Integrand size = 33, antiderivative size = 536 \[ \int \frac {x^3 \sqrt {a-a \sin (e+f x)}}{(c+c \sin (e+f x))^{3/2}} \, dx=-\frac {3 a x^2}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {3 i a x^2 \cos (e+f x)}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {12 i a x \arctan \left (e^{i (e+f x)}\right ) \cos (e+f x)}{c f^3 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {6 a x \cos (e+f x) \log \left (1+e^{2 i (e+f x)}\right )}{c f^3 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {6 i a \cos (e+f x) \operatorname {PolyLog}\left (2,-i e^{i (e+f x)}\right )}{c f^4 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {6 i a \cos (e+f x) \operatorname {PolyLog}\left (2,i e^{i (e+f x)}\right )}{c f^4 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {3 i a \cos (e+f x) \operatorname {PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{c f^4 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {a x^3 \sec (e+f x)}{c f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {3 a x^2 \sin (e+f x)}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {a x^3 \tan (e+f x)}{c f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}} \]
[Out]
Time = 4.17 (sec) , antiderivative size = 536, normalized size of antiderivative = 1.00, number of steps used = 51, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.515, Rules used = {4700, 6873, 12, 6874, 4271, 4266, 2317, 2438, 2611, 6744, 2320, 6724, 3842, 4269, 3800, 2221, 4498} \[ \int \frac {x^3 \sqrt {a-a \sin (e+f x)}}{(c+c \sin (e+f x))^{3/2}} \, dx=-\frac {12 i a x \arctan \left (e^{i (e+f x)}\right ) \cos (e+f x)}{c f^3 \sqrt {a-a \sin (e+f x)} \sqrt {c \sin (e+f x)+c}}+\frac {6 i a \operatorname {PolyLog}\left (2,-i e^{i (e+f x)}\right ) \cos (e+f x)}{c f^4 \sqrt {a-a \sin (e+f x)} \sqrt {c \sin (e+f x)+c}}-\frac {6 i a \operatorname {PolyLog}\left (2,i e^{i (e+f x)}\right ) \cos (e+f x)}{c f^4 \sqrt {a-a \sin (e+f x)} \sqrt {c \sin (e+f x)+c}}-\frac {3 i a \operatorname {PolyLog}\left (2,-e^{2 i (e+f x)}\right ) \cos (e+f x)}{c f^4 \sqrt {a-a \sin (e+f x)} \sqrt {c \sin (e+f x)+c}}+\frac {6 a x \log \left (1+e^{2 i (e+f x)}\right ) \cos (e+f x)}{c f^3 \sqrt {a-a \sin (e+f x)} \sqrt {c \sin (e+f x)+c}}+\frac {3 a x^2 \sin (e+f x)}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c \sin (e+f x)+c}}-\frac {3 a x^2}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c \sin (e+f x)+c}}-\frac {3 i a x^2 \cos (e+f x)}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c \sin (e+f x)+c}}+\frac {a x^3 \tan (e+f x)}{c f \sqrt {a-a \sin (e+f x)} \sqrt {c \sin (e+f x)+c}}-\frac {a x^3 \sec (e+f x)}{c f \sqrt {a-a \sin (e+f x)} \sqrt {c \sin (e+f x)+c}} \]
[In]
[Out]
Rule 12
Rule 2221
Rule 2317
Rule 2320
Rule 2438
Rule 2611
Rule 3800
Rule 3842
Rule 4266
Rule 4269
Rule 4271
Rule 4498
Rule 4700
Rule 6724
Rule 6744
Rule 6873
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \frac {\cos (e+f x) \int x^3 \sec ^3(e+f x) (a-a \sin (e+f x))^2 \, dx}{a c \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}} \\ & = \frac {\cos (e+f x) \int a^2 x^3 \sec ^3(e+f x) (1-\sin (e+f x))^2 \, dx}{a c \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}} \\ & = \frac {(a \cos (e+f x)) \int x^3 \sec ^3(e+f x) (1-\sin (e+f x))^2 \, dx}{c \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}} \\ & = \frac {(a \cos (e+f x)) \int \left (x^3 \sec ^3(e+f x)-2 x^3 \sec ^2(e+f x) \tan (e+f x)+x^3 \sec (e+f x) \tan ^2(e+f x)\right ) \, dx}{c \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}} \\ & = \frac {(a \cos (e+f x)) \int x^3 \sec ^3(e+f x) \, dx}{c \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {(a \cos (e+f x)) \int x^3 \sec (e+f x) \tan ^2(e+f x) \, dx}{c \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {(2 a \cos (e+f x)) \int x^3 \sec ^2(e+f x) \tan (e+f x) \, dx}{c \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}} \\ & = -\frac {3 a x^2}{2 c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {a x^3 \sec (e+f x)}{c f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {a x^3 \tan (e+f x)}{2 c f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {(a \cos (e+f x)) \int x^3 \sec (e+f x) \, dx}{2 c \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {(a \cos (e+f x)) \int x^3 \sec (e+f x) \, dx}{c \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {(a \cos (e+f x)) \int x^3 \sec ^3(e+f x) \, dx}{c \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {(3 a \cos (e+f x)) \int x \sec (e+f x) \, dx}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {(3 a \cos (e+f x)) \int x^2 \sec ^2(e+f x) \, dx}{c f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}} \\ & = -\frac {3 a x^2}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {6 i a x \arctan \left (e^{i (e+f x)}\right ) \cos (e+f x)}{c f^3 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {i a x^3 \arctan \left (e^{i (e+f x)}\right ) \cos (e+f x)}{c f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {a x^3 \sec (e+f x)}{c f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {3 a x^2 \sin (e+f x)}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {a x^3 \tan (e+f x)}{c f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {(a \cos (e+f x)) \int x^3 \sec (e+f x) \, dx}{2 c \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {(3 a \cos (e+f x)) \int \log \left (1-i e^{i (e+f x)}\right ) \, dx}{c f^3 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {(3 a \cos (e+f x)) \int \log \left (1+i e^{i (e+f x)}\right ) \, dx}{c f^3 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {(3 a \cos (e+f x)) \int x \sec (e+f x) \, dx}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {(6 a \cos (e+f x)) \int x \tan (e+f x) \, dx}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {(3 a \cos (e+f x)) \int x^2 \log \left (1-i e^{i (e+f x)}\right ) \, dx}{2 c f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {(3 a \cos (e+f x)) \int x^2 \log \left (1+i e^{i (e+f x)}\right ) \, dx}{2 c f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {(3 a \cos (e+f x)) \int x^2 \log \left (1-i e^{i (e+f x)}\right ) \, dx}{c f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {(3 a \cos (e+f x)) \int x^2 \log \left (1+i e^{i (e+f x)}\right ) \, dx}{c f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}} \\ & = -\frac {3 a x^2}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {3 i a x^2 \cos (e+f x)}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {12 i a x \arctan \left (e^{i (e+f x)}\right ) \cos (e+f x)}{c f^3 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {3 i a x^2 \cos (e+f x) \operatorname {PolyLog}\left (2,-i e^{i (e+f x)}\right )}{2 c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {3 i a x^2 \cos (e+f x) \operatorname {PolyLog}\left (2,i e^{i (e+f x)}\right )}{2 c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {a x^3 \sec (e+f x)}{c f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {3 a x^2 \sin (e+f x)}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {a x^3 \tan (e+f x)}{c f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {(3 i a \cos (e+f x)) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{i (e+f x)}\right )}{c f^4 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {(3 i a \cos (e+f x)) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{i (e+f x)}\right )}{c f^4 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {(3 a \cos (e+f x)) \int \log \left (1-i e^{i (e+f x)}\right ) \, dx}{c f^3 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {(3 a \cos (e+f x)) \int \log \left (1+i e^{i (e+f x)}\right ) \, dx}{c f^3 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {(3 i a \cos (e+f x)) \int x \operatorname {PolyLog}\left (2,-i e^{i (e+f x)}\right ) \, dx}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {(3 i a \cos (e+f x)) \int x \operatorname {PolyLog}\left (2,i e^{i (e+f x)}\right ) \, dx}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {(6 i a \cos (e+f x)) \int x \operatorname {PolyLog}\left (2,-i e^{i (e+f x)}\right ) \, dx}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {(6 i a \cos (e+f x)) \int x \operatorname {PolyLog}\left (2,i e^{i (e+f x)}\right ) \, dx}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {(12 i a \cos (e+f x)) \int \frac {e^{2 i (e+f x)} x}{1+e^{2 i (e+f x)}} \, dx}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {(3 a \cos (e+f x)) \int x^2 \log \left (1-i e^{i (e+f x)}\right ) \, dx}{2 c f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {(3 a \cos (e+f x)) \int x^2 \log \left (1+i e^{i (e+f x)}\right ) \, dx}{2 c f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}} \\ & = -\frac {3 a x^2}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {3 i a x^2 \cos (e+f x)}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {12 i a x \arctan \left (e^{i (e+f x)}\right ) \cos (e+f x)}{c f^3 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {6 a x \cos (e+f x) \log \left (1+e^{2 i (e+f x)}\right )}{c f^3 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {3 i a \cos (e+f x) \operatorname {PolyLog}\left (2,-i e^{i (e+f x)}\right )}{c f^4 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {3 i a \cos (e+f x) \operatorname {PolyLog}\left (2,i e^{i (e+f x)}\right )}{c f^4 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {3 a x \cos (e+f x) \operatorname {PolyLog}\left (3,-i e^{i (e+f x)}\right )}{c f^3 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {3 a x \cos (e+f x) \operatorname {PolyLog}\left (3,i e^{i (e+f x)}\right )}{c f^3 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {a x^3 \sec (e+f x)}{c f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {3 a x^2 \sin (e+f x)}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {a x^3 \tan (e+f x)}{c f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {(3 i a \cos (e+f x)) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{i (e+f x)}\right )}{c f^4 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {(3 i a \cos (e+f x)) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{i (e+f x)}\right )}{c f^4 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {(3 a \cos (e+f x)) \int \operatorname {PolyLog}\left (3,-i e^{i (e+f x)}\right ) \, dx}{c f^3 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {(3 a \cos (e+f x)) \int \operatorname {PolyLog}\left (3,i e^{i (e+f x)}\right ) \, dx}{c f^3 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {(6 a \cos (e+f x)) \int \log \left (1+e^{2 i (e+f x)}\right ) \, dx}{c f^3 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {(6 a \cos (e+f x)) \int \operatorname {PolyLog}\left (3,-i e^{i (e+f x)}\right ) \, dx}{c f^3 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {(6 a \cos (e+f x)) \int \operatorname {PolyLog}\left (3,i e^{i (e+f x)}\right ) \, dx}{c f^3 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {(3 i a \cos (e+f x)) \int x \operatorname {PolyLog}\left (2,-i e^{i (e+f x)}\right ) \, dx}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {(3 i a \cos (e+f x)) \int x \operatorname {PolyLog}\left (2,i e^{i (e+f x)}\right ) \, dx}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}} \\ & = -\frac {3 a x^2}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {3 i a x^2 \cos (e+f x)}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {12 i a x \arctan \left (e^{i (e+f x)}\right ) \cos (e+f x)}{c f^3 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {6 a x \cos (e+f x) \log \left (1+e^{2 i (e+f x)}\right )}{c f^3 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {6 i a \cos (e+f x) \operatorname {PolyLog}\left (2,-i e^{i (e+f x)}\right )}{c f^4 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {6 i a \cos (e+f x) \operatorname {PolyLog}\left (2,i e^{i (e+f x)}\right )}{c f^4 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {a x^3 \sec (e+f x)}{c f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {3 a x^2 \sin (e+f x)}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {a x^3 \tan (e+f x)}{c f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {(3 i a \cos (e+f x)) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 i (e+f x)}\right )}{c f^4 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {(3 i a \cos (e+f x)) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,-i x)}{x} \, dx,x,e^{i (e+f x)}\right )}{c f^4 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {(3 i a \cos (e+f x)) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,i x)}{x} \, dx,x,e^{i (e+f x)}\right )}{c f^4 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {(6 i a \cos (e+f x)) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,-i x)}{x} \, dx,x,e^{i (e+f x)}\right )}{c f^4 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {(6 i a \cos (e+f x)) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,i x)}{x} \, dx,x,e^{i (e+f x)}\right )}{c f^4 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {(3 a \cos (e+f x)) \int \operatorname {PolyLog}\left (3,-i e^{i (e+f x)}\right ) \, dx}{c f^3 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {(3 a \cos (e+f x)) \int \operatorname {PolyLog}\left (3,i e^{i (e+f x)}\right ) \, dx}{c f^3 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}} \\ & = -\frac {3 a x^2}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {3 i a x^2 \cos (e+f x)}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {12 i a x \arctan \left (e^{i (e+f x)}\right ) \cos (e+f x)}{c f^3 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {6 a x \cos (e+f x) \log \left (1+e^{2 i (e+f x)}\right )}{c f^3 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {6 i a \cos (e+f x) \operatorname {PolyLog}\left (2,-i e^{i (e+f x)}\right )}{c f^4 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {6 i a \cos (e+f x) \operatorname {PolyLog}\left (2,i e^{i (e+f x)}\right )}{c f^4 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {3 i a \cos (e+f x) \operatorname {PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{c f^4 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {3 i a \cos (e+f x) \operatorname {PolyLog}\left (4,-i e^{i (e+f x)}\right )}{c f^4 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {3 i a \cos (e+f x) \operatorname {PolyLog}\left (4,i e^{i (e+f x)}\right )}{c f^4 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {a x^3 \sec (e+f x)}{c f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {3 a x^2 \sin (e+f x)}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {a x^3 \tan (e+f x)}{c f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {(3 i a \cos (e+f x)) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,-i x)}{x} \, dx,x,e^{i (e+f x)}\right )}{c f^4 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {(3 i a \cos (e+f x)) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,i x)}{x} \, dx,x,e^{i (e+f x)}\right )}{c f^4 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}} \\ & = -\frac {3 a x^2}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {3 i a x^2 \cos (e+f x)}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {12 i a x \arctan \left (e^{i (e+f x)}\right ) \cos (e+f x)}{c f^3 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {6 a x \cos (e+f x) \log \left (1+e^{2 i (e+f x)}\right )}{c f^3 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {6 i a \cos (e+f x) \operatorname {PolyLog}\left (2,-i e^{i (e+f x)}\right )}{c f^4 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {6 i a \cos (e+f x) \operatorname {PolyLog}\left (2,i e^{i (e+f x)}\right )}{c f^4 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {3 i a \cos (e+f x) \operatorname {PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{c f^4 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {a x^3 \sec (e+f x)}{c f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {3 a x^2 \sin (e+f x)}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {a x^3 \tan (e+f x)}{c f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}} \\ \end{align*}
Time = 2.30 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.36 \[ \int \frac {x^3 \sqrt {a-a \sin (e+f x)}}{(c+c \sin (e+f x))^{3/2}} \, dx=\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {a-a \sin (e+f x)} \left (12 i \operatorname {PolyLog}\left (2,-i e^{-i (e+f x)}\right ) (1+\sin (e+f x))+f x \left (3 i f x-f^2 x^2-3 f x \cos (e+f x)+12 \log \left (1+i e^{-i (e+f x)}\right )+3 \left (i f x+4 \log \left (1+i e^{-i (e+f x)}\right )\right ) \sin (e+f x)\right )\right )}{f^4 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) (c (1+\sin (e+f x)))^{3/2}} \]
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\[\int \frac {x^{3} \sqrt {a -\sin \left (f x +e \right ) a}}{\left (c +c \sin \left (f x +e \right )\right )^{\frac {3}{2}}}d x\]
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\[ \int \frac {x^3 \sqrt {a-a \sin (e+f x)}}{(c+c \sin (e+f x))^{3/2}} \, dx=\int { \frac {\sqrt {-a \sin \left (f x + e\right ) + a} x^{3}}{{\left (c \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {x^3 \sqrt {a-a \sin (e+f x)}}{(c+c \sin (e+f x))^{3/2}} \, dx=\int \frac {x^{3} \sqrt {- a \left (\sin {\left (e + f x \right )} - 1\right )}}{\left (c \left (\sin {\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {x^3 \sqrt {a-a \sin (e+f x)}}{(c+c \sin (e+f x))^{3/2}} \, dx=\int { \frac {\sqrt {-a \sin \left (f x + e\right ) + a} x^{3}}{{\left (c \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {x^3 \sqrt {a-a \sin (e+f x)}}{(c+c \sin (e+f x))^{3/2}} \, dx=\int { \frac {\sqrt {-a \sin \left (f x + e\right ) + a} x^{3}}{{\left (c \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {x^3 \sqrt {a-a \sin (e+f x)}}{(c+c \sin (e+f x))^{3/2}} \, dx=\int \frac {x^3\,\sqrt {a-a\,\sin \left (e+f\,x\right )}}{{\left (c+c\,\sin \left (e+f\,x\right )\right )}^{3/2}} \,d x \]
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