Optimal. Leaf size=197 \[ \frac{2 c^2 (A-B) \cos (e+f x) \sqrt{c-c \sin (e+f x)}}{f \sqrt{a \sin (e+f x)+a}}+\frac{4 c^3 (A-B) \cos (e+f x) \log (\sin (e+f x)+1)}{f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{c (A-B) \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{2 f \sqrt{a \sin (e+f x)+a}}-\frac{B \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{3 f \sqrt{a \sin (e+f x)+a}} \]
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Rubi [A] time = 0.462934, antiderivative size = 197, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {2973, 2740, 2737, 2667, 31} \[ \frac{2 c^2 (A-B) \cos (e+f x) \sqrt{c-c \sin (e+f x)}}{f \sqrt{a \sin (e+f x)+a}}+\frac{4 c^3 (A-B) \cos (e+f x) \log (\sin (e+f x)+1)}{f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{c (A-B) \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{2 f \sqrt{a \sin (e+f x)+a}}-\frac{B \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{3 f \sqrt{a \sin (e+f x)+a}} \]
Antiderivative was successfully verified.
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Rule 2973
Rule 2740
Rule 2737
Rule 2667
Rule 31
Rubi steps
\begin{align*} \int \frac{(A+B \sin (e+f x)) (c-c \sin (e+f x))^{5/2}}{\sqrt{a+a \sin (e+f x)}} \, dx &=-\frac{B \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{3 f \sqrt{a+a \sin (e+f x)}}+(A-B) \int \frac{(c-c \sin (e+f x))^{5/2}}{\sqrt{a+a \sin (e+f x)}} \, dx\\ &=\frac{(A-B) c \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{2 f \sqrt{a+a \sin (e+f x)}}-\frac{B \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{3 f \sqrt{a+a \sin (e+f x)}}+(2 (A-B) c) \int \frac{(c-c \sin (e+f x))^{3/2}}{\sqrt{a+a \sin (e+f x)}} \, dx\\ &=\frac{2 (A-B) c^2 \cos (e+f x) \sqrt{c-c \sin (e+f x)}}{f \sqrt{a+a \sin (e+f x)}}+\frac{(A-B) c \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{2 f \sqrt{a+a \sin (e+f x)}}-\frac{B \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{3 f \sqrt{a+a \sin (e+f x)}}+\left (4 (A-B) c^2\right ) \int \frac{\sqrt{c-c \sin (e+f x)}}{\sqrt{a+a \sin (e+f x)}} \, dx\\ &=\frac{2 (A-B) c^2 \cos (e+f x) \sqrt{c-c \sin (e+f x)}}{f \sqrt{a+a \sin (e+f x)}}+\frac{(A-B) c \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{2 f \sqrt{a+a \sin (e+f x)}}-\frac{B \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{3 f \sqrt{a+a \sin (e+f x)}}+\frac{\left (4 a (A-B) c^3 \cos (e+f x)\right ) \int \frac{\cos (e+f x)}{a+a \sin (e+f x)} \, dx}{\sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}\\ &=\frac{2 (A-B) c^2 \cos (e+f x) \sqrt{c-c \sin (e+f x)}}{f \sqrt{a+a \sin (e+f x)}}+\frac{(A-B) c \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{2 f \sqrt{a+a \sin (e+f x)}}-\frac{B \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{3 f \sqrt{a+a \sin (e+f x)}}+\frac{\left (4 (A-B) c^3 \cos (e+f x)\right ) \operatorname{Subst}\left (\int \frac{1}{a+x} \, dx,x,a \sin (e+f x)\right )}{f \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}\\ &=\frac{4 (A-B) c^3 \cos (e+f x) \log (1+\sin (e+f x))}{f \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}+\frac{2 (A-B) c^2 \cos (e+f x) \sqrt{c-c \sin (e+f x)}}{f \sqrt{a+a \sin (e+f x)}}+\frac{(A-B) c \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{2 f \sqrt{a+a \sin (e+f x)}}-\frac{B \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{3 f \sqrt{a+a \sin (e+f x)}}\\ \end{align*}
Mathematica [A] time = 1.31412, size = 185, normalized size = 0.94 \[ -\frac{c^2 (\sin (e+f x)-1)^2 \sqrt{c-c \sin (e+f x)} \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right ) \left ((36 A-51 B) \sin (e+f x)+3 (A-3 B) \cos (2 (e+f x))-96 A \log \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )+B \sin (3 (e+f x))+96 B \log \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )\right )}{12 f \sqrt{a (\sin (e+f x)+1)} \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^5} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.378, size = 595, normalized size = 3. \begin{align*} \text{result too large to display} \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 (-c \sin \left (f x + e\right ) + c\right )}^{\frac{5}{2}}}{\sqrt{a \sin \left (f x + e\right ) + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{{\left ({\left (A - 2 \, B\right )} c^{2} \cos \left (f x + e\right )^{2} - 2 \,{\left (A - B\right )} c^{2} +{\left (B c^{2} \cos \left (f x + e\right )^{2} + 2 \,{\left (A - B\right )} c^{2}\right )} \sin \left (f x + e\right )\right )} \sqrt{-c \sin \left (f x + e\right ) + c}}{\sqrt{a \sin \left (f x + e\right ) + a}}, x\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*} \int \frac{{\left (B \sin \left (f x + e\right ) + A\right )}{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac{5}{2}}}{\sqrt{a \sin \left (f x + e\right ) + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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