Optimal. Leaf size=168 \[ \frac{(2 A-B (p+3)) (c-c \sin (e+f x))^{-p-1} (g \cos (e+f x))^{p+1}}{c^2 f g (p+1) (p+3) (p+5)}+\frac{(A+B) (c-c \sin (e+f x))^{-p-3} (g \cos (e+f x))^{p+1}}{f g (p+5)}+\frac{(2 A-B (p+3)) (c-c \sin (e+f x))^{-p-2} (g \cos (e+f x))^{p+1}}{c f g (p+3) (p+5)} \]
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Rubi [A] time = 0.306062, antiderivative size = 168, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.079, Rules used = {2859, 2672, 2671} \[ \frac{(2 A-B (p+3)) (c-c \sin (e+f x))^{-p-1} (g \cos (e+f x))^{p+1}}{c^2 f g (p+1) (p+3) (p+5)}+\frac{(A+B) (c-c \sin (e+f x))^{-p-3} (g \cos (e+f x))^{p+1}}{f g (p+5)}+\frac{(2 A-B (p+3)) (c-c \sin (e+f x))^{-p-2} (g \cos (e+f x))^{p+1}}{c f g (p+3) (p+5)} \]
Antiderivative was successfully verified.
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Rule 2859
Rule 2672
Rule 2671
Rubi steps
\begin{align*} \int (g \cos (e+f x))^p (A+B \sin (e+f x)) (c-c \sin (e+f x))^{-3-p} \, dx &=\frac{(A+B) (g \cos (e+f x))^{1+p} (c-c \sin (e+f x))^{-3-p}}{f g (5+p)}+\frac{(2 A-B (3+p)) \int (g \cos (e+f x))^p (c-c \sin (e+f x))^{-2-p} \, dx}{c (5+p)}\\ &=\frac{(A+B) (g \cos (e+f x))^{1+p} (c-c \sin (e+f x))^{-3-p}}{f g (5+p)}+\frac{(2 A-B (3+p)) (g \cos (e+f x))^{1+p} (c-c \sin (e+f x))^{-2-p}}{c f g (3+p) (5+p)}+\frac{(2 A-B (3+p)) \int (g \cos (e+f x))^p (c-c \sin (e+f x))^{-1-p} \, dx}{c^2 (3+p) (5+p)}\\ &=\frac{(A+B) (g \cos (e+f x))^{1+p} (c-c \sin (e+f x))^{-3-p}}{f g (5+p)}+\frac{(2 A-B (3+p)) (g \cos (e+f x))^{1+p} (c-c \sin (e+f x))^{-2-p}}{c f g (3+p) (5+p)}+\frac{(2 A-B (3+p)) (g \cos (e+f x))^{1+p} (c-c \sin (e+f x))^{-1-p}}{c^2 f g (1+p) (3+p) (5+p)}\\ \end{align*}
Mathematica [A] time = 0.261668, size = 119, normalized size = 0.71 \[ -\frac{\cos (e+f x) (c-c \sin (e+f x))^{-p} (g \cos (e+f x))^p \left ((2 A-B (p+3)) \sin ^2(e+f x)+(p+3) (B (p+3)-2 A) \sin (e+f x)+A \left (p^2+6 p+7\right )-B (p+3)\right )}{c^3 f (p+1) (p+3) (p+5) (\sin (e+f x)-1)^3} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.632, size = 0, normalized size = 0. \begin{align*} \int \left ( g\cos \left ( fx+e \right ) \right ) ^{p} \left ( A+B\sin \left ( fx+e \right ) \right ) \left ( c-c\sin \left ( fx+e \right ) \right ) ^{-3-p}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [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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Fricas [A] time = 1.43571, size = 317, normalized size = 1.89 \begin{align*} \frac{{\left ({\left (B p - 2 \, A + 3 \, B\right )} \cos \left (f x + e\right )^{3} +{\left (B p^{2} - 2 \,{\left (A - 3 \, B\right )} p - 6 \, A + 9 \, B\right )} \cos \left (f x + e\right ) \sin \left (f x + e\right ) +{\left (A p^{2} + 2 \,{\left (3 \, A - B\right )} p + 9 \, A - 6 \, B\right )} \cos \left (f x + e\right )\right )} \left (g \cos \left (f x + e\right )\right )^{p}{\left (-c \sin \left (f x + e\right ) + c\right )}^{-p - 3}}{f p^{3} + 9 \, f p^{2} + 23 \, f p + 15 \, 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] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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