Optimal. Leaf size=22 \[ \frac{\sin ^2(x)}{2}-\frac{1}{2} \csc ^2(x)-2 \log (\sin (x)) \]
[Out]
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Rubi [A] time = 0.0539437, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{\sin ^2(x)}{2}-\frac{1}{2} \csc ^2(x)-2 \log (\sin (x)) \]
Antiderivative was successfully verified.
[In] Int[Cos[x]^2*Cot[x]^3,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\cos ^{2}{\left (x \right )}}{\tan ^{3}{\left (x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(cos(x)**2*cot(x)**3,x)
[Out]
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Mathematica [A] time = 0.00876849, size = 22, normalized size = 1. \[ -\frac{1}{4} \cos (2 x)-\frac{1}{2} \csc ^2(x)-2 \log (\sin (x)) \]
Antiderivative was successfully verified.
[In] Integrate[Cos[x]^2*Cot[x]^3,x]
[Out]
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Maple [A] time = 0.02, size = 29, normalized size = 1.3 \[ -{\frac{ \left ( \cos \left ( x \right ) \right ) ^{6}}{2\, \left ( \sin \left ( x \right ) \right ) ^{2}}}-{\frac{ \left ( \cos \left ( x \right ) \right ) ^{4}}{2}}- \left ( \cos \left ( x \right ) \right ) ^{2}-2\,\ln \left ( \sin \left ( x \right ) \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(cos(x)^2*cot(x)^3,x)
[Out]
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Maxima [A] time = 1.3562, size = 27, normalized size = 1.23 \[ \frac{1}{2} \, \sin \left (x\right )^{2} - \frac{1}{2 \, \sin \left (x\right )^{2}} - \log \left (\sin \left (x\right )^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(cos(x)^2*cot(x)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.23225, size = 50, normalized size = 2.27 \[ -\frac{2 \, \cos \left (x\right )^{4} - 3 \, \cos \left (x\right )^{2} + 8 \,{\left (\cos \left (x\right )^{2} - 1\right )} \log \left (\frac{1}{2} \, \sin \left (x\right )\right ) - 1}{4 \,{\left (\cos \left (x\right )^{2} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(cos(x)^2*cot(x)^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.099241, size = 20, normalized size = 0.91 \[ - 2 \log{\left (\sin{\left (x \right )} \right )} + \frac{\sin ^{2}{\left (x \right )}}{2} - \frac{1}{2 \sin ^{2}{\left (x \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(cos(x)**2*cot(x)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.205748, size = 49, normalized size = 2.23 \[ -\frac{1}{2} \, \cos \left (x\right )^{2} + \frac{2 \, \cos \left (x\right )^{2} - 1}{2 \,{\left (\cos \left (x\right )^{2} - 1\right )}} -{\rm ln}\left (-\cos \left (x\right )^{2} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(cos(x)^2*cot(x)^3,x, algorithm="giac")
[Out]