Optimal. Leaf size=202 \[ \frac{(n+1) (A-B) \cos (e+f x) (d \sin (e+f x))^{n+2} \, _2F_1\left (\frac{1}{2},\frac{n+2}{2};\frac{n+4}{2};\sin ^2(e+f x)\right )}{a d^2 f (n+2) \sqrt{\cos ^2(e+f x)}}+\frac{(-A n+B n+B) \cos (e+f x) (d \sin (e+f x))^{n+1} \, _2F_1\left (\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\sin ^2(e+f x)\right )}{a d f (n+1) \sqrt{\cos ^2(e+f x)}}+\frac{(A-B) \cos (e+f x) (d \sin (e+f x))^{n+1}}{d f (a \sin (e+f x)+a)} \]
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Rubi [A] time = 0.223897, antiderivative size = 202, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {2978, 2748, 2643} \[ \frac{(n+1) (A-B) \cos (e+f x) (d \sin (e+f x))^{n+2} \, _2F_1\left (\frac{1}{2},\frac{n+2}{2};\frac{n+4}{2};\sin ^2(e+f x)\right )}{a d^2 f (n+2) \sqrt{\cos ^2(e+f x)}}+\frac{(-A n+B n+B) \cos (e+f x) (d \sin (e+f x))^{n+1} \, _2F_1\left (\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\sin ^2(e+f x)\right )}{a d f (n+1) \sqrt{\cos ^2(e+f x)}}+\frac{(A-B) \cos (e+f x) (d \sin (e+f x))^{n+1}}{d f (a \sin (e+f x)+a)} \]
Antiderivative was successfully verified.
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Rule 2978
Rule 2748
Rule 2643
Rubi steps
\begin{align*} \int \frac{(d \sin (e+f x))^n (A+B \sin (e+f x))}{a+a \sin (e+f x)} \, dx &=\frac{(A-B) \cos (e+f x) (d \sin (e+f x))^{1+n}}{d f (a+a \sin (e+f x))}+\frac{\int (d \sin (e+f x))^n (a d (B-A n+B n)+a (A-B) d (1+n) \sin (e+f x)) \, dx}{a^2 d}\\ &=\frac{(A-B) \cos (e+f x) (d \sin (e+f x))^{1+n}}{d f (a+a \sin (e+f x))}+\frac{((A-B) (1+n)) \int (d \sin (e+f x))^{1+n} \, dx}{a d}+\frac{(B-A n+B n) \int (d \sin (e+f x))^n \, dx}{a}\\ &=\frac{(B-A n+B 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}}{a d f (1+n) \sqrt{\cos ^2(e+f x)}}+\frac{(A-B) (1+n) \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}}{a d^2 f (2+n) \sqrt{\cos ^2(e+f x)}}+\frac{(A-B) \cos (e+f x) (d \sin (e+f x))^{1+n}}{d f (a+a \sin (e+f x))}\\ \end{align*}
Mathematica [A] time = 0.866731, size = 157, normalized size = 0.78 \[ \frac{\sin (e+f x) \cos (e+f x) (d \sin (e+f x))^n \left (\frac{(n+1) (A-B) \sin (e+f x) \, _2F_1\left (\frac{1}{2},\frac{n+2}{2};\frac{n+4}{2};\sin ^2(e+f x)\right )}{(n+2) \sqrt{\cos ^2(e+f x)}}+\frac{(-A n+B n+B) \, _2F_1\left (\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\sin ^2(e+f x)\right )}{(n+1) \sqrt{\cos ^2(e+f x)}}+\frac{A-B}{\sin (e+f x)+1}\right )}{a f} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.033, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( d\sin \left ( fx+e \right ) \right ) ^{n} \left ( A+B\sin \left ( fx+e \right ) \right ) }{a+a\sin \left ( fx+e \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 \frac{{\left (B \sin \left (f x + e\right ) + A\right )} \left (d \sin \left (f x + e\right )\right )^{n}}{a \sin \left (f x + e\right ) + a}\,{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 (\frac{{\left (B \sin \left (f x + e\right ) + A\right )} \left (d \sin \left (f x + e\right )\right )^{n}}{a \sin \left (f x + e\right ) + a}, 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 \frac{{\left (B \sin \left (f x + e\right ) + A\right )} \left (d \sin \left (f x + e\right )\right )^{n}}{a \sin \left (f x + e\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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