Optimal. Leaf size=239 \[ \frac{8 c^3 \cos (e+f x) \sqrt{c-c \sin (e+f x)}}{a f \sqrt{a \sin (e+f x)+a}}+\frac{2 c^2 \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{a f \sqrt{a \sin (e+f x)+a}}+\frac{16 c^4 \cos (e+f x) \log (\sin (e+f x)+1)}{a f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{2 c \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{3 a f \sqrt{a \sin (e+f x)+a}}+\frac{\cos (e+f x) (c-c \sin (e+f x))^{7/2}}{4 a f \sqrt{a \sin (e+f x)+a}} \]
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Rubi [A] time = 0.744696, antiderivative size = 239, 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, 2740, 2737, 2667, 31} \[ \frac{8 c^3 \cos (e+f x) \sqrt{c-c \sin (e+f x)}}{a f \sqrt{a \sin (e+f x)+a}}+\frac{2 c^2 \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{a f \sqrt{a \sin (e+f x)+a}}+\frac{16 c^4 \cos (e+f x) \log (\sin (e+f x)+1)}{a f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{2 c \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{3 a f \sqrt{a \sin (e+f x)+a}}+\frac{\cos (e+f x) (c-c \sin (e+f x))^{7/2}}{4 a f \sqrt{a \sin (e+f x)+a}} \]
Antiderivative was successfully verified.
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Rule 2841
Rule 2740
Rule 2737
Rule 2667
Rule 31
Rubi steps
\begin{align*} \int \frac{\cos ^2(e+f x) (c-c \sin (e+f x))^{7/2}}{(a+a \sin (e+f x))^{3/2}} \, dx &=\frac{\int \frac{(c-c \sin (e+f x))^{9/2}}{\sqrt{a+a \sin (e+f x)}} \, dx}{a c}\\ &=\frac{\cos (e+f x) (c-c \sin (e+f x))^{7/2}}{4 a f \sqrt{a+a \sin (e+f x)}}+\frac{2 \int \frac{(c-c \sin (e+f x))^{7/2}}{\sqrt{a+a \sin (e+f x)}} \, dx}{a}\\ &=\frac{2 c \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{3 a f \sqrt{a+a \sin (e+f x)}}+\frac{\cos (e+f x) (c-c \sin (e+f x))^{7/2}}{4 a f \sqrt{a+a \sin (e+f x)}}+\frac{(4 c) \int \frac{(c-c \sin (e+f x))^{5/2}}{\sqrt{a+a \sin (e+f x)}} \, dx}{a}\\ &=\frac{2 c^2 \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{a f \sqrt{a+a \sin (e+f x)}}+\frac{2 c \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{3 a f \sqrt{a+a \sin (e+f x)}}+\frac{\cos (e+f x) (c-c \sin (e+f x))^{7/2}}{4 a f \sqrt{a+a \sin (e+f x)}}+\frac{\left (8 c^2\right ) \int \frac{(c-c \sin (e+f x))^{3/2}}{\sqrt{a+a \sin (e+f x)}} \, dx}{a}\\ &=\frac{8 c^3 \cos (e+f x) \sqrt{c-c \sin (e+f x)}}{a f \sqrt{a+a \sin (e+f x)}}+\frac{2 c^2 \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{a f \sqrt{a+a \sin (e+f x)}}+\frac{2 c \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{3 a f \sqrt{a+a \sin (e+f x)}}+\frac{\cos (e+f x) (c-c \sin (e+f x))^{7/2}}{4 a f \sqrt{a+a \sin (e+f x)}}+\frac{\left (16 c^3\right ) \int \frac{\sqrt{c-c \sin (e+f x)}}{\sqrt{a+a \sin (e+f x)}} \, dx}{a}\\ &=\frac{8 c^3 \cos (e+f x) \sqrt{c-c \sin (e+f x)}}{a f \sqrt{a+a \sin (e+f x)}}+\frac{2 c^2 \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{a f \sqrt{a+a \sin (e+f x)}}+\frac{2 c \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{3 a f \sqrt{a+a \sin (e+f x)}}+\frac{\cos (e+f x) (c-c \sin (e+f x))^{7/2}}{4 a f \sqrt{a+a \sin (e+f x)}}+\frac{\left (16 c^4 \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{8 c^3 \cos (e+f x) \sqrt{c-c \sin (e+f x)}}{a f \sqrt{a+a \sin (e+f x)}}+\frac{2 c^2 \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{a f \sqrt{a+a \sin (e+f x)}}+\frac{2 c \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{3 a f \sqrt{a+a \sin (e+f x)}}+\frac{\cos (e+f x) (c-c \sin (e+f x))^{7/2}}{4 a f \sqrt{a+a \sin (e+f x)}}+\frac{\left (16 c^4 \cos (e+f x)\right ) \operatorname{Subst}\left (\int \frac{1}{a+x} \, dx,x,a \sin (e+f x)\right )}{a f \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}\\ &=\frac{16 c^4 \cos (e+f x) \log (1+\sin (e+f x))}{a f \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}+\frac{8 c^3 \cos (e+f x) \sqrt{c-c \sin (e+f x)}}{a f \sqrt{a+a \sin (e+f x)}}+\frac{2 c^2 \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{a f \sqrt{a+a \sin (e+f x)}}+\frac{2 c \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{3 a f \sqrt{a+a \sin (e+f x)}}+\frac{\cos (e+f x) (c-c \sin (e+f x))^{7/2}}{4 a f \sqrt{a+a \sin (e+f x)}}\\ \end{align*}
Mathematica [A] time = 6.4823, size = 471, normalized size = 1.97 \[ \frac{5 \sin (3 (e+f x)) (c-c \sin (e+f x))^{7/2} \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^3}{12 f (a (\sin (e+f x)+1))^{3/2} \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^7}-\frac{23 \cos (2 (e+f x)) (c-c \sin (e+f x))^{7/2} \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^3}{8 f (a (\sin (e+f x)+1))^{3/2} \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^7}+\frac{\cos (4 (e+f x)) (c-c \sin (e+f x))^{7/2} \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^3}{32 f (a (\sin (e+f x)+1))^{3/2} \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^7}-\frac{65 \sin (e+f x) (c-c \sin (e+f x))^{7/2} \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^3}{4 f (a (\sin (e+f x)+1))^{3/2} \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^7}+\frac{32 (c-c \sin (e+f x))^{7/2} \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^3 \log \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )}{f (a (\sin (e+f x)+1))^{3/2} \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^7} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.238, size = 252, normalized size = 1.1 \begin{align*}{\frac{\sin \left ( fx+e \right ) \cos \left ( fx+e \right ) + \left ( \cos \left ( fx+e \right ) \right ) ^{2}-2\,\sin \left ( fx+e \right ) +\cos \left ( fx+e \right ) -2}{12\,f \left ( \sin \left ( fx+e \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{3}- \left ( \cos \left ( fx+e \right ) \right ) ^{4}-4\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}\sin \left ( fx+e \right ) -3\, \left ( \cos \left ( fx+e \right ) \right ) ^{3}-4\,\sin \left ( fx+e \right ) \cos \left ( fx+e \right ) +8\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}+8\,\sin \left ( fx+e \right ) +4\,\cos \left ( fx+e \right ) -8 \right ) } \left ( -3\, \left ( \cos \left ( fx+e \right ) \right ) ^{4}-20\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}\sin \left ( fx+e \right ) +72\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}+192\,\ln \left ( 2\, \left ( \cos \left ( fx+e \right ) +1 \right ) ^{-1} \right ) +200\,\sin \left ( fx+e \right ) -384\,\ln \left ({\frac{1-\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }} \right ) -69 \right ) \left ( -c \left ( -1+\sin \left ( fx+e \right ) \right ) \right ) ^{{\frac{7}{2}}} \left ( a \left ( 1+\sin \left ( fx+e \right ) \right ) \right ) ^{-{\frac{3}{2}}}} \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 (-c \sin \left (f x + e\right ) + c\right )}^{\frac{7}{2}} \cos \left (f x + e\right )^{2}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac{3}{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 \, c^{3} \cos \left (f x + e\right )^{4} - 4 \, c^{3} \cos \left (f x + e\right )^{2} -{\left (c^{3} \cos \left (f x + e\right )^{4} - 4 \, c^{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}}{a^{2} \cos \left (f x + e\right )^{2} - 2 \, a^{2} \sin \left (f x + e\right ) - 2 \, a^{2}}, 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 (-c \sin \left (f x + e\right ) + c\right )}^{\frac{7}{2}} \cos \left (f x + e\right )^{2}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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