Optimal. Leaf size=23 \[ -\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {1-x}}{\sqrt {2}}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {641, 65, 212}
\begin {gather*} -\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {1-x}}{\sqrt {2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 212
Rule 641
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {1+x} \sqrt {1-x^2}} \, dx &=\int \frac {1}{\sqrt {1-x} (1+x)} \, dx\\ &=-\left (2 \text {Subst}\left (\int \frac {1}{2-x^2} \, dx,x,\sqrt {1-x}\right )\right )\\ &=-\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {1-x}}{\sqrt {2}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 32, normalized size = 1.39 \begin {gather*} -\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {1+x}}{\sqrt {1-x^2}}\right ) \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. \(39\) vs.
\(2(18)=36\).
time = 0.47, size = 40, normalized size = 1.74
method | result | size |
default | \(-\frac {\sqrt {-x^{2}+1}\, \arctanh \left (\frac {\sqrt {1-x}\, \sqrt {2}}{2}\right ) \sqrt {2}}{\sqrt {x +1}\, \sqrt {1-x}}\) | \(40\) |
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 45 vs.
\(2 (18) = 36\).
time = 3.74, size = 45, normalized size = 1.96 \begin {gather*} \frac {1}{2} \, \sqrt {2} \log \left (-\frac {x^{2} + 2 \, \sqrt {2} \sqrt {-x^{2} + 1} \sqrt {x + 1} - 2 \, x - 3}{x^{2} + 2 \, x + 1}\right ) \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}{\sqrt {- \left (x - 1\right ) \left (x + 1\right )} \sqrt {x + 1}}\, 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. 37 vs.
\(2 (18) = 36\).
time = 1.52, size = 37, normalized size = 1.61 \begin {gather*} -\frac {1}{2} \, \sqrt {2} \log \left (\sqrt {2} + \sqrt {-x + 1}\right ) + \frac {1}{2} \, \sqrt {2} \log \left (\sqrt {2} - \sqrt {-x + 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {1}{\sqrt {1-x^2}\,\sqrt {x+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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