\(\int F^{c (a+b x)} \sqrt {f \cosh (d+e x)} \sqrt {g \coth (d+e x)} \, dx\) [155]

Optimal result

Integrand size = 34, antiderivative size = 207 \[ \int F^{c (a+b x)} \sqrt {f \cosh (d+e x)} \sqrt {g \coth (d+e x)} \, dx=-\frac {2 \left (1-e^{2 d+2 e x}\right ) F^{c (a+b x)} \sqrt {f \cosh (d+e x)} \sqrt {g \coth (d+e x)}}{\left (1+e^{2 d+2 e x}\right ) (e+2 b c \log (F))}-\frac {8 b c \sqrt {1-e^{2 d+2 e x}} F^{c (a+b x)} \sqrt {f \cosh (d+e x)} \sqrt {g \coth (d+e x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{4} \left (-1+\frac {2 b c \log (F)}{e}\right ),\frac {1}{4} \left (3+\frac {2 b c \log (F)}{e}\right ),e^{2 d+2 e x}\right ) \log (F)}{\left (1+e^{2 d+2 e x}\right ) \left (e^2-4 b^2 c^2 \log ^2(F)\right )} \] Output:

-2*(1-exp(2*e*x+2*d))*F^(c*(b*x+a))*(f*cosh(e*x+d))^(1/2)*(g*coth(e*x+d))^ 
(1/2)/(1+exp(2*e*x+2*d))/(e+2*b*c*ln(F))-8*b*c*(1-exp(2*e*x+2*d))^(1/2)*F^ 
(c*(b*x+a))*(f*cosh(e*x+d))^(1/2)*(g*coth(e*x+d))^(1/2)*hypergeom([1/2, -1 
/4+1/2*b*c*ln(F)/e],[3/4+1/2*b*c*ln(F)/e],exp(2*e*x+2*d))*ln(F)/(1+exp(2*e 
*x+2*d))/(e^2-4*b^2*c^2*ln(F)^2)
 

Mathematica [A] (warning: unable to verify)

Time = 2.11 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.94 \[ \int F^{c (a+b x)} \sqrt {f \cosh (d+e x)} \sqrt {g \coth (d+e x)} \, dx=\frac {2 F^{c (a+b x)} g \sqrt {f \cosh (d+e x)} \left (\operatorname {Hypergeometric2F1}\left (1,\frac {e+2 b c \log (F)}{4 e},\frac {3}{4}+\frac {b c \log (F)}{2 e},\cosh (2 (d+e x))+\sinh (2 (d+e x))\right ) (3 e+2 b c \log (F))-\operatorname {Hypergeometric2F1}\left (1,\frac {5}{4}+\frac {b c \log (F)}{2 e},\frac {7}{4}+\frac {b c \log (F)}{2 e},\cosh (2 (d+e x))+\sinh (2 (d+e x))\right ) (e-2 b c \log (F)) (\cosh (2 (d+e x))+\sinh (2 (d+e x)))\right )}{\sqrt {g \coth (d+e x)} (e-2 b c \log (F)) (3 e+2 b c \log (F))} \] Input:

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

Output:

(2*F^(c*(a + b*x))*g*Sqrt[f*Cosh[d + e*x]]*(Hypergeometric2F1[1, (e + 2*b* 
c*Log[F])/(4*e), 3/4 + (b*c*Log[F])/(2*e), Cosh[2*(d + e*x)] + Sinh[2*(d + 
 e*x)]]*(3*e + 2*b*c*Log[F]) - Hypergeometric2F1[1, 5/4 + (b*c*Log[F])/(2* 
e), 7/4 + (b*c*Log[F])/(2*e), Cosh[2*(d + e*x)] + Sinh[2*(d + e*x)]]*(e - 
2*b*c*Log[F])*(Cosh[2*(d + e*x)] + Sinh[2*(d + e*x)])))/(Sqrt[g*Coth[d + e 
*x]]*(e - 2*b*c*Log[F])*(3*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.

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

\(\Big \downarrow \) 7271

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

\(\Big \downarrow \) 7271

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

\(\Big \downarrow \) 7292

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

\(\Big \downarrow \) 7299

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

Input:

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

Output:

$Aborted
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(sqrt(g)*f**((2*a*c + 1)/2)*(2*f**(b*c*x)*sqrt(coth(d + e*x))*sqrt(cosh(d 
+ e*x)) - int((f**(b*c*x)*sqrt(coth(d + e*x))*sqrt(cosh(d + e*x))*sinh(d + 
 e*x))/cosh(d + e*x),x)*e - int((f**(b*c*x)*sqrt(coth(d + e*x))*sqrt(cosh( 
d + e*x)))/coth(d + e*x),x)*e + int(f**(b*c*x)*sqrt(coth(d + e*x))*sqrt(co 
sh(d + e*x))*coth(d + e*x),x)*e))/(2*log(f)*b*c)