Optimal. Leaf size=170 \[ -\frac {a 2^{\frac {1}{2} (2 m+p+1)} (A (m+p+1)+B m) (a \sin (e+f x)+a)^{m-1} (g \cos (e+f x))^{p+1} (\sin (e+f x)+1)^{\frac {1}{2} (-2 m-p+1)} \, _2F_1\left (\frac {1}{2} (-2 m-p+1),\frac {p+1}{2};\frac {p+3}{2};\frac {1}{2} (1-\sin (e+f x))\right )}{f g (p+1) (m+p+1)}-\frac {B (a \sin (e+f x)+a)^m (g \cos (e+f x))^{p+1}}{f g (m+p+1)} \]
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Rubi [A] time = 0.27, antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {2860, 2689, 70, 69} \[ -\frac {a 2^{\frac {1}{2} (2 m+p+1)} (A (m+p+1)+B m) (a \sin (e+f x)+a)^{m-1} (g \cos (e+f x))^{p+1} (\sin (e+f x)+1)^{\frac {1}{2} (-2 m-p+1)} \, _2F_1\left (\frac {1}{2} (-2 m-p+1),\frac {p+1}{2};\frac {p+3}{2};\frac {1}{2} (1-\sin (e+f x))\right )}{f g (p+1) (m+p+1)}-\frac {B (a \sin (e+f x)+a)^m (g \cos (e+f x))^{p+1}}{f g (m+p+1)} \]
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+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx &=-\frac {B (g \cos (e+f x))^{1+p} (a+a \sin (e+f x))^m}{f g (1+m+p)}+\left (A+\frac {B m}{1+m+p}\right ) \int (g \cos (e+f x))^p (a+a \sin (e+f x))^m \, dx\\ &=-\frac {B (g \cos (e+f x))^{1+p} (a+a \sin (e+f x))^m}{f g (1+m+p)}+\frac {\left (a^2 \left (A+\frac {B m}{1+m+p}\right ) (g \cos (e+f x))^{1+p} (a-a \sin (e+f x))^{\frac {1}{2} (-1-p)} (a+a \sin (e+f x))^{\frac {1}{2} (-1-p)}\right ) \operatorname {Subst}\left (\int (a-a x)^{\frac {1}{2} (-1+p)} (a+a x)^{m+\frac {1}{2} (-1+p)} \, dx,x,\sin (e+f x)\right )}{f g}\\ &=-\frac {B (g \cos (e+f x))^{1+p} (a+a \sin (e+f x))^m}{f g (1+m+p)}+\frac {\left (2^{-\frac {1}{2}+m+\frac {p}{2}} a^2 \left (A+\frac {B m}{1+m+p}\right ) (g \cos (e+f x))^{1+p} (a-a \sin (e+f x))^{\frac {1}{2} (-1-p)} (a+a \sin (e+f x))^{-\frac {1}{2}+m+\frac {1}{2} (-1-p)+\frac {p}{2}} \left (\frac {a+a \sin (e+f x)}{a}\right )^{\frac {1}{2}-m-\frac {p}{2}}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{2}+\frac {x}{2}\right )^{m+\frac {1}{2} (-1+p)} (a-a x)^{\frac {1}{2} (-1+p)} \, dx,x,\sin (e+f x)\right )}{f g}\\ &=-\frac {2^{\frac {1}{2} (1+2 m+p)} a \left (A+\frac {B m}{1+m+p}\right ) (g \cos (e+f x))^{1+p} \, _2F_1\left (\frac {1}{2} (1-2 m-p),\frac {1+p}{2};\frac {3+p}{2};\frac {1}{2} (1-\sin (e+f x))\right ) (1+\sin (e+f x))^{\frac {1}{2} (1-2 m-p)} (a+a \sin (e+f x))^{-1+m}}{f g (1+p)}-\frac {B (g \cos (e+f x))^{1+p} (a+a \sin (e+f x))^m}{f g (1+m+p)}\\ \end {align*}
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Mathematica [A] time = 0.46, size = 154, normalized size = 0.91 \[ -\frac {\cos (e+f x) (a (\sin (e+f x)+1))^m (g \cos (e+f x))^p (\sin (e+f x)+1)^{\frac {1}{2} (-2 m-p-1)} \left (2^{\frac {1}{2} (2 m+p+1)} (A (m+p+1)+B m) \, _2F_1\left (\frac {1}{2} (-2 m-p+1),\frac {p+1}{2};\frac {p+3}{2};\frac {1}{2} (1-\sin (e+f x))\right )+B (p+1) (\sin (e+f x)+1)^{\frac {1}{2} (2 m+p+1)}\right )}{f (p+1) (m+p+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.75, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (B \sin \left (f x + e\right ) + A\right )} \left (g \cos \left (f x + e\right )\right )^{p} {\left (a \sin \left (f x + e\right ) + a\right )}^{m}, x\right ) \]
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 \[ \int {\left (B \sin \left (f x + e\right ) + A\right )} \left (g \cos \left (f x + e\right )\right )^{p} {\left (a \sin \left (f x + e\right ) + a\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 7.63, size = 0, normalized size = 0.00 \[ \int \left (g \cos \left (f x +e \right )\right )^{p} \left (a +a \sin \left (f x +e \right )\right )^{m} \left (A +B \sin \left (f x +e \right )\right )\, dx \]
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 \[ \int {\left (B \sin \left (f x + e\right ) + A\right )} \left (g \cos \left (f x + e\right )\right )^{p} {\left (a \sin \left (f x + e\right ) + a\right )}^{m}\,{d x} \]
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 \[ \int {\left (g\,\cos \left (e+f\,x\right )\right )}^p\,\left (A+B\,\sin \left (e+f\,x\right )\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{m} \left (g \cos {\left (e + f x \right )}\right )^{p} \left (A + B \sin {\left (e + f x \right )}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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