Integrand size = 13, antiderivative size = 46 \[ \int \frac {\cot ^3(x)}{a+a \cos (x)} \, dx=\frac {3 \text {arctanh}(\cos (x))}{8 a}-\frac {\cot ^4(x)}{4 a}-\frac {3 \cot (x) \csc (x)}{8 a}+\frac {\cot ^3(x) \csc (x)}{4 a} \]
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Time = 0.14 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {2785, 2687, 30, 2691, 3855} \[ \int \frac {\cot ^3(x)}{a+a \cos (x)} \, dx=\frac {3 \text {arctanh}(\cos (x))}{8 a}-\frac {\cot ^4(x)}{4 a}+\frac {\cot ^3(x) \csc (x)}{4 a}-\frac {3 \cot (x) \csc (x)}{8 a} \]
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Rule 30
Rule 2687
Rule 2691
Rule 2785
Rule 3855
Rubi steps \begin{align*} \text {integral}& = -\frac {\int \cot ^4(x) \csc (x) \, dx}{a}+\frac {\int \cot ^3(x) \csc ^2(x) \, dx}{a} \\ & = \frac {\cot ^3(x) \csc (x)}{4 a}+\frac {3 \int \cot ^2(x) \csc (x) \, dx}{4 a}-\frac {\text {Subst}\left (\int x^3 \, dx,x,-\cot (x)\right )}{a} \\ & = -\frac {\cot ^4(x)}{4 a}-\frac {3 \cot (x) \csc (x)}{8 a}+\frac {\cot ^3(x) \csc (x)}{4 a}-\frac {3 \int \csc (x) \, dx}{8 a} \\ & = \frac {3 \text {arctanh}(\cos (x))}{8 a}-\frac {\cot ^4(x)}{4 a}-\frac {3 \cot (x) \csc (x)}{8 a}+\frac {\cot ^3(x) \csc (x)}{4 a} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.30 \[ \int \frac {\cot ^3(x)}{a+a \cos (x)} \, dx=-\frac {-8+2 \cot ^2\left (\frac {x}{2}\right )-12 \cos ^2\left (\frac {x}{2}\right ) \left (\log \left (\cos \left (\frac {x}{2}\right )\right )-\log \left (\sin \left (\frac {x}{2}\right )\right )\right )+\sec ^2\left (\frac {x}{2}\right )}{16 a (1+\cos (x))} \]
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Time = 0.54 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.96
method | result | size |
default | \(\frac {\frac {1}{8 \cos \left (x \right )-8}-\frac {3 \ln \left (\cos \left (x \right )-1\right )}{16}-\frac {1}{8 \left (\cos \left (x \right )+1\right )^{2}}+\frac {1}{2 \cos \left (x \right )+2}+\frac {3 \ln \left (\cos \left (x \right )+1\right )}{16}}{a}\) | \(44\) |
risch | \(\frac {5 \,{\mathrm e}^{5 i x}+2 \,{\mathrm e}^{4 i x}+2 \,{\mathrm e}^{3 i x}+2 \,{\mathrm e}^{2 i x}+5 \,{\mathrm e}^{i x}}{4 \left ({\mathrm e}^{i x}+1\right )^{4} a \left ({\mathrm e}^{i x}-1\right )^{2}}-\frac {3 \ln \left ({\mathrm e}^{i x}-1\right )}{8 a}+\frac {3 \ln \left ({\mathrm e}^{i x}+1\right )}{8 a}\) | \(87\) |
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Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (38) = 76\).
Time = 0.28 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.80 \[ \int \frac {\cot ^3(x)}{a+a \cos (x)} \, dx=\frac {10 \, \cos \left (x\right )^{2} + 3 \, {\left (\cos \left (x\right )^{3} + \cos \left (x\right )^{2} - \cos \left (x\right ) - 1\right )} \log \left (\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) - 3 \, {\left (\cos \left (x\right )^{3} + \cos \left (x\right )^{2} - \cos \left (x\right ) - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) + 2 \, \cos \left (x\right ) - 4}{16 \, {\left (a \cos \left (x\right )^{3} + a \cos \left (x\right )^{2} - a \cos \left (x\right ) - a\right )}} \]
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\[ \int \frac {\cot ^3(x)}{a+a \cos (x)} \, dx=\frac {\int \frac {\cot ^{3}{\left (x \right )}}{\cos {\left (x \right )} + 1}\, dx}{a} \]
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none
Time = 0.24 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.22 \[ \int \frac {\cot ^3(x)}{a+a \cos (x)} \, dx=\frac {5 \, \cos \left (x\right )^{2} + \cos \left (x\right ) - 2}{8 \, {\left (a \cos \left (x\right )^{3} + a \cos \left (x\right )^{2} - a \cos \left (x\right ) - a\right )}} + \frac {3 \, \log \left (\cos \left (x\right ) + 1\right )}{16 \, a} - \frac {3 \, \log \left (\cos \left (x\right ) - 1\right )}{16 \, a} \]
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Time = 0.31 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.09 \[ \int \frac {\cot ^3(x)}{a+a \cos (x)} \, dx=\frac {3 \, \log \left (\cos \left (x\right ) + 1\right )}{16 \, a} - \frac {3 \, \log \left (-\cos \left (x\right ) + 1\right )}{16 \, a} + \frac {5 \, \cos \left (x\right )^{2} + \cos \left (x\right ) - 2}{8 \, a {\left (\cos \left (x\right ) + 1\right )}^{2} {\left (\cos \left (x\right ) - 1\right )}} \]
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Time = 13.84 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.87 \[ \int \frac {\cot ^3(x)}{a+a \cos (x)} \, dx=-\frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^6-6\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4+12\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2\,\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )+2}{32\,a\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2} \]
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