Integrand size = 24, antiderivative size = 33 \[ \int \frac {a+a \sin (e+f x)}{c-c \sin (e+f x)} \, dx=-\frac {a x}{c}+\frac {2 a \cos (e+f x)}{f (c-c \sin (e+f x))} \]
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Time = 0.03 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {2814, 2727} \[ \int \frac {a+a \sin (e+f x)}{c-c \sin (e+f x)} \, dx=\frac {2 a \cos (e+f x)}{f (c-c \sin (e+f x))}-\frac {a x}{c} \]
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Rule 2727
Rule 2814
Rubi steps \begin{align*} \text {integral}& = -\frac {a x}{c}+(2 a) \int \frac {1}{c-c \sin (e+f x)} \, dx \\ & = -\frac {a x}{c}+\frac {2 a \cos (e+f x)}{f (c-c \sin (e+f x))} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(83\) vs. \(2(33)=66\).
Time = 1.74 (sec) , antiderivative size = 83, normalized size of antiderivative = 2.52 \[ \int \frac {a+a \sin (e+f x)}{c-c \sin (e+f x)} \, dx=\frac {a \left (-f x \cos \left (\frac {f x}{2}\right )+4 \sin \left (\frac {f x}{2}\right )+f x \sin \left (e+\frac {f x}{2}\right )\right )}{c f \left (\cos \left (\frac {e}{2}\right )-\sin \left (\frac {e}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )} \]
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Result contains complex when optimal does not.
Time = 0.66 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.97
method | result | size |
risch | \(-\frac {a x}{c}+\frac {4 a}{f c \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )}\) | \(32\) |
derivativedivides | \(\frac {2 a \left (-\arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {2}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}\right )}{f c}\) | \(38\) |
default | \(\frac {2 a \left (-\arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {2}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}\right )}{f c}\) | \(38\) |
parallelrisch | \(-\frac {a \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right ) f x -f x +4\right )}{f c \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}\) | \(41\) |
norman | \(\frac {\frac {a x}{c}+\frac {a x \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c}-\frac {4 a}{c f}-\frac {a x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{c}-\frac {a x \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c}-\frac {4 a \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c f}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}\) | \(117\) |
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Time = 0.26 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.00 \[ \int \frac {a+a \sin (e+f x)}{c-c \sin (e+f x)} \, dx=-\frac {a f x + {\left (a f x - 2 \, a\right )} \cos \left (f x + e\right ) - {\left (a f x + 2 \, a\right )} \sin \left (f x + e\right ) - 2 \, a}{c f \cos \left (f x + e\right ) - c f \sin \left (f x + e\right ) + c f} \]
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Leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (26) = 52\).
Time = 0.58 (sec) , antiderivative size = 88, normalized size of antiderivative = 2.67 \[ \int \frac {a+a \sin (e+f x)}{c-c \sin (e+f x)} \, dx=\begin {cases} - \frac {a f x \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )}}{c f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - c f} + \frac {a f x}{c f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - c f} - \frac {4 a}{c f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - c f} & \text {for}\: f \neq 0 \\\frac {x \left (a \sin {\left (e \right )} + a\right )}{- c \sin {\left (e \right )} + c} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (34) = 68\).
Time = 0.31 (sec) , antiderivative size = 82, normalized size of antiderivative = 2.48 \[ \int \frac {a+a \sin (e+f x)}{c-c \sin (e+f x)} \, dx=-\frac {2 \, {\left (a {\left (\frac {\arctan \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{c} - \frac {1}{c - \frac {c \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}}\right )} - \frac {a}{c - \frac {c \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}}\right )}}{f} \]
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Time = 0.29 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.06 \[ \int \frac {a+a \sin (e+f x)}{c-c \sin (e+f x)} \, dx=-\frac {\frac {{\left (f x + e\right )} a}{c} + \frac {4 \, a}{c {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}}}{f} \]
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Time = 6.42 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.39 \[ \int \frac {a+a \sin (e+f x)}{c-c \sin (e+f x)} \, dx=-\frac {a\,x}{c}-\frac {a\,\left (e+f\,x\right )-a\,\left (e+f\,x-4\right )}{c\,f\,\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )-1\right )} \]
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