Integrand size = 13, antiderivative size = 11 \[ \int \cot (a+b x) \csc (a+b x) \, dx=-\frac {\csc (a+b x)}{b} \]
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Time = 0.01 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2686, 8} \[ \int \cot (a+b x) \csc (a+b x) \, dx=-\frac {\csc (a+b x)}{b} \]
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Rule 8
Rule 2686
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}(\int 1 \, dx,x,\csc (a+b x))}{b} \\ & = -\frac {\csc (a+b x)}{b} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \cot (a+b x) \csc (a+b x) \, dx=-\frac {\csc (a+b x)}{b} \]
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Time = 0.03 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.27
method | result | size |
derivativedivides | \(-\frac {1}{\sin \left (b x +a \right ) b}\) | \(14\) |
default | \(-\frac {1}{\sin \left (b x +a \right ) b}\) | \(14\) |
parallelrisch | \(-\frac {\sec \left (\frac {b x}{2}+\frac {a}{2}\right ) \csc \left (\frac {b x}{2}+\frac {a}{2}\right )}{2 b}\) | \(24\) |
risch | \(-\frac {2 i {\mathrm e}^{i \left (b x +a \right )}}{b \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )}\) | \(29\) |
norman | \(\frac {-\frac {1}{2 b}-\frac {\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )}{2 b}}{\tan \left (\frac {b x}{2}+\frac {a}{2}\right )}\) | \(35\) |
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Time = 0.29 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.18 \[ \int \cot (a+b x) \csc (a+b x) \, dx=-\frac {1}{b \sin \left (b x + a\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 20 vs. \(2 (8) = 16\).
Time = 0.31 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.82 \[ \int \cot (a+b x) \csc (a+b x) \, dx=\begin {cases} - \frac {1}{b \sin {\left (a + b x \right )}} & \text {for}\: b \neq 0 \\\frac {x \cos {\left (a \right )}}{\sin ^{2}{\left (a \right )}} & \text {otherwise} \end {cases} \]
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Time = 0.19 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.18 \[ \int \cot (a+b x) \csc (a+b x) \, dx=-\frac {1}{b \sin \left (b x + a\right )} \]
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Time = 0.35 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.18 \[ \int \cot (a+b x) \csc (a+b x) \, dx=-\frac {1}{b \sin \left (b x + a\right )} \]
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Time = 0.16 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.18 \[ \int \cot (a+b x) \csc (a+b x) \, dx=-\frac {1}{b\,\sin \left (a+b\,x\right )} \]
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