\(\int \cot ^4(e+f x) (b \sec (e+f x))^m \, dx\) [360]

Optimal result

Integrand size = 19, antiderivative size = 63 \[ \int \cot ^4(e+f x) (b \sec (e+f x))^m \, dx=-\frac {\cos ^2(e+f x)^{\frac {1}{2} (-3+m)} \cot ^3(e+f x) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1}{2} (-3+m),-\frac {1}{2},\sin ^2(e+f x)\right ) (b \sec (e+f x))^m}{3 f} \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2697} \[ \int \cot ^4(e+f x) (b \sec (e+f x))^m \, dx=-\frac {\cot ^3(e+f x) \cos ^2(e+f x)^{\frac {m-3}{2}} (b \sec (e+f x))^m \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {m-3}{2},-\frac {1}{2},\sin ^2(e+f x)\right )}{3 f} \]

[In]

Int[Cot[e + f*x]^4*(b*Sec[e + f*x])^m,x]

[Out]

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

Rule 2697

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(a*Sec[e + f
*x])^m*(b*Tan[e + f*x])^(n + 1)*((Cos[e + f*x]^2)^((m + n + 1)/2)/(b*f*(n + 1)))*Hypergeometric2F1[(n + 1)/2,
(m + n + 1)/2, (n + 3)/2, Sin[e + f*x]^2], x] /; FreeQ[{a, b, e, f, m, n}, x] &&  !IntegerQ[(n - 1)/2] &&  !In
tegerQ[m/2]

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

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.97 \[ \int \cot ^4(e+f x) (b \sec (e+f x))^m \, dx=\frac {\cot (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {5}{2},\frac {m}{2},\frac {2+m}{2},\sec ^2(e+f x)\right ) (b \sec (e+f x))^m \sqrt {-\tan ^2(e+f x)}}{f m} \]

[In]

Integrate[Cot[e + f*x]^4*(b*Sec[e + f*x])^m,x]

[Out]

(Cot[e + f*x]*Hypergeometric2F1[5/2, m/2, (2 + m)/2, Sec[e + f*x]^2]*(b*Sec[e + f*x])^m*Sqrt[-Tan[e + f*x]^2])
/(f*m)

Maple [F]

\[\int \left (\cot ^{4}\left (f x +e \right )\right ) \left (b \sec \left (f x +e \right )\right )^{m}d x\]

[In]

int(cot(f*x+e)^4*(b*sec(f*x+e))^m,x)

[Out]

int(cot(f*x+e)^4*(b*sec(f*x+e))^m,x)

Fricas [F]

\[ \int \cot ^4(e+f x) (b \sec (e+f x))^m \, dx=\int { \left (b \sec \left (f x + e\right )\right )^{m} \cot \left (f x + e\right )^{4} \,d x } \]

[In]

integrate(cot(f*x+e)^4*(b*sec(f*x+e))^m,x, algorithm="fricas")

[Out]

integral((b*sec(f*x + e))^m*cot(f*x + e)^4, x)

Sympy [F]

\[ \int \cot ^4(e+f x) (b \sec (e+f x))^m \, dx=\int \left (b \sec {\left (e + f x \right )}\right )^{m} \cot ^{4}{\left (e + f x \right )}\, dx \]

[In]

integrate(cot(f*x+e)**4*(b*sec(f*x+e))**m,x)

[Out]

Integral((b*sec(e + f*x))**m*cot(e + f*x)**4, x)

Maxima [F]

\[ \int \cot ^4(e+f x) (b \sec (e+f x))^m \, dx=\int { \left (b \sec \left (f x + e\right )\right )^{m} \cot \left (f x + e\right )^{4} \,d x } \]

[In]

integrate(cot(f*x+e)^4*(b*sec(f*x+e))^m,x, algorithm="maxima")

[Out]

integrate((b*sec(f*x + e))^m*cot(f*x + e)^4, x)

Giac [F]

\[ \int \cot ^4(e+f x) (b \sec (e+f x))^m \, dx=\int { \left (b \sec \left (f x + e\right )\right )^{m} \cot \left (f x + e\right )^{4} \,d x } \]

[In]

integrate(cot(f*x+e)^4*(b*sec(f*x+e))^m,x, algorithm="giac")

[Out]

integrate((b*sec(f*x + e))^m*cot(f*x + e)^4, x)

Mupad [F(-1)]

Timed out. \[ \int \cot ^4(e+f x) (b \sec (e+f x))^m \, dx=\int {\mathrm {cot}\left (e+f\,x\right )}^4\,{\left (\frac {b}{\cos \left (e+f\,x\right )}\right )}^m \,d x \]

[In]

int(cot(e + f*x)^4*(b/cos(e + f*x))^m,x)

[Out]

int(cot(e + f*x)^4*(b/cos(e + f*x))^m, x)