\(\int \csc ^2(e+f x) (b \sec (e+f x))^n \, dx\) [501]

Optimal result

Integrand size = 19, antiderivative size = 73 \[ \int \csc ^2(e+f x) (b \sec (e+f x))^n \, dx=-\frac {b \csc (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {1-n}{2},\frac {3-n}{2},\cos ^2(e+f x)\right ) (b \sec (e+f x))^{-1+n} \sqrt {\sin ^2(e+f x)}}{f (1-n)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2712, 2656} \[ \int \csc ^2(e+f x) (b \sec (e+f x))^n \, dx=-\frac {b \sqrt {\sin ^2(e+f x)} \csc (e+f x) (b \sec (e+f x))^{n-1} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {1-n}{2},\frac {3-n}{2},\cos ^2(e+f x)\right )}{f (1-n)} \]

[In]

Int[Csc[e + f*x]^2*(b*Sec[e + f*x])^n,x]

[Out]

-((b*Csc[e + f*x]*Hypergeometric2F1[3/2, (1 - n)/2, (3 - n)/2, Cos[e + f*x]^2]*(b*Sec[e + f*x])^(-1 + n)*Sqrt[
Sin[e + f*x]^2])/(f*(1 - 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 2712

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Dist[(a^2/b^2)*(a*
Sec[e + f*x])^(m - 1)*(b*Csc[e + f*x])^(n + 1)*(a*Cos[e + f*x])^(m - 1)*(b*Sin[e + f*x])^(n + 1), Int[1/((a*Co
s[e + f*x])^m*(b*Sin[e + f*x])^n), x], x] /; FreeQ[{a, b, e, f, m, n}, x]

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

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.78 \[ \int \csc ^2(e+f x) (b \sec (e+f x))^n \, dx=-\frac {\cot (e+f x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-\frac {n}{2},\frac {1}{2},-\tan ^2(e+f x)\right ) (b \sec (e+f x))^n \sec ^2(e+f x)^{-n/2}}{f} \]

[In]

Integrate[Csc[e + f*x]^2*(b*Sec[e + f*x])^n,x]

[Out]

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

Maple [F]

\[\int \left (\csc ^{2}\left (f x +e \right )\right ) \left (b \sec \left (f x +e \right )\right )^{n}d x\]

[In]

int(csc(f*x+e)^2*(b*sec(f*x+e))^n,x)

[Out]

int(csc(f*x+e)^2*(b*sec(f*x+e))^n,x)

Fricas [F]

\[ \int \csc ^2(e+f x) (b \sec (e+f x))^n \, dx=\int { \left (b \sec \left (f x + e\right )\right )^{n} \csc \left (f x + e\right )^{2} \,d x } \]

[In]

integrate(csc(f*x+e)^2*(b*sec(f*x+e))^n,x, algorithm="fricas")

[Out]

integral((b*sec(f*x + e))^n*csc(f*x + e)^2, x)

Sympy [F]

\[ \int \csc ^2(e+f x) (b \sec (e+f x))^n \, dx=\int \left (b \sec {\left (e + f x \right )}\right )^{n} \csc ^{2}{\left (e + f x \right )}\, dx \]

[In]

integrate(csc(f*x+e)**2*(b*sec(f*x+e))**n,x)

[Out]

Integral((b*sec(e + f*x))**n*csc(e + f*x)**2, x)

Maxima [F]

\[ \int \csc ^2(e+f x) (b \sec (e+f x))^n \, dx=\int { \left (b \sec \left (f x + e\right )\right )^{n} \csc \left (f x + e\right )^{2} \,d x } \]

[In]

integrate(csc(f*x+e)^2*(b*sec(f*x+e))^n,x, algorithm="maxima")

[Out]

integrate((b*sec(f*x + e))^n*csc(f*x + e)^2, x)

Giac [F]

\[ \int \csc ^2(e+f x) (b \sec (e+f x))^n \, dx=\int { \left (b \sec \left (f x + e\right )\right )^{n} \csc \left (f x + e\right )^{2} \,d x } \]

[In]

integrate(csc(f*x+e)^2*(b*sec(f*x+e))^n,x, algorithm="giac")

[Out]

integrate((b*sec(f*x + e))^n*csc(f*x + e)^2, x)

Mupad [F(-1)]

Timed out. \[ \int \csc ^2(e+f x) (b \sec (e+f x))^n \, dx=\int \frac {{\left (\frac {b}{\cos \left (e+f\,x\right )}\right )}^n}{{\sin \left (e+f\,x\right )}^2} \,d x \]

[In]

int((b/cos(e + f*x))^n/sin(e + f*x)^2,x)

[Out]

int((b/cos(e + f*x))^n/sin(e + f*x)^2, x)