3.9.99 \(\int \frac {F^{c (a+b x)}}{f+f \cosh (d+e x)} \, dx\) [899]

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

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 61, normalized size of antiderivative = 1.00, 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 = {5605, 5600} \begin {gather*} \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)} \end {gather*}

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 5600

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))/(e*n + b*c*Log[F]))*Hypergeometric2F1[n, n/2 + b*c*(Log[F]/(2*e)), 1 + n/2 + b*c*(Log[F]/(2*e))
, -E^(2*(d + e*x))], x] /; FreeQ[{F, a, b, c, d, e}, x] && IntegerQ[n]

Rule 5605

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]

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*}

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Mathematica [A]
time = 0.03, size = 61, normalized size = 1.00 \begin {gather*} \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 )}{e f+b c f \log (F)} \end {gather*}

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 = 0.76, size = 0, normalized size = 0.00 \[\int \frac {F^{c \left (b x +a \right )}}{f +f \cosh \left (e x +d \right )}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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*integrate(e^(b*c*x*log(F) + 1)/(b^2*c^2*f*log(F)^2 - 3*b*c*f*e*log(F) + 2*f*e^2 + (b^2*c^2*f*e^(
3*d)*log(F)^2 - 3*b*c*f*e^(3*d + 1)*log(F) + 2*f*e^(3*d + 2))*e^(3*x*e) + 3*(b^2*c^2*f*e^(2*d)*log(F)^2 - 3*b*
c*f*e^(2*d + 1)*log(F) + 2*f*e^(2*d + 2))*e^(2*x*e) + 3*(b^2*c^2*f*e^d*log(F)^2 - 3*b*c*f*e^(d + 1)*log(F) + 2
*f*e^(d + 2))*e^(x*e)), x)*log(F) - 2*(2*F^(a*c)*e - (F^(a*c)*b*c*e^d*log(F) - 2*F^(a*c)*e^(d + 1))*e^(x*e))*F
^(b*c*x)/(b^2*c^2*f*log(F)^2 - 3*b*c*f*e*log(F) + 2*f*e^2 + (b^2*c^2*f*e^(2*d)*log(F)^2 - 3*b*c*f*e^(2*d + 1)*
log(F) + 2*f*e^(2*d + 2))*e^(2*x*e) + 2*(b^2*c^2*f*e^d*log(F)^2 - 3*b*c*f*e^(d + 1)*log(F) + 2*f*e^(d + 2))*e^
(x*e))

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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(x*e + d) + f), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {F^{a c} F^{b c x}}{\cosh {\left (d + e x \right )} + 1}\, dx}{f} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {F^{c\,\left (a+b\,x\right )}}{f+f\,\mathrm {cosh}\left (d+e\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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