Integrand size = 30, antiderivative size = 40 \[ \int \frac {\sqrt {3+3 \sin (e+f x)}}{(c-c \sin (e+f x))^{3/2}} \, dx=\frac {3 \cos (e+f x)}{f \sqrt {3+3 \sin (e+f x)} (c-c \sin (e+f x))^{3/2}} \]
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Time = 0.07 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {2817} \[ \int \frac {\sqrt {3+3 \sin (e+f x)}}{(c-c \sin (e+f x))^{3/2}} \, dx=\frac {a \cos (e+f x)}{f \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}} \]
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Rule 2817
Rubi steps \begin{align*} \text {integral}& = \frac {a \cos (e+f x)}{f \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(87\) vs. \(2(40)=80\).
Time = 0.72 (sec) , antiderivative size = 87, normalized size of antiderivative = 2.18 \[ \int \frac {\sqrt {3+3 \sin (e+f x)}}{(c-c \sin (e+f x))^{3/2}} \, dx=\frac {\sqrt {3} \sqrt {1+\sin (e+f x)} \sqrt {c-c \sin (e+f x)}}{c^2 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^3 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )} \]
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Time = 3.21 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.98
method | result | size |
default | \(\frac {\tan \left (f x +e \right ) \sqrt {a \left (\sin \left (f x +e \right )+1\right )}}{f c \sqrt {-c \left (\sin \left (f x +e \right )-1\right )}}\) | \(39\) |
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Time = 0.25 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.48 \[ \int \frac {\sqrt {3+3 \sin (e+f x)}}{(c-c \sin (e+f x))^{3/2}} \, dx=-\frac {\sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{c^{2} f \cos \left (f x + e\right ) \sin \left (f x + e\right ) - c^{2} f \cos \left (f x + e\right )} \]
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\[ \int \frac {\sqrt {3+3 \sin (e+f x)}}{(c-c \sin (e+f x))^{3/2}} \, dx=\int \frac {\sqrt {a \left (\sin {\left (e + f x \right )} + 1\right )}}{\left (- c \left (\sin {\left (e + f x \right )} - 1\right )\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {\sqrt {3+3 \sin (e+f x)}}{(c-c \sin (e+f x))^{3/2}} \, dx=\int { \frac {\sqrt {a \sin \left (f x + e\right ) + a}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}} \,d x } \]
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Time = 0.31 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.32 \[ \int \frac {\sqrt {3+3 \sin (e+f x)}}{(c-c \sin (e+f x))^{3/2}} \, dx=-\frac {\sqrt {a} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{2 \, c^{\frac {3}{2}} f \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2}} \]
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Timed out. \[ \int \frac {\sqrt {3+3 \sin (e+f x)}}{(c-c \sin (e+f x))^{3/2}} \, dx=\int \frac {\sqrt {a+a\,\sin \left (e+f\,x\right )}}{{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{3/2}} \,d x \]
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