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 = 0.217322, antiderivative size = 191, normalized size of antiderivative = 1., 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 2968
Rule 3023
Rule 2748
Rule 2643
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*}
Mathematica [C] time = 3.77841, 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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Maple [F] time = 1.806, size = 0, normalized size = 0. \begin{align*} \int \left ( d\sin \left ( fx+e \right ) \right ) ^{n} \left ( a+a\sin \left ( fx+e \right ) \right ) \left ( A+B\sin \left ( fx+e \right ) \right ) \, 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 )} \left (d \sin \left (f x + e\right )\right )^{n}\,{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 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 ) \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 )} \left (d \sin \left (f x + e\right )\right )^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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