\(\int \frac {F^{c (a+b x)}}{(g \cosh (d+e x)+f \sinh (d+e x))^2} \, dx\) [26]

Optimal result

Integrand size = 29, antiderivative size = 90 \[ \int \frac {F^{c (a+b x)}}{(g \cosh (d+e x)+f \sinh (d+e x))^2} \, dx=\frac {4 e^{2 d+2 e x} F^{c (a+b x)} \operatorname {Hypergeometric2F1}\left (2,\frac {1}{2} \left (2+\frac {b c \log (F)}{e}\right ),\frac {1}{2} \left (4+\frac {b c \log (F)}{e}\right ),\frac {e^{2 d+2 e x} (f+g)}{f-g}\right )}{(f-g)^2 (2 e+b c \log (F))} \] Output:

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

Mathematica [F]

\[ \int \frac {F^{c (a+b x)}}{(g \cosh (d+e x)+f \sinh (d+e x))^2} \, dx=\int \frac {F^{c (a+b x)}}{(g \cosh (d+e x)+f \sinh (d+e x))^2} \, dx \] Input:

Integrate[F^(c*(a + b*x))/(g*Cosh[d + e*x] + f*Sinh[d + e*x])^2,x]
 

Output:

Integrate[F^(c*(a + b*x))/(g*Cosh[d + e*x] + f*Sinh[d + e*x])^2, 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.

\(\displaystyle \int \frac {F^{c (a+b x)}}{(f \sinh (d+e x)+g \cosh (d+e x))^2} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {F^{a c+b c x}}{(f \sinh (d+e x)+g \cosh (d+e x))^2}dx\)

\(\Big \downarrow \) 7299

\(\displaystyle \int \frac {F^{a c+b c x}}{(f \sinh (d+e x)+g \cosh (d+e x))^2}dx\)

Input:

Int[F^(c*(a + b*x))/(g*Cosh[d + e*x] + f*Sinh[d + e*x])^2,x]
 

Output:

$Aborted
 

Maple [F]

\[\int \frac {F^{c \left (b x +a \right )}}{\left (g \cosh \left (e x +d \right )+f \sinh \left (e x +d \right )\right )^{2}}d x\]

Input:

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

Output:

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

Fricas [F]

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

integrate(F^(c*(b*x+a))/(g*cosh(e*x+d)+f*sinh(e*x+d))^2,x, algorithm="fric 
as")
                                                                                    
                                                                                    
 

Output:

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

Sympy [F]

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

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

Output:

Integral(F**(c*(a + b*x))/(f*sinh(d + e*x) + g*cosh(d + e*x))**2, x)
 

Maxima [F]

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

integrate(F^(c*(b*x+a))/(g*cosh(e*x+d)+f*sinh(e*x+d))^2,x, algorithm="maxi 
ma")
 

Output:

16*(f^2 - 2*f*g + g^2)*F^(a*c)*b*c*e*integrate(-F^(b*c*x)/((f^4 - 2*f^3*g 
+ 2*f*g^3 - g^4)*b^2*c^2*log(F)^2 - 6*(f^4 - 2*f^3*g + 2*f*g^3 - g^4)*b*c* 
e*log(F) + 8*(f^4 - 2*f^3*g + 2*f*g^3 - g^4)*e^2 - ((f^4 + 4*f^3*g + 6*f^2 
*g^2 + 4*f*g^3 + g^4)*b^2*c^2*e^(6*d)*log(F)^2 - 6*(f^4 + 4*f^3*g + 6*f^2* 
g^2 + 4*f*g^3 + g^4)*b*c*e*e^(6*d)*log(F) + 8*(f^4 + 4*f^3*g + 6*f^2*g^2 + 
 4*f*g^3 + g^4)*e^2*e^(6*d))*e^(6*e*x) + 3*((f^4 + 2*f^3*g - 2*f*g^3 - g^4 
)*b^2*c^2*e^(4*d)*log(F)^2 - 6*(f^4 + 2*f^3*g - 2*f*g^3 - g^4)*b*c*e*e^(4* 
d)*log(F) + 8*(f^4 + 2*f^3*g - 2*f*g^3 - g^4)*e^2*e^(4*d))*e^(4*e*x) - 3*( 
(f^4 - 2*f^2*g^2 + g^4)*b^2*c^2*e^(2*d)*log(F)^2 - 6*(f^4 - 2*f^2*g^2 + g^ 
4)*b*c*e*e^(2*d)*log(F) + 8*(f^4 - 2*f^2*g^2 + g^4)*e^2*e^(2*d))*e^(2*e*x) 
), x)*log(F) + 4*(4*F^(a*c)*e*(f - g) + (F^(a*c)*b*c*(f + g)*e^(2*d)*log(F 
) - 4*F^(a*c)*e*(f + g)*e^(2*d))*e^(2*e*x))*F^(b*c*x)/((f^3 - f^2*g - f*g^ 
2 + g^3)*b^2*c^2*log(F)^2 - 6*(f^3 - f^2*g - f*g^2 + g^3)*b*c*e*log(F) + 8 
*(f^3 - f^2*g - f*g^2 + g^3)*e^2 + ((f^3 + 3*f^2*g + 3*f*g^2 + g^3)*b^2*c^ 
2*e^(4*d)*log(F)^2 - 6*(f^3 + 3*f^2*g + 3*f*g^2 + g^3)*b*c*e*e^(4*d)*log(F 
) + 8*(f^3 + 3*f^2*g + 3*f*g^2 + g^3)*e^2*e^(4*d))*e^(4*e*x) - 2*((f^3 + f 
^2*g - f*g^2 - g^3)*b^2*c^2*e^(2*d)*log(F)^2 - 6*(f^3 + f^2*g - f*g^2 - g^ 
3)*b*c*e*e^(2*d)*log(F) + 8*(f^3 + f^2*g - f*g^2 - g^3)*e^2*e^(2*d))*e^(2* 
e*x))
 

Giac [F]

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

integrate(F^(c*(b*x+a))/(g*cosh(e*x+d)+f*sinh(e*x+d))^2,x, algorithm="giac 
")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {F^{c (a+b x)}}{(g \cosh (d+e x)+f \sinh (d+e x))^2} \, dx=\int \frac {F^{c\,\left (a+b\,x\right )}}{{\left (g\,\mathrm {cosh}\left (d+e\,x\right )+f\,\mathrm {sinh}\left (d+e\,x\right )\right )}^2} \,d x \] Input:

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

Output:

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

Reduce [F]

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

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

Output:

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