Optimal. Leaf size=18 \[ -\frac {3}{8} \left (a^2-4 \sin ^2(x)\right )^{2/3} \]
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Rubi [A] time = 0.04, 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, 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*}
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Mathematica [A] time = 0.02, size = 18, normalized size = 1.00 \[ -\frac {3}{8} \left (a^2-4 \sin ^2(x)\right )^{2/3} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin (2 x)}{\sqrt [3]{a^2-4 \sin ^2(x)}} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 1.00, size = 15, normalized size = 0.83 \[ -\frac {3}{8} \, {\left (a^{2} + 4 \, \cos \relax (x)^{2} - 4\right )}^{\frac {2}{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.92, size = 14, normalized size = 0.78 \[ -\frac {3}{8} \, {\left (a^{2} - 4 \, \sin \relax (x)^{2}\right )}^{\frac {2}{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.13, size = 15, normalized size = 0.83
method | result | size |
derivativedivides | \(-\frac {3 \left (a^{2}-4 \left (\sin ^{2}\relax (x )\right )\right )^{\frac {2}{3}}}{8}\) | \(15\) |
default | \(-\frac {3 \left (a^{2}-4 \left (\sin ^{2}\relax (x )\right )\right )^{\frac {2}{3}}}{8}\) | \(15\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 14, normalized size = 0.78 \[ -\frac {3}{8} \, {\left (a^{2} - 4 \, \sin \relax (x)^{2}\right )}^{\frac {2}{3}} \]
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 \[ -\frac {3\,{\left (a^2-4\,{\sin \relax (x)}^2\right )}^{2/3}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.29, size = 17, normalized size = 0.94 \[ - \frac {3 \left (a^{2} - 4 \sin ^{2}{\relax (x )}\right )^{\frac {2}{3}}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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