Integrand size = 23, antiderivative size = 48 \[ \int \csc ^n(c+d x) \sqrt {a+a \csc (c+d x)} \, dx=-\frac {2 a \cot (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},1-n,\frac {3}{2},1-\csc (c+d x)\right )}{d \sqrt {a+a \csc (c+d x)}} \]
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Time = 0.08 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {3891, 67} \[ \int \csc ^n(c+d x) \sqrt {a+a \csc (c+d x)} \, dx=-\frac {2 a \cot (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},1-n,\frac {3}{2},1-\csc (c+d x)\right )}{d \sqrt {a \csc (c+d x)+a}} \]
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Rule 67
Rule 3891
Rubi steps \begin{align*} \text {integral}& = \frac {\left (a^2 \cot (c+d x)\right ) \text {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) \operatorname {Hypergeometric2F1}\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*}
Time = 1.18 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00 \[ \int \csc ^n(c+d x) \sqrt {a+a \csc (c+d x)} \, dx=-\frac {2 a \cot (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},1-n,\frac {3}{2},1-\csc (c+d x)\right )}{d \sqrt {a (1+\csc (c+d x))}} \]
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\[\int \csc \left (d x +c \right )^{n} \sqrt {a +a \csc \left (d x +c \right )}d x\]
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\[ \int \csc ^n(c+d x) \sqrt {a+a \csc (c+d x)} \, dx=\int { \sqrt {a \csc \left (d x + c\right ) + a} \csc \left (d x + c\right )^{n} \,d x } \]
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\[ \int \csc ^n(c+d x) \sqrt {a+a \csc (c+d x)} \, dx=\int \sqrt {a \left (\csc {\left (c + d x \right )} + 1\right )} \csc ^{n}{\left (c + d x \right )}\, dx \]
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\[ \int \csc ^n(c+d x) \sqrt {a+a \csc (c+d x)} \, dx=\int { \sqrt {a \csc \left (d x + c\right ) + a} \csc \left (d x + c\right )^{n} \,d x } \]
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\[ \int \csc ^n(c+d x) \sqrt {a+a \csc (c+d x)} \, dx=\int { \sqrt {a \csc \left (d x + c\right ) + a} \csc \left (d x + c\right )^{n} \,d x } \]
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Timed out. \[ \int \csc ^n(c+d x) \sqrt {a+a \csc (c+d x)} \, dx=\int \sqrt {a+\frac {a}{\sin \left (c+d\,x\right )}}\,{\left (\frac {1}{\sin \left (c+d\,x\right )}\right )}^n \,d x \]
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