Integrand size = 30, antiderivative size = 42 \[ \int \frac {(3+3 \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{7/2}} \, dx=\frac {\cos (e+f x) (3+3 \sin (e+f x))^{5/2}}{6 f (c-c \sin (e+f x))^{7/2}} \]
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Time = 0.07 (sec) , antiderivative size = 42, 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 = {2821} \[ \int \frac {(3+3 \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{7/2}} \, dx=\frac {\cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{6 f (c-c \sin (e+f x))^{7/2}} \]
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Rule 2821
Rubi steps \begin{align*} \text {integral}& = \frac {\cos (e+f x) (a+a \sin (e+f x))^{5/2}}{6 f (c-c \sin (e+f x))^{7/2}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(110\) vs. \(2(42)=84\).
Time = 12.74 (sec) , antiderivative size = 110, normalized size of antiderivative = 2.62 \[ \int \frac {(3+3 \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{7/2}} \, dx=\frac {3 \sqrt {3} (-5+3 \cos (2 (e+f x))) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) (1+\sin (e+f x))^{5/2}}{2 c^3 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5 (-1+\sin (e+f x))^3 \sqrt {c-c \sin (e+f x)}} \]
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Time = 3.02 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.74
method | result | size |
default | \(\frac {\tan \left (f x +e \right ) a^{2} \left (\cos ^{2}\left (f x +e \right )-4\right ) \sqrt {a \left (\sin \left (f x +e \right )+1\right )}}{3 f \left (2 \sin \left (f x +e \right )+\cos ^{2}\left (f x +e \right )-2\right ) \sqrt {-c \left (\sin \left (f x +e \right )-1\right )}\, c^{3}}\) | \(73\) |
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Leaf count of result is larger than twice the leaf count of optimal. 109 vs. \(2 (36) = 72\).
Time = 0.27 (sec) , antiderivative size = 109, normalized size of antiderivative = 2.60 \[ \int \frac {(3+3 \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{7/2}} \, dx=\frac {{\left (3 \, a^{2} \cos \left (f x + e\right )^{2} - 4 \, a^{2}\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{3 \, {\left (3 \, c^{4} f \cos \left (f x + e\right )^{3} - 4 \, c^{4} f \cos \left (f x + e\right ) - {\left (c^{4} f \cos \left (f x + e\right )^{3} - 4 \, c^{4} f \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )}} \]
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Timed out. \[ \int \frac {(3+3 \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{7/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {(3+3 \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{7/2}} \, dx=\int { \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {7}{2}}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 131 vs. \(2 (36) = 72\).
Time = 0.32 (sec) , antiderivative size = 131, normalized size of antiderivative = 3.12 \[ \int \frac {(3+3 \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{7/2}} \, dx=-\frac {{\left (3 \, a^{2} \sqrt {c} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 3 \, a^{2} \sqrt {c} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a^{2} \sqrt {c} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \sqrt {a}}{6 \, c^{4} f \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6}} \]
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Timed out. \[ \int \frac {(3+3 \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{7/2}} \, dx=\int \frac {{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{5/2}}{{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{7/2}} \,d x \]
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