\(\int \csc ^n(e+f x) (a \sec (e+f x))^m \, dx\) [281]

Optimal result

Integrand size = 19, antiderivative size = 86 \[ \int \csc ^n(e+f x) (a \sec (e+f x))^m \, dx=\frac {\cos ^2(e+f x)^{\frac {1+m}{2}} \csc ^{-1+n}(e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1+m}{2},\frac {1-n}{2},\frac {3-n}{2},\sin ^2(e+f x)\right ) (a \sec (e+f x))^{1+m}}{a f (1-n)} \] Output:

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

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.

Time = 0.84 (sec) , antiderivative size = 280, normalized size of antiderivative = 3.26 \[ \int \csc ^n(e+f x) (a \sec (e+f x))^m \, dx=-\frac {(-3+n) \operatorname {AppellF1}\left (\frac {1}{2}-\frac {n}{2},m,1-m-n,\frac {3}{2}-\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \csc ^{-1+n}(e+f x) (a \sec (e+f x))^m}{f (-1+n) \left ((-3+n) \operatorname {AppellF1}\left (\frac {1}{2}-\frac {n}{2},m,1-m-n,\frac {3}{2}-\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )-2 \left ((-1+m+n) \operatorname {AppellF1}\left (\frac {3}{2}-\frac {n}{2},m,2-m-n,\frac {5}{2}-\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )+m \operatorname {AppellF1}\left (\frac {3}{2}-\frac {n}{2},1+m,1-m-n,\frac {5}{2}-\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )\right ) \tan ^2\left (\frac {1}{2} (e+f x)\right )\right )} \] Input:

Integrate[Csc[e + f*x]^n*(a*Sec[e + f*x])^m,x]
 

Output:

-(((-3 + n)*AppellF1[1/2 - n/2, m, 1 - m - n, 3/2 - n/2, Tan[(e + f*x)/2]^ 
2, -Tan[(e + f*x)/2]^2]*Csc[e + f*x]^(-1 + n)*(a*Sec[e + f*x])^m)/(f*(-1 + 
 n)*((-3 + n)*AppellF1[1/2 - n/2, m, 1 - m - n, 3/2 - n/2, Tan[(e + f*x)/2 
]^2, -Tan[(e + f*x)/2]^2] - 2*((-1 + m + n)*AppellF1[3/2 - n/2, m, 2 - m - 
 n, 5/2 - n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + m*AppellF1[3/2 - 
 n/2, 1 + m, 1 - m - n, 5/2 - n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2 
])*Tan[(e + f*x)/2]^2)))
 

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {3042, 3111, 3042, 3057}

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.

\(\displaystyle \int \csc ^n(e+f x) (a \sec (e+f x))^m \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \csc (e+f x)^n (a \sec (e+f x))^mdx\)

\(\Big \downarrow \) 3111

\(\displaystyle \frac {\sin ^{n-1}(e+f x) \csc ^{n-1}(e+f x) (a \cos (e+f x))^{m+1} (a \sec (e+f x))^{m+1} \int (a \cos (e+f x))^{-m} \sin ^{-n}(e+f x)dx}{a^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\sin ^{n-1}(e+f x) \csc ^{n-1}(e+f x) (a \cos (e+f x))^{m+1} (a \sec (e+f x))^{m+1} \int (a \cos (e+f x))^{-m} \sin (e+f x)^{-n}dx}{a^2}\)

\(\Big \downarrow \) 3057

\(\displaystyle \frac {\cos ^2(e+f x)^{\frac {m+1}{2}} \csc ^{n-1}(e+f x) (a \sec (e+f x))^{m+1} \operatorname {Hypergeometric2F1}\left (\frac {m+1}{2},\frac {1-n}{2},\frac {3-n}{2},\sin ^2(e+f x)\right )}{a f (1-n)}\)

Input:

Int[Csc[e + f*x]^n*(a*Sec[e + f*x])^m,x]
 

Output:

((Cos[e + f*x]^2)^((1 + m)/2)*Csc[e + f*x]^(-1 + n)*Hypergeometric2F1[(1 + 
 m)/2, (1 - n)/2, (3 - n)/2, Sin[e + f*x]^2]*(a*Sec[e + f*x])^(1 + m))/(a* 
f*(1 - n))
 

Defintions of rubi rules used

rule 3042

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

rule 3057

Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m 
_), x_Symbol] :> Simp[b^(2*IntPart[(n - 1)/2] + 1)*(b*Cos[e + f*x])^(2*Frac 
Part[(n - 1)/2])*((a*Sin[e + f*x])^(m + 1)/(a*f*(m + 1)*(Cos[e + f*x]^2)^Fr 
acPart[(n - 1)/2]))*Hypergeometric2F1[(1 + m)/2, (1 - n)/2, (3 + m)/2, Sin[ 
e + f*x]^2], x] /; FreeQ[{a, b, e, f, m, n}, x]
 

rule 3111

Int[(csc[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*sec[(e_.) + (f_.)*(x_)])^(n 
_), x_Symbol] :> Simp[(a^2/b^2)*(a*Csc[e + f*x])^(m - 1)*(b*Sec[e + f*x])^( 
n + 1)*(a*Sin[e + f*x])^(m - 1)*(b*Cos[e + f*x])^(n + 1)   Int[1/((a*Sin[e 
+ f*x])^m*(b*Cos[e + f*x])^n), x], x] /; FreeQ[{a, b, e, f, m, n}, x] &&  ! 
SimplerQ[-m, -n]
 

Maple [F]

\[\int \csc \left (f x +e \right )^{n} \left (a \sec \left (f x +e \right )\right )^{m}d x\]

Input:

int(csc(f*x+e)^n*(a*sec(f*x+e))^m,x)
 

Output:

int(csc(f*x+e)^n*(a*sec(f*x+e))^m,x)
 

Fricas [F]

\[ \int \csc ^n(e+f x) (a \sec (e+f x))^m \, dx=\int { \left (a \sec \left (f x + e\right )\right )^{m} \csc \left (f x + e\right )^{n} \,d x } \] Input:

integrate(csc(f*x+e)^n*(a*sec(f*x+e))^m,x, algorithm="fricas")
 

Output:

integral((a*sec(f*x + e))^m*csc(f*x + e)^n, x)
 

Sympy [F]

\[ \int \csc ^n(e+f x) (a \sec (e+f x))^m \, dx=\int \left (a \sec {\left (e + f x \right )}\right )^{m} \csc ^{n}{\left (e + f x \right )}\, dx \] Input:

integrate(csc(f*x+e)**n*(a*sec(f*x+e))**m,x)
 

Output:

Integral((a*sec(e + f*x))**m*csc(e + f*x)**n, x)
 

Maxima [F]

\[ \int \csc ^n(e+f x) (a \sec (e+f x))^m \, dx=\int { \left (a \sec \left (f x + e\right )\right )^{m} \csc \left (f x + e\right )^{n} \,d x } \] Input:

integrate(csc(f*x+e)^n*(a*sec(f*x+e))^m,x, algorithm="maxima")
 

Output:

integrate((a*sec(f*x + e))^m*csc(f*x + e)^n, x)
 

Giac [F]

\[ \int \csc ^n(e+f x) (a \sec (e+f x))^m \, dx=\int { \left (a \sec \left (f x + e\right )\right )^{m} \csc \left (f x + e\right )^{n} \,d x } \] Input:

integrate(csc(f*x+e)^n*(a*sec(f*x+e))^m,x, algorithm="giac")
 

Output:

integrate((a*sec(f*x + e))^m*csc(f*x + e)^n, x)
 

Mupad [F(-1)]

Timed out. \[ \int \csc ^n(e+f x) (a \sec (e+f x))^m \, dx=\int {\left (\frac {a}{\cos \left (e+f\,x\right )}\right )}^m\,{\left (\frac {1}{\sin \left (e+f\,x\right )}\right )}^n \,d x \] Input:

int((a/cos(e + f*x))^m*(1/sin(e + f*x))^n,x)
 

Output:

int((a/cos(e + f*x))^m*(1/sin(e + f*x))^n, x)
 

Reduce [F]

\[ \int \csc ^n(e+f x) (a \sec (e+f x))^m \, dx=a^{m} \left (\int \sec \left (f x +e \right )^{m} \csc \left (f x +e \right )^{n}d x \right ) \] Input:

int(csc(f*x+e)^n*(a*sec(f*x+e))^m,x)
 

Output:

a**m*int(sec(e + f*x)**m*csc(e + f*x)**n,x)