Integrand size = 12, antiderivative size = 31 \[ \int \frac {1}{5+3 \sin (c+d x)} \, dx=\frac {x}{4}+\frac {\arctan \left (\frac {\cos (c+d x)}{3+\sin (c+d x)}\right )}{2 d} \]
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Time = 0.01 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {2736} \[ \int \frac {1}{5+3 \sin (c+d x)} \, dx=\frac {\arctan \left (\frac {\cos (c+d x)}{\sin (c+d x)+3}\right )}{2 d}+\frac {x}{4} \]
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Rule 2736
Rubi steps \begin{align*} \text {integral}& = \frac {x}{4}+\frac {\arctan \left (\frac {\cos (c+d x)}{3+\sin (c+d x)}\right )}{2 d} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.81 \[ \int \frac {1}{5+3 \sin (c+d x)} \, dx=\frac {\arctan \left (\frac {2 \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}{\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )}\right )}{2 d} \]
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Time = 0.18 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.65
method | result | size |
derivativedivides | \(\frac {\arctan \left (\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4}+\frac {3}{4}\right )}{2 d}\) | \(20\) |
default | \(\frac {\arctan \left (\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4}+\frac {3}{4}\right )}{2 d}\) | \(20\) |
risch | \(-\frac {i \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i}{3}\right )}{4 d}+\frac {i \ln \left ({\mathrm e}^{i \left (d x +c \right )}+3 i\right )}{4 d}\) | \(40\) |
parallelrisch | \(-\frac {i \left (\ln \left (5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+3-4 i\right )-\ln \left (5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+3+4 i\right )\right )}{4 d}\) | \(42\) |
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none
Time = 0.27 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.84 \[ \int \frac {1}{5+3 \sin (c+d x)} \, dx=\frac {\arctan \left (\frac {5 \, \sin \left (d x + c\right ) + 3}{4 \, \cos \left (d x + c\right )}\right )}{4 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (22) = 44\).
Time = 0.37 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.48 \[ \int \frac {1}{5+3 \sin (c+d x)} \, dx=\begin {cases} \frac {\operatorname {atan}{\left (\frac {5 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{4} + \frac {3}{4} \right )} + \pi \left \lfloor {\frac {\frac {c}{2} + \frac {d x}{2} - \frac {\pi }{2}}{\pi }}\right \rfloor }{2 d} & \text {for}\: d \neq 0 \\\frac {x}{3 \sin {\left (c \right )} + 5} & \text {otherwise} \end {cases} \]
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none
Time = 0.27 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.84 \[ \int \frac {1}{5+3 \sin (c+d x)} \, dx=\frac {\arctan \left (\frac {5 \, \sin \left (d x + c\right )}{4 \, {\left (\cos \left (d x + c\right ) + 1\right )}} + \frac {3}{4}\right )}{2 \, d} \]
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none
Time = 0.31 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.58 \[ \int \frac {1}{5+3 \sin (c+d x)} \, dx=\frac {d x + c + 2 \, \arctan \left (-\frac {3 \, \cos \left (d x + c\right ) + \sin \left (d x + c\right ) + 3}{\cos \left (d x + c\right ) - 3 \, \sin \left (d x + c\right ) - 9}\right )}{4 \, d} \]
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Time = 6.11 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.29 \[ \int \frac {1}{5+3 \sin (c+d x)} \, dx=\frac {\mathrm {atan}\left (\frac {5\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}+\frac {3}{4}\right )}{2\,d}-\frac {\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}}{2\,d} \]
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