Optimal. Leaf size=164 \[ \frac {2 \cos (e+f x) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-1}}{c^2 f (2 m+5) \left (4 m^2+8 m+3\right )}+\frac {2 \cos (e+f x) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-2}}{c f \left (4 m^2+16 m+15\right )}+\frac {\cos (e+f x) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-3}}{f (2 m+5)} \]
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Rubi [A] time = 0.22, antiderivative size = 164, normalized size of antiderivative = 1.00, 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 = {2743, 2742} \[ \frac {2 \cos (e+f x) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-1}}{c^2 f (2 m+5) \left (4 m^2+8 m+3\right )}+\frac {2 \cos (e+f x) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-2}}{c f \left (4 m^2+16 m+15\right )}+\frac {\cos (e+f x) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-3}}{f (2 m+5)} \]
Antiderivative was successfully verified.
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Rule 2742
Rule 2743
Rubi steps
\begin {align*} \int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-3-m} \, dx &=\frac {\cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-3-m}}{f (5+2 m)}+\frac {2 \int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-2-m} \, dx}{c (5+2 m)}\\ &=\frac {\cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-3-m}}{f (5+2 m)}+\frac {2 \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-2-m}}{c f \left (15+16 m+4 m^2\right )}+\frac {2 \int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-1-m} \, dx}{c^2 (3+2 m) (5+2 m)}\\ &=\frac {\cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-3-m}}{f (5+2 m)}+\frac {2 \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-2-m}}{c f \left (15+16 m+4 m^2\right )}+\frac {2 \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-1-m}}{c^2 f (1+2 m) (3+2 m) (5+2 m)}\\ \end {align*}
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Mathematica [A] time = 8.53, size = 174, normalized size = 1.06 \[ \frac {2^{-m-2} \cos \left (\frac {1}{2} \left (-e-f x+\frac {\pi }{2}\right )\right ) \sin ^{-2 m-5}\left (\frac {1}{2} \left (-e-f x+\frac {\pi }{2}\right )\right ) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-3} \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^{-2 (-m-3)} \left (-2 (2 m+3) \sin (e+f x)+\cos \left (2 \left (-e-f x+\frac {\pi }{2}\right )\right )+4 \left (m^2+3 m+2\right )\right )}{f (2 m+1) (2 m+3) (2 m+5)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 101, normalized size = 0.62 \[ -\frac {{\left (2 \, \cos \left (f x + e\right )^{3} + 2 \, {\left (2 \, m + 3\right )} \cos \left (f x + e\right ) \sin \left (f x + e\right ) - {\left (4 \, m^{2} + 12 \, m + 9\right )} \cos \left (f x + e\right )\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} {\left (-c \sin \left (f x + e\right ) + c\right )}^{-m - 3}}{8 \, f m^{3} + 36 \, f m^{2} + 46 \, f m + 15 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (a \sin \left (f x + e\right ) + a\right )}^{m} {\left (-c \sin \left (f x + e\right ) + c\right )}^{-m - 3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.02, size = 0, normalized size = 0.00 \[ \int \left (a +a \sin \left (f x +e \right )\right )^{m} \left (c -c \sin \left (f x +e \right )\right )^{-3-m}\, dx \]
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 \[ \int {\left (a \sin \left (f x + e\right ) + a\right )}^{m} {\left (-c \sin \left (f x + e\right ) + c\right )}^{-m - 3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.25, size = 149, normalized size = 0.91 \[ -\frac {2\,{\left (a\,\left (\sin \left (e+f\,x\right )+1\right )\right )}^m\,\left (15\,\cos \left (e+f\,x\right )-\cos \left (3\,e+3\,f\,x\right )-6\,\sin \left (2\,e+2\,f\,x\right )+24\,m\,\cos \left (e+f\,x\right )+8\,m^2\,\cos \left (e+f\,x\right )-4\,m\,\sin \left (2\,e+2\,f\,x\right )\right )}{c^3\,f\,{\left (-c\,\left (\sin \left (e+f\,x\right )-1\right )\right )}^m\,\left (8\,m^3+36\,m^2+46\,m+15\right )\,\left (15\,\sin \left (e+f\,x\right )+6\,\cos \left (2\,e+2\,f\,x\right )-\sin \left (3\,e+3\,f\,x\right )-10\right )} \]
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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