Integrand size = 30, antiderivative size = 42 \[ \int \frac {\sqrt {3+3 \sin (e+f x)}}{(c-c \sin (e+f x))^{5/2}} \, dx=\frac {3 \cos (e+f x)}{2 f \sqrt {3+3 \sin (e+f x)} (c-c \sin (e+f x))^{5/2}} \]
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Time = 0.07 (sec) , antiderivative size = 43, 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} \[ \int \frac {\sqrt {3+3 \sin (e+f x)}}{(c-c \sin (e+f x))^{5/2}} \, dx=\frac {a \cos (e+f x)}{2 f \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}} \]
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Rule 2817
Rubi steps \begin{align*} \text {integral}& = \frac {a \cos (e+f x)}{2 f \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(90\) vs. \(2(42)=84\).
Time = 0.80 (sec) , antiderivative size = 90, normalized size of antiderivative = 2.14 \[ \int \frac {\sqrt {3+3 \sin (e+f x)}}{(c-c \sin (e+f x))^{5/2}} \, dx=\frac {\sqrt {3} \sqrt {1+\sin (e+f x)} \sqrt {c-c \sin (e+f x)}}{2 c^3 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )} \]
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Time = 2.47 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.60
method | result | size |
default | \(-\frac {\sqrt {a \left (\sin \left (f x +e \right )+1\right )}\, \left (\cos \left (f x +e \right )+2 \tan \left (f x +e \right )-\sec \left (f x +e \right )\right )}{2 f \left (\sin \left (f x +e \right )-1\right ) \sqrt {-c \left (\sin \left (f x +e \right )-1\right )}\, c^{2}}\) | \(67\) |
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Time = 0.27 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.74 \[ \int \frac {\sqrt {3+3 \sin (e+f x)}}{(c-c \sin (e+f x))^{5/2}} \, dx=-\frac {\sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{2 \, {\left (c^{3} f \cos \left (f x + e\right )^{3} + 2 \, c^{3} f \cos \left (f x + e\right ) \sin \left (f x + e\right ) - 2 \, c^{3} f \cos \left (f x + e\right )\right )}} \]
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\[ \int \frac {\sqrt {3+3 \sin (e+f x)}}{(c-c \sin (e+f x))^{5/2}} \, dx=\int \frac {\sqrt {a \left (\sin {\left (e + f x \right )} + 1\right )}}{\left (- c \left (\sin {\left (e + f x \right )} - 1\right )\right )^{\frac {5}{2}}}\, dx \]
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\[ \int \frac {\sqrt {3+3 \sin (e+f x)}}{(c-c \sin (e+f x))^{5/2}} \, dx=\int { \frac {\sqrt {a \sin \left (f x + e\right ) + a}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}} \,d x } \]
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Time = 0.33 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.26 \[ \int \frac {\sqrt {3+3 \sin (e+f x)}}{(c-c \sin (e+f x))^{5/2}} \, dx=-\frac {\sqrt {a} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{8 \, c^{\frac {5}{2}} f \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4}} \]
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Time = 8.97 (sec) , antiderivative size = 142, normalized size of antiderivative = 3.38 \[ \int \frac {\sqrt {3+3 \sin (e+f x)}}{(c-c \sin (e+f x))^{5/2}} \, dx=\frac {4\,\sqrt {a\,\left (\sin \left (e+f\,x\right )+1\right )}\,\sqrt {-c\,\left (\sin \left (e+f\,x\right )-1\right )}\,\left (10\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2-2\,{\sin \left (\frac {3\,e}{2}+\frac {3\,f\,x}{2}\right )}^2+4\,\sin \left (2\,e+2\,f\,x\right )-4\right )}{c^3\,f\,\left (30\,{\sin \left (e+f\,x\right )}^2+48\,\sin \left (e+f\,x\right )-52\,{\sin \left (2\,e+2\,f\,x\right )}^2+2\,{\sin \left (3\,e+3\,f\,x\right )}^2+40\,\sin \left (3\,e+3\,f\,x\right )-8\,\sin \left (5\,e+5\,f\,x\right )-32\right )} \]
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