Integrand size = 15, antiderivative size = 23 \[ \int \cos (a+b x) \cot ^2(a+b x) \, dx=-\frac {\csc (a+b x)}{b}-\frac {\sin (a+b x)}{b} \]
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Time = 0.01 (sec) , antiderivative size = 23, 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} \[ \int \cos (a+b x) \cot ^2(a+b x) \, dx=-\frac {\sin (a+b x)}{b}-\frac {\csc (a+b x)}{b} \]
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Rule 14
Rule 2670
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {1-x^2}{x^2} \, dx,x,-\sin (a+b x)\right )}{b} \\ & = -\frac {\text {Subst}\left (\int \left (-1+\frac {1}{x^2}\right ) \, dx,x,-\sin (a+b x)\right )}{b} \\ & = -\frac {\csc (a+b x)}{b}-\frac {\sin (a+b x)}{b} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \cos (a+b x) \cot ^2(a+b x) \, dx=-\frac {\csc (a+b x)}{b}-\frac {\sin (a+b x)}{b} \]
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Time = 0.17 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.52
method | result | size |
parallelrisch | \(\frac {\left (-3+\cos \left (2 b x +2 a \right )\right ) \sec \left (\frac {b x}{2}+\frac {a}{2}\right ) \csc \left (\frac {b x}{2}+\frac {a}{2}\right )}{4 b}\) | \(35\) |
derivativedivides | \(\frac {-\frac {\cos ^{4}\left (b x +a \right )}{\sin \left (b x +a \right )}-\left (2+\cos ^{2}\left (b x +a \right )\right ) \sin \left (b x +a \right )}{b}\) | \(42\) |
default | \(\frac {-\frac {\cos ^{4}\left (b x +a \right )}{\sin \left (b x +a \right )}-\left (2+\cos ^{2}\left (b x +a \right )\right ) \sin \left (b x +a \right )}{b}\) | \(42\) |
risch | \(\frac {i \left ({\mathrm e}^{3 i \left (b x +a \right )}-5 \cos \left (b x +a \right )-7 i \sin \left (b x +a \right )\right )}{2 b \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )}\) | \(47\) |
norman | \(\frac {-\frac {1}{2 b}-\frac {3 \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{b}-\frac {\tan ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )}{2 b}}{\tan \left (\frac {b x}{2}+\frac {a}{2}\right ) \left (1+\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}\) | \(66\) |
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Time = 0.29 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \cos (a+b x) \cot ^2(a+b x) \, dx=\frac {\cos \left (b x + a\right )^{2} - 2}{b \sin \left (b x + a\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 39 vs. \(2 (17) = 34\).
Time = 0.36 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.70 \[ \int \cos (a+b x) \cot ^2(a+b x) \, dx=\begin {cases} - \frac {2 \sin {\left (a + b x \right )}}{b} - \frac {\cos ^{2}{\left (a + b x \right )}}{b \sin {\left (a + b x \right )}} & \text {for}\: b \neq 0 \\\frac {x \cos ^{3}{\left (a \right )}}{\sin ^{2}{\left (a \right )}} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int \cos (a+b x) \cot ^2(a+b x) \, dx=-\frac {\frac {1}{\sin \left (b x + a\right )} + \sin \left (b x + a\right )}{b} \]
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Time = 0.32 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int \cos (a+b x) \cot ^2(a+b x) \, dx=-\frac {\frac {1}{\sin \left (b x + a\right )} + \sin \left (b x + a\right )}{b} \]
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Time = 0.17 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \cos (a+b x) \cot ^2(a+b x) \, dx=-\frac {{\sin \left (a+b\,x\right )}^2+1}{b\,\sin \left (a+b\,x\right )} \]
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