Integrand size = 30, antiderivative size = 40 \[ \int \sqrt {3+3 \sin (e+f x)} \sqrt {c-c \sin (e+f x)} \, dx=-\frac {3 \cos (e+f x) \sqrt {c-c \sin (e+f x)}}{f \sqrt {3+3 \sin (e+f x)}} \]
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Time = 0.06 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.02, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {2817} \[ \int \sqrt {3+3 \sin (e+f x)} \sqrt {c-c \sin (e+f x)} \, dx=-\frac {a \cos (e+f x) \sqrt {c-c \sin (e+f x)}}{f \sqrt {a \sin (e+f x)+a}} \]
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Rule 2817
Rubi steps \begin{align*} \text {integral}& = -\frac {a \cos (e+f x) \sqrt {c-c \sin (e+f x)}}{f \sqrt {a+a \sin (e+f x)}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(82\) vs. \(2(40)=80\).
Time = 0.06 (sec) , antiderivative size = 82, normalized size of antiderivative = 2.05 \[ \int \sqrt {3+3 \sin (e+f x)} \sqrt {c-c \sin (e+f x)} \, dx=\frac {\sqrt {3} \left (-\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {1+\sin (e+f x)} \sqrt {c-c \sin (e+f x)}}{f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )} \]
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Time = 3.26 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.90
method | result | size |
default | \(\frac {\tan \left (f x +e \right ) \sqrt {a \left (\sin \left (f x +e \right )+1\right )}\, \sqrt {-c \left (\sin \left (f x +e \right )-1\right )}}{f}\) | \(36\) |
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Time = 0.27 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.08 \[ \int \sqrt {3+3 \sin (e+f x)} \sqrt {c-c \sin (e+f x)} \, dx=\frac {\sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c} \sin \left (f x + e\right )}{f \cos \left (f x + e\right )} \]
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\[ \int \sqrt {3+3 \sin (e+f x)} \sqrt {c-c \sin (e+f x)} \, dx=\int \sqrt {a \left (\sin {\left (e + f x \right )} + 1\right )} \sqrt {- c \left (\sin {\left (e + f x \right )} - 1\right )}\, dx \]
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\[ \int \sqrt {3+3 \sin (e+f x)} \sqrt {c-c \sin (e+f x)} \, dx=\int { \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c} \,d x } \]
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Time = 0.30 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.28 \[ \int \sqrt {3+3 \sin (e+f x)} \sqrt {c-c \sin (e+f x)} \, dx=\frac {2 \, \sqrt {a} \sqrt {c} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2}}{f} \]
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Time = 6.76 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.18 \[ \int \sqrt {3+3 \sin (e+f x)} \sqrt {c-c \sin (e+f x)} \, dx=\frac {\sin \left (2\,e+2\,f\,x\right )\,\sqrt {a\,\left (\sin \left (e+f\,x\right )+1\right )}\,\sqrt {-c\,\left (\sin \left (e+f\,x\right )-1\right )}}{2\,f\,{\cos \left (e+f\,x\right )}^2} \]
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