Optimal. Leaf size=152 \[ -\frac{(A-B) \cos (e+f x) \sin ^{-n}(e+f x) F_1\left (\frac{1}{2};-n,1;\frac{3}{2};1-\sin (e+f x),\frac{1}{2} (1-\sin (e+f x))\right ) (d \sin (e+f x))^n}{f \sqrt{a \sin (e+f x)+a}}-\frac{2 B \cos (e+f x) \sin ^{-n}(e+f x) (d \sin (e+f x))^n \, _2F_1\left (\frac{1}{2},-n;\frac{3}{2};1-\sin (e+f x)\right )}{f \sqrt{a \sin (e+f x)+a}} \]
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Rubi [A] time = 0.396167, antiderivative size = 152, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.257, Rules used = {2987, 2787, 2786, 2785, 130, 429, 2776, 67, 65} \[ -\frac{(A-B) \cos (e+f x) \sin ^{-n}(e+f x) F_1\left (\frac{1}{2};-n,1;\frac{3}{2};1-\sin (e+f x),\frac{1}{2} (1-\sin (e+f x))\right ) (d \sin (e+f x))^n}{f \sqrt{a \sin (e+f x)+a}}-\frac{2 B \cos (e+f x) \sin ^{-n}(e+f x) (d \sin (e+f x))^n \, _2F_1\left (\frac{1}{2},-n;\frac{3}{2};1-\sin (e+f x)\right )}{f \sqrt{a \sin (e+f x)+a}} \]
Antiderivative was successfully verified.
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Rule 2987
Rule 2787
Rule 2786
Rule 2785
Rule 130
Rule 429
Rule 2776
Rule 67
Rule 65
Rubi steps
\begin{align*} \int \frac{(d \sin (e+f x))^n (A+B \sin (e+f x))}{\sqrt{a+a \sin (e+f x)}} \, dx &=(A-B) \int \frac{(d \sin (e+f x))^n}{\sqrt{a+a \sin (e+f x)}} \, dx+\frac{B \int (d \sin (e+f x))^n \sqrt{a+a \sin (e+f x)} \, dx}{a}\\ &=\frac{\left ((A-B) \sqrt{1+\sin (e+f x)}\right ) \int \frac{(d \sin (e+f x))^n}{\sqrt{1+\sin (e+f x)}} \, dx}{\sqrt{a+a \sin (e+f x)}}+\frac{(a B \cos (e+f x)) \operatorname{Subst}\left (\int \frac{(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-B) \sin ^{-n}(e+f x) (d \sin (e+f x))^n \sqrt{1+\sin (e+f x)}\right ) \int \frac{\sin ^n(e+f x)}{\sqrt{1+\sin (e+f x)}} \, dx}{\sqrt{a+a \sin (e+f x)}}+\frac{\left (a B \cos (e+f x) \sin ^{-n}(e+f x) (d \sin (e+f x))^n\right ) \operatorname{Subst}\left (\int \frac{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{2 B \cos (e+f x) \, _2F_1\left (\frac{1}{2},-n;\frac{3}{2};1-\sin (e+f x)\right ) \sin ^{-n}(e+f x) (d \sin (e+f x))^n}{f \sqrt{a+a \sin (e+f x)}}-\frac{\left ((A-B) \cos (e+f x) \sin ^{-n}(e+f x) (d \sin (e+f x))^n\right ) \operatorname{Subst}\left (\int \frac{(1-x)^n}{(2-x) \sqrt{x}} \, dx,x,1-\sin (e+f x)\right )}{f \sqrt{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};1-\sin (e+f x)\right ) \sin ^{-n}(e+f x) (d \sin (e+f x))^n}{f \sqrt{a+a \sin (e+f x)}}-\frac{\left (2 (A-B) \cos (e+f x) \sin ^{-n}(e+f x) (d \sin (e+f x))^n\right ) \operatorname{Subst}\left (\int \frac{\left (1-x^2\right )^n}{2-x^2} \, dx,x,\sqrt{1-\sin (e+f x)}\right )}{f \sqrt{1-\sin (e+f x)} \sqrt{a+a \sin (e+f x)}}\\ &=-\frac{(A-B) F_1\left (\frac{1}{2};-n,1;\frac{3}{2};1-\sin (e+f x),\frac{1}{2} (1-\sin (e+f x))\right ) \cos (e+f x) \sin ^{-n}(e+f x) (d \sin (e+f x))^n}{f \sqrt{a+a \sin (e+f x)}}-\frac{2 B \cos (e+f x) \, _2F_1\left (\frac{1}{2},-n;\frac{3}{2};1-\sin (e+f x)\right ) \sin ^{-n}(e+f x) (d \sin (e+f x))^n}{f \sqrt{a+a \sin (e+f x)}}\\ \end{align*}
Mathematica [A] time = 4.70877, size = 250, normalized size = 1.64 \[ \frac{\cos (e+f x) \sqrt{a (\sin (e+f x)+1)} \sin ^n(e+f x) \left (-\sin ^2(e+f x)\right )^{-n} \left (1-\frac{1}{\sin (e+f x)+1}\right )^{-n} (d \sin (e+f x))^n \left (4 (A-B) \sqrt{\frac{\sin (e+f x)-1}{\sin (e+f x)+1}} (-\sin (e+f x))^n F_1\left (-n-\frac{1}{2};-\frac{1}{2},-n;\frac{1}{2}-n;\frac{2}{\sin (e+f x)+1},\frac{1}{\sin (e+f x)+1}\right )-(2 n+1) (A+B) \sqrt{2-2 \sin (e+f x)} \left (1-\frac{1}{\sin (e+f x)+1}\right )^n F_1\left (1;\frac{1}{2},-n;2;\frac{1}{2} (\sin (e+f x)+1),\sin (e+f x)+1\right )\right )}{4 a f (2 n+1) (\sin (e+f x)-1)} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.383, size = 0, normalized size = 0. \begin{align*} \int{ \left ( d\sin \left ( fx+e \right ) \right ) ^{n} \left ( A+B\sin \left ( fx+e \right ) \right ){\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 )\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 )\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 (d \sin{\left (e + f x \right )}\right )^{n} \left (A + B \sin{\left (e + f x \right )}\right )}{\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 )\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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