\(\int F^{c (a+b x)} \sqrt {f \text {sech}(d+e x)} \sqrt {g \text {sech}(d+e x)} \, dx\) [142]

Optimal result

Integrand size = 34, antiderivative size = 94 \[ \int F^{c (a+b x)} \sqrt {f \text {sech}(d+e x)} \sqrt {g \text {sech}(d+e x)} \, dx=\frac {2 e^{d+e x} f F^{c (a+b x)} \operatorname {Hypergeometric2F1}\left (1,\frac {e+b c \log (F)}{2 e},\frac {1}{2} \left (3+\frac {b c \log (F)}{e}\right ),-e^{2 d+2 e x}\right ) \sqrt {g \text {sech}(d+e x)}}{(e+b c \log (F)) \sqrt {f \text {sech}(d+e x)}} \] Output:

2*exp(e*x+d)*f*F^(c*(b*x+a))*hypergeom([1, 1/2*(e+b*c*ln(F))/e],[3/2+1/2*b 
*c*ln(F)/e],-exp(2*e*x+2*d))*(g*sech(e*x+d))^(1/2)/(e+b*c*ln(F))/(f*sech(e 
*x+d))^(1/2)
 

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.01 \[ \int F^{c (a+b x)} \sqrt {f \text {sech}(d+e x)} \sqrt {g \text {sech}(d+e x)} \, dx=\frac {\left (1+e^{2 (d+e x)}\right ) F^{c (a+b x)} \operatorname {Hypergeometric2F1}\left (1,\frac {e+b c \log (F)}{2 e},\frac {1}{2} \left (3+\frac {b c \log (F)}{e}\right ),-e^{2 (d+e x)}\right ) \sqrt {f \text {sech}(d+e x)} \sqrt {g \text {sech}(d+e x)}}{e+b c \log (F)} \] Input:

Integrate[F^(c*(a + b*x))*Sqrt[f*Sech[d + e*x]]*Sqrt[g*Sech[d + e*x]],x]
 

Output:

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

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.99, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {2031, 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.

\(\displaystyle \int F^{c (a+b x)} \sqrt {f \text {sech}(d+e x)} \sqrt {g \text {sech}(d+e x)} \, dx\)

\(\Big \downarrow \) 2031

\(\displaystyle \frac {f \sqrt {g \text {sech}(d+e x)} \int F^{c (a+b x)} \text {sech}(d+e x)dx}{\sqrt {f \text {sech}(d+e x)}}\)

\(\Big \downarrow \) 6015

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

Input:

Int[F^(c*(a + b*x))*Sqrt[f*Sech[d + e*x]]*Sqrt[g*Sech[d + e*x]],x]
 

Output:

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

Defintions of rubi rules used

rule 2031

Int[(Fx_.)*((a_.)*(v_))^(m_)*((b_.)*(v_))^(n_), x_Symbol] :> Simp[a^(m + 1/ 
2)*b^(n - 1/2)*(Sqrt[b*v]/Sqrt[a*v])   Int[v^(m + n)*Fx, x], x] /; FreeQ[{a 
, b, m}, x] &&  !IntegerQ[m] && IGtQ[n + 1/2, 0] && IntegerQ[m + n]
 

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]
 

Maple [F]

\[\int F^{c \left (b x +a \right )} \sqrt {f \,\operatorname {sech}\left (e x +d \right )}\, \sqrt {g \,\operatorname {sech}\left (e x +d \right )}d x\]

Input:

int(F^(c*(b*x+a))*(f*sech(e*x+d))^(1/2)*(g*sech(e*x+d))^(1/2),x)
 

Output:

int(F^(c*(b*x+a))*(f*sech(e*x+d))^(1/2)*(g*sech(e*x+d))^(1/2),x)
 

Fricas [F]

\[ \int F^{c (a+b x)} \sqrt {f \text {sech}(d+e x)} \sqrt {g \text {sech}(d+e x)} \, dx=\int { \sqrt {f \operatorname {sech}\left (e x + d\right )} \sqrt {g \operatorname {sech}\left (e x + d\right )} F^{{\left (b x + a\right )} c} \,d x } \] Input:

integrate(F^(c*(b*x+a))*(f*sech(e*x+d))^(1/2)*(g*sech(e*x+d))^(1/2),x, alg 
orithm="fricas")
 

Output:

integral(sqrt(f*sech(e*x + d))*sqrt(g*sech(e*x + d))*F^(b*c*x + a*c), x)
 

Sympy [F]

\[ \int F^{c (a+b x)} \sqrt {f \text {sech}(d+e x)} \sqrt {g \text {sech}(d+e x)} \, dx=\int F^{c \left (a + b x\right )} \sqrt {f \operatorname {sech}{\left (d + e x \right )}} \sqrt {g \operatorname {sech}{\left (d + e x \right )}}\, dx \] Input:

integrate(F**(c*(b*x+a))*(f*sech(e*x+d))**(1/2)*(g*sech(e*x+d))**(1/2),x)
 

Output:

Integral(F**(c*(a + b*x))*sqrt(f*sech(d + e*x))*sqrt(g*sech(d + e*x)), x)
 

Maxima [F]

\[ \int F^{c (a+b x)} \sqrt {f \text {sech}(d+e x)} \sqrt {g \text {sech}(d+e x)} \, dx=\int { \sqrt {f \operatorname {sech}\left (e x + d\right )} \sqrt {g \operatorname {sech}\left (e x + d\right )} F^{{\left (b x + a\right )} c} \,d x } \] Input:

integrate(F^(c*(b*x+a))*(f*sech(e*x+d))^(1/2)*(g*sech(e*x+d))^(1/2),x, alg 
orithm="maxima")
 

Output:

integrate(sqrt(f*sech(e*x + d))*sqrt(g*sech(e*x + d))*F^((b*x + a)*c), x)
 

Giac [F]

\[ \int F^{c (a+b x)} \sqrt {f \text {sech}(d+e x)} \sqrt {g \text {sech}(d+e x)} \, dx=\int { \sqrt {f \operatorname {sech}\left (e x + d\right )} \sqrt {g \operatorname {sech}\left (e x + d\right )} F^{{\left (b x + a\right )} c} \,d x } \] Input:

integrate(F^(c*(b*x+a))*(f*sech(e*x+d))^(1/2)*(g*sech(e*x+d))^(1/2),x, alg 
orithm="giac")
 

Output:

integrate(sqrt(f*sech(e*x + d))*sqrt(g*sech(e*x + d))*F^((b*x + a)*c), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

int(F^(c*(b*x+a))*(f*sech(e*x+d))^(1/2)*(g*sech(e*x+d))^(1/2),x)
 

Output:

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