Optimal. Leaf size=188 \[ \frac{c^2 \cos (e+f x) (a \sin (e+f x)+a)^{9/2} \sqrt{c-c \sin (e+f x)}}{14 a f}+\frac{c^3 \cos (e+f x) (a \sin (e+f x)+a)^{9/2}}{35 a f \sqrt{c-c \sin (e+f x)}}+\frac{\cos (e+f x) (a \sin (e+f x)+a)^{9/2} (c-c \sin (e+f x))^{5/2}}{8 a f}+\frac{3 c \cos (e+f x) (a \sin (e+f x)+a)^{9/2} (c-c \sin (e+f x))^{3/2}}{28 a f} \]
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Rubi [A] time = 0.619513, antiderivative size = 188, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.079, Rules used = {2841, 2740, 2738} \[ \frac{c^2 \cos (e+f x) (a \sin (e+f x)+a)^{9/2} \sqrt{c-c \sin (e+f x)}}{14 a f}+\frac{c^3 \cos (e+f x) (a \sin (e+f x)+a)^{9/2}}{35 a f \sqrt{c-c \sin (e+f x)}}+\frac{\cos (e+f x) (a \sin (e+f x)+a)^{9/2} (c-c \sin (e+f x))^{5/2}}{8 a f}+\frac{3 c \cos (e+f x) (a \sin (e+f x)+a)^{9/2} (c-c \sin (e+f x))^{3/2}}{28 a f} \]
Antiderivative was successfully verified.
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Rule 2841
Rule 2740
Rule 2738
Rubi steps
\begin{align*} \int \cos ^2(e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{5/2} \, dx &=\frac{\int (a+a \sin (e+f x))^{9/2} (c-c \sin (e+f x))^{7/2} \, dx}{a c}\\ &=\frac{\cos (e+f x) (a+a \sin (e+f x))^{9/2} (c-c \sin (e+f x))^{5/2}}{8 a f}+\frac{3 \int (a+a \sin (e+f x))^{9/2} (c-c \sin (e+f x))^{5/2} \, dx}{4 a}\\ &=\frac{3 c \cos (e+f x) (a+a \sin (e+f x))^{9/2} (c-c \sin (e+f x))^{3/2}}{28 a f}+\frac{\cos (e+f x) (a+a \sin (e+f x))^{9/2} (c-c \sin (e+f x))^{5/2}}{8 a f}+\frac{(3 c) \int (a+a \sin (e+f x))^{9/2} (c-c \sin (e+f x))^{3/2} \, dx}{7 a}\\ &=\frac{c^2 \cos (e+f x) (a+a \sin (e+f x))^{9/2} \sqrt{c-c \sin (e+f x)}}{14 a f}+\frac{3 c \cos (e+f x) (a+a \sin (e+f x))^{9/2} (c-c \sin (e+f x))^{3/2}}{28 a f}+\frac{\cos (e+f x) (a+a \sin (e+f x))^{9/2} (c-c \sin (e+f x))^{5/2}}{8 a f}+\frac{c^2 \int (a+a \sin (e+f x))^{9/2} \sqrt{c-c \sin (e+f x)} \, dx}{7 a}\\ &=\frac{c^3 \cos (e+f x) (a+a \sin (e+f x))^{9/2}}{35 a f \sqrt{c-c \sin (e+f x)}}+\frac{c^2 \cos (e+f x) (a+a \sin (e+f x))^{9/2} \sqrt{c-c \sin (e+f x)}}{14 a f}+\frac{3 c \cos (e+f x) (a+a \sin (e+f x))^{9/2} (c-c \sin (e+f x))^{3/2}}{28 a f}+\frac{\cos (e+f x) (a+a \sin (e+f x))^{9/2} (c-c \sin (e+f x))^{5/2}}{8 a f}\\ \end{align*}
Mathematica [A] time = 1.90371, size = 127, normalized size = 0.68 \[ \frac{a^3 c^2 \sec (e+f x) \sqrt{a (\sin (e+f x)+1)} \sqrt{c-c \sin (e+f x)} (19600 \sin (e+f x)+3920 \sin (3 (e+f x))+784 \sin (5 (e+f x))+80 \sin (7 (e+f x))-1960 \cos (2 (e+f x))-980 \cos (4 (e+f x))-280 \cos (6 (e+f x))-35 \cos (8 (e+f x)))}{35840 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.23, size = 143, normalized size = 0.8 \begin{align*} -{\frac{\sin \left ( fx+e \right ) \left ( -35\, \left ( \cos \left ( fx+e \right ) \right ) ^{8}+5\, \left ( \cos \left ( fx+e \right ) \right ) ^{6}\sin \left ( fx+e \right ) -40\, \left ( \cos \left ( fx+e \right ) \right ) ^{6}+13\,\sin \left ( fx+e \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{4}-48\, \left ( \cos \left ( fx+e \right ) \right ) ^{4}+29\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}\sin \left ( fx+e \right ) -64\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}+93\,\sin \left ( fx+e \right ) -93 \right ) }{280\,f \left ( \cos \left ( fx+e \right ) \right ) ^{7}} \left ( -c \left ( -1+\sin \left ( fx+e \right ) \right ) \right ) ^{{\frac{5}{2}}} \left ( a \left ( 1+\sin \left ( fx+e \right ) \right ) \right ) ^{{\frac{7}{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{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac{7}{2}}{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac{5}{2}} \cos \left (f x + e\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.91788, size = 308, normalized size = 1.64 \begin{align*} -\frac{{\left (35 \, a^{3} c^{2} \cos \left (f x + e\right )^{8} - 35 \, a^{3} c^{2} - 8 \,{\left (5 \, a^{3} c^{2} \cos \left (f x + e\right )^{6} + 6 \, a^{3} c^{2} \cos \left (f x + e\right )^{4} + 8 \, a^{3} c^{2} \cos \left (f x + e\right )^{2} + 16 \, a^{3} c^{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}}{280 \, f \cos \left (f x + e\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{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac{7}{2}}{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac{5}{2}} \cos \left (f x + e\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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