Optimal. Leaf size=13 \[ \frac {\sin ^{-1}(2 x)}{2 \sqrt {6}} \]
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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 = {41, 222}
\begin {gather*} \frac {\sin ^{-1}(2 x)}{2 \sqrt {6}} \end {gather*}
Antiderivative was successfully verified.
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Rule 41
Rule 222
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {3-6 x} \sqrt {2+4 x}} \, dx &=\int \frac {1}{\sqrt {6-24 x^2}} \, dx\\ &=\frac {\sin ^{-1}(2 x)}{2 \sqrt {6}}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(27\) vs. \(2(13)=26\).
time = 0.04, size = 27, normalized size = 2.08 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\sqrt {1-4 x^2}}{1+2 x}\right )}{\sqrt {6}} \end {gather*}
Antiderivative was successfully verified.
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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 2.80, size = 38, normalized size = 2.92 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\left (-\frac {I}{6}\right ) \sqrt {6} \text {ArcCosh}\left [\frac {\sqrt {2} \sqrt {1+2 x}}{2}\right ],\text {Abs}\left [\frac {1}{2}+x\right ]>1\right \}\right \},\frac {\sqrt {6} \text {ArcSin}\left [\sqrt {\frac {1}{2}+x}\right ]}{6}\right ] \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(36\) vs.
\(2(9)=18\).
time = 0.16, size = 37, normalized size = 2.85
method | result | size |
default | \(\frac {\sqrt {\left (2+4 x \right ) \left (3-6 x \right )}\, \arcsin \left (2 x \right ) \sqrt {6}}{12 \sqrt {2+4 x}\, \sqrt {3-6 x}}\) | \(37\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.35, size = 9, normalized size = 0.69 \begin {gather*} \frac {1}{12} \, \sqrt {6} \arcsin \left (2 \, x\right ) \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. 28 vs.
\(2 (9) = 18\).
time = 0.30, size = 28, normalized size = 2.15 \begin {gather*} -\frac {1}{12} \, \sqrt {6} \arctan \left (\frac {\sqrt {6} \sqrt {4 \, x + 2} \sqrt {-6 \, x + 3}}{12 \, x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.06, size = 41, normalized size = 3.15 \begin {gather*} \begin {cases} - \frac {\sqrt {6} i \operatorname {acosh}{\left (\sqrt {x + \frac {1}{2}} \right )}}{6} & \text {for}\: \left |{x + \frac {1}{2}}\right | > 1 \\\frac {\sqrt {6} \operatorname {asin}{\left (\sqrt {x + \frac {1}{2}} \right )}}{6} & \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. 21 vs.
\(2 (9) = 18\).
time = 0.00, size = 32, normalized size = 2.46 \begin {gather*} -\frac {\arcsin \left (\frac {\sqrt {-2 x+1}}{\sqrt {2}}\right )}{\sqrt {3} \sqrt {2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 40, normalized size = 3.08 \begin {gather*} -\frac {\sqrt {6}\,\mathrm {atan}\left (\frac {\sqrt {24}\,\left (\sqrt {3}-\sqrt {3-6\,x}\right )}{6\,\left (\sqrt {2}-\sqrt {4\,x+2}\right )}\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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