\(\int F^{c (a+b x)} \cos ^n(d+e x) \, dx\) [10]

Optimal result

Integrand size = 18, antiderivative size = 107 \[ \int F^{c (a+b x)} \cos ^n(d+e x) \, dx=-\frac {\left (1+e^{2 i (d+e x)}\right )^{-n} F^{c (a+b x)} \cos ^n(d+e x) \operatorname {Hypergeometric2F1}\left (-n,-\frac {e n+i b c \log (F)}{2 e},\frac {1}{2} \left (2-n-\frac {i b c \log (F)}{e}\right ),-e^{2 i (d+e x)}\right )}{i e n-b c \log (F)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {4526, 2291} \[ \int F^{c (a+b x)} \cos ^n(d+e x) \, dx=-\frac {\left (1+e^{2 i (d+e x)}\right )^{-n} F^{c (a+b x)} \cos ^n(d+e x) \operatorname {Hypergeometric2F1}\left (-n,-\frac {e n+i b c \log (F)}{2 e},\frac {1}{2} \left (-n-\frac {i b c \log (F)}{e}+2\right ),-e^{2 i (d+e x)}\right )}{-b c \log (F)+i e n} \]

[In]

Int[F^(c*(a + b*x))*Cos[d + e*x]^n,x]

[Out]

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

Rule 2291

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*Log[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, Simplify[(-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 4526

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

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

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.03 \[ \int F^{c (a+b x)} \cos ^n(d+e x) \, dx=\frac {\left (1+e^{2 i (d+e x)}\right )^{-n} F^{c (a+b x)} \cos ^n(d+e x) \operatorname {Hypergeometric2F1}\left (-n,-\frac {i (-i e n+b c \log (F))}{2 e},1-\frac {i (-i e n+b c \log (F))}{2 e},-e^{2 i (d+e x)}\right )}{-i e n+b c \log (F)} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int F^{c (a+b x)} \cos ^n(d+e x) \, dx=\int F^{c\,\left (a+b\,x\right )}\,{\cos \left (d+e\,x\right )}^n \,d x \]

[In]

int(F^(c*(a + b*x))*cos(d + e*x)^n,x)

[Out]

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