Optimal. Leaf size=114 \[ \frac{(A+B) \cos (e+f x) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-2}}{f (2 m+3)}+\frac{(A-2 B (m+1)) \cos (e+f x) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-1}}{c f (2 m+1) (2 m+3)} \]
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Rubi [A] time = 0.221648, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {2972, 2742} \[ \frac{(A+B) \cos (e+f x) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-2}}{f (2 m+3)}+\frac{(A-2 B (m+1)) \cos (e+f x) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-1}}{c f (2 m+1) (2 m+3)} \]
Antiderivative was successfully verified.
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Rule 2972
Rule 2742
Rubi steps
\begin{align*} \int (a+a \sin (e+f x))^m (A+B \sin (e+f x)) (c-c \sin (e+f x))^{-2-m} \, dx &=\frac{(A+B) \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-2-m}}{f (3+2 m)}+\frac{(A-2 B (1+m)) \int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-1-m} \, dx}{c (3+2 m)}\\ &=\frac{(A+B) \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-2-m}}{f (3+2 m)}+\frac{(A-2 B (1+m)) \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-1-m}}{c f (1+2 m) (3+2 m)}\\ \end{align*}
Mathematica [A] time = 8.50224, size = 211, normalized size = 1.85 \[ -\frac{2^{-m-7} \cos \left (\frac{1}{2} \left (-e-f x+\frac{\pi }{2}\right )\right ) \csc ^9\left (\frac{1}{8} \left (-e-f x+\frac{\pi }{2}\right )\right ) \sec ^3\left (\frac{1}{8} \left (-e-f x+\frac{\pi }{2}\right )\right ) \sin ^{-2 m}\left (\frac{1}{2} \left (-e-f x+\frac{\pi }{2}\right )\right ) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-2} \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^{-2 (-m-2)} ((A-2 B (m+1)) \sin (e+f x)-2 A (m+1)+B)}{f \left (4 m^2+8 m+3\right ) \left (\cot ^2\left (\frac{1}{8} \left (-e-f x+\frac{\pi }{2}\right )\right )-1\right )^3} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.515, size = 0, normalized size = 0. \begin{align*} \int \left ( a+a\sin \left ( fx+e \right ) \right ) ^{m} \left ( A+B\sin \left ( fx+e \right ) \right ) \left ( c-c\sin \left ( fx+e \right ) \right ) ^{-2-m}\, dx \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 )}^{m}{\left (-c \sin \left (f x + e\right ) + c\right )}^{-m - 2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.01434, size = 213, normalized size = 1.87 \begin{align*} \frac{{\left ({\left (2 \, B m - A + 2 \, B\right )} \cos \left (f x + e\right ) \sin \left (f x + e\right ) +{\left (2 \, A m + 2 \, A - B\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 - 2}}{4 \, f m^{2} + 8 \, f m + 3 \, f} \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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