\(\int F^{c (a+b x)} (f \csc (d+e x))^n \, dx\) [116]

Optimal result

Integrand size = 20, antiderivative size = 99 \[ \int F^{c (a+b x)} (f \csc (d+e x))^n \, dx=\frac {\left (1-e^{2 i (d+e x)}\right )^n F^{c (a+b x)} (f \csc (d+e x))^n \operatorname {Hypergeometric2F1}\left (n,\frac {1}{2} \left (n-\frac {i b c \log (F)}{e}\right ),\frac {1}{2} \left (2+n-\frac {i b c \log (F)}{e}\right ),e^{2 i (d+e x)}\right )}{i e n+b c \log (F)} \] Output:

(1-exp(2*I*(e*x+d)))^n*F^(c*(b*x+a))*(f*csc(e*x+d))^n*hypergeom([n, 1/2*n- 
1/2*I*b*c*ln(F)/e],[1+1/2*n-1/2*I*b*c*ln(F)/e],exp(2*I*(e*x+d)))/(I*e*n+b* 
c*ln(F))
 

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.05 \[ \int F^{c (a+b x)} (f \csc (d+e x))^n \, dx=\frac {i \left (1-e^{-2 i (d+e x)}\right )^n F^{c (a+b x)} (f \csc (d+e x))^n \operatorname {Hypergeometric2F1}\left (n,\frac {e n+i b c \log (F)}{2 e},\frac {1}{2} \left (2+n+\frac {i b c \log (F)}{e}\right ),e^{-2 i (d+e x)}\right )}{e n+i b c \log (F)} \] Input:

Integrate[F^(c*(a + b*x))*(f*Csc[d + e*x])^n,x]
 

Output:

(I*(1 - E^((-2*I)*(d + e*x)))^n*F^(c*(a + b*x))*(f*Csc[d + e*x])^n*Hyperge 
ometric2F1[n, (e*n + I*b*c*Log[F])/(2*e), (2 + n + (I*b*c*Log[F])/e)/2, E^ 
((-2*I)*(d + e*x))])/(e*n + I*b*c*Log[F])
 

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.31, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {7271, 4955, 2689}

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.

\(\displaystyle \int F^{c (a+b x)} (f \csc (d+e x))^n \, dx\)

\(\Big \downarrow \) 7271

\(\displaystyle \csc ^{-n}(d+e x) (f \csc (d+e x))^n \int F^{c (a+b x)} \csc ^n(d+e x)dx\)

\(\Big \downarrow \) 4955

\(\displaystyle e^{i n (d+e x)} \left (1-e^{-2 i (d+e x)}\right )^n (f \csc (d+e x))^n \int e^{-i d n-i e x n} \left (1-e^{-2 i (d+e x)}\right )^{-n} F^{a c+b x c}dx\)

\(\Big \downarrow \) 2689

\(\displaystyle -\frac {e^{i n (d+e x)-i d n-i e n x} \left (1-e^{-2 i (d+e x)}\right )^n F^{a c+b c x} (f \csc (d+e x))^n \operatorname {Hypergeometric2F1}\left (n,\frac {e n+i b c \log (F)}{2 e},\frac {1}{2} \left (n+\frac {i b c \log (F)}{e}+2\right ),e^{-2 i (d+e x)}\right )}{-b c \log (F)+i e n}\)

Input:

Int[F^(c*(a + b*x))*(f*Csc[d + e*x])^n,x]
 

Output:

-((E^((-I)*d*n - I*e*n*x + I*n*(d + e*x))*(1 - E^((-2*I)*(d + e*x)))^n*F^( 
a*c + b*c*x)*(f*Csc[d + e*x])^n*Hypergeometric2F1[n, (e*n + I*b*c*Log[F])/ 
(2*e), (2 + n + (I*b*c*Log[F])/e)/2, E^((-2*I)*(d + e*x))])/(I*e*n - b*c*L 
og[F]))
 

Defintions of rubi rules used

rule 2689

Int[((a_) + (b_.)*(F_)^((e_.)*((c_.) + (d_.)*(x_))))^(p_)*(G_)^((h_.)*((f_. 
) + (g_.)*(x_)))*(H_)^((t_.)*((r_.) + (s_.)*(x_))), x_Symbol] :> Simp[G^(h* 
(f + g*x))*H^(t*(r + s*x))*((a + b*F^(e*(c + d*x)))^p/((g*h*Log[G] + s*t*Lo 
g[H])*((a + b*F^(e*(c + d*x)))/a)^p))*Hypergeometric2F1[-p, (g*h*Log[G] + s 
*t*Log[H])/(d*e*Log[F]), (g*h*Log[G] + s*t*Log[H])/(d*e*Log[F]) + 1, Simpli 
fy[(-b/a)*F^(e*(c + d*x))]], x] /; FreeQ[{F, G, H, a, b, c, d, e, f, g, h, 
r, s, t, p}, x] &&  !IntegerQ[p]
 

rule 4955

Int[Csc[(d_.) + (e_.)*(x_)]^(n_.)*(F_)^((c_.)*((a_.) + (b_.)*(x_))), x_Symb 
ol] :> Simp[(1 - E^(-2*I*(d + e*x)))^n*(Csc[d + e*x]^n/E^((-I)*n*(d + e*x)) 
)   Int[SimplifyIntegrand[F^(c*(a + b*x))*(1/(E^(I*n*(d + e*x))*(1 - E^(-2* 
I*(d + e*x)))^n)), x], x], x] /; FreeQ[{F, a, b, c, d, e}, x] &&  !IntegerQ 
[n]
 

rule 7271

Int[(u_.)*((a_.)*(v_)^(m_.))^(p_), x_Symbol] :> Simp[a^IntPart[p]*((a*v^m)^ 
FracPart[p]/v^(m*FracPart[p]))   Int[u*v^(m*p), x], x] /; FreeQ[{a, m, p}, 
x] &&  !IntegerQ[p] &&  !FreeQ[v, x] &&  !(EqQ[a, 1] && EqQ[m, 1]) &&  !(Eq 
Q[v, x] && EqQ[m, 1])
 

Maple [F]

\[\int F^{c \left (b x +a \right )} \left (f \csc \left (e x +d \right )\right )^{n}d x\]

Input:

int(F^(c*(b*x+a))*(f*csc(e*x+d))^n,x)
 

Output:

int(F^(c*(b*x+a))*(f*csc(e*x+d))^n,x)
 

Fricas [F]

\[ \int F^{c (a+b x)} (f \csc (d+e x))^n \, dx=\int { \left (f \csc \left (e x + d\right )\right )^{n} F^{{\left (b x + a\right )} c} \,d x } \] Input:

integrate(F^(c*(b*x+a))*(f*csc(e*x+d))^n,x, algorithm="fricas")
 

Output:

integral((f*csc(e*x + d))^n*F^(b*c*x + a*c), x)
 

Sympy [F]

\[ \int F^{c (a+b x)} (f \csc (d+e x))^n \, dx=\int F^{c \left (a + b x\right )} \left (f \csc {\left (d + e x \right )}\right )^{n}\, dx \] Input:

integrate(F**(c*(b*x+a))*(f*csc(e*x+d))**n,x)
 

Output:

Integral(F**(c*(a + b*x))*(f*csc(d + e*x))**n, x)
 

Maxima [F]

\[ \int F^{c (a+b x)} (f \csc (d+e x))^n \, dx=\int { \left (f \csc \left (e x + d\right )\right )^{n} F^{{\left (b x + a\right )} c} \,d x } \] Input:

integrate(F^(c*(b*x+a))*(f*csc(e*x+d))^n,x, algorithm="maxima")
 

Output:

integrate((f*csc(e*x + d))^n*F^((b*x + a)*c), x)
 

Giac [F]

\[ \int F^{c (a+b x)} (f \csc (d+e x))^n \, dx=\int { \left (f \csc \left (e x + d\right )\right )^{n} F^{{\left (b x + a\right )} c} \,d x } \] Input:

integrate(F^(c*(b*x+a))*(f*csc(e*x+d))^n,x, algorithm="giac")
 

Output:

integrate((f*csc(e*x + d))^n*F^((b*x + a)*c), x)
 

Mupad [F(-1)]

Timed out. \[ \int F^{c (a+b x)} (f \csc (d+e x))^n \, dx=\int F^{c\,\left (a+b\,x\right )}\,{\left (\frac {f}{\sin \left (d+e\,x\right )}\right )}^n \,d x \] Input:

int(F^(c*(a + b*x))*(f/sin(d + e*x))^n,x)
 

Output:

int(F^(c*(a + b*x))*(f/sin(d + e*x))^n, x)
 

Reduce [F]

\[ \int F^{c (a+b x)} (f \csc (d+e x))^n \, dx=f^{a c +n} \left (\int f^{b c x} \csc \left (e x +d \right )^{n}d x \right ) \] Input:

int(F^(c*(b*x+a))*(f*csc(e*x+d))^n,x)
 

Output:

f**(a*c + n)*int(f**(b*c*x)*csc(d + e*x)**n,x)