Optimal. Leaf size=16 \[ -\frac {2}{9} \left (4-3 \sin ^2(x)\right )^{3/2} \]
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Rubi [A]
time = 0.04, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {12, 267}
\begin {gather*} -\frac {2}{9} \left (4-3 \sin ^2(x)\right )^{3/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 267
Rubi steps
\begin {align*} \int \sqrt {1+3 \cos ^2(x)} \sin (2 x) \, dx &=\text {Subst}\left (\int 2 x \sqrt {4-3 x^2} \, dx,x,\sin (x)\right )\\ &=2 \text {Subst}\left (\int x \sqrt {4-3 x^2} \, dx,x,\sin (x)\right )\\ &=-\frac {2}{9} \left (4-3 \sin ^2(x)\right )^{3/2}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 16, normalized size = 1.00 \begin {gather*} -\frac {2}{9} \left (4-3 \sin ^2(x)\right )^{3/2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 13, normalized size = 0.81
method | result | size |
derivativedivides | \(-\frac {2 \left (1+3 \left (\cos ^{2}\left (x \right )\right )\right )^{\frac {3}{2}}}{9}\) | \(13\) |
default | \(-\frac {2 \left (1+3 \left (\cos ^{2}\left (x \right )\right )\right )^{\frac {3}{2}}}{9}\) | \(13\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.83, size = 12, normalized size = 0.75 \begin {gather*} -\frac {2}{9} \, {\left (3 \, \cos \left (x\right )^{2} + 1\right )}^{\frac {3}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.19, size = 12, normalized size = 0.75 \begin {gather*} -\frac {2}{9} \, {\left (3 \, \cos \left (x\right )^{2} + 1\right )}^{\frac {3}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.98, size = 15, normalized size = 0.94 \begin {gather*} - \frac {2 \left (3 \cos ^{2}{\left (x \right )} + 1\right )^{\frac {3}{2}}}{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.44, size = 12, normalized size = 0.75 \begin {gather*} -\frac {2}{9} \, {\left (3 \, \cos \left (x\right )^{2} + 1\right )}^{\frac {3}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.19, size = 12, normalized size = 0.75 \begin {gather*} -\frac {2\,{\left (3\,{\cos \left (x\right )}^2+1\right )}^{3/2}}{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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