Optimal. Leaf size=117 \[ -\frac{2^{m+\frac{1}{2}} (A m+A+B m) \cos (e+f x) (\sin (e+f x)+1)^{-m-\frac{1}{2}} (a \sin (e+f x)+a)^m \, _2F_1\left (\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x))\right )}{f (m+1)}-\frac{B \cos (e+f x) (a \sin (e+f x)+a)^m}{f (m+1)} \]
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Rubi [A] time = 0.0817716, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {2751, 2652, 2651} \[ -\frac{2^{m+\frac{1}{2}} (A m+A+B m) \cos (e+f x) (\sin (e+f x)+1)^{-m-\frac{1}{2}} (a \sin (e+f x)+a)^m \, _2F_1\left (\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x))\right )}{f (m+1)}-\frac{B \cos (e+f x) (a \sin (e+f x)+a)^m}{f (m+1)} \]
Antiderivative was successfully verified.
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Rule 2751
Rule 2652
Rule 2651
Rubi steps
\begin{align*} \int (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx &=-\frac{B \cos (e+f x) (a+a \sin (e+f x))^m}{f (1+m)}+\frac{(A+A m+B m) \int (a+a \sin (e+f x))^m \, dx}{1+m}\\ &=-\frac{B \cos (e+f x) (a+a \sin (e+f x))^m}{f (1+m)}+\frac{\left ((A+A m+B m) (1+\sin (e+f x))^{-m} (a+a \sin (e+f x))^m\right ) \int (1+\sin (e+f x))^m \, dx}{1+m}\\ &=-\frac{B \cos (e+f x) (a+a \sin (e+f x))^m}{f (1+m)}-\frac{2^{\frac{1}{2}+m} (A+A m+B m) \cos (e+f x) \, _2F_1\left (\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x))\right ) (1+\sin (e+f x))^{-\frac{1}{2}-m} (a+a \sin (e+f x))^m}{f (1+m)}\\ \end{align*}
Mathematica [C] time = 1.82332, size = 275, normalized size = 2.35 \[ -\frac{\sin ^{-2 m}\left (\frac{1}{4} (2 e+2 f x+\pi )\right ) (a (\sin (e+f x)+1))^m \left (\frac{2 \sqrt{2} A \sin \left (\frac{1}{4} (2 e+2 f x-\pi )\right ) \cos ^{2 m+1}\left (\frac{1}{4} (2 e+2 f x-\pi )\right ) \, _2F_1\left (\frac{1}{2},m+\frac{1}{2};m+\frac{3}{2};\sin ^2\left (\frac{1}{4} (2 e+2 f x+\pi )\right )\right )}{(2 m+1) \sqrt{1-\sin (e+f x)}}+\frac{\sqrt [4]{-1} B 2^{-2 m-1} e^{-\frac{3}{2} i (e+f x)} \left (-(-1)^{3/4} e^{-\frac{1}{2} i (e+f x)} \left (e^{i (e+f x)}+i\right )\right )^{2 m+1} \left ((m-1) e^{2 i (e+f x)} \, _2F_1\left (1,m;-m;-i e^{-i (e+f x)}\right )-(m+1) \, _2F_1\left (1,m+2;2-m;-i e^{-i (e+f x)}\right )\right )}{m^2-1}\right )}{f} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 1.133, 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 ) \, 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}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (B \sin \left (f x + e\right ) + A\right )}{\left (a \sin \left (f x + e\right ) + a\right )}^{m}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a \left (\sin{\left (e + f x \right )} + 1\right )\right )^{m} \left (A + B \sin{\left (e + f x \right )}\right )\, dx \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*} \int{\left (B \sin \left (f x + e\right ) + A\right )}{\left (a \sin \left (f x + e\right ) + a\right )}^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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