Integrand size = 17, antiderivative size = 48 \[ \int (b \csc (e+f x))^n \sec (e+f x) \, dx=\frac {(b \csc (e+f x))^{1+n} \operatorname {Hypergeometric2F1}\left (1,\frac {1+n}{2},\frac {3+n}{2},\csc ^2(e+f x)\right )}{b f (1+n)} \] Output:
(b*csc(f*x+e))^(1+n)*hypergeom([1, 1/2+1/2*n],[3/2+1/2*n],csc(f*x+e)^2)/b/ f/(1+n)
Time = 0.02 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.06 \[ \int (b \csc (e+f x))^n \sec (e+f x) \, dx=-\frac {b (b \csc (e+f x))^{-1+n} \operatorname {Hypergeometric2F1}\left (1,\frac {1-n}{2},\frac {3-n}{2},\sin ^2(e+f x)\right )}{f (-1+n)} \] Input:
Integrate[(b*Csc[e + f*x])^n*Sec[e + f*x],x]
Output:
-((b*(b*Csc[e + f*x])^(-1 + n)*Hypergeometric2F1[1, (1 - n)/2, (3 - n)/2, Sin[e + f*x]^2])/(f*(-1 + n)))
Time = 0.22 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {3042, 3101, 25, 27, 278}
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.
\(\displaystyle \int \sec (e+f x) (b \csc (e+f x))^n \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sec (e+f x) (b \csc (e+f x))^ndx\) |
\(\Big \downarrow \) 3101 |
\(\displaystyle -\frac {\int -\frac {b^2 (b \csc (e+f x))^n}{b^2-b^2 \csc ^2(e+f x)}d(b \csc (e+f x))}{b f}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\int \frac {b^2 (b \csc (e+f x))^n}{b^2-b^2 \csc ^2(e+f x)}d(b \csc (e+f x))}{b f}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {b \int \frac {(b \csc (e+f x))^n}{b^2-b^2 \csc ^2(e+f x)}d(b \csc (e+f x))}{f}\) |
\(\Big \downarrow \) 278 |
\(\displaystyle \frac {(b \csc (e+f x))^{n+1} \operatorname {Hypergeometric2F1}\left (1,\frac {n+1}{2},\frac {n+3}{2},\csc ^2(e+f x)\right )}{b f (n+1)}\) |
Input:
Int[(b*Csc[e + f*x])^n*Sec[e + f*x],x]
Output:
((b*Csc[e + f*x])^(1 + n)*Hypergeometric2F1[1, (1 + n)/2, (3 + n)/2, Csc[e + f*x]^2])/(b*f*(1 + n))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*(( c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/2, (m + 1)/2 + 1, ( -b)*(x^2/a)], x] /; FreeQ[{a, b, c, m, p}, x] && !IGtQ[p, 0] && (ILtQ[p, 0 ] || GtQ[a, 0])
Int[(csc[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*sec[(e_.) + (f_.)*(x_)]^(n_.), x_S ymbol] :> Simp[-(f*a^n)^(-1) Subst[Int[x^(m + n - 1)/(-1 + x^2/a^2)^((n + 1)/2), x], x, a*Csc[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n + 1)/2] && !(IntegerQ[(m + 1)/2] && LtQ[0, m, n])
\[\int \left (b \csc \left (f x +e \right )\right )^{n} \sec \left (f x +e \right )d x\]
Input:
int((b*csc(f*x+e))^n*sec(f*x+e),x)
Output:
int((b*csc(f*x+e))^n*sec(f*x+e),x)
\[ \int (b \csc (e+f x))^n \sec (e+f x) \, dx=\int { \left (b \csc \left (f x + e\right )\right )^{n} \sec \left (f x + e\right ) \,d x } \] Input:
integrate((b*csc(f*x+e))^n*sec(f*x+e),x, algorithm="fricas")
Output:
integral((b*csc(f*x + e))^n*sec(f*x + e), x)
\[ \int (b \csc (e+f x))^n \sec (e+f x) \, dx=\int \left (b \csc {\left (e + f x \right )}\right )^{n} \sec {\left (e + f x \right )}\, dx \] Input:
integrate((b*csc(f*x+e))**n*sec(f*x+e),x)
Output:
Integral((b*csc(e + f*x))**n*sec(e + f*x), x)
\[ \int (b \csc (e+f x))^n \sec (e+f x) \, dx=\int { \left (b \csc \left (f x + e\right )\right )^{n} \sec \left (f x + e\right ) \,d x } \] Input:
integrate((b*csc(f*x+e))^n*sec(f*x+e),x, algorithm="maxima")
Output:
integrate((b*csc(f*x + e))^n*sec(f*x + e), x)
\[ \int (b \csc (e+f x))^n \sec (e+f x) \, dx=\int { \left (b \csc \left (f x + e\right )\right )^{n} \sec \left (f x + e\right ) \,d x } \] Input:
integrate((b*csc(f*x+e))^n*sec(f*x+e),x, algorithm="giac")
Output:
integrate((b*csc(f*x + e))^n*sec(f*x + e), x)
Timed out. \[ \int (b \csc (e+f x))^n \sec (e+f x) \, dx=\int \frac {{\left (\frac {b}{\sin \left (e+f\,x\right )}\right )}^n}{\cos \left (e+f\,x\right )} \,d x \] Input:
int((b/sin(e + f*x))^n/cos(e + f*x),x)
Output:
int((b/sin(e + f*x))^n/cos(e + f*x), x)
\[ \int (b \csc (e+f x))^n \sec (e+f x) \, dx=b^{n} \left (\int \csc \left (f x +e \right )^{n} \sec \left (f x +e \right )d x \right ) \] Input:
int((b*csc(f*x+e))^n*sec(f*x+e),x)
Output:
b**n*int(csc(e + f*x)**n*sec(e + f*x),x)