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 = 0.34, antiderivative size = 173, normalized size of antiderivative = 1.00, 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 69
Rule 70
Rule 2689
Rule 2745
Rule 2973
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*}
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Mathematica [C] time = 49.22, size = 5163, normalized size = 29.84 \[ \text {Result too large to show} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} {\left (-c \sin \left (f x + e\right ) + c\right )}^{-m + 2}, x\right ) \]
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 \[ \int {\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} {\left (-c \sin \left (f x + e\right ) + c\right )}^{-m + 2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 6.07, size = 0, normalized size = 0.00 \[ \int \left (a +a \sin \left (f x +e \right )\right )^{m} \left (A +B \sin \left (f x +e \right )\right ) \left (c -c \sin \left (f x +e \right )\right )^{2-m}\, dx \]
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 \[ \int {\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} {\left (-c \sin \left (f x + e\right ) + c\right )}^{-m + 2}\,{d x} \]
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 \[ \int \left (A+B\,\sin \left (e+f\,x\right )\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{2-m} \,d x \]
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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