Optimal. Leaf size=54 \[ -\frac{a \log (a+b \cos (x))}{a^2-b^2}+\frac{\log (1-\cos (x))}{2 (a+b)}+\frac{\log (\cos (x)+1)}{2 (a-b)} \]
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Rubi [A] time = 0.0579725, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {2721, 801} \[ -\frac{a \log (a+b \cos (x))}{a^2-b^2}+\frac{\log (1-\cos (x))}{2 (a+b)}+\frac{\log (\cos (x)+1)}{2 (a-b)} \]
Antiderivative was successfully verified.
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Rule 2721
Rule 801
Rubi steps
\begin{align*} \int \frac{\cot (x)}{a+b \cos (x)} \, dx &=-\operatorname{Subst}\left (\int \frac{x}{(a+x) \left (b^2-x^2\right )} \, dx,x,b \cos (x)\right )\\ &=-\operatorname{Subst}\left (\int \left (\frac{1}{2 (a+b) (b-x)}+\frac{a}{(a-b) (a+b) (a+x)}-\frac{1}{2 (a-b) (b+x)}\right ) \, dx,x,b \cos (x)\right )\\ &=\frac{\log (1-\cos (x))}{2 (a+b)}+\frac{\log (1+\cos (x))}{2 (a-b)}-\frac{a \log (a+b \cos (x))}{a^2-b^2}\\ \end{align*}
Mathematica [A] time = 0.0713264, size = 50, normalized size = 0.93 \[ -\frac{a \log (a+b \cos (x))}{a^2-b^2}+\frac{\log \left (\sin \left (\frac{x}{2}\right )\right )}{a+b}+\frac{\log \left (\cos \left (\frac{x}{2}\right )\right )}{a-b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 54, normalized size = 1. \begin{align*} -{\frac{a\ln \left ( a+b\cos \left ( x \right ) \right ) }{ \left ( a-b \right ) \left ( a+b \right ) }}+{\frac{\ln \left ( -1+\cos \left ( x \right ) \right ) }{2\,a+2\,b}}+{\frac{\ln \left ( \cos \left ( x \right ) +1 \right ) }{2\,a-2\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.14711, size = 65, normalized size = 1.2 \begin{align*} -\frac{a \log \left (b \cos \left (x\right ) + a\right )}{a^{2} - b^{2}} + \frac{\log \left (\cos \left (x\right ) + 1\right )}{2 \,{\left (a - b\right )}} + \frac{\log \left (\cos \left (x\right ) - 1\right )}{2 \,{\left (a + b\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.49816, size = 146, normalized size = 2.7 \begin{align*} -\frac{2 \, a \log \left (-b \cos \left (x\right ) - a\right ) -{\left (a + b\right )} \log \left (\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) -{\left (a - b\right )} \log \left (-\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right )}{2 \,{\left (a^{2} - b^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cot{\left (x \right )}}{a + b \cos{\left (x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.39519, size = 73, normalized size = 1.35 \begin{align*} -\frac{a b \log \left ({\left | b \cos \left (x\right ) + a \right |}\right )}{a^{2} b - b^{3}} + \frac{\log \left (\cos \left (x\right ) + 1\right )}{2 \,{\left (a - b\right )}} + \frac{\log \left (-\cos \left (x\right ) + 1\right )}{2 \,{\left (a + b\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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