Integrand size = 23, antiderivative size = 78 \[ \int \frac {(a \cos (e+f x))^m}{(b \csc (e+f x))^{3/2}} \, dx=-\frac {(a \cos (e+f x))^{1+m} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1+m}{2},\frac {3+m}{2},\cos ^2(e+f x)\right )}{a b f (1+m) \sqrt {b \csc (e+f x)} \sqrt [4]{\sin ^2(e+f x)}} \]
[Out]
Time = 0.13 (sec) , antiderivative size = 78, 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 = {2666, 2656} \[ \int \frac {(a \cos (e+f x))^m}{(b \csc (e+f x))^{3/2}} \, dx=-\frac {(a \cos (e+f x))^{m+1} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {m+1}{2},\frac {m+3}{2},\cos ^2(e+f x)\right )}{a b f (m+1) \sqrt [4]{\sin ^2(e+f x)} \sqrt {b \csc (e+f x)}} \]
[In]
[Out]
Rule 2656
Rule 2666
Rubi steps \begin{align*} \text {integral}& = \frac {\int (a \cos (e+f x))^m (b \sin (e+f x))^{3/2} \, dx}{b^2 \sqrt {b \csc (e+f x)} \sqrt {b \sin (e+f x)}} \\ & = -\frac {(a \cos (e+f x))^{1+m} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1+m}{2},\frac {3+m}{2},\cos ^2(e+f x)\right )}{a b f (1+m) \sqrt {b \csc (e+f x)} \sqrt [4]{\sin ^2(e+f x)}} \\ \end{align*}
Time = 11.13 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.49 \[ \int \frac {(a \cos (e+f x))^m}{(b \csc (e+f x))^{3/2}} \, dx=\frac {2 a (a \cos (e+f x))^{-1+m} \cos (2 (e+f x)) \left (-\cot ^2(e+f x)\right )^{\frac {1-m}{2}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4} (-3-2 m),\frac {1-m}{2},\frac {1}{4} (1-2 m),\csc ^2(e+f x)\right )}{b f (3+2 m) \sqrt {b \csc (e+f x)} \left (-2+\csc ^2(e+f x)\right )} \]
[In]
[Out]
\[\int \frac {\left (\cos \left (f x +e \right ) a \right )^{m}}{\left (b \csc \left (f x +e \right )\right )^{\frac {3}{2}}}d x\]
[In]
[Out]
\[ \int \frac {(a \cos (e+f x))^m}{(b \csc (e+f x))^{3/2}} \, dx=\int { \frac {\left (a \cos \left (f x + e\right )\right )^{m}}{\left (b \csc \left (f x + e\right )\right )^{\frac {3}{2}}} \,d x } \]
[In]
[Out]
\[ \int \frac {(a \cos (e+f x))^m}{(b \csc (e+f x))^{3/2}} \, dx=\int \frac {\left (a \cos {\left (e + f x \right )}\right )^{m}}{\left (b \csc {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]
[In]
[Out]
\[ \int \frac {(a \cos (e+f x))^m}{(b \csc (e+f x))^{3/2}} \, dx=\int { \frac {\left (a \cos \left (f x + e\right )\right )^{m}}{\left (b \csc \left (f x + e\right )\right )^{\frac {3}{2}}} \,d x } \]
[In]
[Out]
\[ \int \frac {(a \cos (e+f x))^m}{(b \csc (e+f x))^{3/2}} \, dx=\int { \frac {\left (a \cos \left (f x + e\right )\right )^{m}}{\left (b \csc \left (f x + e\right )\right )^{\frac {3}{2}}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {(a \cos (e+f x))^m}{(b \csc (e+f x))^{3/2}} \, dx=\int \frac {{\left (a\,\cos \left (e+f\,x\right )\right )}^m}{{\left (\frac {b}{\sin \left (e+f\,x\right )}\right )}^{3/2}} \,d x \]
[In]
[Out]