Optimal. Leaf size=10 \[ \frac {1}{3} \sin ^{-1}\left (\frac {3 x}{2}\right ) \]
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Rubi [A] time = 0.00, antiderivative size = 10, 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 = {41, 216} \[ \frac {1}{3} \sin ^{-1}\left (\frac {3 x}{2}\right ) \]
Antiderivative was successfully verified.
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Rule 41
Rule 216
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {2-3 x} \sqrt {2+3 x}} \, dx &=\int \frac {1}{\sqrt {4-9 x^2}} \, dx\\ &=\frac {1}{3} \sin ^{-1}\left (\frac {3 x}{2}\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 10, normalized size = 1.00 \[ \frac {1}{3} \sin ^{-1}\left (\frac {3 x}{2}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 25, normalized size = 2.50 \[ -\frac {2}{3} \, \arctan \left (\frac {\sqrt {3 \, x + 2} \sqrt {-3 \, x + 2} - 2}{3 \, x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.37, size = 12, normalized size = 1.20 \[ \frac {2}{3} \, \arcsin \left (\frac {1}{2} \, \sqrt {3 \, x + 2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 34, normalized size = 3.40 \[ \frac {\sqrt {\left (-3 x +2\right ) \left (3 x +2\right )}\, \arcsin \left (\frac {3 x}{2}\right )}{3 \sqrt {-3 x +2}\, \sqrt {3 x +2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.93, size = 6, normalized size = 0.60 \[ \frac {1}{3} \, \arcsin \left (\frac {3}{2} \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.15, size = 32, normalized size = 3.20 \[ -\frac {4\,\mathrm {atan}\left (\frac {\sqrt {2}-\sqrt {2-3\,x}}{\sqrt {2}-\sqrt {3\,x+2}}\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.04, size = 51, normalized size = 5.10 \[ \begin {cases} - \frac {2 i \operatorname {acosh}{\left (\frac {\sqrt {3} \sqrt {x + \frac {2}{3}}}{2} \right )}}{3} & \text {for}\: \frac {3 \left |{x + \frac {2}{3}}\right |}{4} > 1 \\\frac {2 \operatorname {asin}{\left (\frac {\sqrt {3} \sqrt {x + \frac {2}{3}}}{2} \right )}}{3} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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