Optimal. Leaf size=173 \[ \frac{c^3 2^{\frac{5}{2}-m} (3 A-2 B (1-m)) \cos (e+f x) (1-\sin (e+f x))^{m+\frac{1}{2}} (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-1} \, _2F_1\left (\frac{1}{2} (2 m-3),\frac{1}{2} (2 m+1);\frac{1}{2} (2 m+3);\frac{1}{2} (\sin (e+f x)+1)\right )}{3 f (2 m+1)}-\frac{B \cos (e+f x) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{2-m}}{3 f} \]
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Rubi [A] time = 0.336136, antiderivative size = 173, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {2973, 2745, 2689, 70, 69} \[ \frac{c^3 2^{\frac{5}{2}-m} (3 A-2 B (1-m)) \cos (e+f x) (1-\sin (e+f x))^{m+\frac{1}{2}} (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-1} \, _2F_1\left (\frac{1}{2} (2 m-3),\frac{1}{2} (2 m+1);\frac{1}{2} (2 m+3);\frac{1}{2} (\sin (e+f x)+1)\right )}{3 f (2 m+1)}-\frac{B \cos (e+f x) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{2-m}}{3 f} \]
Antiderivative was successfully verified.
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Rule 2973
Rule 2745
Rule 2689
Rule 70
Rule 69
Rubi steps
\begin{align*} \int (a+a \sin (e+f x))^m (A+B \sin (e+f x)) (c-c \sin (e+f x))^{2-m} \, dx &=-\frac{B \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{2-m}}{3 f}+\frac{1}{3} (3 A-2 B (1-m)) \int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{2-m} \, dx\\ &=-\frac{B \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{2-m}}{3 f}+\frac{1}{3} \left ((3 A-2 B (1-m)) \cos ^{-2 m}(e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^m\right ) \int \cos ^{2 m}(e+f x) (c-c \sin (e+f x))^{2-2 m} \, dx\\ &=-\frac{B \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{2-m}}{3 f}+\frac{\left (c^2 (3 A-2 B (1-m)) \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{\frac{1}{2} (-1-2 m)+m} (c+c \sin (e+f x))^{\frac{1}{2} (-1-2 m)}\right ) \operatorname{Subst}\left (\int (c-c x)^{2-2 m+\frac{1}{2} (-1+2 m)} (c+c x)^{\frac{1}{2} (-1+2 m)} \, dx,x,\sin (e+f x)\right )}{3 f}\\ &=-\frac{B \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{2-m}}{3 f}+\frac{\left (2^{\frac{3}{2}-m} c^4 (3 A-2 B (1-m)) \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-\frac{1}{2}+\frac{1}{2} (-1-2 m)} \left (\frac{c-c \sin (e+f x)}{c}\right )^{\frac{1}{2}+m} (c+c \sin (e+f x))^{\frac{1}{2} (-1-2 m)}\right ) \operatorname{Subst}\left (\int \left (\frac{1}{2}-\frac{x}{2}\right )^{2-2 m+\frac{1}{2} (-1+2 m)} (c+c x)^{\frac{1}{2} (-1+2 m)} \, dx,x,\sin (e+f x)\right )}{3 f}\\ &=\frac{2^{\frac{5}{2}-m} c^3 (3 A-2 B (1-m)) \cos (e+f x) \, _2F_1\left (\frac{1}{2} (-3+2 m),\frac{1}{2} (1+2 m);\frac{1}{2} (3+2 m);\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 (c-c \sin (e+f x))^{-1-m}}{3 f (1+2 m)}-\frac{B \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{2-m}}{3 f}\\ \end{align*}
Mathematica [C] time = 53.1319, size = 5163, normalized size = 29.84 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.522, 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 ) \left ( c-c\sin \left ( fx+e \right ) \right ) ^{2-m}\, 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}{\left (-c \sin \left (f x + e\right ) + c\right )}^{-m + 2}\,{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}{\left (-c \sin \left (f x + e\right ) + c\right )}^{-m + 2}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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}{\left (-c \sin \left (f x + e\right ) + c\right )}^{-m + 2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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