Optimal. Leaf size=159 \[ -\frac {32 c^3 (5 A-7 B) \cos (e+f x)}{15 a f \sqrt {c-c \sin (e+f x)}}-\frac {8 c^2 (5 A-7 B) \cos (e+f x) \sqrt {c-c \sin (e+f x)}}{15 a f}-\frac {c (5 A-7 B) \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{5 a f}-\frac {(A-B) \sec (e+f x) (c-c \sin (e+f x))^{7/2}}{a c f} \]
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Rubi [A] time = 0.35, antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2967, 2855, 2647, 2646} \[ -\frac {8 c^2 (5 A-7 B) \cos (e+f x) \sqrt {c-c \sin (e+f x)}}{15 a f}-\frac {32 c^3 (5 A-7 B) \cos (e+f x)}{15 a f \sqrt {c-c \sin (e+f x)}}-\frac {c (5 A-7 B) \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{5 a f}-\frac {(A-B) \sec (e+f x) (c-c \sin (e+f x))^{7/2}}{a c f} \]
Antiderivative was successfully verified.
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Rule 2646
Rule 2647
Rule 2855
Rule 2967
Rubi steps
\begin {align*} \int \frac {(A+B \sin (e+f x)) (c-c \sin (e+f x))^{5/2}}{a+a \sin (e+f x)} \, dx &=\frac {\int \sec ^2(e+f x) (A+B \sin (e+f x)) (c-c \sin (e+f x))^{7/2} \, dx}{a c}\\ &=-\frac {(A-B) \sec (e+f x) (c-c \sin (e+f x))^{7/2}}{a c f}-\frac {(5 A-7 B) \int (c-c \sin (e+f x))^{5/2} \, dx}{2 a}\\ &=-\frac {(5 A-7 B) c \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{5 a f}-\frac {(A-B) \sec (e+f x) (c-c \sin (e+f x))^{7/2}}{a c f}-\frac {(4 (5 A-7 B) c) \int (c-c \sin (e+f x))^{3/2} \, dx}{5 a}\\ &=-\frac {8 (5 A-7 B) c^2 \cos (e+f x) \sqrt {c-c \sin (e+f x)}}{15 a f}-\frac {(5 A-7 B) c \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{5 a f}-\frac {(A-B) \sec (e+f x) (c-c \sin (e+f x))^{7/2}}{a c f}-\frac {\left (16 (5 A-7 B) c^2\right ) \int \sqrt {c-c \sin (e+f x)} \, dx}{15 a}\\ &=-\frac {32 (5 A-7 B) c^3 \cos (e+f x)}{15 a f \sqrt {c-c \sin (e+f x)}}-\frac {8 (5 A-7 B) c^2 \cos (e+f x) \sqrt {c-c \sin (e+f x)}}{15 a f}-\frac {(5 A-7 B) c \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{5 a f}-\frac {(A-B) \sec (e+f x) (c-c \sin (e+f x))^{7/2}}{a c f}\\ \end {align*}
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Mathematica [A] time = 1.76, size = 134, normalized size = 0.84 \[ -\frac {c^2 \sqrt {c-c \sin (e+f x)} \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right ) (25 (8 A-13 B) \sin (e+f x)+2 (5 A-16 B) \cos (2 (e+f x))+450 A+3 B \sin (3 (e+f x))-600 B)}{30 a f (\sin (e+f x)+1) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 95, normalized size = 0.60 \[ -\frac {2 \, {\left ({\left (5 \, A - 16 \, B\right )} c^{2} \cos \left (f x + e\right )^{2} + 2 \, {\left (55 \, A - 71 \, B\right )} c^{2} + {\left (3 \, B c^{2} \cos \left (f x + e\right )^{2} + 2 \, {\left (25 \, A - 41 \, B\right )} c^{2}\right )} \sin \left (f x + e\right )\right )} \sqrt {-c \sin \left (f x + e\right ) + c}}{15 \, a f \cos \left (f x + e\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.10, size = 95, normalized size = 0.60 \[ -\frac {2 c^{3} \left (\sin \left (f x +e \right )-1\right ) \left (-3 B \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )+\left (-50 A +82 B \right ) \sin \left (f x +e \right )+\left (-5 A +16 B \right ) \left (\cos ^{2}\left (f x +e \right )\right )-110 A +142 B \right )}{15 a \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.46, size = 386, normalized size = 2.43 \[ \frac {2 \, {\left (\frac {5 \, {\left (23 \, c^{\frac {5}{2}} + \frac {20 \, c^{\frac {5}{2}} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {65 \, c^{\frac {5}{2}} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {40 \, c^{\frac {5}{2}} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {65 \, c^{\frac {5}{2}} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {20 \, c^{\frac {5}{2}} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {23 \, c^{\frac {5}{2}} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}}\right )} A}{{\left (a + \frac {a \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )} {\left (\frac {\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 1\right )}^{\frac {5}{2}}} - \frac {2 \, {\left (79 \, c^{\frac {5}{2}} + \frac {79 \, c^{\frac {5}{2}} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {205 \, c^{\frac {5}{2}} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {170 \, c^{\frac {5}{2}} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {205 \, c^{\frac {5}{2}} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {79 \, c^{\frac {5}{2}} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {79 \, c^{\frac {5}{2}} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}}\right )} B}{{\left (a + \frac {a \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )} {\left (\frac {\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 1\right )}^{\frac {5}{2}}}\right )}}{15 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\left (A+B\,\sin \left (e+f\,x\right )\right )\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{5/2}}{a+a\,\sin \left (e+f\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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