Optimal. Leaf size=23 \[ -\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {1-x}}{\sqrt {2}}\right ) \]
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Rubi [A]
time = 0.00, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {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
Rubi steps
\begin {align*} \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 = 0.03, size = 23, normalized size = 1.00 \begin {gather*} -\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {1-x}}{\sqrt {2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.47, size = 19, normalized size = 0.83
method | result | size |
derivativedivides | \(-\arctanh \left (\frac {\sqrt {1-x}\, \sqrt {2}}{2}\right ) \sqrt {2}\) | \(19\) |
default | \(-\arctanh \left (\frac {\sqrt {1-x}\, \sqrt {2}}{2}\right ) \sqrt {2}\) | \(19\) |
trager | \(-\frac {\RootOf \left (\textit {\_Z}^{2}-2\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}-2\right ) x -3 \RootOf \left (\textit {\_Z}^{2}-2\right )-4 \sqrt {1-x}}{x +1}\right )}{2}\) | \(43\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 34, normalized size = 1.48 \begin {gather*} \frac {1}{2} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - \sqrt {-x + 1}}{\sqrt {2} + \sqrt {-x + 1}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.62, size = 27, normalized size = 1.17 \begin {gather*} \frac {1}{2} \, \sqrt {2} \log \left (\frac {x + 2 \, \sqrt {2} \sqrt {-x + 1} - 3}{x + 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 0.46, size = 46, normalized size = 2.00 \begin {gather*} \begin {cases} - \sqrt {2} \operatorname {acosh}{\left (\frac {\sqrt {2}}{\sqrt {x + 1}} \right )} & \text {for}\: \frac {1}{\left |{x + 1}\right |} > \frac {1}{2} \\\sqrt {2} i \operatorname {asin}{\left (\frac {\sqrt {2}}{\sqrt {x + 1}} \right )} & \text {otherwise} \end {cases} \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. 38 vs.
\(2 (18) = 36\).
time = 1.65, size = 38, normalized size = 1.65 \begin {gather*} -\frac {1}{2} \, \sqrt {2} \log \left (\sqrt {2} + \sqrt {-x + 1}\right ) + \frac {1}{2} \, \sqrt {2} \log \left ({\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 = 0.09, size = 18, normalized size = 0.78 \begin {gather*} -\sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {1-x}}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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