\(\int (a \cos (e+f x))^m (b \csc (e+f x))^{7/2} \, dx\) [288]

Optimal result

Integrand size = 23, antiderivative size = 78 \[ \int (a \cos (e+f x))^m (b \csc (e+f x))^{7/2} \, dx=-\frac {b^3 (a \cos (e+f x))^{1+m} \sqrt {b \csc (e+f x)} \operatorname {Hypergeometric2F1}\left (\frac {9}{4},\frac {1+m}{2},\frac {3+m}{2},\cos ^2(e+f x)\right ) \sqrt [4]{\sin ^2(e+f x)}}{a f (1+m)} \] Output:

-b^3*(a*cos(f*x+e))^(1+m)*(b*csc(f*x+e))^(1/2)*hypergeom([9/4, 1/2+1/2*m], 
[3/2+1/2*m],cos(f*x+e)^2)*(sin(f*x+e)^2)^(1/4)/a/f/(1+m)
 

Mathematica [A] (verified)

Time = 17.40 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.21 \[ \int (a \cos (e+f x))^m (b \csc (e+f x))^{7/2} \, dx=\frac {2 a b (a \cos (e+f x))^{-1+m} \left (-\cot ^2(e+f x)\right )^{\frac {1-m}{2}} (b \csc (e+f x))^{5/2} \operatorname {Hypergeometric2F1}\left (\frac {1}{4} (7-2 m),\frac {1-m}{2},\frac {1}{4} (11-2 m),\csc ^2(e+f x)\right )}{f (-7+2 m)} \] Input:

Integrate[(a*Cos[e + f*x])^m*(b*Csc[e + f*x])^(7/2),x]
 

Output:

(2*a*b*(a*Cos[e + f*x])^(-1 + m)*(-Cot[e + f*x]^2)^((1 - m)/2)*(b*Csc[e + 
f*x])^(5/2)*Hypergeometric2F1[(7 - 2*m)/4, (1 - m)/2, (11 - 2*m)/4, Csc[e 
+ f*x]^2])/(f*(-7 + 2*m))
 

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.97, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3042, 3067, 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.

\(\displaystyle \int (b \csc (e+f x))^{7/2} (a \cos (e+f x))^m \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \left (-b \sec \left (e+f x+\frac {\pi }{2}\right )\right )^{7/2} \left (a \sin \left (e+f x+\frac {\pi }{2}\right )\right )^mdx\)

\(\Big \downarrow \) 3067

\(\displaystyle b^2 (b \sin (e+f x))^{5/2} (b \csc (e+f x))^{5/2} \int \frac {(a \cos (e+f x))^m}{(b \sin (e+f x))^{7/2}}dx\)

\(\Big \downarrow \) 3042

\(\displaystyle b^2 (b \sin (e+f x))^{5/2} (b \csc (e+f x))^{5/2} \int \frac {(a \cos (e+f x))^m}{(b \sin (e+f x))^{7/2}}dx\)

\(\Big \downarrow \) 3056

\(\displaystyle -\frac {b \sin ^2(e+f x)^{5/4} (b \csc (e+f x))^{5/2} (a \cos (e+f x))^{m+1} \operatorname {Hypergeometric2F1}\left (\frac {9}{4},\frac {m+1}{2},\frac {m+3}{2},\cos ^2(e+f x)\right )}{a f (m+1)}\)

Input:

Int[(a*Cos[e + f*x])^m*(b*Csc[e + f*x])^(7/2),x]
 

Output:

-((b*(a*Cos[e + f*x])^(1 + m)*(b*Csc[e + f*x])^(5/2)*Hypergeometric2F1[9/4 
, (1 + m)/2, (3 + m)/2, Cos[e + f*x]^2]*(Sin[e + f*x]^2)^(5/4))/(a*f*(1 + 
m)))
 

Defintions of rubi rules used

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3056

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]
 

rule 3067

Int[((b_.)*sec[(e_.) + (f_.)*(x_)])^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m 
_), x_Symbol] :> Simp[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]
 

Maple [F]

\[\int \left (\cos \left (f x +e \right ) a \right )^{m} \left (b \csc \left (f x +e \right )\right )^{\frac {7}{2}}d x\]

Input:

int((cos(f*x+e)*a)^m*(b*csc(f*x+e))^(7/2),x)
 

Output:

int((cos(f*x+e)*a)^m*(b*csc(f*x+e))^(7/2),x)
 

Fricas [F]

\[ \int (a \cos (e+f x))^m (b \csc (e+f x))^{7/2} \, dx=\int { \left (b \csc \left (f x + e\right )\right )^{\frac {7}{2}} \left (a \cos \left (f x + e\right )\right )^{m} \,d x } \] Input:

integrate((a*cos(f*x+e))^m*(b*csc(f*x+e))^(7/2),x, algorithm="fricas")
 

Output:

integral(sqrt(b*csc(f*x + e))*(a*cos(f*x + e))^m*b^3*csc(f*x + e)^3, x)
 

Sympy [F(-1)]

Timed out. \[ \int (a \cos (e+f x))^m (b \csc (e+f x))^{7/2} \, dx=\text {Timed out} \] Input:

integrate((a*cos(f*x+e))**m*(b*csc(f*x+e))**(7/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int (a \cos (e+f x))^m (b \csc (e+f x))^{7/2} \, dx=\int { \left (b \csc \left (f x + e\right )\right )^{\frac {7}{2}} \left (a \cos \left (f x + e\right )\right )^{m} \,d x } \] Input:

integrate((a*cos(f*x+e))^m*(b*csc(f*x+e))^(7/2),x, algorithm="maxima")
 

Output:

integrate((b*csc(f*x + e))^(7/2)*(a*cos(f*x + e))^m, x)
 

Giac [F]

\[ \int (a \cos (e+f x))^m (b \csc (e+f x))^{7/2} \, dx=\int { \left (b \csc \left (f x + e\right )\right )^{\frac {7}{2}} \left (a \cos \left (f x + e\right )\right )^{m} \,d x } \] Input:

integrate((a*cos(f*x+e))^m*(b*csc(f*x+e))^(7/2),x, algorithm="giac")
 

Output:

integrate((b*csc(f*x + e))^(7/2)*(a*cos(f*x + e))^m, x)
 

Mupad [F(-1)]

Timed out. \[ \int (a \cos (e+f x))^m (b \csc (e+f x))^{7/2} \, dx=\int {\left (a\,\cos \left (e+f\,x\right )\right )}^m\,{\left (\frac {b}{\sin \left (e+f\,x\right )}\right )}^{7/2} \,d x \] Input:

int((a*cos(e + f*x))^m*(b/sin(e + f*x))^(7/2),x)
 

Output:

int((a*cos(e + f*x))^m*(b/sin(e + f*x))^(7/2), x)
 

Reduce [F]

\[ \int (a \cos (e+f x))^m (b \csc (e+f x))^{7/2} \, dx=\sqrt {b}\, a^{m} \left (\int \sqrt {\csc \left (f x +e \right )}\, \cos \left (f x +e \right )^{m} \csc \left (f x +e \right )^{3}d x \right ) b^{3} \] Input:

int((a*cos(f*x+e))^m*(b*csc(f*x+e))^(7/2),x)
 

Output:

sqrt(b)*a**m*int(sqrt(csc(e + f*x))*cos(e + f*x)**m*csc(e + f*x)**3,x)*b** 
3