3.337 \(\int \frac{\sin (2 x)}{\sqrt{9-\cos ^4(x)}} \, dx\)

Optimal. Leaf size=11 \[ -\sin ^{-1}\left (\frac{\cos ^2(x)}{3}\right ) \]

[Out]

-ArcSin[Cos[x]^2/3]

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Rubi [A]  time = 0.082698, 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 \[ -\sin ^{-1}\left (\frac{\cos ^2(x)}{3}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sin[2*x]/Sqrt[9 - Cos[x]^4],x]

[Out]

-ArcSin[Cos[x]^2/3]

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(sin(2*x)/(9-cos(x)**4)**(1/2),x)

[Out]

Timed out

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Mathematica [C]  time = 0.120085, size = 26, normalized size = 2.36 \[ i \log \left (\sqrt{9-\cos ^4(x)}+i \cos ^2(x)\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Sin[2*x]/Sqrt[9 - Cos[x]^4],x]

[Out]

I*Log[I*Cos[x]^2 + Sqrt[9 - Cos[x]^4]]

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Maple [A]  time = 0.045, size = 10, normalized size = 0.9 \[ -\arcsin \left ({\frac{ \left ( \cos \left ( x \right ) \right ) ^{2}}{3}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(sin(2*x)/(9-cos(x)^4)^(1/2),x)

[Out]

-arcsin(1/3*cos(x)^2)

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Maxima [A]  time = 6.50359, size = 61, normalized size = 5.55 \[ \log \left (\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} + 2 \, \sin \left (2 \, x\right ) + 1\right ) - \log \left (\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} - 2 \, \sin \left (2 \, x\right ) + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sin(2*x)/sqrt(-cos(x)^4 + 9),x, algorithm="maxima")

[Out]

log(cos(2*x)^2 + sin(2*x)^2 + 2*sin(2*x) + 1) - log(cos(2*x)^2 + sin(2*x)^2 - 2*
sin(2*x) + 1)

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Fricas [A]  time = 0.271522, size = 24, normalized size = 2.18 \[ -\arctan \left (\frac{\cos \left (x\right )^{2}}{\sqrt{-\cos \left (x\right )^{4} + 9}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sin(2*x)/sqrt(-cos(x)^4 + 9),x, algorithm="fricas")

[Out]

-arctan(cos(x)^2/sqrt(-cos(x)^4 + 9))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sin(2*x)/(9-cos(x)**4)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sin \left (2 \, x\right )}{\sqrt{-\cos \left (x\right )^{4} + 9}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sin(2*x)/sqrt(-cos(x)^4 + 9),x, algorithm="giac")

[Out]

integrate(sin(2*x)/sqrt(-cos(x)^4 + 9), x)