Optimal. Leaf size=191 \[ \frac {a (A+B) \cos (e+f x) (d \sin (e+f x))^{n+2} \, _2F_1\left (\frac {1}{2},\frac {n+2}{2};\frac {n+4}{2};\sin ^2(e+f x)\right )}{d^2 f (n+2) \sqrt {\cos ^2(e+f x)}}+\frac {a (A (n+2)+B (n+1)) \cos (e+f x) (d \sin (e+f x))^{n+1} \, _2F_1\left (\frac {1}{2},\frac {n+1}{2};\frac {n+3}{2};\sin ^2(e+f x)\right )}{d f (n+1) (n+2) \sqrt {\cos ^2(e+f x)}}-\frac {a B \cos (e+f x) (d \sin (e+f x))^{n+1}}{d f (n+2)} \]
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Rubi [A] time = 0.22, antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {2968, 3023, 2748, 2643} \[ \frac {a (A+B) \cos (e+f x) (d \sin (e+f x))^{n+2} \, _2F_1\left (\frac {1}{2},\frac {n+2}{2};\frac {n+4}{2};\sin ^2(e+f x)\right )}{d^2 f (n+2) \sqrt {\cos ^2(e+f x)}}+\frac {a (A (n+2)+B (n+1)) \cos (e+f x) (d \sin (e+f x))^{n+1} \, _2F_1\left (\frac {1}{2},\frac {n+1}{2};\frac {n+3}{2};\sin ^2(e+f x)\right )}{d f (n+1) (n+2) \sqrt {\cos ^2(e+f x)}}-\frac {a B \cos (e+f x) (d \sin (e+f x))^{n+1}}{d f (n+2)} \]
Antiderivative was successfully verified.
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Rule 2643
Rule 2748
Rule 2968
Rule 3023
Rubi steps
\begin {align*} \int (d \sin (e+f x))^n (a+a \sin (e+f x)) (A+B \sin (e+f x)) \, dx &=\int (d \sin (e+f x))^n \left (a A+(a A+a B) \sin (e+f x)+a B \sin ^2(e+f x)\right ) \, dx\\ &=-\frac {a B \cos (e+f x) (d \sin (e+f x))^{1+n}}{d f (2+n)}+\frac {\int (d \sin (e+f x))^n (a d (B (1+n)+A (2+n))+a (A+B) d (2+n) \sin (e+f x)) \, dx}{d (2+n)}\\ &=-\frac {a B \cos (e+f x) (d \sin (e+f x))^{1+n}}{d f (2+n)}+\frac {(a (A+B)) \int (d \sin (e+f x))^{1+n} \, dx}{d}+\frac {(a (B (1+n)+A (2+n))) \int (d \sin (e+f x))^n \, dx}{2+n}\\ &=-\frac {a B \cos (e+f x) (d \sin (e+f x))^{1+n}}{d f (2+n)}+\frac {a (B (1+n)+A (2+n)) \cos (e+f x) \, _2F_1\left (\frac {1}{2},\frac {1+n}{2};\frac {3+n}{2};\sin ^2(e+f x)\right ) (d \sin (e+f x))^{1+n}}{d f (1+n) (2+n) \sqrt {\cos ^2(e+f x)}}+\frac {a (A+B) \cos (e+f x) \, _2F_1\left (\frac {1}{2},\frac {2+n}{2};\frac {4+n}{2};\sin ^2(e+f x)\right ) (d \sin (e+f x))^{2+n}}{d^2 f (2+n) \sqrt {\cos ^2(e+f x)}}\\ \end {align*}
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Mathematica [C] time = 3.85, size = 392, normalized size = 2.05 \[ -\frac {a 2^{-n-2} e^{i f n x} \left (1-e^{2 i (e+f x)}\right )^{-n} \left (-i e^{-i (e+f x)} \left (-1+e^{2 i (e+f x)}\right )\right )^n (\sin (e+f x)+1) \sin ^{-n}(e+f x) (d \sin (e+f x))^n \left (\frac {2 (A+B) e^{-i (e+f (n+1) x)} \, _2F_1\left (\frac {1}{2} (-n-1),-n;\frac {1-n}{2};e^{2 i (e+f x)}\right )}{n+1}-\frac {2 (A+B) e^{i (e-f (n-1) x)} \, _2F_1\left (\frac {1-n}{2},-n;\frac {3-n}{2};e^{2 i (e+f x)}\right )}{n-1}+i \left (\frac {e^{-i f n x} \left (B n e^{2 i (e+f x)} \, _2F_1\left (1-\frac {n}{2},-n;2-\frac {n}{2};e^{2 i (e+f x)}\right )-2 (n-2) (2 A+B) \, _2F_1\left (-n,-\frac {n}{2};1-\frac {n}{2};e^{2 i (e+f x)}\right )\right )}{(n-2) n}+\frac {B e^{-i (2 e+f (n+2) x)} \, _2F_1\left (-\frac {n}{2}-1,-n;-\frac {n}{2};e^{2 i (e+f x)}\right )}{n+2}\right )\right )}{f \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (B a \cos \left (f x + e\right )^{2} - {\left (A + B\right )} a \sin \left (f x + e\right ) - {\left (A + B\right )} a\right )} \left (d \sin \left (f x + e\right )\right )^{n}, x\right ) \]
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 \[ \int {\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )} \left (d \sin \left (f x + e\right )\right )^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 5.40, size = 0, normalized size = 0.00 \[ \int \left (d \sin \left (f x +e \right )\right )^{n} \left (a +a \sin \left (f x +e \right )\right ) \left (A +B \sin \left (f x +e \right )\right )\, dx \]
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 \[ \int {\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )} \left (d \sin \left (f x + e\right )\right )^{n}\,{d x} \]
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 \[ \int {\left (d\,\sin \left (e+f\,x\right )\right )}^n\,\left (A+B\,\sin \left (e+f\,x\right )\right )\,\left (a+a\,\sin \left (e+f\,x\right )\right ) \,d x \]
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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