right ) \]
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Rubi [A] time = 0.0129731, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35, Rules used = {98, 21, 105, 41, 216, 92, 206} \[ \frac{4 \sqrt{x+1}}{\sqrt{1-x}}-\sin ^{-1}(x)-\tanh ^{-1}\left (\sqrt{1-x} \sqrt{x+1}\right ) \]
Antiderivative was successfully verified.
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Rule 98
Rule 21
Rule 105
Rule 41
Rule 216
Rule 92
Rule 206
Rubi steps
\begin{align*} \int \frac{(1+x)^{3/2}}{(1-x)^{3/2} x} \, dx &=\frac{4 \sqrt{1+x}}{\sqrt{1-x}}-2 \int \frac{-\frac{1}{2}+\frac{x}{2}}{\sqrt{1-x} x \sqrt{1+x}} \, dx\\ &=\frac{4 \sqrt{1+x}}{\sqrt{1-x}}+\int \frac{\sqrt{1-x}}{x \sqrt{1+x}} \, dx\\ &=\frac{4 \sqrt{1+x}}{\sqrt{1-x}}-\int \frac{1}{\sqrt{1-x} \sqrt{1+x}} \, dx+\int \frac{1}{\sqrt{1-x} x \sqrt{1+x}} \, dx\\ &=\frac{4 \sqrt{1+x}}{\sqrt{1-x}}-\int \frac{1}{\sqrt{1-x^2}} \, dx-\operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\sqrt{1-x} \sqrt{1+x}\right )\\ &=\frac{4 \sqrt{1+x}}{\sqrt{1-x}}-\sin ^{-1}(x)-\tanh ^{-1}\left (\sqrt{1-x} \sqrt{1+x}\right )\\ \end{align*}
Mathematica [A] time = 0.0381312, size = 61, normalized size = 1.42 \[ \frac{2 \left (\sqrt{1-x^2} \sin ^{-1}\left (\frac{\sqrt{1-x}}{\sqrt{2}}\right )+2 x+2\right )}{\sqrt{1-x^2}}-\tanh ^{-1}\left (\sqrt{1-x^2}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.015, size = 70, normalized size = 1.6 \begin{align*}{\frac{1}{x-1} \left ( -\arcsin \left ( x \right ) x-{\it Artanh} \left ({\frac{1}{\sqrt{-{x}^{2}+1}}} \right ) x+\arcsin \left ( x \right ) +{\it Artanh} \left ({\frac{1}{\sqrt{-{x}^{2}+1}}} \right ) -4\,\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 = 1.56226, size = 72, normalized size = 1.67 \begin{align*} \frac{4 \, x}{\sqrt{-x^{2} + 1}} + \frac{4}{\sqrt{-x^{2} + 1}} - \arcsin \left (x\right ) - \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 [B] time = 1.50558, size = 201, normalized size = 4.67 \begin{align*} \frac{2 \,{\left (x - 1\right )} \arctan \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) +{\left (x - 1\right )} \log \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) + 4 \, x - 4 \, \sqrt{x + 1} \sqrt{-x + 1} - 4}{x - 1} \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 \left (1 - x\right )^{\frac{3}{2}}}\, 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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