Integrand size = 15, antiderivative size = 27 \[ \int \cos ^2(a+b x) \cot (a+b x) \, dx=\frac {\log (\sin (a+b x))}{b}-\frac {\sin ^2(a+b x)}{2 b} \]
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Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2670, 14} \[ \int \cos ^2(a+b x) \cot (a+b x) \, dx=\frac {\log (\sin (a+b x))}{b}-\frac {\sin ^2(a+b x)}{2 b} \]
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Rule 14
Rule 2670
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1-x^2}{x} \, dx,x,-\sin (a+b x)\right )}{b} \\ & = \frac {\text {Subst}\left (\int \left (\frac {1}{x}-x\right ) \, dx,x,-\sin (a+b x)\right )}{b} \\ & = \frac {\log (\sin (a+b x))}{b}-\frac {\sin ^2(a+b x)}{2 b} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \cos ^2(a+b x) \cot (a+b x) \, dx=\frac {\log (\sin (a+b x))}{b}-\frac {\sin ^2(a+b x)}{2 b} \]
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Time = 0.08 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85
method | result | size |
derivativedivides | \(\frac {\frac {\left (\cos ^{2}\left (b x +a \right )\right )}{2}+\ln \left (\sin \left (b x +a \right )\right )}{b}\) | \(23\) |
default | \(\frac {\frac {\left (\cos ^{2}\left (b x +a \right )\right )}{2}+\ln \left (\sin \left (b x +a \right )\right )}{b}\) | \(23\) |
parallelrisch | \(\frac {4 \ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )\right )-4 \ln \left (\sec ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1+\cos \left (2 b x +2 a \right )}{4 b}\) | \(43\) |
risch | \(-i x +\frac {{\mathrm e}^{2 i \left (b x +a \right )}}{8 b}+\frac {{\mathrm e}^{-2 i \left (b x +a \right )}}{8 b}-\frac {2 i a}{b}+\frac {\ln \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )}{b}\) | \(57\) |
norman | \(-\frac {2 \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{b \left (1+\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )^{2}}+\frac {\ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{b}-\frac {\ln \left (1+\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{b}\) | \(66\) |
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Time = 0.30 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93 \[ \int \cos ^2(a+b x) \cot (a+b x) \, dx=\frac {\cos \left (b x + a\right )^{2} + 2 \, \log \left (\frac {1}{2} \, \sin \left (b x + a\right )\right )}{2 \, b} \]
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Leaf count of result is larger than twice the leaf count of optimal. 369 vs. \(2 (20) = 40\).
Time = 0.61 (sec) , antiderivative size = 369, normalized size of antiderivative = 13.67 \[ \int \cos ^2(a+b x) \cot (a+b x) \, dx=\begin {cases} - \frac {\log {\left (\tan ^{2}{\left (\frac {a}{2} + \frac {b x}{2} \right )} + 1 \right )} \tan ^{4}{\left (\frac {a}{2} + \frac {b x}{2} \right )}}{b \tan ^{4}{\left (\frac {a}{2} + \frac {b x}{2} \right )} + 2 b \tan ^{2}{\left (\frac {a}{2} + \frac {b x}{2} \right )} + b} - \frac {2 \log {\left (\tan ^{2}{\left (\frac {a}{2} + \frac {b x}{2} \right )} + 1 \right )} \tan ^{2}{\left (\frac {a}{2} + \frac {b x}{2} \right )}}{b \tan ^{4}{\left (\frac {a}{2} + \frac {b x}{2} \right )} + 2 b \tan ^{2}{\left (\frac {a}{2} + \frac {b x}{2} \right )} + b} - \frac {\log {\left (\tan ^{2}{\left (\frac {a}{2} + \frac {b x}{2} \right )} + 1 \right )}}{b \tan ^{4}{\left (\frac {a}{2} + \frac {b x}{2} \right )} + 2 b \tan ^{2}{\left (\frac {a}{2} + \frac {b x}{2} \right )} + b} + \frac {\log {\left (\tan {\left (\frac {a}{2} + \frac {b x}{2} \right )} \right )} \tan ^{4}{\left (\frac {a}{2} + \frac {b x}{2} \right )}}{b \tan ^{4}{\left (\frac {a}{2} + \frac {b x}{2} \right )} + 2 b \tan ^{2}{\left (\frac {a}{2} + \frac {b x}{2} \right )} + b} + \frac {2 \log {\left (\tan {\left (\frac {a}{2} + \frac {b x}{2} \right )} \right )} \tan ^{2}{\left (\frac {a}{2} + \frac {b x}{2} \right )}}{b \tan ^{4}{\left (\frac {a}{2} + \frac {b x}{2} \right )} + 2 b \tan ^{2}{\left (\frac {a}{2} + \frac {b x}{2} \right )} + b} + \frac {\log {\left (\tan {\left (\frac {a}{2} + \frac {b x}{2} \right )} \right )}}{b \tan ^{4}{\left (\frac {a}{2} + \frac {b x}{2} \right )} + 2 b \tan ^{2}{\left (\frac {a}{2} + \frac {b x}{2} \right )} + b} - \frac {2 \tan ^{2}{\left (\frac {a}{2} + \frac {b x}{2} \right )}}{b \tan ^{4}{\left (\frac {a}{2} + \frac {b x}{2} \right )} + 2 b \tan ^{2}{\left (\frac {a}{2} + \frac {b x}{2} \right )} + b} & \text {for}\: b \neq 0 \\\frac {x \cos ^{3}{\left (a \right )}}{\sin {\left (a \right )}} & \text {otherwise} \end {cases} \]
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Time = 0.18 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93 \[ \int \cos ^2(a+b x) \cot (a+b x) \, dx=-\frac {\sin \left (b x + a\right )^{2} - \log \left (\sin \left (b x + a\right )^{2}\right )}{2 \, b} \]
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Time = 0.30 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93 \[ \int \cos ^2(a+b x) \cot (a+b x) \, dx=-\frac {\sin \left (b x + a\right )^{2} - \log \left (\sin \left (b x + a\right )^{2}\right )}{2 \, b} \]
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Time = 0.15 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.30 \[ \int \cos ^2(a+b x) \cot (a+b x) \, dx=\frac {\frac {{\cos \left (a+b\,x\right )}^2}{2}-\frac {\ln \left ({\mathrm {tan}\left (a+b\,x\right )}^2+1\right )}{2}+\ln \left (\mathrm {tan}\left (a+b\,x\right )\right )}{b} \]
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