Optimal. Leaf size=123 \[ \frac{2^{m+\frac{1}{2}} (A m+B m+B) \sec (e+f x) (\sin (e+f x)+1)^{\frac{1}{2}-m} (a \sin (e+f x)+a)^m \, _2F_1\left (-\frac{1}{2},\frac{1}{2}-m;\frac{1}{2};\frac{1}{2} (1-\sin (e+f x))\right )}{c f m}-\frac{B \sec (e+f x) (a \sin (e+f x)+a)^{m+1}}{a c f m} \]
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Rubi [A] time = 0.303142, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.139, Rules used = {2967, 2860, 2689, 70, 69} \[ \frac{2^{m+\frac{1}{2}} (A m+B m+B) \sec (e+f x) (\sin (e+f x)+1)^{\frac{1}{2}-m} (a \sin (e+f x)+a)^m \, _2F_1\left (-\frac{1}{2},\frac{1}{2}-m;\frac{1}{2};\frac{1}{2} (1-\sin (e+f x))\right )}{c f m}-\frac{B \sec (e+f x) (a \sin (e+f x)+a)^{m+1}}{a c f m} \]
Antiderivative was successfully verified.
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Rule 2967
Rule 2860
Rule 2689
Rule 70
Rule 69
Rubi steps
\begin{align*} \int \frac{(a+a \sin (e+f x))^m (A+B \sin (e+f x))}{c-c \sin (e+f x)} \, dx &=\frac{\int \sec ^2(e+f x) (a+a \sin (e+f x))^{1+m} (A+B \sin (e+f x)) \, dx}{a c}\\ &=-\frac{B \sec (e+f x) (a+a \sin (e+f x))^{1+m}}{a c f m}+\frac{(B+A m+B m) \int \sec ^2(e+f x) (a+a \sin (e+f x))^{1+m} \, dx}{a c m}\\ &=-\frac{B \sec (e+f x) (a+a \sin (e+f x))^{1+m}}{a c f m}+\frac{\left (a (B+A m+B m) \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{a+a \sin (e+f x)}\right ) \operatorname{Subst}\left (\int \frac{(a+a x)^{-\frac{1}{2}+m}}{(a-a x)^{3/2}} \, dx,x,\sin (e+f x)\right )}{c f m}\\ &=-\frac{B \sec (e+f x) (a+a \sin (e+f x))^{1+m}}{a c f m}+\frac{\left (2^{-\frac{1}{2}+m} a (B+A m+B m) \sec (e+f x) \sqrt{a-a \sin (e+f x)} (a+a \sin (e+f x))^m \left (\frac{a+a \sin (e+f x)}{a}\right )^{\frac{1}{2}-m}\right ) \operatorname{Subst}\left (\int \frac{\left (\frac{1}{2}+\frac{x}{2}\right )^{-\frac{1}{2}+m}}{(a-a x)^{3/2}} \, dx,x,\sin (e+f x)\right )}{c f m}\\ &=\frac{2^{\frac{1}{2}+m} (B+A m+B m) \, _2F_1\left (-\frac{1}{2},\frac{1}{2}-m;\frac{1}{2};\frac{1}{2} (1-\sin (e+f x))\right ) \sec (e+f x) (1+\sin (e+f x))^{\frac{1}{2}-m} (a+a \sin (e+f x))^m}{c f m}-\frac{B \sec (e+f x) (a+a \sin (e+f x))^{1+m}}{a c f m}\\ \end{align*}
Mathematica [C] time = 25.6499, size = 7409, normalized size = 60.24 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.296, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+a\sin \left ( fx+e \right ) \right ) ^{m} \left ( A+B\sin \left ( fx+e \right ) \right ) }{c-c\sin \left ( fx+e \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 \frac{{\left (B \sin \left (f x + e\right ) + A\right )}{\left (a \sin \left (f x + e\right ) + a\right )}^{m}}{c \sin \left (f x + e\right ) - c}\,{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 (-\frac{{\left (B \sin \left (f x + e\right ) + A\right )}{\left (a \sin \left (f x + e\right ) + a\right )}^{m}}{c \sin \left (f x + e\right ) - c}, 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*} - \frac{\int \frac{A \left (a \sin{\left (e + f x \right )} + a\right )^{m}}{\sin{\left (e + f x \right )} - 1}\, dx + \int \frac{B \left (a \sin{\left (e + f x \right )} + a\right )^{m} \sin{\left (e + f x \right )}}{\sin{\left (e + f x \right )} - 1}\, dx}{c} \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 -\frac{{\left (B \sin \left (f x + e\right ) + A\right )}{\left (a \sin \left (f x + e\right ) + a\right )}^{m}}{c \sin \left (f x + e\right ) - c}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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