\(\int \csc ^n(e+f x) \sec ^m(e+f x) \, dx\) [280]

Optimal result

Integrand size = 17, antiderivative size = 81 \[ \int \csc ^n(e+f x) \sec ^m(e+f x) \, dx=\frac {\cos ^2(e+f x)^{\frac {1+m}{2}} \csc ^{-1+n}(e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1+m}{2},\frac {1-n}{2},\frac {3-n}{2},\sin ^2(e+f x)\right ) \sec ^{1+m}(e+f x)}{f (1-n)} \] 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)*sec(f*x+e)^(1+m)/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 = 1.76 (sec) , antiderivative size = 278, normalized size of antiderivative = 3.43 \[ \int \csc ^n(e+f x) \sec ^m(e+f x) \, dx=-\frac {(-3+n) \operatorname {AppellF1}\left (\frac {1}{2}-\frac {n}{2},m,1-m-n,\frac {3}{2}-\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \csc ^{-1+n}(e+f x) \sec ^m(e+f x)}{f (-1+n) \left ((-3+n) \operatorname {AppellF1}\left (\frac {1}{2}-\frac {n}{2},m,1-m-n,\frac {3}{2}-\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )-2 \left ((-1+m+n) \operatorname {AppellF1}\left (\frac {3}{2}-\frac {n}{2},m,2-m-n,\frac {5}{2}-\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )+m \operatorname {AppellF1}\left (\frac {3}{2}-\frac {n}{2},1+m,1-m-n,\frac {5}{2}-\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )\right ) \tan ^2\left (\frac {1}{2} (e+f x)\right )\right )} \] Input:

Integrate[Csc[e + f*x]^n*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)*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])*T 
an[(e + f*x)/2]^2)))
 

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, 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.

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

\(\Big \downarrow \) 3042

\(\displaystyle \int \sec (e+f x)^m \csc (e+f x)^ndx\)

\(\Big \downarrow \) 3111

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3057

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

Input:

Int[Csc[e + f*x]^n*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]*Sec[e + f*x]^(1 + m))/(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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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