Integrand size = 19, antiderivative size = 73 \[ \int \csc ^2(e+f x) (b \sec (e+f x))^n \, dx=-\frac {b \csc (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {1-n}{2},\frac {3-n}{2},\cos ^2(e+f x)\right ) (b \sec (e+f x))^{-1+n} \sqrt {\sin ^2(e+f x)}}{f (1-n)} \] Output:
-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)
Time = 0.48 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.78 \[ \int \csc ^2(e+f x) (b \sec (e+f x))^n \, dx=-\frac {\cot (e+f x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-\frac {n}{2},\frac {1}{2},-\tan ^2(e+f x)\right ) (b \sec (e+f x))^n \sec ^2(e+f x)^{-n/2}}{f} \] Input:
Integrate[Csc[e + f*x]^2*(b*Sec[e + f*x])^n,x]
Output:
-((Cot[e + f*x]*Hypergeometric2F1[-1/2, -1/2*n, 1/2, -Tan[e + f*x]^2]*(b*S ec[e + f*x])^n)/(f*(Sec[e + f*x]^2)^(n/2)))
Time = 0.50 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {3042, 3112, 3042, 3056}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \csc ^2(e+f x) (b \sec (e+f x))^n \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \csc (e+f x)^2 (b \sec (e+f x))^ndx\) |
\(\Big \downarrow \) 3112 |
\(\displaystyle b^2 (b \cos (e+f x))^{n-1} (b \sec (e+f x))^{n-1} \int (b \cos (e+f x))^{-n} \csc ^2(e+f x)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle b^2 (b \cos (e+f x))^{n-1} (b \sec (e+f x))^{n-1} \int \frac {(b \cos (e+f x))^{-n}}{\sin (e+f x)^2}dx\) |
\(\Big \downarrow \) 3056 |
\(\displaystyle -\frac {b \sqrt {\sin ^2(e+f x)} \csc (e+f x) (b \sec (e+f x))^{n-1} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {1-n}{2},\frac {3-n}{2},\cos ^2(e+f x)\right )}{f (1-n)}\) |
Input:
Int[Csc[e + f*x]^2*(b*Sec[e + f*x])^n,x]
Output:
-((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)))
Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n _), x_Symbol] :> Simp[(-b^(2*IntPart[(n - 1)/2] + 1))*(b*Sin[e + f*x])^(2*F racPart[(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, C os[e + f*x]^2], x] /; FreeQ[{a, b, e, f, m, n}, x] && SimplerQ[n, m]
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sec[(e_.) + (f_.)*(x_)])^(m _), x_Symbol] :> Simp[(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*Cos[e + f*x])^m*(b*Sin[e + f*x])^n), x], x] /; FreeQ[{a, b, e, f, m, n}, x]
\[\int \csc \left (f x +e \right )^{2} \left (b \sec \left (f x +e \right )\right )^{n}d x\]
Input:
int(csc(f*x+e)^2*(b*sec(f*x+e))^n,x)
Output:
int(csc(f*x+e)^2*(b*sec(f*x+e))^n,x)
\[ \int \csc ^2(e+f x) (b \sec (e+f x))^n \, dx=\int { \left (b \sec \left (f x + e\right )\right )^{n} \csc \left (f x + e\right )^{2} \,d x } \] Input:
integrate(csc(f*x+e)^2*(b*sec(f*x+e))^n,x, algorithm="fricas")
Output:
integral((b*sec(f*x + e))^n*csc(f*x + e)^2, x)
\[ \int \csc ^2(e+f x) (b \sec (e+f x))^n \, dx=\int \left (b \sec {\left (e + f x \right )}\right )^{n} \csc ^{2}{\left (e + f x \right )}\, dx \] Input:
integrate(csc(f*x+e)**2*(b*sec(f*x+e))**n,x)
Output:
Integral((b*sec(e + f*x))**n*csc(e + f*x)**2, x)
\[ \int \csc ^2(e+f x) (b \sec (e+f x))^n \, dx=\int { \left (b \sec \left (f x + e\right )\right )^{n} \csc \left (f x + e\right )^{2} \,d x } \] Input:
integrate(csc(f*x+e)^2*(b*sec(f*x+e))^n,x, algorithm="maxima")
Output:
integrate((b*sec(f*x + e))^n*csc(f*x + e)^2, x)
\[ \int \csc ^2(e+f x) (b \sec (e+f x))^n \, dx=\int { \left (b \sec \left (f x + e\right )\right )^{n} \csc \left (f x + e\right )^{2} \,d x } \] Input:
integrate(csc(f*x+e)^2*(b*sec(f*x+e))^n,x, algorithm="giac")
Output:
integrate((b*sec(f*x + e))^n*csc(f*x + e)^2, x)
Timed out. \[ \int \csc ^2(e+f x) (b \sec (e+f x))^n \, dx=\int \frac {{\left (\frac {b}{\cos \left (e+f\,x\right )}\right )}^n}{{\sin \left (e+f\,x\right )}^2} \,d x \] Input:
int((b/cos(e + f*x))^n/sin(e + f*x)^2,x)
Output:
int((b/cos(e + f*x))^n/sin(e + f*x)^2, x)
\[ \int \csc ^2(e+f x) (b \sec (e+f x))^n \, dx=b^{n} \left (\int \sec \left (f x +e \right )^{n} \csc \left (f x +e \right )^{2}d x \right ) \] Input:
int(csc(f*x+e)^2*(b*sec(f*x+e))^n,x)
Output:
b**n*int(sec(e + f*x)**n*csc(e + f*x)**2,x)