Optimal. Leaf size=126 \[ -\frac{a (A-7 B) \tanh ^{-1}\left (\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right )}{8 \sqrt{2} c^{5/2} f}-\frac{a (A+9 B) \cos (e+f x)}{8 c f (c-c \sin (e+f x))^{3/2}}+\frac{a (A+B) \cos (e+f x)}{2 f (c-c \sin (e+f x))^{5/2}} \]
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Rubi [A] time = 0.334508, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.139, Rules used = {2967, 2857, 2750, 2649, 206} \[ -\frac{a (A-7 B) \tanh ^{-1}\left (\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right )}{8 \sqrt{2} c^{5/2} f}-\frac{a (A+9 B) \cos (e+f x)}{8 c f (c-c \sin (e+f x))^{3/2}}+\frac{a (A+B) \cos (e+f x)}{2 f (c-c \sin (e+f x))^{5/2}} \]
Antiderivative was successfully verified.
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Rule 2967
Rule 2857
Rule 2750
Rule 2649
Rule 206
Rubi steps
\begin{align*} \int \frac{(a+a \sin (e+f x)) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{5/2}} \, dx &=(a c) \int \frac{\cos ^2(e+f x) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{7/2}} \, dx\\ &=\frac{a (A+B) \cos (e+f x)}{2 f (c-c \sin (e+f x))^{5/2}}+\frac{a \int \frac{-A c-5 B c-4 B c \sin (e+f x)}{(c-c \sin (e+f x))^{3/2}} \, dx}{4 c^2}\\ &=\frac{a (A+B) \cos (e+f x)}{2 f (c-c \sin (e+f x))^{5/2}}-\frac{a (A+9 B) \cos (e+f x)}{8 c f (c-c \sin (e+f x))^{3/2}}-\frac{(a (A-7 B)) \int \frac{1}{\sqrt{c-c \sin (e+f x)}} \, dx}{16 c^2}\\ &=\frac{a (A+B) \cos (e+f x)}{2 f (c-c \sin (e+f x))^{5/2}}-\frac{a (A+9 B) \cos (e+f x)}{8 c f (c-c \sin (e+f x))^{3/2}}+\frac{(a (A-7 B)) \operatorname{Subst}\left (\int \frac{1}{2 c-x^2} \, dx,x,-\frac{c \cos (e+f x)}{\sqrt{c-c \sin (e+f x)}}\right )}{8 c^2 f}\\ &=-\frac{a (A-7 B) \tanh ^{-1}\left (\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right )}{8 \sqrt{2} c^{5/2} f}+\frac{a (A+B) \cos (e+f x)}{2 f (c-c \sin (e+f x))^{5/2}}-\frac{a (A+9 B) \cos (e+f x)}{8 c f (c-c \sin (e+f x))^{3/2}}\\ \end{align*}
Mathematica [A] time = 2.17522, size = 199, normalized size = 1.58 \[ -\frac{a (\sin (e+f x)-1) (\sin (e+f x)+1) \left (\frac{2 \sqrt{c} \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right ) ((A+9 B) \sin (e+f x)+3 A-5 B)}{\left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^5}+\sqrt{2} (A-7 B) \sec (e+f x) \sqrt{-c (\sin (e+f x)+1)} \tan ^{-1}\left (\frac{\sqrt{-c (\sin (e+f x)+1)}}{\sqrt{2} \sqrt{c}}\right )\right )}{16 c^{5/2} f \sqrt{c-c \sin (e+f x)} \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 1.45, size = 268, normalized size = 2.1 \begin{align*}{\frac{a}{ \left ( -16+16\,\sin \left ( fx+e \right ) \right ) \cos \left ( fx+e \right ) f} \left ( -2\,\sin \left ( fx+e \right ) \sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{c+c\sin \left ( fx+e \right ) }\sqrt{2}}{\sqrt{c}}} \right ){c}^{2} \left ( A-7\,B \right ) -\sqrt{2}{\it Artanh} \left ({\frac{\sqrt{2}}{2}\sqrt{c+c\sin \left ( fx+e \right ) }{\frac{1}{\sqrt{c}}}} \right ){c}^{2} \left ( A-7\,B \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{2}+2\,A\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{c+c\sin \left ( fx+e \right ) }\sqrt{2}}{\sqrt{c}}} \right ){c}^{2}-4\,A\sqrt{c+c\sin \left ( fx+e \right ) }{c}^{3/2}-2\,A \left ( c+c\sin \left ( fx+e \right ) \right ) ^{3/2}\sqrt{c}-14\,B\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{c+c\sin \left ( fx+e \right ) }\sqrt{2}}{\sqrt{c}}} \right ){c}^{2}+28\,B\sqrt{c+c\sin \left ( fx+e \right ) }{c}^{3/2}-18\,B \left ( c+c\sin \left ( fx+e \right ) \right ) ^{3/2}\sqrt{c} \right ) \sqrt{c \left ( 1+\sin \left ( fx+e \right ) \right ) }{c}^{-{\frac{9}{2}}}{\frac{1}{\sqrt{c-c\sin \left ( fx+e \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (B \sin \left (f x + e\right ) + A\right )}{\left (a \sin \left (f x + e\right ) + a\right )}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.7798, size = 1035, normalized size = 8.21 \begin{align*} -\frac{\sqrt{2}{\left ({\left (A - 7 \, B\right )} a \cos \left (f x + e\right )^{3} + 3 \,{\left (A - 7 \, B\right )} a \cos \left (f x + e\right )^{2} - 2 \,{\left (A - 7 \, B\right )} a \cos \left (f x + e\right ) - 4 \,{\left (A - 7 \, B\right )} a -{\left ({\left (A - 7 \, B\right )} a \cos \left (f x + e\right )^{2} - 2 \,{\left (A - 7 \, B\right )} a \cos \left (f x + e\right ) - 4 \,{\left (A - 7 \, B\right )} a\right )} \sin \left (f x + e\right )\right )} \sqrt{c} \log \left (-\frac{c \cos \left (f x + e\right )^{2} + 2 \, \sqrt{2} \sqrt{-c \sin \left (f x + e\right ) + c} \sqrt{c}{\left (\cos \left (f x + e\right ) + \sin \left (f x + e\right ) + 1\right )} + 3 \, c \cos \left (f x + e\right ) +{\left (c \cos \left (f x + e\right ) - 2 \, c\right )} \sin \left (f x + e\right ) + 2 \, c}{\cos \left (f x + e\right )^{2} +{\left (\cos \left (f x + e\right ) + 2\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 2}\right ) - 4 \,{\left ({\left (A + 9 \, B\right )} a \cos \left (f x + e\right )^{2} -{\left (3 \, A - 5 \, B\right )} a \cos \left (f x + e\right ) - 4 \,{\left (A + B\right )} a -{\left ({\left (A + 9 \, B\right )} a \cos \left (f x + e\right ) + 4 \,{\left (A + B\right )} a\right )} \sin \left (f x + e\right )\right )} \sqrt{-c \sin \left (f x + e\right ) + c}}{32 \,{\left (c^{3} f \cos \left (f x + e\right )^{3} + 3 \, c^{3} f \cos \left (f x + e\right )^{2} - 2 \, c^{3} f \cos \left (f x + e\right ) - 4 \, c^{3} f -{\left (c^{3} f \cos \left (f x + e\right )^{2} - 2 \, c^{3} f \cos \left (f x + e\right ) - 4 \, c^{3} f\right )} \sin \left (f x + e\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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