\(\int \frac {F^{c (a+b x)}}{g \cosh (d+e x)+f \sinh (d+e x)} \, dx\) [25]

Optimal result

Integrand size = 29, antiderivative size = 85 \[ \int \frac {F^{c (a+b x)}}{g \cosh (d+e x)+f \sinh (d+e x)} \, dx=-\frac {2 e^{d+e x} F^{c (a+b x)} \operatorname {Hypergeometric2F1}\left (1,\frac {e+b c \log (F)}{2 e},\frac {1}{2} \left (3+\frac {b c \log (F)}{e}\right ),\frac {e^{2 d+2 e x} (f+g)}{f-g}\right )}{(f-g) (e+b c \log (F))} \] Output:

-2*exp(e*x+d)*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)*(f+g)/(f-g))/(f-g)/(e+b*c*ln(F))
 

Mathematica [F]

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

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

Output:

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

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.

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

\(\Big \downarrow \) 7292

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

\(\Big \downarrow \) 7299

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

Input:

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

Output:

$Aborted
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

4*F^(a*c)*e*(f - g)*integrate(e^(b*c*x*log(F) + e*x + d)/((f^2 - 2*f*g + g 
^2)*b*c*log(F) - (f^2 - 2*f*g + g^2)*e + ((f^2 + 2*f*g + g^2)*b*c*e^(4*d)* 
log(F) - (f^2 + 2*f*g + g^2)*e*e^(4*d))*e^(4*e*x) - 2*((f^2 - g^2)*b*c*e^( 
2*d)*log(F) - (f^2 - g^2)*e*e^(2*d))*e^(2*e*x)), x) - 2*F^(a*c)*e^(b*c*x*l 
og(F) + e*x + d)/(b*c*(f - g)*log(F) - e*(f - g) - (b*c*(f + g)*e^(2*d)*lo 
g(F) - e*(f + g)*e^(2*d))*e^(2*e*x))
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

f**(a*c)*int(f**(b*c*x)/(cosh(d + e*x)*g + sinh(d + e*x)*f),x)