Integrand size = 22, antiderivative size = 11 \[ \int \frac {\cot (c+d x)}{\csc (c+d x)+\sin (c+d x)} \, dx=\frac {\arctan (\sin (c+d x))}{d} \]
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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, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {4423, 209} \[ \int \frac {\cot (c+d x)}{\csc (c+d x)+\sin (c+d x)} \, dx=\frac {\arctan (\sin (c+d x))}{d} \]
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Rule 209
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 {\arctan (\sin (c+d x))}{d} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \frac {\cot (c+d x)}{\csc (c+d x)+\sin (c+d x)} \, dx=\frac {\arctan (\sin (c+d x))}{d} \]
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Time = 0.33 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.09
method | result | size |
derivativedivides | \(\frac {\arctan \left (\sin \left (d x +c \right )\right )}{d}\) | \(12\) |
default | \(\frac {\arctan \left (\sin \left (d x +c \right )\right )}{d}\) | \(12\) |
risch | \(-\frac {i \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+2 \,{\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 d}+\frac {i \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-2 \,{\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 d}\) | \(60\) |
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Time = 0.24 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \frac {\cot (c+d x)}{\csc (c+d x)+\sin (c+d x)} \, dx=\frac {\arctan \left (\sin \left (d x + c\right )\right )}{d} \]
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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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Time = 0.30 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \frac {\cot (c+d x)}{\csc (c+d x)+\sin (c+d x)} \, dx=\frac {\arctan \left (\sin \left (d x + c\right )\right )}{d} \]
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Time = 0.31 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \frac {\cot (c+d x)}{\csc (c+d x)+\sin (c+d x)} \, dx=\frac {\arctan \left (\sin \left (d x + c\right )\right )}{d} \]
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Time = 23.94 (sec) , antiderivative size = 45, normalized size of antiderivative = 4.09 \[ \int \frac {\cot (c+d x)}{\csc (c+d x)+\sin (c+d x)} \, dx=\frac {\mathrm {atan}\left (\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{2}+\frac {5\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2}\right )-\mathrm {atan}\left (\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2}\right )}{d} \]
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