Integrand size = 12, antiderivative size = 28 \[ \int \frac {1}{a+a \csc (c+d x)} \, dx=\frac {x}{a}+\frac {\cot (c+d x)}{d (a+a \csc (c+d x))} \]
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Time = 0.02 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3862, 8} \[ \int \frac {1}{a+a \csc (c+d x)} \, dx=\frac {\cot (c+d x)}{d (a \csc (c+d x)+a)}+\frac {x}{a} \]
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Rule 8
Rule 3862
Rubi steps \begin{align*} \text {integral}& = \frac {\cot (c+d x)}{d (a+a \csc (c+d x))}+\frac {\int a \, dx}{a^2} \\ & = \frac {x}{a}+\frac {\cot (c+d x)}{d (a+a \csc (c+d x))} \\ \end{align*}
Time = 0.26 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.68 \[ \int \frac {1}{a+a \csc (c+d x)} \, dx=\frac {c+d x-\frac {2 \sin \left (\frac {1}{2} (c+d x)\right )}{\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )}}{a d} \]
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Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04
method | result | size |
risch | \(\frac {x}{a}+\frac {2}{d a \left (i+{\mathrm e}^{i \left (d x +c \right )}\right )}\) | \(29\) |
derivativedivides | \(\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {4}{2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2}}{d a}\) | \(37\) |
default | \(\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {4}{2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2}}{d a}\) | \(37\) |
parallelrisch | \(\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) x d +d x -2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d a \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}\) | \(48\) |
norman | \(\frac {\frac {x}{a}+\frac {x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a}-\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d a}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}\) | \(52\) |
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none
Time = 0.24 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.93 \[ \int \frac {1}{a+a \csc (c+d x)} \, dx=\frac {d x + {\left (d x + 1\right )} \cos \left (d x + c\right ) + {\left (d x - 1\right )} \sin \left (d x + c\right ) + 1}{a d \cos \left (d x + c\right ) + a d \sin \left (d x + c\right ) + a d} \]
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\[ \int \frac {1}{a+a \csc (c+d x)} \, dx=\frac {\int \frac {1}{\csc {\left (c + d x \right )} + 1}\, dx}{a} \]
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none
Time = 0.31 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.79 \[ \int \frac {1}{a+a \csc (c+d x)} \, dx=\frac {2 \, {\left (\frac {\arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} + \frac {1}{a + \frac {a \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}}\right )}}{d} \]
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none
Time = 0.27 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.14 \[ \int \frac {1}{a+a \csc (c+d x)} \, dx=\frac {\frac {d x + c}{a} + \frac {2}{a {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}}}{d} \]
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Time = 18.40 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96 \[ \int \frac {1}{a+a \csc (c+d x)} \, dx=\frac {x}{a}+\frac {2}{a\,d\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )} \]
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