Optimal. Leaf size=58 \[ -\frac{\cos (e+f x) F_1\left (\frac{1}{2};-n,1;\frac{3}{2};1-\sin (e+f x),\frac{1}{2} (1-\sin (e+f x))\right )}{f \sqrt{\sin (e+f x)+1}} \]
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Rubi [A] time = 0.0676877, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2785, 130, 429} \[ -\frac{\cos (e+f x) F_1\left (\frac{1}{2};-n,1;\frac{3}{2};1-\sin (e+f x),\frac{1}{2} (1-\sin (e+f x))\right )}{f \sqrt{\sin (e+f x)+1}} \]
Antiderivative was successfully verified.
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Rule 2785
Rule 130
Rule 429
Rubi steps
\begin{align*} \int \frac{\sin ^n(e+f x)}{\sqrt{1+\sin (e+f x)}} \, dx &=-\frac{\cos (e+f x) \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{1+\sin (e+f x)}}\\ &=-\frac{(2 \cos (e+f x)) \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{1+\sin (e+f x)}}\\ &=-\frac{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)}{f \sqrt{1+\sin (e+f x)}}\\ \end{align*}
Mathematica [B] time = 1.53659, size = 225, normalized size = 3.88 \[ \frac{\sqrt{\sin (e+f x)+1} \cos (e+f x) (-\sin (e+f x))^{-n} \sin ^n(e+f x) \left (1-\frac{1}{\sin (e+f x)+1}\right )^{-n} \left (4 \sqrt{\frac{\sin (e+f x)-1}{\sin (e+f x)+1}} (-\sin (e+f x))^n F_1\left (-n-\frac{1}{2};-\frac{1}{2},-n;\frac{1}{2}-n;\frac{2}{\sin (e+f x)+1},\frac{1}{\sin (e+f x)+1}\right )-(2 n+1) \sqrt{2-2 \sin (e+f x)} \left (1-\frac{1}{\sin (e+f x)+1}\right )^n F_1\left (1;\frac{1}{2},-n;2;\frac{1}{2} (\sin (e+f x)+1),\sin (e+f x)+1\right )\right )}{4 f (2 n+1) (\sin (e+f x)-1)} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.088, size = 0, normalized size = 0. \begin{align*} \int{ \left ( \sin \left ( fx+e \right ) \right ) ^{n}{\frac{1}{\sqrt{1+\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{\sin \left (f x + e\right )^{n}}{\sqrt{\sin \left (f x + e\right ) + 1}}\,{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{\sin \left (f x + e\right )^{n}}{\sqrt{\sin \left (f x + e\right ) + 1}}, 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{\sin ^{n}{\left (e + f x \right )}}{\sqrt{\sin{\left (e + f x \right )} + 1}}\, 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{\sin \left (f x + e\right )^{n}}{\sqrt{\sin \left (f x + e\right ) + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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