Integrand size = 18, antiderivative size = 28 \[ \int \cos ^3(a+b x) \csc (2 a+2 b x) \, dx=-\frac {\text {arctanh}(\cos (a+b x))}{2 b}+\frac {\cos (a+b x)}{2 b} \]
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Time = 0.05 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {4372, 2672, 327, 212} \[ \int \cos ^3(a+b x) \csc (2 a+2 b x) \, dx=\frac {\cos (a+b x)}{2 b}-\frac {\text {arctanh}(\cos (a+b x))}{2 b} \]
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Rule 212
Rule 327
Rule 2672
Rule 4372
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \int \cos (a+b x) \cot (a+b x) \, dx \\ & = -\frac {\text {Subst}\left (\int \frac {x^2}{1-x^2} \, dx,x,\cos (a+b x)\right )}{2 b} \\ & = \frac {\cos (a+b x)}{2 b}-\frac {\text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\cos (a+b x)\right )}{2 b} \\ & = -\frac {\text {arctanh}(\cos (a+b x))}{2 b}+\frac {\cos (a+b x)}{2 b} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.64 \[ \int \cos ^3(a+b x) \csc (2 a+2 b x) \, dx=\frac {1}{2} \left (\frac {\cos (a+b x)}{b}-\frac {\log \left (\cos \left (\frac {1}{2} (a+b x)\right )\right )}{b}+\frac {\log \left (\sin \left (\frac {1}{2} (a+b x)\right )\right )}{b}\right ) \]
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Time = 1.06 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04
method | result | size |
default | \(\frac {\cos \left (x b +a \right )+\ln \left (\csc \left (x b +a \right )-\cot \left (x b +a \right )\right )}{2 b}\) | \(29\) |
risch | \(\frac {{\mathrm e}^{i \left (x b +a \right )}}{4 b}+\frac {{\mathrm e}^{-i \left (x b +a \right )}}{4 b}+\frac {\ln \left ({\mathrm e}^{i \left (x b +a \right )}-1\right )}{2 b}-\frac {\ln \left ({\mathrm e}^{i \left (x b +a \right )}+1\right )}{2 b}\) | \(64\) |
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Time = 0.25 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.36 \[ \int \cos ^3(a+b x) \csc (2 a+2 b x) \, dx=\frac {2 \, \cos \left (b x + a\right ) - \log \left (\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2}\right ) + \log \left (-\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2}\right )}{4 \, b} \]
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Timed out. \[ \int \cos ^3(a+b x) \csc (2 a+2 b x) \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 92 vs. \(2 (24) = 48\).
Time = 0.22 (sec) , antiderivative size = 92, normalized size of antiderivative = 3.29 \[ \int \cos ^3(a+b x) \csc (2 a+2 b x) \, dx=\frac {2 \, \cos \left (b x + a\right ) - \log \left (\cos \left (b x\right )^{2} + 2 \, \cos \left (b x\right ) \cos \left (a\right ) + \cos \left (a\right )^{2} + \sin \left (b x\right )^{2} - 2 \, \sin \left (b x\right ) \sin \left (a\right ) + \sin \left (a\right )^{2}\right ) + \log \left (\cos \left (b x\right )^{2} - 2 \, \cos \left (b x\right ) \cos \left (a\right ) + \cos \left (a\right )^{2} + \sin \left (b x\right )^{2} + 2 \, \sin \left (b x\right ) \sin \left (a\right ) + \sin \left (a\right )^{2}\right )}{4 \, b} \]
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Time = 0.28 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.29 \[ \int \cos ^3(a+b x) \csc (2 a+2 b x) \, dx=\frac {2 \, \cos \left (b x + a\right ) - \log \left (\cos \left (b x + a\right ) + 1\right ) + \log \left (-\cos \left (b x + a\right ) + 1\right )}{4 \, b} \]
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Time = 0.06 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.79 \[ \int \cos ^3(a+b x) \csc (2 a+2 b x) \, dx=\frac {\frac {\cos \left (a+b\,x\right )}{2}-\frac {\mathrm {atanh}\left (\cos \left (a+b\,x\right )\right )}{2}}{b} \]
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