\(\int \cot (e+f x) (b \sec (e+f x))^m \, dx\) [354]

Optimal result

Integrand size = 17, antiderivative size = 40 \[ \int \cot (e+f x) (b \sec (e+f x))^m \, dx=-\frac {\operatorname {Hypergeometric2F1}\left (1,\frac {m}{2},\frac {2+m}{2},\sec ^2(e+f x)\right ) (b \sec (e+f x))^m}{f m} \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 40, 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 = {2686, 371} \[ \int \cot (e+f x) (b \sec (e+f x))^m \, dx=-\frac {(b \sec (e+f x))^m \operatorname {Hypergeometric2F1}\left (1,\frac {m}{2},\frac {m+2}{2},\sec ^2(e+f x)\right )}{f m} \]

[In]

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

[Out]

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

Rule 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 2686

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a/f, Subst[
Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n -
1)/2] &&  !(IntegerQ[m/2] && LtQ[0, m, n + 1])

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

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00 \[ \int \cot (e+f x) (b \sec (e+f x))^m \, dx=-\frac {\operatorname {Hypergeometric2F1}\left (1,\frac {m}{2},\frac {2+m}{2},\sec ^2(e+f x)\right ) (b \sec (e+f x))^m}{f m} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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