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

Optimal. Leaf size=61 \[ \frac{2 e^{d+e x} F^{c (a+b x)} \, _2F_1\left (2,\frac{b c \log (F)}{e}+1;\frac{b c \log (F)}{e}+2;-e^{d+e x}\right )}{f (b c \log (F)+e)} \]

[Out]

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

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Rubi [A]  time = 0.0627048, antiderivative size = 61, 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 = {5497, 5492} \[ \frac{2 e^{d+e x} F^{c (a+b x)} \, _2F_1\left (2,\frac{b c \log (F)}{e}+1;\frac{b c \log (F)}{e}+2;-e^{d+e x}\right )}{f (b c \log (F)+e)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5497

Int[(Cosh[(d_.) + (e_.)*(x_)]*(g_.) + (f_))^(n_.)*(F_)^((c_.)*((a_.) + (b_.)*(x_))), x_Symbol] :> Dist[2^n*g^n
, Int[F^(c*(a + b*x))*Cosh[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 5492

Int[(F_)^((c_.)*((a_.) + (b_.)*(x_)))*Sech[(d_.) + (e_.)*(x_)]^(n_.), x_Symbol] :> Simp[(2^n*E^(n*(d + e*x))*F
^(c*(a + b*x))*Hypergeometric2F1[n, n/2 + (b*c*Log[F])/(2*e), 1 + n/2 + (b*c*Log[F])/(2*e), -E^(2*(d + e*x))])
/(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 \cosh (d+e x)} \, dx &=\frac{\int F^{c (a+b x)} \text{sech}^2\left (\frac{d}{2}+\frac{e x}{2}\right ) \, dx}{2 f}\\ &=\frac{2 e^{d+e x} F^{c (a+b x)} \, _2F_1\left (2,1+\frac{b c \log (F)}{e};2+\frac{b c \log (F)}{e};-e^{d+e x}\right )}{f (e+b c \log (F))}\\ \end{align*}

Mathematica [A]  time = 0.0461783, size = 61, normalized size = 1. \[ \frac{2 e^{d+e x} F^{c (a+b x)} \, _2F_1\left (2,\frac{b c \log (F)}{e}+1;\frac{b c \log (F)}{e}+2;-e^{d+e x}\right )}{b c f \log (F)+e f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.033, size = 0, normalized size = 0. \begin{align*} \int{\frac{{F}^{c \left ( bx+a \right ) }}{f+f\cosh \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*cosh(e*x+d)),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} 4 \, F^{a c} b c e \int \frac{F^{b c x}}{b^{2} c^{2} f \log \left (F\right )^{2} - 3 \, b c e f \log \left (F\right ) + 2 \, e^{2} f +{\left (b^{2} c^{2} f e^{\left (3 \, d\right )} \log \left (F\right )^{2} - 3 \, b c e f e^{\left (3 \, d\right )} \log \left (F\right ) + 2 \, e^{2} f e^{\left (3 \, d\right )}\right )} e^{\left (3 \, e x\right )} + 3 \,{\left (b^{2} c^{2} f e^{\left (2 \, d\right )} \log \left (F\right )^{2} - 3 \, b c e f e^{\left (2 \, d\right )} \log \left (F\right ) + 2 \, e^{2} f e^{\left (2 \, d\right )}\right )} e^{\left (2 \, e x\right )} + 3 \,{\left (b^{2} c^{2} f e^{d} \log \left (F\right )^{2} - 3 \, b c e f e^{d} \log \left (F\right ) + 2 \, e^{2} f e^{d}\right )} e^{\left (e x\right )}}\,{d x} \log \left (F\right ) - \frac{2 \,{\left (2 \, F^{a c} e -{\left (F^{a c} b c e^{d} \log \left (F\right ) - 2 \, F^{a c} e e^{d}\right )} e^{\left (e x\right )}\right )} F^{b c x}}{b^{2} c^{2} f \log \left (F\right )^{2} - 3 \, b c e f \log \left (F\right ) + 2 \, e^{2} f +{\left (b^{2} c^{2} f e^{\left (2 \, d\right )} \log \left (F\right )^{2} - 3 \, b c e f e^{\left (2 \, d\right )} \log \left (F\right ) + 2 \, e^{2} f e^{\left (2 \, d\right )}\right )} e^{\left (2 \, e x\right )} + 2 \,{\left (b^{2} c^{2} f e^{d} \log \left (F\right )^{2} - 3 \, b c e f e^{d} \log \left (F\right ) + 2 \, e^{2} f e^{d}\right )} e^{\left (e x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*F^(a*c)*b*c*e*integrate(F^(b*c*x)/(b^2*c^2*f*log(F)^2 - 3*b*c*e*f*log(F) + 2*e^2*f + (b^2*c^2*f*e^(3*d)*log(
F)^2 - 3*b*c*e*f*e^(3*d)*log(F) + 2*e^2*f*e^(3*d))*e^(3*e*x) + 3*(b^2*c^2*f*e^(2*d)*log(F)^2 - 3*b*c*e*f*e^(2*
d)*log(F) + 2*e^2*f*e^(2*d))*e^(2*e*x) + 3*(b^2*c^2*f*e^d*log(F)^2 - 3*b*c*e*f*e^d*log(F) + 2*e^2*f*e^d)*e^(e*
x)), x)*log(F) - 2*(2*F^(a*c)*e - (F^(a*c)*b*c*e^d*log(F) - 2*F^(a*c)*e*e^d)*e^(e*x))*F^(b*c*x)/(b^2*c^2*f*log
(F)^2 - 3*b*c*e*f*log(F) + 2*e^2*f + (b^2*c^2*f*e^(2*d)*log(F)^2 - 3*b*c*e*f*e^(2*d)*log(F) + 2*e^2*f*e^(2*d))
*e^(2*e*x) + 2*(b^2*c^2*f*e^d*log(F)^2 - 3*b*c*e*f*e^d*log(F) + 2*e^2*f*e^d)*e^(e*x))

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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 \cosh \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*cosh(e*x+d)),x, algorithm="fricas")

[Out]

integral(F^(b*c*x + a*c)/(f*cosh(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}}{\cosh{\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*cosh(e*x+d)),x)

[Out]

Integral(F**(a*c)*F**(b*c*x)/(cosh(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 \cosh \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*cosh(e*x+d)),x, algorithm="giac")

[Out]

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