Optimal. Leaf size=32 \[ \frac{\sqrt{1-x^2}}{2 x}-\frac{1}{2 x}+\frac{1}{2} \sin ^{-1}(x) \]
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Rubi [A] time = 0.156235, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {6690, 277, 216} \[ \frac{\sqrt{1-x^2}}{2 x}-\frac{1}{2 x}+\frac{1}{2} \sin ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 6690
Rule 277
Rule 216
Rubi steps
\begin{align*} \int \frac{1}{\left (\sqrt{1-x}+\sqrt{1+x}\right )^2} \, dx &=\frac{1}{4} \int \left (\frac{2}{x^2}-\frac{2 \sqrt{1-x^2}}{x^2}\right ) \, dx\\ &=-\frac{1}{2 x}-\frac{1}{2} \int \frac{\sqrt{1-x^2}}{x^2} \, dx\\ &=-\frac{1}{2 x}+\frac{\sqrt{1-x^2}}{2 x}+\frac{1}{2} \int \frac{1}{\sqrt{1-x^2}} \, dx\\ &=-\frac{1}{2 x}+\frac{\sqrt{1-x^2}}{2 x}+\frac{1}{2} \sin ^{-1}(x)\\ \end{align*}
Mathematica [A] time = 0.0171405, size = 24, normalized size = 0.75 \[ \frac{\sqrt{1-x^2}+x \sin ^{-1}(x)-1}{2 x} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.009, size = 50, normalized size = 1.6 \begin{align*} -{\frac{1}{2\,x}}-{\frac{1}{2\,x} \left ( -\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (\sqrt{x + 1} + \sqrt{-x + 1}\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.11214, size = 116, normalized size = 3.62 \begin{align*} -\frac{2 \, x \arctan \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) - \sqrt{x + 1} \sqrt{-x + 1} + 1}{2 \, 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{1}{\left (\sqrt{1 - x} + \sqrt{x + 1}\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.14937, size = 198, normalized size = 6.19 \begin{align*} \frac{1}{2} \, \pi + \frac{2 \,{\left (\frac{\sqrt{2} - \sqrt{-x + 1}}{\sqrt{x + 1}} - \frac{\sqrt{x + 1}}{\sqrt{2} - \sqrt{-x + 1}}\right )}}{{\left (\frac{\sqrt{2} - \sqrt{-x + 1}}{\sqrt{x + 1}} - \frac{\sqrt{x + 1}}{\sqrt{2} - \sqrt{-x + 1}}\right )}^{2} - 4} - \frac{1}{2 \, x} + \arctan \left (\frac{\sqrt{x + 1}{\left (\frac{{\left (\sqrt{2} - \sqrt{-x + 1}\right )}^{2}}{x + 1} - 1\right )}}{2 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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