Optimal. Leaf size=18 \[ -\frac {3}{8} \left (a^2-4 \sin ^2(x)\right )^{2/3} \]
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Rubi [A]
time = 0.03, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {12, 267}
\begin {gather*} -\frac {3}{8} \left (a^2-4 \sin ^2(x)\right )^{2/3} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 267
Rubi steps
\begin {align*} \int \frac {\sin (2 x)}{\sqrt [3]{a^2-4 \sin ^2(x)}} \, dx &=\text {Subst}\left (\int \frac {2 x}{\sqrt [3]{a^2-4 x^2}} \, dx,x,\sin (x)\right )\\ &=2 \text {Subst}\left (\int \frac {x}{\sqrt [3]{a^2-4 x^2}} \, dx,x,\sin (x)\right )\\ &=-\frac {3}{8} \left (a^2-4 \sin ^2(x)\right )^{2/3}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 18, normalized size = 1.00 \begin {gather*} -\frac {3}{8} \left (a^2-4 \sin ^2(x)\right )^{2/3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 15, normalized size = 0.83
method | result | size |
derivativedivides | \(-\frac {3 \left (a^{2}-4 \left (\sin ^{2}\left (x \right )\right )\right )^{\frac {2}{3}}}{8}\) | \(15\) |
default | \(-\frac {3 \left (a^{2}-4 \left (\sin ^{2}\left (x \right )\right )\right )^{\frac {2}{3}}}{8}\) | \(15\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 2.26, size = 14, normalized size = 0.78 \begin {gather*} -\frac {3}{8} \, {\left (a^{2} - 4 \, \sin \left (x\right )^{2}\right )}^{\frac {2}{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.79, size = 15, normalized size = 0.83 \begin {gather*} -\frac {3}{8} \, {\left (a^{2} + 4 \, \cos \left (x\right )^{2} - 4\right )}^{\frac {2}{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.04, size = 17, normalized size = 0.94 \begin {gather*} - \frac {3 \left (a^{2} - 4 \sin ^{2}{\left (x \right )}\right )^{\frac {2}{3}}}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.16, size = 14, normalized size = 0.78 \begin {gather*} -\frac {3}{8} \, {\left (a^{2} - 4 \, \sin \left (x\right )^{2}\right )}^{\frac {2}{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.30, size = 14, normalized size = 0.78 \begin {gather*} -\frac {3\,{\left (a^2-4\,{\sin \left (x\right )}^2\right )}^{2/3}}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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