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

Optimal result

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

[Out]

-b^3*(a*cos(f*x+e))^(1+m)*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)*(b*csc(f*x
+e))^(1/2)/a/f/(1+m)

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.97, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2667, 2656} \[ \int (a \cos (e+f x))^m (b \csc (e+f x))^{7/2} \, dx=-\frac {b \sin ^2(e+f x)^{5/4} (b \csc (e+f x))^{5/2} (a \cos (e+f x))^{m+1} \operatorname {Hypergeometric2F1}\left (\frac {9}{4},\frac {m+1}{2},\frac {m+3}{2},\cos ^2(e+f x)\right )}{a f (m+1)} \]

[In]

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

[Out]

-((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)))

Rule 2656

Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b^(2*IntPar
t[(n - 1)/2] + 1))*(b*Sin[e + f*x])^(2*FracPart[(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, Cos[e + f*x]^2], x] /; FreeQ[{a
, b, e, f, m, n}, x] && SimplerQ[n, m]

Rule 2667

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

Rubi steps \begin{align*} \text {integral}& = \left (b^2 (b \csc (e+f x))^{5/2} (b \sin (e+f x))^{5/2}\right ) \int \frac {(a \cos (e+f x))^m}{(b \sin (e+f x))^{7/2}} \, dx \\ & = -\frac {b (a \cos (e+f x))^{1+m} (b \csc (e+f x))^{5/2} \operatorname {Hypergeometric2F1}\left (\frac {9}{4},\frac {1+m}{2},\frac {3+m}{2},\cos ^2(e+f x)\right ) \sin ^2(e+f x)^{5/4}}{a f (1+m)} \\ \end{align*}

Mathematica [A] (verified)

Time = 16.30 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.21 \[ \int (a \cos (e+f x))^m (b \csc (e+f x))^{7/2} \, dx=\frac {2 a b (a \cos (e+f x))^{-1+m} \left (-\cot ^2(e+f x)\right )^{\frac {1-m}{2}} (b \csc (e+f x))^{5/2} \operatorname {Hypergeometric2F1}\left (\frac {1}{4} (7-2 m),\frac {1-m}{2},\frac {1}{4} (11-2 m),\csc ^2(e+f x)\right )}{f (-7+2 m)} \]

[In]

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

[Out]

(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))

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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