Optimal. Leaf size=32 \[ -\frac {1}{2 x}+\frac {\sqrt {1-x^2}}{2 x}+\frac {1}{2} \sin ^{-1}(x) \]
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Rubi [A]
time = 0.02, antiderivative size = 32, normalized size of antiderivative = 1.00, 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 = {6822, 283, 222}
\begin {gather*} \frac {\text {ArcSin}(x)}{2}+\frac {\sqrt {1-x^2}}{2 x}-\frac {1}{2 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 222
Rule 283
Rule 6822
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*}
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Mathematica [A]
time = 0.10, size = 41, normalized size = 1.28 \begin {gather*} \frac {-1+\sqrt {1-x^2}+2 x \tan ^{-1}\left (\frac {\sqrt {1+x}}{\sqrt {1-x}}\right )}{2 x} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(49\) vs.
\(2(24)=48\).
time = 0.01, size = 50, normalized size = 1.56
method | result | size |
default | \(-\frac {1}{2 x}-\frac {\left (-\arcsin \left (x \right ) x -\sqrt {-x^{2}+1}\right ) \sqrt {1+x}\, \sqrt {1-x}}{2 x \sqrt {-x^{2}+1}}\) | \(50\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.74, size = 44, normalized size = 1.38 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (\sqrt {1 - x} + \sqrt {x + 1}\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 147 vs.
\(2 (24) = 48\).
time = 0.47, size = 147, normalized size = 4.59 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.62, size = 49, normalized size = 1.53 \begin {gather*} \frac {\left (\frac {x}{2}+\frac {1}{2}\right )\,\sqrt {1-x}}{x\,\sqrt {x+1}}-\frac {1}{2\,x}-2\,\mathrm {atan}\left (\frac {\sqrt {1-x}-1}{\sqrt {x+1}-1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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