Optimal. Leaf size=102 \[ \frac {(A+B) (c-c \sin (e+f x))^{-p-2} (g \cos (e+f x))^{p+1}}{f g (p+3)}+\frac {(A-B (p+2)) (c-c \sin (e+f x))^{-p-1} (g \cos (e+f x))^{p+1}}{c f g (p+1) (p+3)} \]
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Rubi [A] time = 0.21, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2859, 2671} \[ \frac {(A+B) (c-c \sin (e+f x))^{-p-2} (g \cos (e+f x))^{p+1}}{f g (p+3)}+\frac {(A-B (p+2)) (c-c \sin (e+f x))^{-p-1} (g \cos (e+f x))^{p+1}}{c f g (p+1) (p+3)} \]
Antiderivative was successfully verified.
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Rule 2671
Rule 2859
Rubi steps
\begin {align*} \int (g \cos (e+f x))^p (A+B \sin (e+f x)) (c-c \sin (e+f x))^{-2-p} \, dx &=\frac {(A+B) (g \cos (e+f x))^{1+p} (c-c \sin (e+f x))^{-2-p}}{f g (3+p)}+\frac {(A-B (2+p)) \int (g \cos (e+f x))^p (c-c \sin (e+f x))^{-1-p} \, dx}{c (3+p)}\\ &=\frac {(A+B) (g \cos (e+f x))^{1+p} (c-c \sin (e+f x))^{-2-p}}{f g (3+p)}+\frac {(A-B (2+p)) (g \cos (e+f x))^{1+p} (c-c \sin (e+f x))^{-1-p}}{c f g (1+p) (3+p)}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 83, normalized size = 0.81 \[ \frac {\cos (e+f x) (c-c \sin (e+f x))^{-p} (g \cos (e+f x))^p ((B (p+2)-A) \sin (e+f x)+A (p+2)-B)}{c^2 f (p+1) (p+3) (\sin (e+f x)-1)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.53, size = 84, normalized size = 0.82 \[ \frac {{\left ({\left (B p - A + 2 \, B\right )} \cos \left (f x + e\right ) \sin \left (f x + e\right ) + {\left (A p + 2 \, A - B\right )} \cos \left (f x + e\right )\right )} \left (g \cos \left (f x + e\right )\right )^{p} {\left (-c \sin \left (f x + e\right ) + c\right )}^{-p - 2}}{f p^{2} + 4 \, f p + 3 \, f} \]
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 \[ \int {\left (B \sin \left (f x + e\right ) + A\right )} \left (g \cos \left (f x + e\right )\right )^{p} {\left (-c \sin \left (f x + e\right ) + c\right )}^{-p - 2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 7.81, size = 0, normalized size = 0.00 \[ \int \left (g \cos \left (f x +e \right )\right )^{p} \left (A +B \sin \left (f x +e \right )\right ) \left (c -c \sin \left (f x +e \right )\right )^{-2-p}\, dx \]
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 \[ \int {\left (B \sin \left (f x + e\right ) + A\right )} \left (g \cos \left (f x + e\right )\right )^{p} {\left (-c \sin \left (f x + e\right ) + c\right )}^{-p - 2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.28, size = 129, normalized size = 1.26 \[ -\frac {{\left (g\,\cos \left (e+f\,x\right )\right )}^p\,\left (4\,A\,\cos \left (e+f\,x\right )-2\,B\,\cos \left (e+f\,x\right )-A\,\sin \left (2\,e+2\,f\,x\right )+2\,B\,\sin \left (2\,e+2\,f\,x\right )+2\,A\,p\,\cos \left (e+f\,x\right )+B\,p\,\sin \left (2\,e+2\,f\,x\right )\right )}{c^2\,f\,{\left (-c\,\left (\sin \left (e+f\,x\right )-1\right )\right )}^p\,\left (4\,\sin \left (e+f\,x\right )+\cos \left (2\,e+2\,f\,x\right )-3\right )\,\left (p^2+4\,p+3\right )} \]
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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