Optimal. Leaf size=220 \[ -\frac{(A-B) \cos (e+f x) \sqrt{\frac{d (1-\sin (e+f x))}{c+d}} (c+d \sin (e+f x))^{n+1} F_1\left (n+1;\frac{1}{2},1;n+2;\frac{c+d \sin (e+f x)}{c+d},\frac{c+d \sin (e+f x)}{c-d}\right )}{f (n+1) (c-d) (1-\sin (e+f x)) \sqrt{a \sin (e+f x)+a}}-\frac{2 B \cos (e+f x) (c+d \sin (e+f x))^n \left (\frac{c+d \sin (e+f x)}{c+d}\right )^{-n} \, _2F_1\left (\frac{1}{2},-n;\frac{3}{2};\frac{d (1-\sin (e+f x))}{c+d}\right )}{f \sqrt{a \sin (e+f x)+a}} \]
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Rubi [A] time = 0.402729, antiderivative size = 220, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.189, Rules used = {2987, 2788, 137, 136, 2776, 70, 69} \[ -\frac{(A-B) \cos (e+f x) \sqrt{\frac{d (1-\sin (e+f x))}{c+d}} (c+d \sin (e+f x))^{n+1} F_1\left (n+1;\frac{1}{2},1;n+2;\frac{c+d \sin (e+f x)}{c+d},\frac{c+d \sin (e+f x)}{c-d}\right )}{f (n+1) (c-d) (1-\sin (e+f x)) \sqrt{a \sin (e+f x)+a}}-\frac{2 B \cos (e+f x) (c+d \sin (e+f x))^n \left (\frac{c+d \sin (e+f x)}{c+d}\right )^{-n} \, _2F_1\left (\frac{1}{2},-n;\frac{3}{2};\frac{d (1-\sin (e+f x))}{c+d}\right )}{f \sqrt{a \sin (e+f x)+a}} \]
Antiderivative was successfully verified.
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Rule 2987
Rule 2788
Rule 137
Rule 136
Rule 2776
Rule 70
Rule 69
Rubi steps
\begin{align*} \int \frac{(A+B \sin (e+f x)) (c+d \sin (e+f x))^n}{\sqrt{a+a \sin (e+f x)}} \, dx &=(A-B) \int \frac{(c+d \sin (e+f x))^n}{\sqrt{a+a \sin (e+f x)}} \, dx+\frac{B \int \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^n \, dx}{a}\\ &=\frac{\left (a^2 (A-B) \cos (e+f x)\right ) \operatorname{Subst}\left (\int \frac{(c+d x)^n}{\sqrt{a-a x} (a+a x)} \, dx,x,\sin (e+f x)\right )}{f \sqrt{a-a \sin (e+f x)} \sqrt{a+a \sin (e+f x)}}+\frac{(a B \cos (e+f x)) \operatorname{Subst}\left (\int \frac{(c+d x)^n}{\sqrt{a-a x}} \, dx,x,\sin (e+f x)\right )}{f \sqrt{a-a \sin (e+f x)} \sqrt{a+a \sin (e+f x)}}\\ &=\frac{\left (a^2 (A-B) \cos (e+f x) \sqrt{\frac{d (a-a \sin (e+f x))}{a c+a d}}\right ) \operatorname{Subst}\left (\int \frac{(c+d x)^n}{(a+a x) \sqrt{\frac{a d}{a c+a d}-\frac{a d x}{a c+a d}}} \, dx,x,\sin (e+f x)\right )}{f (a-a \sin (e+f x)) \sqrt{a+a \sin (e+f x)}}+\frac{\left (a B \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 )}{f \sqrt{a-a \sin (e+f x)} \sqrt{a+a \sin (e+f x)}}\\ &=-\frac{(A-B) F_1\left (1+n;\frac{1}{2},1;2+n;\frac{c+d \sin (e+f x)}{c+d},\frac{c+d \sin (e+f x)}{c-d}\right ) \cos (e+f x) \sqrt{\frac{d (1-\sin (e+f x))}{c+d}} (c+d \sin (e+f x))^{1+n}}{(c-d) f (1+n) (1-\sin (e+f x)) \sqrt{a+a \sin (e+f x)}}-\frac{2 B \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}}{f \sqrt{a+a \sin (e+f x)}}\\ \end{align*}
Mathematica [A] time = 5.4829, size = 244, normalized size = 1.11 \[ \frac{\cos (e+f x) \sqrt{a (\sin (e+f x)+1)} (c+d \sin (e+f x))^n \left (\frac{4 (A-B) \sqrt{\frac{\sin (e+f x)-1}{\sin (e+f x)+1}} \left (\frac{c-d}{d \sin (e+f x)+d}+1\right )^{-n} F_1\left (-n-\frac{1}{2};-\frac{1}{2},-n;\frac{1}{2}-n;\frac{2}{\sin (e+f x)+1},\frac{d-c}{\sin (e+f x) d+d}\right )}{2 n+1}-(A+B) \sqrt{2-2 \sin (e+f x)} \left (\frac{c+d \sin (e+f x)}{c-d}\right )^{-n} F_1\left (1;\frac{1}{2},-n;2;\frac{1}{2} (\sin (e+f x)+1),\frac{d (\sin (e+f x)+1)}{d-c}\right )\right )}{4 a f (\sin (e+f x)-1)} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.346, size = 0, normalized size = 0. \begin{align*} \int{ \left ( A+B\sin \left ( fx+e \right ) \right ) \left ( c+d\sin \left ( fx+e \right ) \right ) ^{n}{\frac{1}{\sqrt{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 ) + c\right )}^{n}}{\sqrt{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 ) + c\right )}^{n}}{\sqrt{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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (A + B \sin{\left (e + f x \right )}\right ) \left (c + d \sin{\left (e + f x \right )}\right )^{n}}{\sqrt{a \left (\sin{\left (e + f x \right )} + 1\right )}}\, dx \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 ) + c\right )}^{n}}{\sqrt{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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