Integrand size = 23, antiderivative size = 78 \[ \int \frac {(a \cos (e+f x))^m}{\sqrt {b \csc (e+f x)}} \, dx=-\frac {(a \cos (e+f x))^{1+m} \sqrt {b \csc (e+f x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1+m}{2},\frac {3+m}{2},\cos ^2(e+f x)\right ) \sqrt [4]{\sin ^2(e+f x)}}{a b f (1+m)} \]
-(a*cos(f*x+e))^(1+m)*hypergeom([1/4, 1/2+1/2*m],[3/2+1/2*m],cos(f*x+e)^2) *(sin(f*x+e)^2)^(1/4)*(b*csc(f*x+e))^(1/2)/a/b/f/(1+m)
Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.
Time = 28.33 (sec) , antiderivative size = 225, normalized size of antiderivative = 2.88 \[ \int \frac {(a \cos (e+f x))^m}{\sqrt {b \csc (e+f x)}} \, dx=\frac {14 b \operatorname {AppellF1}\left (\frac {3}{4},-m,\frac {3}{2}+m,\frac {7}{4},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) (a \cos (e+f x))^m}{3 f (b \csc (e+f x))^{3/2} \left (7 \operatorname {AppellF1}\left (\frac {3}{4},-m,\frac {3}{2}+m,\frac {7}{4},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )-2 \left (2 m \operatorname {AppellF1}\left (\frac {7}{4},1-m,\frac {3}{2}+m,\frac {11}{4},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )+(3+2 m) \operatorname {AppellF1}\left (\frac {7}{4},-m,\frac {5}{2}+m,\frac {11}{4},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )\right ) \tan ^2\left (\frac {1}{2} (e+f x)\right )\right )} \]
(14*b*AppellF1[3/4, -m, 3/2 + m, 7/4, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2 ]^2]*(a*Cos[e + f*x])^m)/(3*f*(b*Csc[e + f*x])^(3/2)*(7*AppellF1[3/4, -m, 3/2 + m, 7/4, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - 2*(2*m*AppellF1[7 /4, 1 - m, 3/2 + m, 11/4, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (3 + 2*m)*AppellF1[7/4, -m, 5/2 + m, 11/4, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2 ]^2])*Tan[(e + f*x)/2]^2))
Time = 0.33 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3042, 3066, 3042, 3056}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {(a \cos (e+f x))^m}{\sqrt {b \csc (e+f x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\left (a \sin \left (e+f x+\frac {\pi }{2}\right )\right )^m}{\sqrt {-b \sec \left (e+f x+\frac {\pi }{2}\right )}}dx\) |
\(\Big \downarrow \) 3066 |
\(\displaystyle \frac {\sqrt {b \sin (e+f x)} \sqrt {b \csc (e+f x)} \int (a \cos (e+f x))^m \sqrt {b \sin (e+f x)}dx}{b^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sqrt {b \sin (e+f x)} \sqrt {b \csc (e+f x)} \int (a \cos (e+f x))^m \sqrt {b \sin (e+f x)}dx}{b^2}\) |
\(\Big \downarrow \) 3056 |
\(\displaystyle -\frac {\sqrt [4]{\sin ^2(e+f x)} \sqrt {b \csc (e+f x)} (a \cos (e+f x))^{m+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {m+1}{2},\frac {m+3}{2},\cos ^2(e+f x)\right )}{a b f (m+1)}\) |
-(((a*Cos[e + f*x])^(1 + m)*Sqrt[b*Csc[e + f*x]]*Hypergeometric2F1[1/4, (1 + m)/2, (3 + m)/2, Cos[e + f*x]^2]*(Sin[e + f*x]^2)^(1/4))/(a*b*f*(1 + m) ))
3.3.92.3.1 Defintions of rubi rules used
Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n _), x_Symbol] :> Simp[(-b^(2*IntPart[(n - 1)/2] + 1))*(b*Sin[e + f*x])^(2*F racPart[(n - 1)/2])*((a*Cos[e + f*x])^(m + 1)/(a*f*(m + 1)*(Sin[e + f*x]^2) ^FracPart[(n - 1)/2]))*Hypergeometric2F1[(1 + m)/2, (1 - n)/2, (3 + m)/2, C os[e + f*x]^2], x] /; FreeQ[{a, b, e, f, m, n}, x] && SimplerQ[n, m]
Int[((b_.)*sec[(e_.) + (f_.)*(x_)])^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m _), x_Symbol] :> Simp[(1/b^2)*(b*Cos[e + f*x])^(n + 1)*(b*Sec[e + f*x])^(n + 1) Int[(a*Sin[e + f*x])^m/(b*Cos[e + f*x])^n, x], x] /; FreeQ[{a, b, e, f, m, n}, x] && !IntegerQ[m] && !IntegerQ[n] && LtQ[n, 1]
\[\int \frac {\left (\cos \left (f x +e \right ) a \right )^{m}}{\sqrt {b \csc \left (f x +e \right )}}d x\]
\[ \int \frac {(a \cos (e+f x))^m}{\sqrt {b \csc (e+f x)}} \, dx=\int { \frac {\left (a \cos \left (f x + e\right )\right )^{m}}{\sqrt {b \csc \left (f x + e\right )}} \,d x } \]
\[ \int \frac {(a \cos (e+f x))^m}{\sqrt {b \csc (e+f x)}} \, dx=\int \frac {\left (a \cos {\left (e + f x \right )}\right )^{m}}{\sqrt {b \csc {\left (e + f x \right )}}}\, dx \]
\[ \int \frac {(a \cos (e+f x))^m}{\sqrt {b \csc (e+f x)}} \, dx=\int { \frac {\left (a \cos \left (f x + e\right )\right )^{m}}{\sqrt {b \csc \left (f x + e\right )}} \,d x } \]
\[ \int \frac {(a \cos (e+f x))^m}{\sqrt {b \csc (e+f x)}} \, dx=\int { \frac {\left (a \cos \left (f x + e\right )\right )^{m}}{\sqrt {b \csc \left (f x + e\right )}} \,d x } \]
Timed out. \[ \int \frac {(a \cos (e+f x))^m}{\sqrt {b \csc (e+f x)}} \, dx=\int \frac {{\left (a\,\cos \left (e+f\,x\right )\right )}^m}{\sqrt {\frac {b}{\sin \left (e+f\,x\right )}}} \,d x \]