Optimal. Leaf size=13 \[ -\frac {2}{3} (1-x)^{3/2} \]
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Rubi [A]
time = 0.00, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {26, 32}
\begin {gather*} -\frac {2}{3} (1-x)^{3/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 26
Rule 32
Rubi steps
\begin {align*} \int \frac {\sqrt {1-x^2}}{\sqrt {1+x}} \, dx &=\int \sqrt {1-x} \, dx\\ &=-\frac {2}{3} (1-x)^{3/2}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 22, normalized size = 1.69 \begin {gather*} -\frac {2 \left (1-x^2\right )^{3/2}}{3 (1+x)^{3/2}} \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. \(19\) vs.
\(2(9)=18\).
time = 0.60, size = 20, normalized size = 1.54
method | result | size |
gosper | \(\frac {2 \left (-1+x \right ) \sqrt {-x^{2}+1}}{3 \sqrt {1+x}}\) | \(20\) |
default | \(\frac {2 \left (-1+x \right ) \sqrt {-x^{2}+1}}{3 \sqrt {1+x}}\) | \(20\) |
risch | \(-\frac {2 \sqrt {\frac {-x^{2}+1}{1+x}}\, \sqrt {1+x}\, \left (-1+x \right )^{2}}{3 \sqrt {-x^{2}+1}\, \sqrt {1-x}}\) | \(44\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 12, normalized size = 0.92 \begin {gather*} \frac {2}{3} \, {\left (x - 1\right )} \sqrt {-x + 1} \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. 19 vs.
\(2 (9) = 18\).
time = 0.34, size = 19, normalized size = 1.46 \begin {gather*} \frac {2 \, \sqrt {-x^{2} + 1} {\left (x - 1\right )}}{3 \, \sqrt {x + 1}} \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 {\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 [A]
time = 2.30, size = 15, normalized size = 1.15 \begin {gather*} -\frac {2}{3} \, {\left (-x + 1\right )}^{\frac {3}{2}} + \frac {4}{3} \, \sqrt {2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.52, size = 20, normalized size = 1.54 \begin {gather*} \frac {\left (\frac {2\,x}{3}-\frac {2}{3}\right )\,\sqrt {1-x^2}}{\sqrt {x+1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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