\(\int (a \cos (e+f x))^m (b \cot (e+f x))^n \, dx\) [40]

Optimal result

Integrand size = 21, antiderivative size = 84 \[ \int (a \cos (e+f x))^m (b \cot (e+f x))^n \, dx=-\frac {(a \cos (e+f x))^m (b \cot (e+f x))^{1+n} \operatorname {Hypergeometric2F1}\left (\frac {1+n}{2},\frac {1}{2} (1+m+n),\frac {1}{2} (3+m+n),\cos ^2(e+f x)\right ) \sin ^2(e+f x)^{\frac {1+n}{2}}}{b f (1+m+n)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2682, 2656} \[ \int (a \cos (e+f x))^m (b \cot (e+f x))^n \, dx=-\frac {\sin ^2(e+f x)^{\frac {n+1}{2}} (a \cos (e+f x))^m (b \cot (e+f x))^{n+1} \operatorname {Hypergeometric2F1}\left (\frac {n+1}{2},\frac {1}{2} (m+n+1),\frac {1}{2} (m+n+3),\cos ^2(e+f x)\right )}{b f (m+n+1)} \]

[In]

Int[(a*Cos[e + f*x])^m*(b*Cot[e + f*x])^n,x]

[Out]

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

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 2682

Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[a*Cos[e + f*
x]^(n + 1)*((b*Tan[e + f*x])^(n + 1)/(b*(a*Sin[e + f*x])^(n + 1))), Int[(a*Sin[e + f*x])^(m + n)/Cos[e + f*x]^
n, x], x] /; FreeQ[{a, b, e, f, m, n}, x] &&  !IntegerQ[n]

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

Mathematica [A] (verified)

Time = 0.93 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.99 \[ \int (a \cos (e+f x))^m (b \cot (e+f x))^n \, dx=-\frac {b (a \cos (e+f x))^m (b \cot (e+f x))^{-1+n} \operatorname {Hypergeometric2F1}\left (\frac {2+m}{2},\frac {1-n}{2},\frac {3-n}{2},-\tan ^2(e+f x)\right ) \sec ^2(e+f x)^{m/2}}{f (-1+n)} \]

[In]

Integrate[(a*Cos[e + f*x])^m*(b*Cot[e + f*x])^n,x]

[Out]

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

Maple [F]

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

[In]

int((a*cos(f*x+e))^m*(b*cot(f*x+e))^n,x)

[Out]

int((a*cos(f*x+e))^m*(b*cot(f*x+e))^n,x)

Fricas [F]

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

[In]

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

[Out]

integral((a*cos(f*x + e))^m*(b*cot(f*x + e))^n, x)

Sympy [F]

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

[In]

integrate((a*cos(f*x+e))**m*(b*cot(f*x+e))**n,x)

[Out]

Integral((a*cos(e + f*x))**m*(b*cot(e + f*x))**n, x)

Maxima [F]

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

[In]

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

[Out]

integrate((a*cos(f*x + e))^m*(b*cot(f*x + e))^n, x)

Giac [F]

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

[In]

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

[Out]

integrate((a*cos(f*x + e))^m*(b*cot(f*x + e))^n, x)

Mupad [F(-1)]

Timed out. \[ \int (a \cos (e+f x))^m (b \cot (e+f x))^n \, dx=\int {\left (a\,\cos \left (e+f\,x\right )\right )}^m\,{\left (b\,\mathrm {cot}\left (e+f\,x\right )\right )}^n \,d x \]

[In]

int((a*cos(e + f*x))^m*(b*cot(e + f*x))^n,x)

[Out]

int((a*cos(e + f*x))^m*(b*cot(e + f*x))^n, x)