\(\int \cos ^m(e+f x) \csc ^n(e+f x) \, dx\) [284]

Optimal result

Integrand size = 17, antiderivative size = 85 \[ \int \cos ^m(e+f x) \csc ^n(e+f x) \, dx=\frac {\cos ^{-1+m}(e+f x) \cos ^2(e+f x)^{\frac {1-m}{2}} \csc ^{-1+n}(e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1-m}{2},\frac {1-n}{2},\frac {3-n}{2},\sin ^2(e+f x)\right )}{f (1-n)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2667, 2657} \[ \int \cos ^m(e+f x) \csc ^n(e+f x) \, dx=\frac {\cos ^{m-1}(e+f x) \cos ^2(e+f x)^{\frac {1-m}{2}} \csc ^{n-1}(e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1-m}{2},\frac {1-n}{2},\frac {3-n}{2},\sin ^2(e+f x)\right )}{f (1-n)} \]

[In]

Int[Cos[e + f*x]^m*Csc[e + f*x]^n,x]

[Out]

(Cos[e + f*x]^(-1 + m)*(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])/(f*(1 - n))

Rule 2657

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*FracPart[(n - 1)/2])*((a*Sin[e + f*x])^(m + 1)/(a*f*(m + 1)*(Cos[e + f*x]^
2)^FracPart[(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 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 (\csc ^n(e+f x) \sin ^n(e+f x)\right ) \int \cos ^m(e+f x) \sin ^{-n}(e+f x) \, dx \\ & = \frac {\cos ^{-1+m}(e+f x) \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 )}{f (1-n)} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

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

Time = 2.39 (sec) , antiderivative size = 312, normalized size of antiderivative = 3.67 \[ \int \cos ^m(e+f x) \csc ^n(e+f x) \, dx=-\frac {2 (-3+n) \operatorname {AppellF1}\left (\frac {1}{2}-\frac {n}{2},-m,1+m-n,\frac {3}{2}-\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \cos ^3\left (\frac {1}{2} (e+f x)\right ) \cos ^m(e+f x) \csc ^n(e+f x) \sin \left (\frac {1}{2} (e+f x)\right )}{f (-1+n) \left ((-3+n) \operatorname {AppellF1}\left (\frac {1}{2}-\frac {n}{2},-m,1+m-n,\frac {3}{2}-\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \cos ^2\left (\frac {1}{2} (e+f x)\right )+2 \left (m \operatorname {AppellF1}\left (\frac {3}{2}-\frac {n}{2},1-m,1+m-n,\frac {5}{2}-\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )+(1+m-n) \operatorname {AppellF1}\left (\frac {3}{2}-\frac {n}{2},-m,2+m-n,\frac {5}{2}-\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )\right ) \sin ^2\left (\frac {1}{2} (e+f x)\right )\right )} \]

[In]

Integrate[Cos[e + f*x]^m*Csc[e + f*x]^n,x]

[Out]

(-2*(-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]*Cos[(e + f
*x)/2]^3*Cos[e + f*x]^m*Csc[e + f*x]^n*Sin[(e + f*x)/2])/(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]*Cos[(e + f*x)/2]^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] + (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])*Sin[(e + f*x)/2]^2))

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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