Optimal. Leaf size=134 \[ -\frac{2 a^3 \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{15 f \sqrt{a \sin (e+f x)+a}}-\frac{a^2 \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}{5 f}-\frac{a \cos (e+f x) (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{5/2}}{5 f} \]
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Rubi [A] time = 0.271218, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {2740, 2738} \[ -\frac{2 a^3 \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{15 f \sqrt{a \sin (e+f x)+a}}-\frac{a^2 \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}{5 f}-\frac{a \cos (e+f x) (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{5/2}}{5 f} \]
Antiderivative was successfully verified.
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Rule 2740
Rule 2738
Rubi steps
\begin{align*} \int (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2} \, dx &=-\frac{a \cos (e+f x) (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{5/2}}{5 f}+\frac{1}{5} (4 a) \int (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{5/2} \, dx\\ &=-\frac{a^2 \cos (e+f x) \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}{5 f}-\frac{a \cos (e+f x) (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{5/2}}{5 f}+\frac{1}{5} \left (2 a^2\right ) \int \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2} \, dx\\ &=-\frac{2 a^3 \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{15 f \sqrt{a+a \sin (e+f x)}}-\frac{a^2 \cos (e+f x) \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}{5 f}-\frac{a \cos (e+f x) (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{5/2}}{5 f}\\ \end{align*}
Mathematica [A] time = 0.537671, size = 77, normalized size = 0.57 \[ \frac{a^2 c^2 (150 \sin (e+f x)+25 \sin (3 (e+f x))+3 \sin (5 (e+f x))) \sec (e+f x) \sqrt{a (\sin (e+f x)+1)} \sqrt{c-c \sin (e+f x)}}{240 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.145, size = 67, normalized size = 0.5 \begin{align*}{\frac{ \left ( 3\, \left ( \cos \left ( fx+e \right ) \right ) ^{4}+4\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}+8 \right ) \sin \left ( fx+e \right ) }{15\,f \left ( \cos \left ( fx+e \right ) \right ) ^{5}} \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{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{5}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.09305, size = 207, normalized size = 1.54 \begin{align*} \frac{{\left (3 \, a^{2} c^{2} \cos \left (f x + e\right )^{4} + 4 \, a^{2} c^{2} \cos \left (f x + e\right )^{2} + 8 \, a^{2} c^{2}\right )} \sqrt{a \sin \left (f x + e\right ) + a} \sqrt{-c \sin \left (f x + e\right ) + c} \sin \left (f x + e\right )}{15 \, 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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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