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