Integrand size = 20, antiderivative size = 23 \[ \int \csc (c+d x) (\cot (c+d x)+\csc (c+d x)) \, dx=-\frac {\cot (c+d x)}{d}-\frac {\csc (c+d x)}{d} \]
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Time = 0.11 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4482, 2748, 3852, 8} \[ \int \csc (c+d x) (\cot (c+d x)+\csc (c+d x)) \, dx=-\frac {\cot (c+d x)}{d}-\frac {\csc (c+d x)}{d} \]
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Rule 8
Rule 2748
Rule 3852
Rule 4482
Rubi steps \begin{align*} \text {integral}& = \int (1+\cos (c+d x)) \csc ^2(c+d x) \, dx \\ & = -\frac {\csc (c+d x)}{d}+\int \csc ^2(c+d x) \, dx \\ & = -\frac {\csc (c+d x)}{d}-\frac {\text {Subst}(\int 1 \, dx,x,\cot (c+d x))}{d} \\ & = -\frac {\cot (c+d x)}{d}-\frac {\csc (c+d x)}{d} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.65 \[ \int \csc (c+d x) (\cot (c+d x)+\csc (c+d x)) \, dx=-\frac {\cot \left (\frac {1}{2} (c+d x)\right )}{d} \]
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Result contains complex when optimal does not.
Time = 0.53 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87
method | result | size |
risch | \(-\frac {2 i}{d \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}\) | \(20\) |
derivativedivides | \(\frac {-\cot \left (d x +c \right )-\frac {1}{\sin \left (d x +c \right )}}{d}\) | \(24\) |
default | \(\frac {-\cot \left (d x +c \right )-\frac {1}{\sin \left (d x +c \right )}}{d}\) | \(24\) |
parts | \(-\frac {\cot \left (d x +c \right )}{d}-\frac {\csc \left (d x +c \right )}{d}\) | \(24\) |
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none
Time = 0.24 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \csc (c+d x) (\cot (c+d x)+\csc (c+d x)) \, dx=-\frac {\cos \left (d x + c\right ) + 1}{d \sin \left (d x + c\right )} \]
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Time = 0.87 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.17 \[ \int \csc (c+d x) (\cot (c+d x)+\csc (c+d x)) \, dx=\begin {cases} \frac {- \cot {\left (c + d x \right )} - \csc {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (\cot {\left (c \right )} + \csc {\left (c \right )}\right ) \csc {\left (c \right )} & \text {otherwise} \end {cases} \]
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none
Time = 0.22 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \csc (c+d x) (\cot (c+d x)+\csc (c+d x)) \, dx=-\frac {\frac {1}{\sin \left (d x + c\right )} + \frac {1}{\tan \left (d x + c\right )}}{d} \]
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none
Time = 0.29 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.70 \[ \int \csc (c+d x) (\cot (c+d x)+\csc (c+d x)) \, dx=-\frac {1}{d \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )} \]
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Time = 22.54 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.61 \[ \int \csc (c+d x) (\cot (c+d x)+\csc (c+d x)) \, dx=-\frac {\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d} \]
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