Integrand size = 30, antiderivative size = 42 \[ \int \frac {(c-c \sin (e+f x))^{3/2}}{(3+3 \sin (e+f x))^{5/2}} \, dx=-\frac {\cos (e+f x) (c-c \sin (e+f x))^{3/2}}{4 f (3+3 \sin (e+f x))^{5/2}} \]
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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} \[ \int \frac {(c-c \sin (e+f x))^{3/2}}{(3+3 \sin (e+f x))^{5/2}} \, dx=-\frac {\cos (e+f x) (c-c \sin (e+f x))^{3/2}}{4 f (a \sin (e+f x)+a)^{5/2}} \]
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Rule 2821
Rubi steps \begin{align*} \text {integral}& = -\frac {\cos (e+f x) (c-c \sin (e+f x))^{3/2}}{4 f (a+a \sin (e+f x))^{5/2}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(92\) vs. \(2(42)=84\).
Time = 1.19 (sec) , antiderivative size = 92, normalized size of antiderivative = 2.19 \[ \int \frac {(c-c \sin (e+f x))^{3/2}}{(3+3 \sin (e+f x))^{5/2}} \, dx=\frac {c \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sin (e+f x) \sqrt {c-c \sin (e+f x)}}{9 \sqrt {3} f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) (1+\sin (e+f x))^{5/2}} \]
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Time = 3.14 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.19
method | result | size |
default | \(\frac {\tan \left (f x +e \right ) \sqrt {-c \left (\sin \left (f x +e \right )-1\right )}\, c}{f \left (\sin \left (f x +e \right )+1\right ) \sqrt {a \left (\sin \left (f x +e \right )+1\right )}\, a^{2}}\) | \(50\) |
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Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (36) = 72\).
Time = 0.28 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.90 \[ \int \frac {(c-c \sin (e+f x))^{3/2}}{(3+3 \sin (e+f x))^{5/2}} \, dx=-\frac {\sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c} c \sin \left (f x + e\right )}{a^{3} f \cos \left (f x + e\right )^{3} - 2 \, a^{3} f \cos \left (f x + e\right ) \sin \left (f x + e\right ) - 2 \, a^{3} f \cos \left (f x + e\right )} \]
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\[ \int \frac {(c-c \sin (e+f x))^{3/2}}{(3+3 \sin (e+f x))^{5/2}} \, dx=\int \frac {\left (- c \left (\sin {\left (e + f x \right )} - 1\right )\right )^{\frac {3}{2}}}{\left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{\frac {5}{2}}}\, dx \]
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\[ \int \frac {(c-c \sin (e+f x))^{3/2}}{(3+3 \sin (e+f x))^{5/2}} \, dx=\int { \frac {{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 93 vs. \(2 (36) = 72\).
Time = 0.54 (sec) , antiderivative size = 93, normalized size of antiderivative = 2.21 \[ \int \frac {(c-c \sin (e+f x))^{3/2}}{(3+3 \sin (e+f x))^{5/2}} \, dx=-\frac {{\left (2 \, \sqrt {a} c \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - \sqrt {a} c \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \sqrt {c}}{4 \, a^{3} f \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} \]
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Time = 8.05 (sec) , antiderivative size = 118, normalized size of antiderivative = 2.81 \[ \int \frac {(c-c \sin (e+f x))^{3/2}}{(3+3 \sin (e+f x))^{5/2}} \, dx=\frac {2\,c\,\sqrt {-c\,\left (\sin \left (e+f\,x\right )-1\right )}\,\left (-2\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+2\,{\sin \left (\frac {3\,e}{2}+\frac {3\,f\,x}{2}\right )}^2+2\,\sin \left (2\,e+2\,f\,x\right )\right )}{a^2\,f\,\sqrt {a\,\left (\sin \left (e+f\,x\right )+1\right )}\,\left (-8\,{\sin \left (e+f\,x\right )}^2+4\,\sin \left (e+f\,x\right )+2\,{\sin \left (2\,e+2\,f\,x\right )}^2+4\,\sin \left (3\,e+3\,f\,x\right )+8\right )} \]
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