Integrand size = 17, antiderivative size = 24 \[ \int \cot (c+d x) (a+a \sin (c+d x)) \, dx=\frac {a \log (\sin (c+d x))}{d}+\frac {a \sin (c+d x)}{d} \]
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Time = 0.02 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2786, 45} \[ \int \cot (c+d x) (a+a \sin (c+d x)) \, dx=\frac {a \sin (c+d x)}{d}+\frac {a \log (\sin (c+d x))}{d} \]
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Rule 45
Rule 2786
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {a+x}{x} \, dx,x,a \sin (c+d x)\right )}{d} \\ & = \frac {\text {Subst}\left (\int \left (1+\frac {a}{x}\right ) \, dx,x,a \sin (c+d x)\right )}{d} \\ & = \frac {a \log (\sin (c+d x))}{d}+\frac {a \sin (c+d x)}{d} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.96 \[ \int \cot (c+d x) (a+a \sin (c+d x)) \, dx=\frac {a \log (\cos (c+d x))}{d}+\frac {a \log (\tan (c+d x))}{d}+\frac {a \cos (d x) \sin (c)}{d}+\frac {a \cos (c) \sin (d x)}{d} \]
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Time = 0.10 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.04
method | result | size |
derivativedivides | \(-\frac {a \left (\ln \left (\csc \left (d x +c \right )\right )-\frac {1}{\csc \left (d x +c \right )}\right )}{d}\) | \(25\) |
default | \(-\frac {a \left (\ln \left (\csc \left (d x +c \right )\right )-\frac {1}{\csc \left (d x +c \right )}\right )}{d}\) | \(25\) |
risch | \(-i a x -\frac {2 i a c}{d}+\frac {a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}+\frac {a \sin \left (d x +c \right )}{d}\) | \(43\) |
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none
Time = 0.26 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \cot (c+d x) (a+a \sin (c+d x)) \, dx=\frac {a \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) + a \sin \left (d x + c\right )}{d} \]
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\[ \int \cot (c+d x) (a+a \sin (c+d x)) \, dx=a \left (\int \cos {\left (c + d x \right )} \csc {\left (c + d x \right )}\, dx + \int \sin {\left (c + d x \right )} \cos {\left (c + d x \right )} \csc {\left (c + d x \right )}\, dx\right ) \]
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none
Time = 0.19 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \cot (c+d x) (a+a \sin (c+d x)) \, dx=\frac {a \log \left (\sin \left (d x + c\right )\right ) + a \sin \left (d x + c\right )}{d} \]
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none
Time = 0.31 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.96 \[ \int \cot (c+d x) (a+a \sin (c+d x)) \, dx=\frac {a \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) + a \sin \left (d x + c\right )}{d} \]
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Time = 8.88 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.58 \[ \int \cot (c+d x) (a+a \sin (c+d x)) \, dx=\frac {a\,\left (\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )+\sin \left (c+d\,x\right )-\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )\right )}{d} \]
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