Optimal. Leaf size=140 \[ \frac{c^2 \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{15 a f \sqrt{c-c \sin (e+f x)}}+\frac{\cos (e+f x) (a \sin (e+f x)+a)^{7/2} (c-c \sin (e+f x))^{3/2}}{6 a f}+\frac{2 c \cos (e+f x) (a \sin (e+f x)+a)^{7/2} \sqrt{c-c \sin (e+f x)}}{15 a f} \]
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Rubi [A] time = 0.51661, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.079, Rules used = {2841, 2740, 2738} \[ \frac{c^2 \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{15 a f \sqrt{c-c \sin (e+f x)}}+\frac{\cos (e+f x) (a \sin (e+f x)+a)^{7/2} (c-c \sin (e+f x))^{3/2}}{6 a f}+\frac{2 c \cos (e+f x) (a \sin (e+f x)+a)^{7/2} \sqrt{c-c \sin (e+f x)}}{15 a f} \]
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))^{5/2} (c-c \sin (e+f x))^{3/2} \, dx &=\frac{\int (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{5/2} \, dx}{a c}\\ &=\frac{\cos (e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{3/2}}{6 a f}+\frac{2 \int (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{3/2} \, dx}{3 a}\\ &=\frac{2 c \cos (e+f x) (a+a \sin (e+f x))^{7/2} \sqrt{c-c \sin (e+f x)}}{15 a f}+\frac{\cos (e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{3/2}}{6 a f}+\frac{(4 c) \int (a+a \sin (e+f x))^{7/2} \sqrt{c-c \sin (e+f x)} \, dx}{15 a}\\ &=\frac{c^2 \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{15 a f \sqrt{c-c \sin (e+f x)}}+\frac{2 c \cos (e+f x) (a+a \sin (e+f x))^{7/2} \sqrt{c-c \sin (e+f x)}}{15 a f}+\frac{\cos (e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{3/2}}{6 a f}\\ \end{align*}
Mathematica [A] time = 1.01715, size = 152, normalized size = 1.09 \[ -\frac{c (\sin (e+f x)-1) (a (\sin (e+f x)+1))^{5/2} \sqrt{c-c \sin (e+f x)} (600 \sin (e+f x)+100 \sin (3 (e+f x))+12 \sin (5 (e+f x))-75 \cos (2 (e+f x))-30 \cos (4 (e+f x))-5 \cos (6 (e+f x)))}{960 f \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^3 \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.2, size = 116, normalized size = 0.8 \begin{align*} -{\frac{\sin \left ( fx+e \right ) \left ( -5\, \left ( \cos \left ( fx+e \right ) \right ) ^{6}+\sin \left ( fx+e \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{4}-6\, \left ( \cos \left ( fx+e \right ) \right ) ^{4}+3\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}\sin \left ( fx+e \right ) -8\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}+11\,\sin \left ( fx+e \right ) -11 \right ) }{30\,f \left ( \cos \left ( fx+e \right ) \right ) ^{5}} \left ( -c \left ( -1+\sin \left ( fx+e \right ) \right ) \right ) ^{{\frac{3}{2}}} \left ( a \left ( 1+\sin \left ( fx+e \right ) \right ) \right ) ^{{\frac{5}{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{5}{2}}{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac{3}{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 = 2.41294, size = 252, normalized size = 1.8 \begin{align*} -\frac{{\left (5 \, a^{2} c \cos \left (f x + e\right )^{6} - 5 \, a^{2} c - 2 \,{\left (3 \, a^{2} c \cos \left (f x + e\right )^{4} + 4 \, a^{2} c \cos \left (f x + e\right )^{2} + 8 \, a^{2} c\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}}{30 \, 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*} \mathit{sage}_{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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