Optimal. Leaf size=167 \[ -\frac{2 a \cos (e+f x) (A d (2 n+3)-B (c-2 d (n+1))) (c+d \sin (e+f x))^n \left (\frac{c+d \sin (e+f x)}{c+d}\right )^{-n} \, _2F_1\left (\frac{1}{2},-n;\frac{3}{2};\frac{d (1-\sin (e+f x))}{c+d}\right )}{d f (2 n+3) \sqrt{a \sin (e+f x)+a}}-\frac{2 a B \cos (e+f x) (c+d \sin (e+f x))^{n+1}}{d f (2 n+3) \sqrt{a \sin (e+f x)+a}} \]
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Rubi [A] time = 0.26483, antiderivative size = 167, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.108, Rules used = {2981, 2776, 70, 69} \[ \frac{2 a \cos (e+f x) (-A d (2 n+3)+B c-2 B d (n+1)) (c+d \sin (e+f x))^n \left (\frac{c+d \sin (e+f x)}{c+d}\right )^{-n} \, _2F_1\left (\frac{1}{2},-n;\frac{3}{2};\frac{d (1-\sin (e+f x))}{c+d}\right )}{d f (2 n+3) \sqrt{a \sin (e+f x)+a}}-\frac{2 a B \cos (e+f x) (c+d \sin (e+f x))^{n+1}}{d f (2 n+3) \sqrt{a \sin (e+f x)+a}} \]
Antiderivative was successfully verified.
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Rule 2981
Rule 2776
Rule 70
Rule 69
Rubi steps
\begin{align*} \int \sqrt{a+a \sin (e+f x)} (A+B \sin (e+f x)) (c+d \sin (e+f x))^n \, dx &=-\frac{2 a B \cos (e+f x) (c+d \sin (e+f x))^{1+n}}{d f (3+2 n) \sqrt{a+a \sin (e+f x)}}+\frac{(a A d (3+2 n)-B (a c-2 a d (1+n))) \int \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^n \, dx}{a d (3+2 n)}\\ &=-\frac{2 a B \cos (e+f x) (c+d \sin (e+f x))^{1+n}}{d f (3+2 n) \sqrt{a+a \sin (e+f x)}}+\frac{(a (a A d (3+2 n)-B (a c-2 a d (1+n))) \cos (e+f x)) \operatorname{Subst}\left (\int \frac{(c+d x)^n}{\sqrt{a-a x}} \, dx,x,\sin (e+f x)\right )}{d f (3+2 n) \sqrt{a-a \sin (e+f x)} \sqrt{a+a \sin (e+f x)}}\\ &=-\frac{2 a B \cos (e+f x) (c+d \sin (e+f x))^{1+n}}{d f (3+2 n) \sqrt{a+a \sin (e+f x)}}+\frac{\left (a (a A d (3+2 n)-B (a c-2 a d (1+n))) \cos (e+f x) (c+d \sin (e+f x))^n \left (-\frac{a (c+d \sin (e+f x))}{-a c-a d}\right )^{-n}\right ) \operatorname{Subst}\left (\int \frac{\left (\frac{c}{c+d}+\frac{d x}{c+d}\right )^n}{\sqrt{a-a x}} \, dx,x,\sin (e+f x)\right )}{d f (3+2 n) \sqrt{a-a \sin (e+f x)} \sqrt{a+a \sin (e+f x)}}\\ &=-\frac{2 a B \cos (e+f x) (c+d \sin (e+f x))^{1+n}}{d f (3+2 n) \sqrt{a+a \sin (e+f x)}}+\frac{2 a (B c-2 B d (1+n)-A d (3+2 n)) \cos (e+f x) \, _2F_1\left (\frac{1}{2},-n;\frac{3}{2};\frac{d (1-\sin (e+f x))}{c+d}\right ) (c+d \sin (e+f x))^n \left (\frac{c+d \sin (e+f x)}{c+d}\right )^{-n}}{d f (3+2 n) \sqrt{a+a \sin (e+f x)}}\\ \end{align*}
Mathematica [F] time = 8.21614, size = 0, normalized size = 0. \[ \int \sqrt{a+a \sin (e+f x)} (A+B \sin (e+f x)) (c+d \sin (e+f x))^n \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.451, size = 0, normalized size = 0. \begin{align*} \int \sqrt{a+a\sin \left ( fx+e \right ) } \left ( A+B\sin \left ( fx+e \right ) \right ) \left ( c+d\sin \left ( fx+e \right ) \right ) ^{n}\, 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 )} \sqrt{a \sin \left (f x + e\right ) + a}{\left (d \sin \left (f x + e\right ) + c\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 \sin \left (f x + e\right ) + A\right )} \sqrt{a \sin \left (f x + e\right ) + a}{\left (d \sin \left (f x + e\right ) + c\right )}^{n}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a \left (\sin{\left (e + f x \right )} + 1\right )} \left (A + B \sin{\left (e + f x \right )}\right ) \left (c + d \sin{\left (e + f x \right )}\right )^{n}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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