3.141 \(\int \frac{F^{c (a+b x)}}{f+f \cos (d+e x)} \, dx\)

Optimal. Leaf size=79 \[ \frac{2 e^{i (d+e x)} F^{c (a+b x)} \text{Hypergeometric2F1}\left (2,1-\frac{i b c \log (F)}{e},2-\frac{i b c \log (F)}{e},-e^{i (d+e x)}\right )}{f (b c \log (F)+i e)} \]

[Out]

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

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Rubi [A]  time = 0.0602341, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {4457, 4451} \[ \frac{2 e^{i (d+e x)} F^{c (a+b x)} \, _2F_1\left (2,1-\frac{i b c \log (F)}{e};2-\frac{i b c \log (F)}{e};-e^{i (d+e x)}\right )}{f (b c \log (F)+i e)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 4457

Int[(Cos[(d_.) + (e_.)*(x_)]*(g_.) + (f_))^(n_.)*(F_)^((c_.)*((a_.) + (b_.)*(x_))), x_Symbol] :> Dist[2^n*f^n,
Int[F^(c*(a + b*x))*Cos[d/2 + (e*x)/2]^(2*n), x], x] /; FreeQ[{F, a, b, c, d, e, f, g}, x] && EqQ[f - g, 0] &
& ILtQ[n, 0]

Rule 4451

Int[(F_)^((c_.)*((a_.) + (b_.)*(x_)))*Sec[(d_.) + (e_.)*(x_)]^(n_.), x_Symbol] :> Simp[(2^n*E^(I*n*(d + e*x))*
F^(c*(a + b*x))*Hypergeometric2F1[n, n/2 - (I*b*c*Log[F])/(2*e), 1 + n/2 - (I*b*c*Log[F])/(2*e), -E^(2*I*(d +
e*x))])/(I*e*n + b*c*Log[F]), x] /; FreeQ[{F, a, b, c, d, e}, x] && IntegerQ[n]

Rubi steps

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

Mathematica [A]  time = 0.0501889, size = 80, normalized size = 1.01 \[ -\frac{2 i e^{i (d+e x)} F^{c (a+b x)} \text{Hypergeometric2F1}\left (2,1-\frac{i b c \log (F)}{e},2-\frac{i b c \log (F)}{e},-e^{i (d+e x)}\right )}{f (e-i b c \log (F))} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.056, size = 0, normalized size = 0. \begin{align*} \int{\frac{{F}^{c \left ( bx+a \right ) }}{f+f\cos \left ( ex+d \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [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))/(f+f*cos(e*x+d)),x, algorithm="maxima")

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{F^{a c} F^{b c x}}{\cos{\left (d + e x \right )} + 1}\, dx}{f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(F**(a*c)*F**(b*c*x)/(cos(d + e*x) + 1), x)/f

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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