Optimal. Leaf size=217 \[ -\frac{2 \sqrt{2} a (A-B) \cos (e+f x) (c+d \sin (e+f x))^n \left (\frac{c+d \sin (e+f x)}{c+d}\right )^{-n} F_1\left (\frac{1}{2};-\frac{1}{2},-n;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right )}{f \sqrt{\sin (e+f x)+1}}-\frac{4 \sqrt{2} a B \cos (e+f x) (c+d \sin (e+f x))^n \left (\frac{c+d \sin (e+f x)}{c+d}\right )^{-n} F_1\left (\frac{1}{2};-\frac{3}{2},-n;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right )}{f \sqrt{\sin (e+f x)+1}} \]
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Rubi [A] time = 0.300444, antiderivative size = 217, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {2968, 3017, 2755, 139, 138, 2784} \[ -\frac{2 \sqrt{2} a (A-B) \cos (e+f x) (c+d \sin (e+f x))^n \left (\frac{c+d \sin (e+f x)}{c+d}\right )^{-n} F_1\left (\frac{1}{2};-\frac{1}{2},-n;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right )}{f \sqrt{\sin (e+f x)+1}}-\frac{4 \sqrt{2} a B \cos (e+f x) (c+d \sin (e+f x))^n \left (\frac{c+d \sin (e+f x)}{c+d}\right )^{-n} F_1\left (\frac{1}{2};-\frac{3}{2},-n;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right )}{f \sqrt{\sin (e+f x)+1}} \]
Antiderivative was successfully verified.
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Rule 2968
Rule 3017
Rule 2755
Rule 139
Rule 138
Rule 2784
Rubi steps
\begin{align*} \int (a+a \sin (e+f x)) (A+B \sin (e+f x)) (c+d \sin (e+f x))^n \, dx &=\int (c+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\\ &=(a (A-B)) \int (1+\sin (e+f x)) (c+d \sin (e+f x))^n \, dx+(a B) \int (1+\sin (e+f x))^2 (c+d \sin (e+f x))^n \, dx\\ &=\frac{(a (A-B) \cos (e+f x)) \operatorname{Subst}\left (\int \frac{\sqrt{1+x} (c+d x)^n}{\sqrt{1-x}} \, dx,x,\sin (e+f x)\right )}{f \sqrt{1-\sin (e+f x)} \sqrt{1+\sin (e+f x)}}+\frac{(a B \cos (e+f x)) \operatorname{Subst}\left (\int \frac{(1+x)^{3/2} (c+d x)^n}{\sqrt{1-x}} \, dx,x,\sin (e+f x)\right )}{f \sqrt{1-\sin (e+f x)} \sqrt{1+\sin (e+f x)}}\\ &=\frac{\left (a (A-B) \cos (e+f x) (c+d \sin (e+f x))^n \left (-\frac{c+d \sin (e+f x)}{-c-d}\right )^{-n}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+x} \left (-\frac{c}{-c-d}-\frac{d x}{-c-d}\right )^n}{\sqrt{1-x}} \, dx,x,\sin (e+f x)\right )}{f \sqrt{1-\sin (e+f x)} \sqrt{1+\sin (e+f x)}}+\frac{\left (a B \cos (e+f x) (c+d \sin (e+f x))^n \left (-\frac{c+d \sin (e+f x)}{-c-d}\right )^{-n}\right ) \operatorname{Subst}\left (\int \frac{(1+x)^{3/2} \left (-\frac{c}{-c-d}-\frac{d x}{-c-d}\right )^n}{\sqrt{1-x}} \, dx,x,\sin (e+f x)\right )}{f \sqrt{1-\sin (e+f x)} \sqrt{1+\sin (e+f x)}}\\ &=-\frac{4 \sqrt{2} a B F_1\left (\frac{1}{2};-\frac{3}{2},-n;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right ) \cos (e+f x) (c+d \sin (e+f x))^n \left (\frac{c+d \sin (e+f x)}{c+d}\right )^{-n}}{f \sqrt{1+\sin (e+f x)}}-\frac{2 \sqrt{2} a (A-B) F_1\left (\frac{1}{2};-\frac{1}{2},-n;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right ) \cos (e+f x) (c+d \sin (e+f x))^n \left (\frac{c+d \sin (e+f x)}{c+d}\right )^{-n}}{f \sqrt{1+\sin (e+f x)}}\\ \end{align*}
Mathematica [F] time = 9.05416, size = 0, normalized size = 0. \[ \int (a+a \sin (e+f x)) (A+B \sin (e+f x)) (c+d \sin (e+f x))^n \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.48, size = 0, normalized size = 0. \begin{align*} \int \left ( a+a\sin \left ( fx+e \right ) \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 )}{\left (a \sin \left (f x + e\right ) + a\right )}{\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 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 ) + c\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 ) + c\right )}^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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