Integrand size = 11, antiderivative size = 33 \[ \int \frac {\cot (x)}{a+a \cos (x)} \, dx=-\frac {\text {arctanh}(\cos (x))}{2 a}+\frac {\cot (x) \csc (x)}{2 a}-\frac {\csc ^2(x)}{2 a} \]
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Time = 0.06 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {2785, 2686, 30, 2691, 3855} \[ \int \frac {\cot (x)}{a+a \cos (x)} \, dx=-\frac {\text {arctanh}(\cos (x))}{2 a}-\frac {\csc ^2(x)}{2 a}+\frac {\cot (x) \csc (x)}{2 a} \]
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Rule 30
Rule 2686
Rule 2691
Rule 2785
Rule 3855
Rubi steps \begin{align*} \text {integral}& = -\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 {\text {Subst}(\int x \, dx,x,\csc (x))}{a} \\ & = -\frac {\text {arctanh}(\cos (x))}{2 a}+\frac {\cot (x) \csc (x)}{2 a}-\frac {\csc ^2(x)}{2 a} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.27 \[ \int \frac {\cot (x)}{a+a \cos (x)} \, dx=-\frac {1+2 \cos ^2\left (\frac {x}{2}\right ) \left (\log \left (\cos \left (\frac {x}{2}\right )\right )-\log \left (\sin \left (\frac {x}{2}\right )\right )\right )}{2 a (1+\cos (x))} \]
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Time = 0.51 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.85
method | result | size |
default | \(\frac {-\frac {1}{2 \left (\cos \left (x \right )+1\right )}-\frac {\ln \left (\cos \left (x \right )+1\right )}{4}+\frac {\ln \left (\cos \left (x \right )-1\right )}{4}}{a}\) | \(28\) |
risch | \(-\frac {{\mathrm e}^{i x}}{\left ({\mathrm e}^{i x}+1\right )^{2} a}+\frac {\ln \left ({\mathrm e}^{i x}-1\right )}{2 a}-\frac {\ln \left ({\mathrm e}^{i x}+1\right )}{2 a}\) | \(47\) |
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Time = 0.27 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.12 \[ \int \frac {\cot (x)}{a+a \cos (x)} \, dx=-\frac {{\left (\cos \left (x\right ) + 1\right )} \log \left (\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) - {\left (\cos \left (x\right ) + 1\right )} \log \left (-\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) + 2}{4 \, {\left (a \cos \left (x\right ) + a\right )}} \]
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\[ \int \frac {\cot (x)}{a+a \cos (x)} \, dx=\frac {\int \frac {\cot {\left (x \right )}}{\cos {\left (x \right )} + 1}\, dx}{a} \]
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none
Time = 0.23 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.94 \[ \int \frac {\cot (x)}{a+a \cos (x)} \, dx=-\frac {\log \left (\cos \left (x\right ) + 1\right )}{4 \, a} + \frac {\log \left (\cos \left (x\right ) - 1\right )}{4 \, a} - \frac {1}{2 \, {\left (a \cos \left (x\right ) + a\right )}} \]
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Time = 0.30 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.03 \[ \int \frac {\cot (x)}{a+a \cos (x)} \, dx=-\frac {\log \left (\cos \left (x\right ) + 1\right )}{4 \, a} + \frac {\log \left (-\cos \left (x\right ) + 1\right )}{4 \, a} - \frac {1}{2 \, a {\left (\cos \left (x\right ) + 1\right )}} \]
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Time = 14.07 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.64 \[ \int \frac {\cot (x)}{a+a \cos (x)} \, dx=\frac {2\,\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )-{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{4\,a} \]
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