Optimal. Leaf size=48 \[ -\frac{2 a \cot (c+d x) \text{Hypergeometric2F1}\left (\frac{1}{2},1-n,\frac{3}{2},1-\csc (c+d x)\right )}{d \sqrt{a \csc (c+d x)+a}} \]
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Rubi [A] time = 0.0584631, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {3806, 65} \[ -\frac{2 a \cot (c+d x) \, _2F_1\left (\frac{1}{2},1-n;\frac{3}{2};1-\csc (c+d x)\right )}{d \sqrt{a \csc (c+d x)+a}} \]
Antiderivative was successfully verified.
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Rule 3806
Rule 65
Rubi steps
\begin{align*} \int \csc ^n(c+d x) \sqrt{a+a \csc (c+d x)} \, dx &=\frac{\left (a^2 \cot (c+d x)\right ) \operatorname{Subst}\left (\int \frac{x^{-1+n}}{\sqrt{a-a x}} \, dx,x,\csc (c+d x)\right )}{d \sqrt{a-a \csc (c+d x)} \sqrt{a+a \csc (c+d x)}}\\ &=-\frac{2 a \cot (c+d x) \, _2F_1\left (\frac{1}{2},1-n;\frac{3}{2};1-\csc (c+d x)\right )}{d \sqrt{a+a \csc (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.169233, size = 48, normalized size = 1. \[ -\frac{2 a \cot (c+d x) \text{Hypergeometric2F1}\left (\frac{1}{2},1-n,\frac{3}{2},1-\csc (c+d x)\right )}{d \sqrt{a (\csc (c+d x)+1)}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.401, size = 0, normalized size = 0. \begin{align*} \int \left ( \csc \left ( dx+c \right ) \right ) ^{n}\sqrt{a+a\csc \left ( dx+c \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 \sqrt{a \csc \left (d x + c\right ) + a} \csc \left (d x + 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 (\sqrt{a \csc \left (d x + c\right ) + a} \csc \left (d x + 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 (\csc{\left (c + d x \right )} + 1\right )} \csc ^{n}{\left (c + d x \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 \sqrt{a \csc \left (d x + c\right ) + a} \csc \left (d x + c\right )^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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