Integrand size = 28, antiderivative size = 28 \[ \int \frac {\sqrt {c-c \sin (e+f x)}}{3+3 \sin (e+f x)} \, dx=-\frac {2 \sec (e+f x) \sqrt {c-c \sin (e+f x)}}{3 f} \]
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Time = 0.10 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2815, 2752} \[ \int \frac {\sqrt {c-c \sin (e+f x)}}{3+3 \sin (e+f x)} \, dx=-\frac {2 \sec (e+f x) \sqrt {c-c \sin (e+f x)}}{a f} \]
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Rule 2752
Rule 2815
Rubi steps \begin{align*} \text {integral}& = \frac {\int \sec ^2(e+f x) (c-c \sin (e+f x))^{3/2} \, dx}{a c} \\ & = -\frac {2 \sec (e+f x) \sqrt {c-c \sin (e+f x)}}{a f} \\ \end{align*}
Time = 0.47 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {c-c \sin (e+f x)}}{3+3 \sin (e+f x)} \, dx=-\frac {2 \sec (e+f x) \sqrt {c-c \sin (e+f x)}}{3 f} \]
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Time = 0.56 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.39
method | result | size |
default | \(\frac {2 c \left (\sin \left (f x +e \right )-1\right )}{a \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\) | \(39\) |
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none
Time = 0.25 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {\sqrt {c-c \sin (e+f x)}}{3+3 \sin (e+f x)} \, dx=-\frac {2 \, \sqrt {-c \sin \left (f x + e\right ) + c}}{a f \cos \left (f x + e\right )} \]
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\[ \int \frac {\sqrt {c-c \sin (e+f x)}}{3+3 \sin (e+f x)} \, dx=\frac {\int \frac {\sqrt {- c \sin {\left (e + f x \right )} + c}}{\sin {\left (e + f x \right )} + 1}\, dx}{a} \]
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Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (27) = 54\).
Time = 0.33 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.71 \[ \int \frac {\sqrt {c-c \sin (e+f x)}}{3+3 \sin (e+f x)} \, dx=\frac {2 \, {\left (\sqrt {c} + \frac {\sqrt {c} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}}\right )}}{{\left (a + \frac {a \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )} f \sqrt {\frac {\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 1}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (27) = 54\).
Time = 0.31 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.21 \[ \int \frac {\sqrt {c-c \sin (e+f x)}}{3+3 \sin (e+f x)} \, dx=\frac {2 \, \sqrt {2} \sqrt {c} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{a f {\left (\frac {\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1}{\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1} + 1\right )}} \]
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Time = 0.23 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.43 \[ \int \frac {\sqrt {c-c \sin (e+f x)}}{3+3 \sin (e+f x)} \, dx=-\frac {4\,\cos \left (e+f\,x\right )\,\sqrt {-c\,\left (\sin \left (e+f\,x\right )-1\right )}}{a\,f\,\left (\cos \left (2\,e+2\,f\,x\right )+1\right )} \]
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