Optimal. Leaf size=33 \[ -\frac{\csc ^2(x)}{2 a}-\frac{\tanh ^{-1}(\cos (x))}{2 a}+\frac{\cot (x) \csc (x)}{2 a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0548422, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.454, Rules used = {2706, 2606, 30, 2611, 3770} \[ -\frac{\csc ^2(x)}{2 a}-\frac{\tanh ^{-1}(\cos (x))}{2 a}+\frac{\cot (x) \csc (x)}{2 a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2706
Rule 2606
Rule 30
Rule 2611
Rule 3770
Rubi steps
\begin{align*} \int \frac{\cot (x)}{a+a \cos (x)} \, dx &=-\frac{\int \cot ^2(x) \csc (x) \, dx}{a}+\frac{\int \cot (x) \csc ^2(x) \, dx}{a}\\ &=\frac{\cot (x) \csc (x)}{2 a}+\frac{\int \csc (x) \, dx}{2 a}-\frac{\operatorname{Subst}(\int x \, dx,x,\csc (x))}{a}\\ &=-\frac{\tanh ^{-1}(\cos (x))}{2 a}+\frac{\cot (x) \csc (x)}{2 a}-\frac{\csc ^2(x)}{2 a}\\ \end{align*}
Mathematica [A] time = 0.0401624, size = 42, normalized size = 1.27 \[ -\frac{2 \cos ^2\left (\frac{x}{2}\right ) \left (\log \left (\cos \left (\frac{x}{2}\right )\right )-\log \left (\sin \left (\frac{x}{2}\right )\right )\right )+1}{2 a (\cos (x)+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.053, size = 33, normalized size = 1. \begin{align*}{\frac{\ln \left ( -1+\cos \left ( x \right ) \right ) }{4\,a}}-{\frac{1}{2\,a \left ( \cos \left ( x \right ) +1 \right ) }}-{\frac{\ln \left ( \cos \left ( x \right ) +1 \right ) }{4\,a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.14122, size = 42, normalized size = 1.27 \begin{align*} -\frac{\log \left (\cos \left (x\right ) + 1\right )}{4 \, a} + \frac{\log \left (\cos \left (x\right ) - 1\right )}{4 \, a} - \frac{1}{2 \,{\left (a \cos \left (x\right ) + a\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.3867, size = 135, normalized size = 4.09 \begin{align*} -\frac{{\left (\cos \left (x\right ) + 1\right )} \log \left (\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) -{\left (\cos \left (x\right ) + 1\right )} \log \left (-\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) + 2}{4 \,{\left (a \cos \left (x\right ) + a\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\cot{\left (x \right )}}{\cos{\left (x \right )} + 1}\, dx}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.2206, size = 46, normalized size = 1.39 \begin{align*} -\frac{\log \left (\cos \left (x\right ) + 1\right )}{4 \, a} + \frac{\log \left (-\cos \left (x\right ) + 1\right )}{4 \, a} - \frac{1}{2 \, a{\left (\cos \left (x\right ) + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]