Optimal. Leaf size=147 \[ \frac {c 2^{\frac {1}{2}-\frac {p}{2}} (A+B p) (1-\sin (e+f x))^{\frac {p+1}{2}} (c-c \sin (e+f x))^{-p-1} (g \cos (e+f x))^{p+1} \, _2F_1\left (\frac {p+1}{2},\frac {p+1}{2};\frac {p+3}{2};\frac {1}{2} (\sin (e+f x)+1)\right )}{f g (p+1)}-\frac {B (c-c \sin (e+f x))^{-p} (g \cos (e+f x))^{p+1}}{f g} \]
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Rubi [A] time = 0.21, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2860, 2689, 70, 69} \[ \frac {c 2^{\frac {1}{2}-\frac {p}{2}} (A+B p) (1-\sin (e+f x))^{\frac {p+1}{2}} (c-c \sin (e+f x))^{-p-1} (g \cos (e+f x))^{p+1} \, _2F_1\left (\frac {p+1}{2},\frac {p+1}{2};\frac {p+3}{2};\frac {1}{2} (\sin (e+f x)+1)\right )}{f g (p+1)}-\frac {B (c-c \sin (e+f x))^{-p} (g \cos (e+f x))^{p+1}}{f g} \]
Antiderivative was successfully verified.
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Rule 69
Rule 70
Rule 2689
Rule 2860
Rubi steps
\begin {align*} \int (g \cos (e+f x))^p (A+B \sin (e+f x)) (c-c \sin (e+f x))^{-p} \, dx &=-\frac {B (g \cos (e+f x))^{1+p} (c-c \sin (e+f x))^{-p}}{f g}+(A+B p) \int (g \cos (e+f x))^p (c-c \sin (e+f x))^{-p} \, dx\\ &=-\frac {B (g \cos (e+f x))^{1+p} (c-c \sin (e+f x))^{-p}}{f g}+\frac {\left (c^2 (A+B p) (g \cos (e+f x))^{1+p} (c-c \sin (e+f x))^{\frac {1}{2} (-1-p)} (c+c \sin (e+f x))^{\frac {1}{2} (-1-p)}\right ) \operatorname {Subst}\left (\int (c-c x)^{\frac {1}{2} (-1+p)-p} (c+c x)^{\frac {1}{2} (-1+p)} \, dx,x,\sin (e+f x)\right )}{f g}\\ &=-\frac {B (g \cos (e+f x))^{1+p} (c-c \sin (e+f x))^{-p}}{f g}+\frac {\left (2^{-\frac {1}{2}-\frac {p}{2}} c^2 (A+B p) (g \cos (e+f x))^{1+p} (c-c \sin (e+f x))^{-\frac {1}{2}+\frac {1}{2} (-1-p)-\frac {p}{2}} \left (\frac {c-c \sin (e+f x)}{c}\right )^{\frac {1}{2}+\frac {p}{2}} (c+c \sin (e+f x))^{\frac {1}{2} (-1-p)}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{2}-\frac {x}{2}\right )^{\frac {1}{2} (-1+p)-p} (c+c x)^{\frac {1}{2} (-1+p)} \, dx,x,\sin (e+f x)\right )}{f g}\\ &=\frac {2^{\frac {1}{2}-\frac {p}{2}} c (A+B p) (g \cos (e+f x))^{1+p} \, _2F_1\left (\frac {1+p}{2},\frac {1+p}{2};\frac {3+p}{2};\frac {1}{2} (1+\sin (e+f x))\right ) (1-\sin (e+f x))^{\frac {1+p}{2}} (c-c \sin (e+f x))^{-1-p}}{f g (1+p)}-\frac {B (g \cos (e+f x))^{1+p} (c-c \sin (e+f x))^{-p}}{f g}\\ \end {align*}
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Mathematica [A] time = 0.56, size = 144, normalized size = 0.98 \[ -\frac {2^{\frac {1}{2} (-p-1)} \cos (e+f x) (c-c \sin (e+f x))^{-p} (g \cos (e+f x))^p \left (2 (A+B p) (1-\sin (e+f x))^{\frac {p+1}{2}} \, _2F_1\left (\frac {p+1}{2},\frac {p+1}{2};\frac {p+3}{2};\frac {1}{2} (\sin (e+f x)+1)\right )+B 2^{\frac {p+1}{2}} (p+1) (\sin (e+f x)-1)\right )}{f (p+1) (\sin (e+f x)-1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.65, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (B \sin \left (f x + e\right ) + A\right )} \left (g \cos \left (f x + e\right )\right )^{p}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{p}}, x\right ) \]
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 \[ \int \frac {{\left (B \sin \left (f x + e\right ) + A\right )} \left (g \cos \left (f x + e\right )\right )^{p}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{p}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 4.82, size = 0, normalized size = 0.00 \[ \int \left (g \cos \left (f x +e \right )\right )^{p} \left (A +B \sin \left (f x +e \right )\right ) \left (c -c \sin \left (f x +e \right )\right )^{-p}\, dx \]
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 \[ \int \frac {{\left (B \sin \left (f x + e\right ) + A\right )} \left (g \cos \left (f x + e\right )\right )^{p}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{p}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (g\,\cos \left (e+f\,x\right )\right )}^p\,\left (A+B\,\sin \left (e+f\,x\right )\right )}{{\left (c-c\,\sin \left (e+f\,x\right )\right )}^p} \,d x \]
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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