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 = 0.30, antiderivative size = 123, normalized size of antiderivative = 1.00, 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 69
Rule 70
Rule 2689
Rule 2860
Rule 2967
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*}
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Mathematica [C] time = 25.57, size = 7409, normalized size = 60.24 \[ \text {Result too large to show} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m}}{c \sin \left (f x + e\right ) - c}, x\right ) \]
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 \[ \int -\frac {{\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m}}{c \sin \left (f x + e\right ) - c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.09, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +a \sin \left (f x +e \right )\right )^{m} \left (A +B \sin \left (f x +e \right )\right )}{c -c \sin \left (f x +e \right )}\, dx \]
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 \[ -\int \frac {{\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m}}{c \sin \left (f x + e\right ) - c}\,{d x} \]
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 \[ \int \frac {\left (A+B\,\sin \left (e+f\,x\right )\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m}{c-c\,\sin \left (e+f\,x\right )} \,d x \]
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 \[ - \frac {\int \frac {A \left (a \sin {\left (e + f x \right )} + a\right )^{m}}{\sin {\left (e + f x \right )} - 1}\, dx + \int \frac {B \left (a \sin {\left (e + f x \right )} + a\right )^{m} \sin {\left (e + f x \right )}}{\sin {\left (e + f x \right )} - 1}\, dx}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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