Optimal. Leaf size=8 \[ \cos (x)-\tanh ^{-1}(\cos (x)) \]
[Out]
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Rubi [A] time = 0.0217969, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 5, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6 \[ \cos (x)-\tanh ^{-1}(\cos (x)) \]
Antiderivative was successfully verified.
[In] Int[Cos[x]*Cot[x],x]
[Out]
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Rubi in Sympy [A] time = 2.79744, size = 7, normalized size = 0.88 \[ \cos{\left (x \right )} - \operatorname{atanh}{\left (\cos{\left (x \right )} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(cos(x)**2/sin(x),x)
[Out]
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Mathematica [B] time = 0.00678012, size = 19, normalized size = 2.38 \[ \cos (x)+\log \left (\sin \left (\frac{x}{2}\right )\right )-\log \left (\cos \left (\frac{x}{2}\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[Cos[x]*Cot[x],x]
[Out]
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Maple [A] time = 0.013, size = 12, normalized size = 1.5 \[ \cos \left ( x \right ) +\ln \left ( \csc \left ( x \right ) -\cot \left ( x \right ) \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(cos(x)^2/sin(x),x)
[Out]
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Maxima [A] time = 1.34157, size = 23, normalized size = 2.88 \[ \cos \left (x\right ) - \frac{1}{2} \, \log \left (\cos \left (x\right ) + 1\right ) + \frac{1}{2} \, \log \left (\cos \left (x\right ) - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(cos(x)^2/sin(x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.271954, size = 28, normalized size = 3.5 \[ \cos \left (x\right ) - \frac{1}{2} \, \log \left (\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) + \frac{1}{2} \, \log \left (-\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(cos(x)^2/sin(x),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.097092, size = 19, normalized size = 2.38 \[ \frac{\log{\left (\cos{\left (x \right )} - 1 \right )}}{2} - \frac{\log{\left (\cos{\left (x \right )} + 1 \right )}}{2} + \cos{\left (x \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(cos(x)**2/sin(x),x)
[Out]
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GIAC/XCAS [A] time = 0.206512, size = 26, normalized size = 3.25 \[ \cos \left (x\right ) - \frac{1}{2} \,{\rm ln}\left (\cos \left (x\right ) + 1\right ) + \frac{1}{2} \,{\rm ln}\left (-\cos \left (x\right ) + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(cos(x)^2/sin(x),x, algorithm="giac")
[Out]