Optimal. Leaf size=96 \[ \frac{c (A-B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{4 f \sqrt{c-c \sin (e+f x)}}+\frac{B c \cos (e+f x) (a \sin (e+f x)+a)^{9/2}}{5 a f \sqrt{c-c \sin (e+f x)}} \]
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Rubi [A] time = 0.324937, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {2971, 2738} \[ \frac{c (A-B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{4 f \sqrt{c-c \sin (e+f x)}}+\frac{B c \cos (e+f x) (a \sin (e+f x)+a)^{9/2}}{5 a f \sqrt{c-c \sin (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2971
Rule 2738
Rubi steps
\begin{align*} \int (a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x)) \sqrt{c-c \sin (e+f x)} \, dx &=\frac{B \int (a+a \sin (e+f x))^{9/2} \sqrt{c-c \sin (e+f x)} \, dx}{a}-(-A+B) \int (a+a \sin (e+f x))^{7/2} \sqrt{c-c \sin (e+f x)} \, dx\\ &=\frac{(A-B) c \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{4 f \sqrt{c-c \sin (e+f x)}}+\frac{B c \cos (e+f x) (a+a \sin (e+f x))^{9/2}}{5 a f \sqrt{c-c \sin (e+f x)}}\\ \end{align*}
Mathematica [A] time = 1.02277, size = 121, normalized size = 1.26 \[ \frac{a^3 \sec (e+f x) \sqrt{a (\sin (e+f x)+1)} \sqrt{c-c \sin (e+f x)} (4 (60 A+23 B) \sin (e+f x)+\cos (4 (e+f x)) (5 A+4 B \sin (e+f x)+15 B)-4 \cos (2 (e+f x)) (4 (5 A+6 B) \sin (e+f x)+5 (7 A+5 B)))}{160 f} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.359, size = 174, normalized size = 1.8 \begin{align*}{\frac{ \left ( -4\,B \left ( \cos \left ( fx+e \right ) \right ) ^{4}+5\,A \left ( \cos \left ( fx+e \right ) \right ) ^{2}\sin \left ( fx+e \right ) +15\,B \left ( \cos \left ( fx+e \right ) \right ) ^{2}\sin \left ( fx+e \right ) +20\,A \left ( \cos \left ( fx+e \right ) \right ) ^{2}+28\,B \left ( \cos \left ( fx+e \right ) \right ) ^{2}-35\,A\sin \left ( fx+e \right ) -25\,B\sin \left ( fx+e \right ) -40\,A-24\,B \right ) \sin \left ( fx+e \right ) }{20\,f \left ( \left ( \cos \left ( fx+e \right ) \right ) ^{2}\sin \left ( fx+e \right ) +3\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}-4\,\sin \left ( fx+e \right ) -4 \right ) \cos \left ( fx+e \right ) }\sqrt{-c \left ( -1+\sin \left ( fx+e \right ) \right ) } \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 (B \sin \left (f x + e\right ) + A\right )}{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac{7}{2}} \sqrt{-c \sin \left (f x + e\right ) + c}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.42321, size = 340, normalized size = 3.54 \begin{align*} \frac{{\left (5 \,{\left (A + 3 \, B\right )} a^{3} \cos \left (f x + e\right )^{4} - 40 \,{\left (A + B\right )} a^{3} \cos \left (f x + e\right )^{2} + 5 \,{\left (7 \, A + 5 \, B\right )} a^{3} + 4 \,{\left (B a^{3} \cos \left (f x + e\right )^{4} -{\left (5 \, A + 7 \, B\right )} a^{3} \cos \left (f x + e\right )^{2} + 2 \,{\left (5 \, A + 3 \, B\right )} a^{3}\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}}{20 \, 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: AttributeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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