Optimal. Leaf size=18 \[ -\frac{3}{8} \left (a^2-4 \sin ^2(x)\right )^{2/3} \]
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Rubi [A] time = 0.044924, antiderivative size = 18, normalized size of antiderivative = 1., 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, 261} \[ -\frac{3}{8} \left (a^2-4 \sin ^2(x)\right )^{2/3} \]
Antiderivative was successfully verified.
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Rule 12
Rule 261
Rubi steps
\begin{align*} \int \frac{\sin (2 x)}{\sqrt [3]{a^2-4 \sin ^2(x)}} \, dx &=\operatorname{Subst}\left (\int \frac{2 x}{\sqrt [3]{a^2-4 x^2}} \, dx,x,\sin (x)\right )\\ &=2 \operatorname{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*}
Mathematica [A] time = 0.0185391, size = 18, normalized size = 1. \[ -\frac{3}{8} \left (a^2-4 \sin ^2(x)\right )^{2/3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.02, size = 15, normalized size = 0.8 \begin{align*} -{\frac{3}{8} \left ({a}^{2}-4\, \left ( \sin \left ( x \right ) \right ) ^{2} \right ) ^{{\frac{2}{3}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.945078, size = 19, normalized size = 1.06 \begin{align*} -\frac{3}{8} \,{\left (a^{2} - 4 \, \sin \left (x\right )^{2}\right )}^{\frac{2}{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.15758, size = 47, normalized size = 2.61 \begin{align*} -\frac{3}{8} \,{\left (a^{2} + 4 \, \cos \left (x\right )^{2} - 4\right )}^{\frac{2}{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.57448, size = 17, normalized size = 0.94 \begin{align*} - \frac{3 \left (a^{2} - 4 \sin ^{2}{\left (x \right )}\right )^{\frac{2}{3}}}{8} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15055, size = 35, normalized size = 1.94 \begin{align*} -\frac{3}{8} \,{\left (a^{2} - \frac{16 \, \tan \left (\frac{1}{2} \, x\right )^{2}}{{\left (\tan \left (\frac{1}{2} \, x\right )^{2} + 1\right )}^{2}}\right )}^{\frac{2}{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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