Optimal. Leaf size=242 \[ \frac {2048 c^4 (7 A-13 B) \sec (e+f x) \sqrt {c-c \sin (e+f x)}}{105 a^2 f}-\frac {512 c^3 (7 A-13 B) \sec (e+f x) (c-c \sin (e+f x))^{3/2}}{105 a^2 f}-\frac {(A-B) \sec ^3(e+f x) (c-c \sin (e+f x))^{13/2}}{3 a^2 c^2 f}-\frac {64 c^2 (7 A-13 B) \sec (e+f x) (c-c \sin (e+f x))^{5/2}}{105 a^2 f}-\frac {(7 A-13 B) \sec (e+f x) (c-c \sin (e+f x))^{9/2}}{21 a^2 f}-\frac {16 c (7 A-13 B) \sec (e+f x) (c-c \sin (e+f x))^{7/2}}{105 a^2 f} \]
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Rubi [A] time = 0.65, antiderivative size = 242, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2967, 2855, 2674, 2673} \[ -\frac {(A-B) \sec ^3(e+f x) (c-c \sin (e+f x))^{13/2}}{3 a^2 c^2 f}-\frac {64 c^2 (7 A-13 B) \sec (e+f x) (c-c \sin (e+f x))^{5/2}}{105 a^2 f}-\frac {512 c^3 (7 A-13 B) \sec (e+f x) (c-c \sin (e+f x))^{3/2}}{105 a^2 f}+\frac {2048 c^4 (7 A-13 B) \sec (e+f x) \sqrt {c-c \sin (e+f x)}}{105 a^2 f}-\frac {(7 A-13 B) \sec (e+f x) (c-c \sin (e+f x))^{9/2}}{21 a^2 f}-\frac {16 c (7 A-13 B) \sec (e+f x) (c-c \sin (e+f x))^{7/2}}{105 a^2 f} \]
Antiderivative was successfully verified.
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Rule 2673
Rule 2674
Rule 2855
Rule 2967
Rubi steps
\begin {align*} \int \frac {(A+B \sin (e+f x)) (c-c \sin (e+f x))^{9/2}}{(a+a \sin (e+f x))^2} \, dx &=\frac {\int \sec ^4(e+f x) (A+B \sin (e+f x)) (c-c \sin (e+f x))^{13/2} \, dx}{a^2 c^2}\\ &=-\frac {(A-B) \sec ^3(e+f x) (c-c \sin (e+f x))^{13/2}}{3 a^2 c^2 f}-\frac {(7 A-13 B) \int \sec ^2(e+f x) (c-c \sin (e+f x))^{11/2} \, dx}{6 a^2 c}\\ &=-\frac {(7 A-13 B) \sec (e+f x) (c-c \sin (e+f x))^{9/2}}{21 a^2 f}-\frac {(A-B) \sec ^3(e+f x) (c-c \sin (e+f x))^{13/2}}{3 a^2 c^2 f}-\frac {(8 (7 A-13 B)) \int \sec ^2(e+f x) (c-c \sin (e+f x))^{9/2} \, dx}{21 a^2}\\ &=-\frac {16 (7 A-13 B) c \sec (e+f x) (c-c \sin (e+f x))^{7/2}}{105 a^2 f}-\frac {(7 A-13 B) \sec (e+f x) (c-c \sin (e+f x))^{9/2}}{21 a^2 f}-\frac {(A-B) \sec ^3(e+f x) (c-c \sin (e+f x))^{13/2}}{3 a^2 c^2 f}-\frac {(32 (7 A-13 B) c) \int \sec ^2(e+f x) (c-c \sin (e+f x))^{7/2} \, dx}{35 a^2}\\ &=-\frac {64 (7 A-13 B) c^2 \sec (e+f x) (c-c \sin (e+f x))^{5/2}}{105 a^2 f}-\frac {16 (7 A-13 B) c \sec (e+f x) (c-c \sin (e+f x))^{7/2}}{105 a^2 f}-\frac {(7 A-13 B) \sec (e+f x) (c-c \sin (e+f x))^{9/2}}{21 a^2 f}-\frac {(A-B) \sec ^3(e+f x) (c-c \sin (e+f x))^{13/2}}{3 a^2 c^2 f}-\frac {\left (256 (7 A-13 B) c^2\right ) \int \sec ^2(e+f x) (c-c \sin (e+f x))^{5/2} \, dx}{105 a^2}\\ &=-\frac {512 (7 A-13 B) c^3 \sec (e+f x) (c-c \sin (e+f x))^{3/2}}{105 a^2 f}-\frac {64 (7 A-13 B) c^2 \sec (e+f x) (c-c \sin (e+f x))^{5/2}}{105 a^2 f}-\frac {16 (7 A-13 B) c \sec (e+f x) (c-c \sin (e+f x))^{7/2}}{105 a^2 f}-\frac {(7 A-13 B) \sec (e+f x) (c-c \sin (e+f x))^{9/2}}{21 a^2 f}-\frac {(A-B) \sec ^3(e+f x) (c-c \sin (e+f x))^{13/2}}{3 a^2 c^2 f}-\frac {\left (1024 (7 A-13 B) c^3\right ) \int \sec ^2(e+f x) (c-c \sin (e+f x))^{3/2} \, dx}{105 a^2}\\ &=\frac {2048 (7 A-13 B) c^4 \sec (e+f x) \sqrt {c-c \sin (e+f x)}}{105 a^2 f}-\frac {512 (7 A-13 B) c^3 \sec (e+f x) (c-c \sin (e+f x))^{3/2}}{105 a^2 f}-\frac {64 (7 A-13 B) c^2 \sec (e+f x) (c-c \sin (e+f x))^{5/2}}{105 a^2 f}-\frac {16 (7 A-13 B) c \sec (e+f x) (c-c \sin (e+f x))^{7/2}}{105 a^2 f}-\frac {(7 A-13 B) \sec (e+f x) (c-c \sin (e+f x))^{9/2}}{21 a^2 f}-\frac {(A-B) \sec ^3(e+f x) (c-c \sin (e+f x))^{13/2}}{3 a^2 c^2 f}\\ \end {align*}
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Mathematica [B] time = 6.82, size = 953, normalized size = 3.94 \[ -\frac {(26 A-83 B) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4 \sin \left (\frac {3}{2} (e+f x)\right ) (c-c \sin (e+f x))^{9/2}}{12 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^9 (\sin (e+f x) a+a)^2}-\frac {(2 A-13 B) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4 \sin \left (\frac {5}{2} (e+f x)\right ) (c-c \sin (e+f x))^{9/2}}{20 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^9 (\sin (e+f x) a+a)^2}-\frac {B \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4 \sin \left (\frac {7}{2} (e+f x)\right ) (c-c \sin (e+f x))^{9/2}}{28 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^9 (\sin (e+f x) a+a)^2}+\frac {(164 A-351 B) \sin \left (\frac {1}{2} (e+f x)\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4 (c-c \sin (e+f x))^{9/2}}{4 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^9 (\sin (e+f x) a+a)^2}+\frac {(164 A-351 B) \cos \left (\frac {1}{2} (e+f x)\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4 (c-c \sin (e+f x))^{9/2}}{4 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^9 (\sin (e+f x) a+a)^2}+\frac {(26 A-83 B) \cos \left (\frac {3}{2} (e+f x)\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4 (c-c \sin (e+f x))^{9/2}}{12 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^9 (\sin (e+f x) a+a)^2}-\frac {(2 A-13 B) \cos \left (\frac {5}{2} (e+f x)\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4 (c-c \sin (e+f x))^{9/2}}{20 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^9 (\sin (e+f x) a+a)^2}+\frac {B \cos \left (\frac {7}{2} (e+f x)\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4 (c-c \sin (e+f x))^{9/2}}{28 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^9 (\sin (e+f x) a+a)^2}+\frac {32 (2 A-3 B) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3 (c-c \sin (e+f x))^{9/2}}{f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^9 (\sin (e+f x) a+a)^2}-\frac {32 (A-B) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) (c-c \sin (e+f x))^{9/2}}{3 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^9 (\sin (e+f x) a+a)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 153, normalized size = 0.63 \[ \frac {2 \, {\left (3 \, {\left (7 \, A - 38 \, B\right )} c^{4} \cos \left (f x + e\right )^{4} - 12 \, {\left (154 \, A - 311 \, B\right )} c^{4} \cos \left (f x + e\right )^{2} + 24 \, {\left (287 \, A - 543 \, B\right )} c^{4} + {\left (15 \, B c^{4} \cos \left (f x + e\right )^{4} + 4 \, {\left (49 \, A - 136 \, B\right )} c^{4} \cos \left (f x + e\right )^{2} + 8 \, {\left (931 \, A - 1699 \, B\right )} c^{4}\right )} \sin \left (f x + e\right )\right )} \sqrt {-c \sin \left (f x + e\right ) + c}}{105 \, {\left (a^{2} f \cos \left (f x + e\right ) \sin \left (f x + e\right ) + a^{2} f \cos \left (f x + e\right )\right )}} \]
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 = 1.36, size = 143, normalized size = 0.59 \[ -\frac {2 c^{5} \left (\sin \left (f x +e \right )-1\right ) \left (15 B \sin \left (f x +e \right ) \left (\cos ^{4}\left (f x +e \right )\right )+\left (196 A -544 B \right ) \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )+\left (7448 A -13592 B \right ) \sin \left (f x +e \right )+\left (21 A -114 B \right ) \left (\cos ^{4}\left (f x +e \right )\right )+\left (-1848 A +3732 B \right ) \left (\cos ^{2}\left (f x +e \right )\right )+6888 A -13032 B \right )}{105 a^{2} \left (1+\sin \left (f x +e \right )\right ) \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.46, size = 762, normalized size = 3.15 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\left (A+B\,\sin \left (e+f\,x\right )\right )\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{9/2}}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^2} \,d x \]
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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