Integrand size = 18, antiderivative size = 102 \[ \int F^{c (a+b x)} \csc ^n(d+e x) \, dx=-\frac {\left (1-e^{-2 i (d+e x)}\right )^n F^{a c+b c x} \csc ^n(d+e x) \operatorname {Hypergeometric2F1}\left (n,\frac {e n+i b c \log (F)}{2 e},\frac {1}{2} \left (2+n+\frac {i b c \log (F)}{e}\right ),e^{-2 i (d+e x)}\right )}{i e n-b c \log (F)} \] Output:
-(1-exp(-2*I*(e*x+d)))^n*F^(b*c*x+a*c)*csc(e*x+d)^n*hypergeom([n, 1/2*(I*b *c*ln(F)+e*n)/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))
Time = 0.07 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.00 \[ \int F^{c (a+b x)} \csc ^n(d+e x) \, dx=\frac {i \left (1-e^{-2 i (d+e x)}\right )^n F^{c (a+b x)} \csc ^n(d+e x) \operatorname {Hypergeometric2F1}\left (n,\frac {e n+i b c \log (F)}{2 e},\frac {1}{2} \left (2+n+\frac {i b c \log (F)}{e}\right ),e^{-2 i (d+e x)}\right )}{e n+i b c \log (F)} \] Input:
Integrate[F^(c*(a + b*x))*Csc[d + e*x]^n,x]
Output:
(I*(1 - E^((-2*I)*(d + e*x)))^n*F^(c*(a + b*x))*Csc[d + e*x]^n*Hypergeomet ric2F1[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])
Time = 0.37 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.25, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {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.
\(\displaystyle \int F^{c (a+b x)} \csc ^n(d+e x) \, dx\) |
\(\Big \downarrow \) 4955 |
\(\displaystyle e^{i n (d+e x)} \left (1-e^{-2 i (d+e x)}\right )^n \csc ^n(d+e x) \int e^{-i d n-i e x n} \left (1-e^{-2 i (d+e x)}\right )^{-n} F^{a c+b x c}dx\) |
\(\Big \downarrow \) 2689 |
\(\displaystyle -\frac {e^{i n (d+e x)-i d n-i e n x} \left (1-e^{-2 i (d+e x)}\right )^n F^{a c+b c x} \csc ^n(d+e x) \operatorname {Hypergeometric2F1}\left (n,\frac {e n+i b c \log (F)}{2 e},\frac {1}{2} \left (n+\frac {i b c \log (F)}{e}+2\right ),e^{-2 i (d+e x)}\right )}{-b c \log (F)+i e n}\) |
Input:
Int[F^(c*(a + b*x))*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)*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*Log[F ]))
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]
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]
\[\int F^{c \left (b x +a \right )} \csc \left (e x +d \right )^{n}d x\]
Input:
int(F^(c*(b*x+a))*csc(e*x+d)^n,x)
Output:
int(F^(c*(b*x+a))*csc(e*x+d)^n,x)
\[ \int F^{c (a+b x)} \csc ^n(d+e x) \, dx=\int { F^{{\left (b x + a\right )} c} \csc \left (e x + d\right )^{n} \,d x } \] Input:
integrate(F^(c*(b*x+a))*csc(e*x+d)^n,x, algorithm="fricas")
Output:
integral(F^(b*c*x + a*c)*csc(e*x + d)^n, x)
\[ \int F^{c (a+b x)} \csc ^n(d+e x) \, dx=\int F^{c \left (a + b x\right )} \csc ^{n}{\left (d + e x \right )}\, dx \] Input:
integrate(F**(c*(b*x+a))*csc(e*x+d)**n,x)
Output:
Integral(F**(c*(a + b*x))*csc(d + e*x)**n, x)
\[ \int F^{c (a+b x)} \csc ^n(d+e x) \, dx=\int { F^{{\left (b x + a\right )} c} \csc \left (e x + d\right )^{n} \,d x } \] Input:
integrate(F^(c*(b*x+a))*csc(e*x+d)^n,x, algorithm="maxima")
Output:
integrate(F^((b*x + a)*c)*csc(e*x + d)^n, x)
\[ \int F^{c (a+b x)} \csc ^n(d+e x) \, dx=\int { F^{{\left (b x + a\right )} c} \csc \left (e x + d\right )^{n} \,d x } \] Input:
integrate(F^(c*(b*x+a))*csc(e*x+d)^n,x, algorithm="giac")
Output:
integrate(F^((b*x + a)*c)*csc(e*x + d)^n, x)
Timed out. \[ \int F^{c (a+b x)} \csc ^n(d+e x) \, dx=\int F^{c\,\left (a+b\,x\right )}\,{\left (\frac {1}{\sin \left (d+e\,x\right )}\right )}^n \,d x \] Input:
int(F^(c*(a + b*x))*(1/sin(d + e*x))^n,x)
Output:
int(F^(c*(a + b*x))*(1/sin(d + e*x))^n, x)
\[ \int F^{c (a+b x)} \csc ^n(d+e x) \, dx=f^{a c} \left (\int f^{b c x} \csc \left (e x +d \right )^{n}d x \right ) \] Input:
int(F^(c*(b*x+a))*csc(e*x+d)^n,x)
Output:
f**(a*c)*int(f**(b*c*x)*csc(d + e*x)**n,x)