3.212 \(\int \frac{\cos ^2(x) \sin (x)}{5+\cos ^2(x)} \, dx\)

Optimal. Leaf size=20 \[ \sqrt{5} \tan ^{-1}\left (\frac{\cos (x)}{\sqrt{5}}\right )-\cos (x) \]

[Out]

Sqrt[5]*ArcTan[Cos[x]/Sqrt[5]] - Cos[x]

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Rubi [A]  time = 0.079717, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \sqrt{5} \tan ^{-1}\left (\frac{\cos (x)}{\sqrt{5}}\right )-\cos (x) \]

Antiderivative was successfully verified.

[In]  Int[(Cos[x]^2*Sin[x])/(5 + Cos[x]^2),x]

[Out]

Sqrt[5]*ArcTan[Cos[x]/Sqrt[5]] - Cos[x]

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Rubi in Sympy [A]  time = 8.51223, size = 19, normalized size = 0.95 \[ - \cos{\left (x \right )} + \sqrt{5} \operatorname{atan}{\left (\frac{\sqrt{5} \cos{\left (x \right )}}{5} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(cos(x)**2*sin(x)/(5+cos(x)**2),x)

[Out]

-cos(x) + sqrt(5)*atan(sqrt(5)*cos(x)/5)

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Mathematica [B]  time = 0.226915, size = 82, normalized size = 4.1 \[ \frac{1}{20} \left (-20 \cos (x)+21 \sqrt{5} \tan ^{-1}\left (\frac{1}{\sqrt{5}}-\sqrt{\frac{6}{5}} \tan \left (\frac{x}{2}\right )\right )+21 \sqrt{5} \tan ^{-1}\left (\sqrt{\frac{6}{5}} \tan \left (\frac{x}{2}\right )+\frac{1}{\sqrt{5}}\right )-\sqrt{5} \tan ^{-1}\left (\frac{\cos (x)}{\sqrt{5}}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(Cos[x]^2*Sin[x])/(5 + Cos[x]^2),x]

[Out]

(-(Sqrt[5]*ArcTan[Cos[x]/Sqrt[5]]) + 21*Sqrt[5]*ArcTan[1/Sqrt[5] - Sqrt[6/5]*Tan
[x/2]] + 21*Sqrt[5]*ArcTan[1/Sqrt[5] + Sqrt[6/5]*Tan[x/2]] - 20*Cos[x])/20

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Maple [A]  time = 0.02, size = 18, normalized size = 0.9 \[ -\cos \left ( x \right ) +\arctan \left ({\frac{\cos \left ( x \right ) \sqrt{5}}{5}} \right ) \sqrt{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(cos(x)^2*sin(x)/(5+cos(x)^2),x)

[Out]

-cos(x)+arctan(1/5*cos(x)*5^(1/2))*5^(1/2)

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Maxima [A]  time = 1.49503, size = 23, normalized size = 1.15 \[ \sqrt{5} \arctan \left (\frac{1}{5} \, \sqrt{5} \cos \left (x\right )\right ) - \cos \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(x)^2*sin(x)/(cos(x)^2 + 5),x, algorithm="maxima")

[Out]

sqrt(5)*arctan(1/5*sqrt(5)*cos(x)) - cos(x)

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Fricas [A]  time = 0.25907, size = 23, normalized size = 1.15 \[ \sqrt{5} \arctan \left (\frac{1}{5} \, \sqrt{5} \cos \left (x\right )\right ) - \cos \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(x)^2*sin(x)/(cos(x)^2 + 5),x, algorithm="fricas")

[Out]

sqrt(5)*arctan(1/5*sqrt(5)*cos(x)) - cos(x)

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Sympy [A]  time = 1.04585, size = 19, normalized size = 0.95 \[ - \cos{\left (x \right )} + \sqrt{5} \operatorname{atan}{\left (\frac{\sqrt{5} \cos{\left (x \right )}}{5} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(x)**2*sin(x)/(5+cos(x)**2),x)

[Out]

-cos(x) + sqrt(5)*atan(sqrt(5)*cos(x)/5)

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GIAC/XCAS [A]  time = 0.210227, size = 23, normalized size = 1.15 \[ \sqrt{5} \arctan \left (\frac{1}{5} \, \sqrt{5} \cos \left (x\right )\right ) - \cos \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(x)^2*sin(x)/(cos(x)^2 + 5),x, algorithm="giac")

[Out]

sqrt(5)*arctan(1/5*sqrt(5)*cos(x)) - cos(x)