Optimal. Leaf size=243 \[ -\frac{a^3 \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{c^4 f (c-c \sin (e+f x))^{3/2}}+\frac{a^2 \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 c^3 f (c-c \sin (e+f x))^{5/2}}-\frac{a^4 \cos (e+f x) \log (1-\sin (e+f x))}{c^5 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{3 c^2 f (c-c \sin (e+f x))^{7/2}}+\frac{\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{4 c f (c-c \sin (e+f x))^{9/2}} \]
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Rubi [A] time = 0.784987, antiderivative size = 243, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.132, Rules used = {2841, 2739, 2737, 2667, 31} \[ -\frac{a^3 \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{c^4 f (c-c \sin (e+f x))^{3/2}}+\frac{a^2 \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 c^3 f (c-c \sin (e+f x))^{5/2}}-\frac{a^4 \cos (e+f x) \log (1-\sin (e+f x))}{c^5 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{3 c^2 f (c-c \sin (e+f x))^{7/2}}+\frac{\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{4 c f (c-c \sin (e+f x))^{9/2}} \]
Antiderivative was successfully verified.
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Rule 2841
Rule 2739
Rule 2737
Rule 2667
Rule 31
Rubi steps
\begin{align*} \int \frac{\cos ^2(e+f x) (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{11/2}} \, dx &=\frac{\int \frac{(a+a \sin (e+f x))^{9/2}}{(c-c \sin (e+f x))^{9/2}} \, dx}{a c}\\ &=\frac{\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{4 c f (c-c \sin (e+f x))^{9/2}}-\frac{\int \frac{(a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{7/2}} \, dx}{c^2}\\ &=\frac{\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{4 c f (c-c \sin (e+f x))^{9/2}}-\frac{a \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{3 c^2 f (c-c \sin (e+f x))^{7/2}}+\frac{a \int \frac{(a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{5/2}} \, dx}{c^3}\\ &=\frac{\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{4 c f (c-c \sin (e+f x))^{9/2}}-\frac{a \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{3 c^2 f (c-c \sin (e+f x))^{7/2}}+\frac{a^2 \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{2 c^3 f (c-c \sin (e+f x))^{5/2}}-\frac{a^2 \int \frac{(a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{3/2}} \, dx}{c^4}\\ &=\frac{\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{4 c f (c-c \sin (e+f x))^{9/2}}-\frac{a \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{3 c^2 f (c-c \sin (e+f x))^{7/2}}+\frac{a^2 \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{2 c^3 f (c-c \sin (e+f x))^{5/2}}-\frac{a^3 \cos (e+f x) \sqrt{a+a \sin (e+f x)}}{c^4 f (c-c \sin (e+f x))^{3/2}}+\frac{a^3 \int \frac{\sqrt{a+a \sin (e+f x)}}{\sqrt{c-c \sin (e+f x)}} \, dx}{c^5}\\ &=\frac{\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{4 c f (c-c \sin (e+f x))^{9/2}}-\frac{a \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{3 c^2 f (c-c \sin (e+f x))^{7/2}}+\frac{a^2 \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{2 c^3 f (c-c \sin (e+f x))^{5/2}}-\frac{a^3 \cos (e+f x) \sqrt{a+a \sin (e+f x)}}{c^4 f (c-c \sin (e+f x))^{3/2}}+\frac{\left (a^4 \cos (e+f x)\right ) \int \frac{\cos (e+f x)}{c-c \sin (e+f x)} \, dx}{c^4 \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}\\ &=\frac{\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{4 c f (c-c \sin (e+f x))^{9/2}}-\frac{a \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{3 c^2 f (c-c \sin (e+f x))^{7/2}}+\frac{a^2 \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{2 c^3 f (c-c \sin (e+f x))^{5/2}}-\frac{a^3 \cos (e+f x) \sqrt{a+a \sin (e+f x)}}{c^4 f (c-c \sin (e+f x))^{3/2}}-\frac{\left (a^4 \cos (e+f x)\right ) \operatorname{Subst}\left (\int \frac{1}{c+x} \, dx,x,-c \sin (e+f x)\right )}{c^5 f \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}\\ &=\frac{\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{4 c f (c-c \sin (e+f x))^{9/2}}-\frac{a \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{3 c^2 f (c-c \sin (e+f x))^{7/2}}+\frac{a^2 \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{2 c^3 f (c-c \sin (e+f x))^{5/2}}-\frac{a^3 \cos (e+f x) \sqrt{a+a \sin (e+f x)}}{c^4 f (c-c \sin (e+f x))^{3/2}}-\frac{a^4 \cos (e+f x) \log (1-\sin (e+f x))}{c^5 f \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}\\ \end{align*}
Mathematica [A] time = 6.66159, size = 437, normalized size = 1.8 \[ -\frac{8 (a (\sin (e+f x)+1))^{7/2} \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^9}{f (c-c \sin (e+f x))^{11/2} \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^7}+\frac{12 (a (\sin (e+f x)+1))^{7/2} \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^7}{f (c-c \sin (e+f x))^{11/2} \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^7}-\frac{32 (a (\sin (e+f x)+1))^{7/2} \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^5}{3 f (c-c \sin (e+f x))^{11/2} \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^7}+\frac{4 (a (\sin (e+f x)+1))^{7/2} \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^3}{f (c-c \sin (e+f x))^{11/2} \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^7}-\frac{2 (a (\sin (e+f x)+1))^{7/2} \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^{11} \log \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )}{f (c-c \sin (e+f x))^{11/2} \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^7} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.243, size = 490, normalized size = 2. \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 (a \sin \left (f x + e\right ) + a\right )}^{\frac{7}{2}} \cos \left (f x + e\right )^{2}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac{11}{2}}}\,{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 (3 \, a^{3} \cos \left (f x + e\right )^{4} - 4 \, a^{3} \cos \left (f x + e\right )^{2} +{\left (a^{3} \cos \left (f x + e\right )^{4} - 4 \, a^{3} \cos \left (f x + e\right )^{2}\right )} \sin \left (f x + e\right )\right )} \sqrt{a \sin \left (f x + e\right ) + a} \sqrt{-c \sin \left (f x + e\right ) + c}}{c^{6} \cos \left (f x + e\right )^{6} - 18 \, c^{6} \cos \left (f x + e\right )^{4} + 48 \, c^{6} \cos \left (f x + e\right )^{2} - 32 \, c^{6} + 2 \,{\left (3 \, c^{6} \cos \left (f x + e\right )^{4} - 16 \, c^{6} \cos \left (f x + e\right )^{2} + 16 \, c^{6}\right )} \sin \left (f x + e\right )}, 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 (a \sin \left (f x + e\right ) + a\right )}^{\frac{7}{2}} \cos \left (f x + e\right )^{2}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac{11}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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