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

Optimal. Leaf size=107 \[ -\frac{\left (1+e^{2 i (d+e x)}\right )^{-n} F^{c (a+b x)} \cos ^n(d+e x) \text{Hypergeometric2F1}\left (-n,-\frac{e n+i b c \log (F)}{2 e},\frac{1}{2} \left (-\frac{i b c \log (F)}{e}-n+2\right ),-e^{2 i (d+e x)}\right )}{-b c \log (F)+i e n} \]

[Out]

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

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Rubi [A]  time = 0.112139, antiderivative size = 107, normalized size of antiderivative = 1., 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 = {4441, 2259} \[ -\frac{\left (1+e^{2 i (d+e x)}\right )^{-n} F^{c (a+b x)} \cos ^n(d+e x) \, _2F_1\left (-n,-\frac{e n+i b c \log (F)}{2 e};\frac{1}{2} \left (-n-\frac{i b c \log (F)}{e}+2\right );-e^{2 i (d+e x)}\right )}{-b c \log (F)+i e n} \]

Antiderivative was successfully verified.

[In]

Int[F^(c*(a + b*x))*Cos[d + e*x]^n,x]

[Out]

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

Rule 4441

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]

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]

Rubi steps

\begin{align*} \int F^{c (a+b x)} \cos ^n(d+e x) \, dx &=\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) \, _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*}

Mathematica [A]  time = 0.0553799, size = 110, normalized size = 1.03 \[ \frac{\left (1+e^{2 i (d+e x)}\right )^{-n} F^{c (a+b x)} \cos ^n(d+e x) \text{Hypergeometric2F1}\left (-n,-\frac{i (b c \log (F)-i e n)}{2 e},1-\frac{i (b c \log (F)-i e n)}{2 e},-e^{2 i (d+e x)}\right )}{b c \log (F)-i e n} \]

Antiderivative was successfully verified.

[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, ((-I/2)*((-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]))

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Maple [F]  time = 0.498, size = 0, normalized size = 0. \begin{align*} \int{F}^{c \left ( bx+a \right ) } \left ( \cos \left ( ex+d \right ) \right ) ^{n}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int F^{{\left (b x + a\right )} c} \cos \left (e x + d\right )^{n}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (F^{b c x + a c} \cos \left (e x + d\right )^{n}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int F^{{\left (b x + a\right )} c} \cos \left (e x + d\right )^{n}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)