\(\int \frac {F^{c (a+b x)}}{(f+f \cosh (d+e x))^2} \, dx\) [10]

Optimal result

Integrand size = 22, antiderivative size = 66 \[ \int \frac {F^{c (a+b x)}}{(f+f \cosh (d+e x))^2} \, dx=\frac {4 e^{2 d+2 e x} F^{c (a+b x)} \operatorname {Hypergeometric2F1}\left (4,2+\frac {b c \log (F)}{e},3+\frac {b c \log (F)}{e},-e^{d+e x}\right )}{f^2 (2 e+b c \log (F))} \] Output:

4*exp(2*e*x+2*d)*F^(c*(b*x+a))*hypergeom([4, 2+b*c*ln(F)/e],[3+b*c*ln(F)/e 
],-exp(e*x+d))/f^2/(2*e+b*c*ln(F))
 

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.92 \[ \int \frac {F^{c (a+b x)}}{(f+f \cosh (d+e x))^2} \, dx=\frac {2 F^{c (a+b x)} \cosh \left (\frac {1}{2} (d+e x)\right ) \left (b c \cosh \left (\frac {1}{2} (d+e x)\right ) \log (F)+4 e^{d+e x} \cosh ^3\left (\frac {1}{2} (d+e x)\right ) \operatorname {Hypergeometric2F1}\left (2,1+\frac {b c \log (F)}{e},2+\frac {b c \log (F)}{e},-e^{d+e x}\right ) (e-b c \log (F))+e \sinh \left (\frac {1}{2} (d+e x)\right )\right )}{3 e^2 f^2 (1+\cosh (d+e x))^2} \] Input:

Integrate[F^(c*(a + b*x))/(f + f*Cosh[d + e*x])^2,x]
 

Output:

(2*F^(c*(a + b*x))*Cosh[(d + e*x)/2]*(b*c*Cosh[(d + e*x)/2]*Log[F] + 4*E^( 
d + e*x)*Cosh[(d + e*x)/2]^3*Hypergeometric2F1[2, 1 + (b*c*Log[F])/e, 2 + 
(b*c*Log[F])/e, -E^(d + e*x)]*(e - b*c*Log[F]) + e*Sinh[(d + e*x)/2]))/(3* 
e^2*f^2*(1 + Cosh[d + e*x])^2)
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(164\) vs. \(2(66)=132\).

Time = 0.43 (sec) , antiderivative size = 164, normalized size of antiderivative = 2.48, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6020, 6013, 6015}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {F^{c (a+b x)}}{(f \cosh (d+e x)+f)^2} \, dx\)

\(\Big \downarrow \) 6020

\(\displaystyle \frac {\int F^{c (a+b x)} \text {sech}^4\left (\frac {d}{2}+\frac {e x}{2}\right )dx}{4 f^2}\)

\(\Big \downarrow \) 6013

\(\displaystyle \frac {\frac {2}{3} \left (1-\frac {b^2 c^2 \log ^2(F)}{e^2}\right ) \int F^{c (a+b x)} \text {sech}^2\left (\frac {d}{2}+\frac {e x}{2}\right )dx+\frac {2 b c \log (F) \text {sech}^2\left (\frac {d}{2}+\frac {e x}{2}\right ) F^{c (a+b x)}}{3 e^2}+\frac {2 \tanh \left (\frac {d}{2}+\frac {e x}{2}\right ) \text {sech}^2\left (\frac {d}{2}+\frac {e x}{2}\right ) F^{c (a+b x)}}{3 e}}{4 f^2}\)

\(\Big \downarrow \) 6015

\(\displaystyle \frac {\frac {8 e^{d+e x} F^{c (a+b x)} \left (1-\frac {b^2 c^2 \log ^2(F)}{e^2}\right ) \operatorname {Hypergeometric2F1}\left (2,\frac {b c \log (F)}{e}+1,\frac {b c \log (F)}{e}+2,-e^{d+e x}\right )}{3 (b c \log (F)+e)}+\frac {2 b c \log (F) \text {sech}^2\left (\frac {d}{2}+\frac {e x}{2}\right ) F^{c (a+b x)}}{3 e^2}+\frac {2 \tanh \left (\frac {d}{2}+\frac {e x}{2}\right ) \text {sech}^2\left (\frac {d}{2}+\frac {e x}{2}\right ) F^{c (a+b x)}}{3 e}}{4 f^2}\)

Input:

Int[F^(c*(a + b*x))/(f + f*Cosh[d + e*x])^2,x]
 

Output:

((8*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)]*(1 - (b^2*c^2*Log[F]^2)/e^2))/(3*(e + b*c 
*Log[F])) + (2*b*c*F^(c*(a + b*x))*Log[F]*Sech[d/2 + (e*x)/2]^2)/(3*e^2) + 
 (2*F^(c*(a + b*x))*Sech[d/2 + (e*x)/2]^2*Tanh[d/2 + (e*x)/2])/(3*e))/(4*f 
^2)
 

Defintions of rubi rules used

rule 6013

Int[(F_)^((c_.)*((a_.) + (b_.)*(x_)))*Sech[(d_.) + (e_.)*(x_)]^(n_), x_Symb 
ol] :> Simp[b*c*Log[F]*F^(c*(a + b*x))*(Sech[d + e*x]^(n - 2)/(e^2*(n - 1)* 
(n - 2))), x] + (Simp[F^(c*(a + b*x))*Sech[d + e*x]^(n - 1)*(Sinh[d + e*x]/ 
(e*(n - 1))), x] + Simp[(e^2*(n - 2)^2 - b^2*c^2*Log[F]^2)/(e^2*(n - 1)*(n 
- 2))   Int[F^(c*(a + b*x))*Sech[d + e*x]^(n - 2), x], x]) /; FreeQ[{F, a, 
b, c, d, e}, x] && NeQ[e^2*(n - 2)^2 - b^2*c^2*Log[F]^2, 0] && GtQ[n, 1] && 
 NeQ[n, 2]
 

rule 6015

Int[(F_)^((c_.)*((a_.) + (b_.)*(x_)))*Sech[(d_.) + (e_.)*(x_)]^(n_.), x_Sym 
bol] :> Simp[2^n*E^(n*(d + e*x))*(F^(c*(a + b*x))/(e*n + b*c*Log[F]))*Hyper 
geometric2F1[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 6020

Int[(Cosh[(d_.) + (e_.)*(x_)]*(g_.) + (f_))^(n_.)*(F_)^((c_.)*((a_.) + (b_. 
)*(x_))), x_Symbol] :> Simp[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]
 

Maple [F]

\[\int \frac {F^{c \left (b x +a \right )}}{\left (f +f \cosh \left (e x +d \right )\right )^{2}}d x\]

Input:

int(F^(c*(b*x+a))/(f+f*cosh(e*x+d))^2,x)
 

Output:

int(F^(c*(b*x+a))/(f+f*cosh(e*x+d))^2,x)
 

Fricas [F]

\[ \int \frac {F^{c (a+b x)}}{(f+f \cosh (d+e x))^2} \, dx=\int { \frac {F^{{\left (b x + a\right )} c}}{{\left (f \cosh \left (e x + d\right ) + f\right )}^{2}} \,d x } \] Input:

integrate(F^(c*(b*x+a))/(f+f*cosh(e*x+d))^2,x, algorithm="fricas")
 

Output:

integral(F^(b*c*x + a*c)/(f^2*cosh(e*x + d)^2 + 2*f^2*cosh(e*x + d) + f^2) 
, x)
 

Sympy [F]

\[ \int \frac {F^{c (a+b x)}}{(f+f \cosh (d+e x))^2} \, dx=\frac {\int \frac {F^{a c + b c x}}{\cosh ^{2}{\left (d + e x \right )} + 2 \cosh {\left (d + e x \right )} + 1}\, dx}{f^{2}} \] Input:

integrate(F**(c*(b*x+a))/(f+f*cosh(e*x+d))**2,x)
 

Output:

Integral(F**(a*c + b*c*x)/(cosh(d + e*x)**2 + 2*cosh(d + e*x) + 1), x)/f** 
2
 

Maxima [F]

\[ \int \frac {F^{c (a+b x)}}{(f+f \cosh (d+e x))^2} \, dx=\int { \frac {F^{{\left (b x + a\right )} c}}{{\left (f \cosh \left (e x + d\right ) + f\right )}^{2}} \,d x } \] Input:

integrate(F^(c*(b*x+a))/(f+f*cosh(e*x+d))^2,x, algorithm="maxima")
 

Output:

-16*(F^(a*c)*b^2*c^2*e*log(F)^2 + F^(a*c)*b*c*e^2*log(F))*integrate(F^(b*c 
*x)/(b^3*c^3*f^2*log(F)^3 - 9*b^2*c^2*e*f^2*log(F)^2 + 26*b*c*e^2*f^2*log( 
F) - 24*e^3*f^2 + (b^3*c^3*f^2*e^(5*d)*log(F)^3 - 9*b^2*c^2*e*f^2*e^(5*d)* 
log(F)^2 + 26*b*c*e^2*f^2*e^(5*d)*log(F) - 24*e^3*f^2*e^(5*d))*e^(5*e*x) + 
 5*(b^3*c^3*f^2*e^(4*d)*log(F)^3 - 9*b^2*c^2*e*f^2*e^(4*d)*log(F)^2 + 26*b 
*c*e^2*f^2*e^(4*d)*log(F) - 24*e^3*f^2*e^(4*d))*e^(4*e*x) + 10*(b^3*c^3*f^ 
2*e^(3*d)*log(F)^3 - 9*b^2*c^2*e*f^2*e^(3*d)*log(F)^2 + 26*b*c*e^2*f^2*e^( 
3*d)*log(F) - 24*e^3*f^2*e^(3*d))*e^(3*e*x) + 10*(b^3*c^3*f^2*e^(2*d)*log( 
F)^3 - 9*b^2*c^2*e*f^2*e^(2*d)*log(F)^2 + 26*b*c*e^2*f^2*e^(2*d)*log(F) - 
24*e^3*f^2*e^(2*d))*e^(2*e*x) + 5*(b^3*c^3*f^2*e^d*log(F)^3 - 9*b^2*c^2*e* 
f^2*e^d*log(F)^2 + 26*b*c*e^2*f^2*e^d*log(F) - 24*e^3*f^2*e^d)*e^(e*x)), x 
) + 4*(4*F^(a*c)*b*c*e*log(F) + 4*F^(a*c)*e^2 + (F^(a*c)*b^2*c^2*e^(2*d)*l 
og(F)^2 - 7*F^(a*c)*b*c*e*e^(2*d)*log(F) + 12*F^(a*c)*e^2*e^(2*d))*e^(2*e* 
x) - 4*(F^(a*c)*b*c*e*e^d*log(F) - 4*F^(a*c)*e^2*e^d)*e^(e*x))*F^(b*c*x)/( 
b^3*c^3*f^2*log(F)^3 - 9*b^2*c^2*e*f^2*log(F)^2 + 26*b*c*e^2*f^2*log(F) - 
24*e^3*f^2 + (b^3*c^3*f^2*e^(4*d)*log(F)^3 - 9*b^2*c^2*e*f^2*e^(4*d)*log(F 
)^2 + 26*b*c*e^2*f^2*e^(4*d)*log(F) - 24*e^3*f^2*e^(4*d))*e^(4*e*x) + 4*(b 
^3*c^3*f^2*e^(3*d)*log(F)^3 - 9*b^2*c^2*e*f^2*e^(3*d)*log(F)^2 + 26*b*c*e^ 
2*f^2*e^(3*d)*log(F) - 24*e^3*f^2*e^(3*d))*e^(3*e*x) + 6*(b^3*c^3*f^2*e^(2 
*d)*log(F)^3 - 9*b^2*c^2*e*f^2*e^(2*d)*log(F)^2 + 26*b*c*e^2*f^2*e^(2*d...
 

Giac [F]

\[ \int \frac {F^{c (a+b x)}}{(f+f \cosh (d+e x))^2} \, dx=\int { \frac {F^{{\left (b x + a\right )} c}}{{\left (f \cosh \left (e x + d\right ) + f\right )}^{2}} \,d x } \] Input:

integrate(F^(c*(b*x+a))/(f+f*cosh(e*x+d))^2,x, algorithm="giac")
 

Output:

integrate(F^((b*x + a)*c)/(f*cosh(e*x + d) + f)^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {F^{c (a+b x)}}{(f+f \cosh (d+e x))^2} \, dx=\int \frac {F^{c\,\left (a+b\,x\right )}}{{\left (f+f\,\mathrm {cosh}\left (d+e\,x\right )\right )}^2} \,d x \] Input:

int(F^(c*(a + b*x))/(f + f*cosh(d + e*x))^2,x)
 

Output:

int(F^(c*(a + b*x))/(f + f*cosh(d + e*x))^2, x)
 

Reduce [F]

\[ \int \frac {F^{c (a+b x)}}{(f+f \cosh (d+e x))^2} \, dx=\frac {f^{a c} \left (\int \frac {f^{b c x}}{\cosh \left (e x +d \right )^{2}+2 \cosh \left (e x +d \right )+1}d x \right )}{f^{2}} \] Input:

int(F^(c*(b*x+a))/(f+f*cosh(e*x+d))^2,x)
 

Output:

(f**(a*c)*int(f**(b*c*x)/(cosh(d + e*x)**2 + 2*cosh(d + e*x) + 1),x))/f**2