\(\int F^{c (a+b x)} \sqrt {f \text {csch}(d+e x)} \sqrt {g \tanh (d+e x)} \, dx\) [152]

Optimal result

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

2*(1+exp(2*e*x+2*d))^(1/2)*F^(c*(b*x+a))*(f*csch(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))*(g*tanh 
(e*x+d))^(1/2)/(e+2*b*c*ln(F))
 

Mathematica [A] (warning: unable to verify)

Time = 3.42 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.06 \[ \int F^{c (a+b x)} \sqrt {f \text {csch}(d+e x)} \sqrt {g \tanh (d+e x)} \, dx=\frac {4 f F^{c (a+b x)} \cosh (d+e x) (1+\coth (d+e x)) \operatorname {Hypergeometric2F1}\left (1,\frac {3}{4}+\frac {b c \log (F)}{2 e},\frac {5}{4}+\frac {b c \log (F)}{2 e},-\cosh (2 (d+e x))-\sinh (2 (d+e x))\right ) \sqrt {g \tanh (d+e x)}}{\sqrt {f \text {csch}(d+e x)} (e+2 b c \log (F))} \] Input:

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

Output:

(4*f*F^(c*(a + b*x))*Cosh[d + e*x]*(1 + Coth[d + e*x])*Hypergeometric2F1[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[g*Tanh[d + e*x]])/(Sqrt[f*Csch[d + e*x]]*(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.

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

\(\Big \downarrow \) 7271

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

\(\Big \downarrow \) 7271

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

\(\Big \downarrow \) 7292

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

\(\Big \downarrow \) 7299

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

Input:

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

Output:

$Aborted
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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