Optimal. Leaf size=11 \[ -\sin ^{-1}\left (\frac {\cos ^2(x)}{3}\right ) \]
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Rubi [A] time = 0.05, antiderivative size = 11, normalized size of antiderivative = 1.00, 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 216
Rule 619
Rule 1107
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*}
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Mathematica [A] time = 0.02, size = 11, normalized size = 1.00 \[ -\sin ^{-1}\left (\frac {\cos ^2(x)}{3}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 1.31, size = 24, normalized size = 2.18 \[ \arctan \left (\frac {\sqrt {-\cos \relax (x)^{4} + 9} \cos \relax (x)^{2}}{\cos \relax (x)^{4} - 9}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 9, normalized size = 0.82 \[ -\arcsin \left (\frac {1}{3} \, \cos \relax (x)^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 10, normalized size = 0.91 \[ -\arcsin \left (\frac {\left (\cos ^{2}\relax (x )\right )}{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin \left (2 \, x\right )}{\sqrt {-\cos \relax (x)^{4} + 9}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.14, size = 18, normalized size = 1.64 \[ -\mathrm {atan}\left (\frac {{\cos \relax (x)}^2}{\sqrt {9-{\cos \relax (x)}^4}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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