3.55 \(\int \sqrt{1+3 \cos ^2(x)} \sin (2 x) \, dx\)

Optimal. Leaf size=16 \[ -\frac{2}{9} \left (4-3 \sin ^2(x)\right )^{3/2} \]

[Out]

(-2*(4 - 3*Sin[x]^2)^(3/2))/9

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Rubi [A]  time = 0.0545069, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ -\frac{2}{9} \left (4-3 \sin ^2(x)\right )^{3/2} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[1 + 3*Cos[x]^2]*Sin[2*x],x]

[Out]

(-2*(4 - 3*Sin[x]^2)^(3/2))/9

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Rubi in Sympy [A]  time = 2.74074, size = 15, normalized size = 0.94 \[ - \frac{2 \left (3 \cos ^{2}{\left (x \right )} + 1\right )^{\frac{3}{2}}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(sin(2*x)*(1+3*cos(x)**2)**(1/2),x)

[Out]

-2*(3*cos(x)**2 + 1)**(3/2)/9

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Mathematica [B]  time = 0.107486, size = 49, normalized size = 3.06 \[ \frac{-3 \sqrt{3 \cos (2 x)+5} \cos (2 x)-5 \sqrt{3 \cos (2 x)+5}+5 \sqrt{5}}{9 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[1 + 3*Cos[x]^2]*Sin[2*x],x]

[Out]

(5*Sqrt[5] - 5*Sqrt[5 + 3*Cos[2*x]] - 3*Cos[2*x]*Sqrt[5 + 3*Cos[2*x]])/(9*Sqrt[2
])

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Maple [A]  time = 0.043, size = 13, normalized size = 0.8 \[ -{\frac{2}{9} \left ( 1+3\, \left ( \cos \left ( x \right ) \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(sin(2*x)*(1+3*cos(x)^2)^(1/2),x)

[Out]

-2/9*(1+3*cos(x)^2)^(3/2)

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Maxima [A]  time = 1.414, size = 16, normalized size = 1. \[ -\frac{2}{9} \,{\left (3 \, \cos \left (x\right )^{2} + 1\right )}^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(3*cos(x)^2 + 1)*sin(2*x),x, algorithm="maxima")

[Out]

-2/9*(3*cos(x)^2 + 1)^(3/2)

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Fricas [A]  time = 0.227562, size = 16, normalized size = 1. \[ -\frac{2}{9} \,{\left (3 \, \cos \left (x\right )^{2} + 1\right )}^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(3*cos(x)^2 + 1)*sin(2*x),x, algorithm="fricas")

[Out]

-2/9*(3*cos(x)^2 + 1)^(3/2)

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Sympy [A]  time = 3.29817, size = 15, normalized size = 0.94 \[ - \frac{2 \left (3 \cos ^{2}{\left (x \right )} + 1\right )^{\frac{3}{2}}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sin(2*x)*(1+3*cos(x)**2)**(1/2),x)

[Out]

-2*(3*cos(x)**2 + 1)**(3/2)/9

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GIAC/XCAS [A]  time = 0.234006, size = 248, normalized size = 15.5 \[ -\frac{16 \,{\left ({\left (\tan \left (\frac{1}{2} \, x\right )^{2} - \sqrt{\tan \left (\frac{1}{2} \, x\right )^{4} - \tan \left (\frac{1}{2} \, x\right )^{2} + 1}\right )}^{5} -{\left (\tan \left (\frac{1}{2} \, x\right )^{2} - \sqrt{\tan \left (\frac{1}{2} \, x\right )^{4} - \tan \left (\frac{1}{2} \, x\right )^{2} + 1}\right )}^{3} - 2 \,{\left (\tan \left (\frac{1}{2} \, x\right )^{2} - \sqrt{\tan \left (\frac{1}{2} \, x\right )^{4} - \tan \left (\frac{1}{2} \, x\right )^{2} + 1}\right )}^{2} + 3 \, \tan \left (\frac{1}{2} \, x\right )^{2} - 3 \, \sqrt{\tan \left (\frac{1}{2} \, x\right )^{4} - \tan \left (\frac{1}{2} \, x\right )^{2} + 1} - 1\right )}}{{\left ({\left (\tan \left (\frac{1}{2} \, x\right )^{2} - \sqrt{\tan \left (\frac{1}{2} \, x\right )^{4} - \tan \left (\frac{1}{2} \, x\right )^{2} + 1}\right )}^{2} + 2 \, \tan \left (\frac{1}{2} \, x\right )^{2} - 2 \, \sqrt{\tan \left (\frac{1}{2} \, x\right )^{4} - \tan \left (\frac{1}{2} \, x\right )^{2} + 1} - 2\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(3*cos(x)^2 + 1)*sin(2*x),x, algorithm="giac")

[Out]

-16*((tan(1/2*x)^2 - sqrt(tan(1/2*x)^4 - tan(1/2*x)^2 + 1))^5 - (tan(1/2*x)^2 -
sqrt(tan(1/2*x)^4 - tan(1/2*x)^2 + 1))^3 - 2*(tan(1/2*x)^2 - sqrt(tan(1/2*x)^4 -
tan(1/2*x)^2 + 1))^2 + 3*tan(1/2*x)^2 - 3*sqrt(tan(1/2*x)^4 - tan(1/2*x)^2 + 1)
- 1)/((tan(1/2*x)^2 - sqrt(tan(1/2*x)^4 - tan(1/2*x)^2 + 1))^2 + 2*tan(1/2*x)^2
- 2*sqrt(tan(1/2*x)^4 - tan(1/2*x)^2 + 1) - 2)^3