\(\int F^{c (a+b x)} \sqrt {g \coth (d+e x)} \sqrt {f \text {sech}(d+e x)} \, dx\) [157]

Optimal result

Integrand size = 34, antiderivative size = 107 \[ \int F^{c (a+b x)} \sqrt {g \coth (d+e x)} \sqrt {f \text {sech}(d+e x)} \, dx=\frac {2 \sqrt {1-e^{2 d+2 e x}} F^{c (a+b x)} \sqrt {g \coth (d+e x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {e+2 b c \log (F)}{4 e},\frac {1}{4} \left (5+\frac {2 b c \log (F)}{e}\right ),e^{2 d+2 e x}\right ) \sqrt {f \text {sech}(d+e x)}}{e+2 b c \log (F)} \] Output:

2*(1-exp(2*e*x+2*d))^(1/2)*F^(c*(b*x+a))*(g*coth(e*x+d))^(1/2)*hypergeom([ 
1/2, 1/4*(e+2*b*c*ln(F))/e],[5/4+1/2*b*c*ln(F)/e],exp(2*e*x+2*d))*(f*sech( 
e*x+d))^(1/2)/(e+2*b*c*ln(F))
 

Mathematica [A] (warning: unable to verify)

Time = 3.31 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.03 \[ \int F^{c (a+b x)} \sqrt {g \coth (d+e x)} \sqrt {f \text {sech}(d+e x)} \, dx=-\frac {4 F^{c (a+b x)} \sqrt {g \coth (d+e x)} (1+\coth (d+e x)) \operatorname {Hypergeometric2F1}\left (1,\frac {3}{4}+\frac {b c \log (F)}{2 e},\frac {5}{4}+\frac {b c \log (F)}{2 e},\cosh (2 (d+e x))+\sinh (2 (d+e x))\right ) \sqrt {f \text {sech}(d+e x)} \sinh ^2(d+e x)}{e+2 b c \log (F)} \] Input:

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

Output:

(-4*F^(c*(a + b*x))*Sqrt[g*Coth[d + e*x]]*(1 + Coth[d + e*x])*Hypergeometr 
ic2F1[1, 3/4 + (b*c*Log[F])/(2*e), 5/4 + (b*c*Log[F])/(2*e), Cosh[2*(d + e 
*x)] + Sinh[2*(d + e*x)]]*Sqrt[f*Sech[d + e*x]]*Sinh[d + e*x]^2)/(e + 2*b* 
c*Log[F])
 

Rubi [F]

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.

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

\(\Big \downarrow \) 7271

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

\(\Big \downarrow \) 7271

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

\(\Big \downarrow \) 7292

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

\(\Big \downarrow \) 7299

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

Input:

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

Output:

$Aborted
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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