Optimal. Leaf size=96 \[ \frac {c (A-B) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 f \sqrt {c-c \sin (e+f x)}}+\frac {B c \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{3 a f \sqrt {c-c \sin (e+f x)}} \]
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Rubi [A] time = 0.32, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {2971, 2738} \[ \frac {c (A-B) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 f \sqrt {c-c \sin (e+f x)}}+\frac {B c \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{3 a f \sqrt {c-c \sin (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2738
Rule 2971
Rubi steps
\begin {align*} \int (a+a \sin (e+f x))^{3/2} (A+B \sin (e+f x)) \sqrt {c-c \sin (e+f x)} \, dx &=\frac {B \int (a+a \sin (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)} \, dx}{a}-(-A+B) \int (a+a \sin (e+f x))^{3/2} \sqrt {c-c \sin (e+f x)} \, dx\\ &=\frac {(A-B) c \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{2 f \sqrt {c-c \sin (e+f x)}}+\frac {B c \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{3 a f \sqrt {c-c \sin (e+f x)}}\\ \end {align*}
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Mathematica [A] time = 0.56, size = 81, normalized size = 0.84 \[ -\frac {a \sec (e+f x) \sqrt {a (\sin (e+f x)+1)} \sqrt {c-c \sin (e+f x)} (\cos (2 (e+f x)) (3 (A+B)+2 B \sin (e+f x))-2 (6 A+B) \sin (e+f x))}{12 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 87, normalized size = 0.91 \[ -\frac {{\left (3 \, {\left (A + B\right )} a \cos \left (f x + e\right )^{2} - 3 \, {\left (A + B\right )} a + 2 \, {\left (B a \cos \left (f x + e\right )^{2} - {\left (3 \, A + B\right )} a\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{6 \, f \cos \left (f x + e\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.80, size = 91, normalized size = 0.95 \[ \frac {\left (-2 B \left (\cos ^{2}\left (f x +e \right )\right )+3 A \sin \left (f x +e \right )+3 B \sin \left (f x +e \right )+6 A +2 B \right ) \sqrt {-c \left (\sin \left (f x +e \right )-1\right )}\, \sin \left (f x +e \right ) \left (a \left (1+\sin \left (f x +e \right )\right )\right )^{\frac {3}{2}}}{6 f \left (1+\sin \left (f x +e \right )\right ) \cos \left (f x +e \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}} \sqrt {-c \sin \left (f x + e\right ) + c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 14.29, size = 122, normalized size = 1.27 \[ -\frac {a\,\sqrt {a\,\left (\sin \left (e+f\,x\right )+1\right )}\,\sqrt {-c\,\left (\sin \left (e+f\,x\right )-1\right )}\,\left (3\,A\,\cos \left (e+f\,x\right )+3\,B\,\cos \left (e+f\,x\right )+3\,A\,\cos \left (3\,e+3\,f\,x\right )+3\,B\,\cos \left (3\,e+3\,f\,x\right )-12\,A\,\sin \left (2\,e+2\,f\,x\right )-2\,B\,\sin \left (2\,e+2\,f\,x\right )+B\,\sin \left (4\,e+4\,f\,x\right )\right )}{12\,f\,\left (\cos \left (2\,e+2\,f\,x\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}} \sqrt {- c \left (\sin {\left (e + f x \right )} - 1\right )} \left (A + B \sin {\left (e + f x \right )}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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