Integrand size = 24, antiderivative size = 11 \[ \int \frac {\cot (c+d x)}{\csc (c+d x)-\sin (c+d x)} \, dx=\frac {\text {arctanh}(\sin (c+d x))}{d} \]
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Time = 0.03 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {4423, 212} \[ \int \frac {\cot (c+d x)}{\csc (c+d x)-\sin (c+d x)} \, dx=\frac {\text {arctanh}(\sin (c+d x))}{d} \]
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Rule 212
Rule 4423
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sin (c+d x)\right )}{d} \\ & = \frac {\text {arctanh}(\sin (c+d x))}{d} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \frac {\cot (c+d x)}{\csc (c+d x)-\sin (c+d x)} \, dx=\frac {\text {arctanh}(\sin (c+d x))}{d} \]
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Time = 0.46 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.09
method | result | size |
derivativedivides | \(\frac {\operatorname {arctanh}\left (\sin \left (d x +c \right )\right )}{d}\) | \(12\) |
default | \(\frac {\operatorname {arctanh}\left (\sin \left (d x +c \right )\right )}{d}\) | \(12\) |
risch | \(\frac {\ln \left (i+{\mathrm e}^{i \left (d x +c \right )}\right )}{d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}\) | \(37\) |
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Leaf count of result is larger than twice the leaf count of optimal. 28 vs. \(2 (11) = 22\).
Time = 0.25 (sec) , antiderivative size = 28, normalized size of antiderivative = 2.55 \[ \int \frac {\cot (c+d x)}{\csc (c+d x)-\sin (c+d x)} \, dx=\frac {\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (-\sin \left (d x + c\right ) + 1\right )}{2 \, d} \]
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\[ \int \frac {\cot (c+d x)}{\csc (c+d x)-\sin (c+d x)} \, dx=\int \frac {\cot {\left (c + d x \right )}}{- \sin {\left (c + d x \right )} + \csc {\left (c + d x \right )}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 26 vs. \(2 (11) = 22\).
Time = 0.21 (sec) , antiderivative size = 26, normalized size of antiderivative = 2.36 \[ \int \frac {\cot (c+d x)}{\csc (c+d x)-\sin (c+d x)} \, dx=\frac {\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )}{2 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 28 vs. \(2 (11) = 22\).
Time = 0.29 (sec) , antiderivative size = 28, normalized size of antiderivative = 2.55 \[ \int \frac {\cot (c+d x)}{\csc (c+d x)-\sin (c+d x)} \, dx=\frac {\log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) - \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{2 \, d} \]
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Time = 22.12 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.36 \[ \int \frac {\cot (c+d x)}{\csc (c+d x)-\sin (c+d x)} \, dx=\frac {2\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d} \]
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