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

Optimal. Leaf size=102 \[ -\frac {\left (1-e^{-2 i (d+e x)}\right )^n F^{a c+b c x} \csc ^n(d+e x) \, _2F_1\left (n,\frac {e n+i b c \log (F)}{2 e};\frac {1}{2} \left (n+\frac {i b c \log (F)}{e}+2\right );e^{-2 i (d+e x)}\right )}{-b c \log (F)+i e n} \]

[Out]

-(1-1/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))

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Rubi [A]  time = 0.16, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {4455, 2259} \[ -\frac {\left (1-e^{-2 i (d+e x)}\right )^n F^{a c+b c x} \csc ^n(d+e x) \, _2F_1\left (n,\frac {e n+i b c \log (F)}{2 e};\frac {1}{2} \left (n+\frac {i b c \log (F)}{e}+2\right );e^{-2 i (d+e x)}\right )}{-b c \log (F)+i e n} \]

Antiderivative was successfully verified.

[In]

Int[F^(c*(a + b*x))*Csc[d + e*x]^n,x]

[Out]

-(((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]))

Rule 2259

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*Hypergeometric2F
1[-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, Simplify[-((b*F^(e*(
c + d*x)))/a)]])/((g*h*Log[G] + s*t*Log[H])*((a + b*F^(e*(c + d*x)))/a)^p), x] /; FreeQ[{F, G, H, a, b, c, d,
e, f, g, h, r, s, t, p}, x] &&  !IntegerQ[p]

Rule 4455

Int[Csc[(d_.) + (e_.)*(x_)]^(n_.)*(F_)^((c_.)*((a_.) + (b_.)*(x_))), x_Symbol] :> Dist[((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))/(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]

Rubi steps

\begin {align*} \int F^{c (a+b x)} \csc ^n(d+e x) \, dx &=\left (e^{i n (d+e x)} \left (1-e^{-2 i (d+e x)}\right )^n \csc ^n(d+e x)\right ) \int e^{-i d n-i e n x} \left (1-e^{-2 i (d+e x)}\right )^{-n} F^{a c+b c x} \, dx\\ &=-\frac {\left (1-e^{-2 i (d+e x)}\right )^n F^{a c+b c x} \csc ^n(d+e x) \, _2F_1\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)}\\ \end {align*}

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Mathematica [A]  time = 0.10, size = 102, normalized size = 1.00 \[ \frac {i \left (1-e^{-2 i (d+e x)}\right )^n F^{c (a+b x)} \csc ^n(d+e x) \, _2F_1\left (n,\frac {e n+i b c \log (F)}{2 e};\frac {1}{2} \left (n+\frac {i b c \log (F)}{e}+2\right );e^{-2 i (d+e x)}\right )}{e n+i b c \log (F)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F]  time = 0.58, size = 0, normalized size = 0.00 \[ {\rm integral}\left (F^{b c x + a c} \csc \left (e x + d\right )^{n}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int F^{{\left (b x + a\right )} c} \csc \left (e x + d\right )^{n}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.98, size = 0, normalized size = 0.00 \[ \int F^{c \left (b x +a \right )} \left (\csc ^{n}\left (e x +d \right )\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int F^{{\left (b x + a\right )} c} \csc \left (e x + d\right )^{n}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int F^{c\,\left (a+b\,x\right )}\,{\left (\frac {1}{\sin \left (d+e\,x\right )}\right )}^n \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(c*(a + b*x))*(1/sin(d + e*x))^n,x)

[Out]

int(F^(c*(a + b*x))*(1/sin(d + e*x))^n, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int F^{c \left (a + b x\right )} \csc ^{n}{\left (d + e x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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