Integrand size = 19, antiderivative size = 48 \[ \int \csc ^3(e+f x) (b \sec (e+f x))^n \, dx=\frac {\operatorname {Hypergeometric2F1}\left (2,\frac {3+n}{2},\frac {5+n}{2},\sec ^2(e+f x)\right ) (b \sec (e+f x))^{3+n}}{b^3 f (3+n)} \] Output:
hypergeom([2, 3/2+1/2*n],[5/2+1/2*n],sec(f*x+e)^2)*(b*sec(f*x+e))^(3+n)/b^ 3/f/(3+n)
Leaf count is larger than twice the leaf count of optimal. \(201\) vs. \(2(48)=96\).
Time = 3.46 (sec) , antiderivative size = 201, normalized size of antiderivative = 4.19 \[ \int \csc ^3(e+f x) (b \sec (e+f x))^n \, dx=\frac {b (b \sec (e+f x))^{-1+n} \left (2 \operatorname {Hypergeometric2F1}(1,1-n,2-n,\cos (e+f x))+2 \operatorname {Hypergeometric2F1}(2,1-n,2-n,\cos (e+f x))+2^n \operatorname {Hypergeometric2F1}\left (1-n,-n,2-n,\frac {1}{2} \cos (e+f x) \sec ^2\left (\frac {1}{2} (e+f x)\right )\right ) \sec ^2\left (\frac {1}{2} (e+f x)\right )^{1-n}+2^n \operatorname {Hypergeometric2F1}\left (1-n,1-n,2-n,\frac {1}{2} \cos (e+f x) \sec ^2\left (\frac {1}{2} (e+f x)\right )\right ) \sec ^{1-n}(e+f x) \left (\cos ^2\left (\frac {1}{2} (e+f x)\right ) \sec (e+f x)\right )^{-1+n}\right )}{8 f (-1+n)} \] Input:
Integrate[Csc[e + f*x]^3*(b*Sec[e + f*x])^n,x]
Output:
(b*(b*Sec[e + f*x])^(-1 + n)*(2*Hypergeometric2F1[1, 1 - n, 2 - n, Cos[e + f*x]] + 2*Hypergeometric2F1[2, 1 - n, 2 - n, Cos[e + f*x]] + 2^n*Hypergeo metric2F1[1 - n, -n, 2 - n, (Cos[e + f*x]*Sec[(e + f*x)/2]^2)/2]*(Sec[(e + f*x)/2]^2)^(1 - n) + 2^n*Hypergeometric2F1[1 - n, 1 - n, 2 - n, (Cos[e + f*x]*Sec[(e + f*x)/2]^2)/2]*Sec[e + f*x]^(1 - n)*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(-1 + n)))/(8*f*(-1 + n))
Time = 0.37 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {3042, 3102, 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.
\(\displaystyle \int \csc ^3(e+f x) (b \sec (e+f x))^n \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \csc (e+f x)^3 (b \sec (e+f x))^ndx\) |
\(\Big \downarrow \) 3102 |
\(\displaystyle \frac {\int \frac {b^4 (b \sec (e+f x))^{n+2}}{\left (b^2-b^2 \sec ^2(e+f x)\right )^2}d(b \sec (e+f x))}{b^3 f}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {b \int \frac {(b \sec (e+f x))^{n+2}}{\left (b^2-b^2 \sec ^2(e+f x)\right )^2}d(b \sec (e+f x))}{f}\) |
\(\Big \downarrow \) 278 |
\(\displaystyle \frac {(b \sec (e+f x))^{n+3} \operatorname {Hypergeometric2F1}\left (2,\frac {n+3}{2},\frac {n+5}{2},\sec ^2(e+f x)\right )}{b^3 f (n+3)}\) |
Input:
Int[Csc[e + f*x]^3*(b*Sec[e + f*x])^n,x]
Output:
(Hypergeometric2F1[2, (3 + n)/2, (5 + n)/2, Sec[e + f*x]^2]*(b*Sec[e + f*x ])^(3 + n))/(b^3*f*(3 + 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_)]^(n_.)*((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_), x_S ymbol] :> Simp[1/(f*a^n) Subst[Int[x^(m + n - 1)/(-1 + x^2/a^2)^((n + 1)/ 2), x], x, a*Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n + 1 )/2] && !(IntegerQ[(m + 1)/2] && LtQ[0, m, n])
\[\int \csc \left (f x +e \right )^{3} \left (b \sec \left (f x +e \right )\right )^{n}d x\]
Input:
int(csc(f*x+e)^3*(b*sec(f*x+e))^n,x)
Output:
int(csc(f*x+e)^3*(b*sec(f*x+e))^n,x)
\[ \int \csc ^3(e+f x) (b \sec (e+f x))^n \, dx=\int { \left (b \sec \left (f x + e\right )\right )^{n} \csc \left (f x + e\right )^{3} \,d x } \] Input:
integrate(csc(f*x+e)^3*(b*sec(f*x+e))^n,x, algorithm="fricas")
Output:
integral((b*sec(f*x + e))^n*csc(f*x + e)^3, x)
\[ \int \csc ^3(e+f x) (b \sec (e+f x))^n \, dx=\int \left (b \sec {\left (e + f x \right )}\right )^{n} \csc ^{3}{\left (e + f x \right )}\, dx \] Input:
integrate(csc(f*x+e)**3*(b*sec(f*x+e))**n,x)
Output:
Integral((b*sec(e + f*x))**n*csc(e + f*x)**3, x)
\[ \int \csc ^3(e+f x) (b \sec (e+f x))^n \, dx=\int { \left (b \sec \left (f x + e\right )\right )^{n} \csc \left (f x + e\right )^{3} \,d x } \] Input:
integrate(csc(f*x+e)^3*(b*sec(f*x+e))^n,x, algorithm="maxima")
Output:
integrate((b*sec(f*x + e))^n*csc(f*x + e)^3, x)
\[ \int \csc ^3(e+f x) (b \sec (e+f x))^n \, dx=\int { \left (b \sec \left (f x + e\right )\right )^{n} \csc \left (f x + e\right )^{3} \,d x } \] Input:
integrate(csc(f*x+e)^3*(b*sec(f*x+e))^n,x, algorithm="giac")
Output:
integrate((b*sec(f*x + e))^n*csc(f*x + e)^3, x)
Timed out. \[ \int \csc ^3(e+f x) (b \sec (e+f x))^n \, dx=\int \frac {{\left (\frac {b}{\cos \left (e+f\,x\right )}\right )}^n}{{\sin \left (e+f\,x\right )}^3} \,d x \] Input:
int((b/cos(e + f*x))^n/sin(e + f*x)^3,x)
Output:
int((b/cos(e + f*x))^n/sin(e + f*x)^3, x)
\[ \int \csc ^3(e+f x) (b \sec (e+f x))^n \, dx=b^{n} \left (\int \sec \left (f x +e \right )^{n} \csc \left (f x +e \right )^{3}d x \right ) \] Input:
int(csc(f*x+e)^3*(b*sec(f*x+e))^n,x)
Output:
b**n*int(sec(e + f*x)**n*csc(e + f*x)**3,x)