Integrand size = 24, antiderivative size = 69 \[ \int \csc ^n(c+d x) \sqrt {a-a \csc (c+d x)} \, dx=-\frac {2 a \cos (c+d x) (-\csc (c+d x))^{-n} \csc ^{1+n}(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},1-n,\frac {3}{2},1+\csc (c+d x)\right )}{d \sqrt {a-a \csc (c+d x)}} \]
[Out]
Time = 0.09 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3891, 69, 67} \[ \int \csc ^n(c+d x) \sqrt {a-a \csc (c+d x)} \, dx=-\frac {2 a \cos (c+d x) (-\csc (c+d x))^{-n} \csc ^{n+1}(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},1-n,\frac {3}{2},\csc (c+d x)+1\right )}{d \sqrt {a-a \csc (c+d x)}} \]
[In]
[Out]
Rule 67
Rule 69
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 {\left (a^2 \cos (c+d x) (-\csc (c+d x))^{-n} \csc ^{1+n}(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 \cos (c+d x) (-\csc (c+d x))^{-n} \csc ^{1+n}(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 = 2.90 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.06 \[ \int \csc ^n(c+d x) \sqrt {a-a \csc (c+d x)} \, dx=-\frac {2 a \cos (c+d x) \csc ^{1+2 n}(c+d x) \left (-\csc ^2(c+d x)\right )^{-n} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},1-n,\frac {3}{2},1+\csc (c+d x)\right )}{d \sqrt {a-a \csc (c+d x)}} \]
[In]
[Out]
\[\int \csc \left (d x +c \right )^{n} \sqrt {a -a \csc \left (d x +c \right )}d x\]
[In]
[Out]
\[ \int \csc ^n(c+d x) \sqrt {a-a \csc (c+d x)} \, dx=\int { \sqrt {-a \csc \left (d x + c\right ) + a} \csc \left (d x + c\right )^{n} \,d x } \]
[In]
[Out]
\[ \int \csc ^n(c+d x) \sqrt {a-a \csc (c+d x)} \, dx=\int \sqrt {- a \left (\csc {\left (c + d x \right )} - 1\right )} \csc ^{n}{\left (c + d x \right )}\, dx \]
[In]
[Out]
\[ \int \csc ^n(c+d x) \sqrt {a-a \csc (c+d x)} \, dx=\int { \sqrt {-a \csc \left (d x + c\right ) + a} \csc \left (d x + c\right )^{n} \,d x } \]
[In]
[Out]
\[ \int \csc ^n(c+d x) \sqrt {a-a \csc (c+d x)} \, dx=\int { \sqrt {-a \csc \left (d x + c\right ) + a} \csc \left (d x + c\right )^{n} \,d x } \]
[In]
[Out]
Timed out. \[ \int \csc ^n(c+d x) \sqrt {a-a \csc (c+d x)} \, dx=\int \sqrt {a-\frac {a}{\sin \left (c+d\,x\right )}}\,{\left (\frac {1}{\sin \left (c+d\,x\right )}\right )}^n \,d x \]
[In]
[Out]