Optimal. Leaf size=11 \[ -\sin ^{-1}\left (\frac{\cos ^2(x)}{3}\right ) \]
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Rubi [A] time = 0.0500845, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {12, 1107, 619, 216} \[ -\sin ^{-1}\left (\frac{\cos ^2(x)}{3}\right ) \]
Antiderivative was successfully verified.
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Rule 12
Rule 1107
Rule 619
Rule 216
Rubi steps
\begin{align*} \int \frac{\sin (2 x)}{\sqrt{9-\cos ^4(x)}} \, dx &=\operatorname{Subst}\left (\int \frac{2 x}{\sqrt{8+2 x^2-x^4}} \, dx,x,\sin (x)\right )\\ &=2 \operatorname{Subst}\left (\int \frac{x}{\sqrt{8+2 x^2-x^4}} \, dx,x,\sin (x)\right )\\ &=\operatorname{Subst}\left (\int \frac{1}{\sqrt{8+2 x-x^2}} \, dx,x,\sin ^2(x)\right )\\ &=-\left (\frac{1}{6} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{36}}} \, dx,x,2 \cos ^2(x)\right )\right )\\ &=-\sin ^{-1}\left (\frac{\cos ^2(x)}{3}\right )\\ \end{align*}
Mathematica [A] time = 0.0157304, size = 11, normalized size = 1. \[ -\sin ^{-1}\left (\frac{\cos ^2(x)}{3}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.03, size = 10, normalized size = 0.9 \begin{align*} -\arcsin \left ({\frac{ \left ( \cos \left ( x \right ) \right ) ^{2}}{3}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin \left (2 \, x\right )}{\sqrt{-\cos \left (x\right )^{4} + 9}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.55318, size = 72, normalized size = 6.55 \begin{align*} \arctan \left (\frac{\sqrt{-\cos \left (x\right )^{4} + 9} \cos \left (x\right )^{2}}{\cos \left (x\right )^{4} - 9}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin \left (2 \, x\right )}{\sqrt{-\cos \left (x\right )^{4} + 9}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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