Integrand size = 19, antiderivative size = 88 \[ \int \cos ^m(e+f x) (b \csc (e+f x))^n \, dx=\frac {b \cos ^{-1+m}(e+f x) \cos ^2(e+f x)^{\frac {1-m}{2}} (b \csc (e+f x))^{-1+n} \operatorname {Hypergeometric2F1}\left (\frac {1-m}{2},\frac {1-n}{2},\frac {3-n}{2},\sin ^2(e+f x)\right )}{f (1-n)} \]
[Out]
Time = 0.15 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2667, 2657} \[ \int \cos ^m(e+f x) (b \csc (e+f x))^n \, dx=\frac {b \cos ^{m-1}(e+f x) \cos ^2(e+f x)^{\frac {1-m}{2}} (b \csc (e+f x))^{n-1} \operatorname {Hypergeometric2F1}\left (\frac {1-m}{2},\frac {1-n}{2},\frac {3-n}{2},\sin ^2(e+f x)\right )}{f (1-n)} \]
[In]
[Out]
Rule 2657
Rule 2667
Rubi steps \begin{align*} \text {integral}& = \left (b^2 (b \csc (e+f x))^{-1+n} (b \sin (e+f x))^{-1+n}\right ) \int \cos ^m(e+f x) (b \sin (e+f x))^{-n} \, dx \\ & = \frac {b \cos ^{-1+m}(e+f x) \cos ^2(e+f x)^{\frac {1-m}{2}} (b \csc (e+f x))^{-1+n} \operatorname {Hypergeometric2F1}\left (\frac {1-m}{2},\frac {1-n}{2},\frac {3-n}{2},\sin ^2(e+f x)\right )}{f (1-n)} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.
Time = 1.01 (sec) , antiderivative size = 314, normalized size of antiderivative = 3.57 \[ \int \cos ^m(e+f x) (b \csc (e+f x))^n \, dx=-\frac {2 (-3+n) \operatorname {AppellF1}\left (\frac {1}{2}-\frac {n}{2},-m,1+m-n,\frac {3}{2}-\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \cos ^3\left (\frac {1}{2} (e+f x)\right ) \cos ^m(e+f x) (b \csc (e+f x))^n \sin \left (\frac {1}{2} (e+f x)\right )}{f (-1+n) \left ((-3+n) \operatorname {AppellF1}\left (\frac {1}{2}-\frac {n}{2},-m,1+m-n,\frac {3}{2}-\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \cos ^2\left (\frac {1}{2} (e+f x)\right )+2 \left (m \operatorname {AppellF1}\left (\frac {3}{2}-\frac {n}{2},1-m,1+m-n,\frac {5}{2}-\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )+(1+m-n) \operatorname {AppellF1}\left (\frac {3}{2}-\frac {n}{2},-m,2+m-n,\frac {5}{2}-\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )\right ) \sin ^2\left (\frac {1}{2} (e+f x)\right )\right )} \]
[In]
[Out]
\[\int \left (\cos ^{m}\left (f x +e \right )\right ) \left (b \csc \left (f x +e \right )\right )^{n}d x\]
[In]
[Out]
\[ \int \cos ^m(e+f x) (b \csc (e+f x))^n \, dx=\int { \left (b \csc \left (f x + e\right )\right )^{n} \cos \left (f x + e\right )^{m} \,d x } \]
[In]
[Out]
\[ \int \cos ^m(e+f x) (b \csc (e+f x))^n \, dx=\int \left (b \csc {\left (e + f x \right )}\right )^{n} \cos ^{m}{\left (e + f x \right )}\, dx \]
[In]
[Out]
\[ \int \cos ^m(e+f x) (b \csc (e+f x))^n \, dx=\int { \left (b \csc \left (f x + e\right )\right )^{n} \cos \left (f x + e\right )^{m} \,d x } \]
[In]
[Out]
\[ \int \cos ^m(e+f x) (b \csc (e+f x))^n \, dx=\int { \left (b \csc \left (f x + e\right )\right )^{n} \cos \left (f x + e\right )^{m} \,d x } \]
[In]
[Out]
Timed out. \[ \int \cos ^m(e+f x) (b \csc (e+f x))^n \, dx=\int {\cos \left (e+f\,x\right )}^m\,{\left (\frac {b}{\sin \left (e+f\,x\right )}\right )}^n \,d x \]
[In]
[Out]