Optimal. Leaf size=127 \[ -\frac {a 2^{m+\frac {3}{2}} (A (m+3)+B m) \cos ^3(e+f x) (\sin (e+f x)+1)^{-m-\frac {1}{2}} (a \sin (e+f x)+a)^{m-1} \, _2F_1\left (\frac {3}{2},-m-\frac {1}{2};\frac {5}{2};\frac {1}{2} (1-\sin (e+f x))\right )}{3 f (m+3)}-\frac {B \cos ^3(e+f x) (a \sin (e+f x)+a)^m}{f (m+3)} \]
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Rubi [A] time = 0.19, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {2860, 2689, 70, 69} \[ -\frac {a 2^{m+\frac {3}{2}} (A (m+3)+B m) \cos ^3(e+f x) (\sin (e+f x)+1)^{-m-\frac {1}{2}} (a \sin (e+f x)+a)^{m-1} \, _2F_1\left (\frac {3}{2},-m-\frac {1}{2};\frac {5}{2};\frac {1}{2} (1-\sin (e+f x))\right )}{3 f (m+3)}-\frac {B \cos ^3(e+f x) (a \sin (e+f x)+a)^m}{f (m+3)} \]
Antiderivative was successfully verified.
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Rule 69
Rule 70
Rule 2689
Rule 2860
Rubi steps
\begin {align*} \int \cos ^2(e+f x) (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx &=-\frac {B \cos ^3(e+f x) (a+a \sin (e+f x))^m}{f (3+m)}+\left (A+\frac {B m}{3+m}\right ) \int \cos ^2(e+f x) (a+a \sin (e+f x))^m \, dx\\ &=-\frac {B \cos ^3(e+f x) (a+a \sin (e+f x))^m}{f (3+m)}+\frac {\left (a^2 \left (A+\frac {B m}{3+m}\right ) \cos ^3(e+f x)\right ) \operatorname {Subst}\left (\int \sqrt {a-a x} (a+a x)^{\frac {1}{2}+m} \, dx,x,\sin (e+f x)\right )}{f (a-a \sin (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2}}\\ &=-\frac {B \cos ^3(e+f x) (a+a \sin (e+f x))^m}{f (3+m)}+\frac {\left (2^{\frac {1}{2}+m} a^2 \left (A+\frac {B m}{3+m}\right ) \cos ^3(e+f x) (a+a \sin (e+f x))^{-1+m} \left (\frac {a+a \sin (e+f x)}{a}\right )^{-\frac {1}{2}-m}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{2}+\frac {x}{2}\right )^{\frac {1}{2}+m} \sqrt {a-a x} \, dx,x,\sin (e+f x)\right )}{f (a-a \sin (e+f x))^{3/2}}\\ &=-\frac {2^{\frac {3}{2}+m} a \left (A+\frac {B m}{3+m}\right ) \cos ^3(e+f x) \, _2F_1\left (\frac {3}{2},-\frac {1}{2}-m;\frac {5}{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))^{-1+m}}{3 f}-\frac {B \cos ^3(e+f x) (a+a \sin (e+f x))^m}{f (3+m)}\\ \end {align*}
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Mathematica [A] time = 0.33, size = 111, normalized size = 0.87 \[ -\frac {\cos ^3(e+f x) (\sin (e+f x)+1)^{-m-\frac {3}{2}} (a (\sin (e+f x)+1))^m \left (2^{m+\frac {3}{2}} (A (m+3)+B m) \, _2F_1\left (\frac {3}{2},-m-\frac {1}{2};\frac {5}{2};\frac {1}{2} (1-\sin (e+f x))\right )+3 B (\sin (e+f x)+1)^{m+\frac {3}{2}}\right )}{3 f (m+3)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.79, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (B \cos \left (f x + e\right )^{2} \sin \left (f x + e\right ) + A \cos \left (f x + e\right )^{2}\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m}, x\right ) \]
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 \[ \int {\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} \cos \left (f x + e\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 4.23, size = 0, normalized size = 0.00 \[ \int \left (\cos ^{2}\left (f x +e \right )\right ) \left (a +a \sin \left (f x +e \right )\right )^{m} \left (A +B \sin \left (f x +e \right )\right )\, dx \]
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 \[ \int {\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} \cos \left (f x + e\right )^{2}\,{d x} \]
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 \[ \int {\cos \left (e+f\,x\right )}^2\,\left (A+B\,\sin \left (e+f\,x\right )\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m \,d x \]
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 \[ \int \left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{m} \left (A + B \sin {\left (e + f x \right )}\right ) \cos ^{2}{\left (e + f x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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