\(\int e^{a+b x} (g \cosh (a+b x)+f \sinh (a+b x))^n \, dx\) [32]

Optimal result

Integrand size = 27, antiderivative size = 108 \[ \int e^{a+b x} (g \cosh (a+b x)+f \sinh (a+b x))^n \, dx=\frac {e^{a+b x} \left (1-\frac {e^{2 a+2 b x} (f+g)}{f-g}\right )^{-n} \operatorname {Hypergeometric2F1}\left (\frac {1-n}{2},-n,\frac {3-n}{2},\frac {e^{2 a+2 b x} (f+g)}{f-g}\right ) (g \cosh (a+b x)+f \sinh (a+b x))^n}{b (1-n)} \] Output:

exp(b*x+a)*hypergeom([-n, 1/2-1/2*n],[3/2-1/2*n],exp(2*b*x+2*a)*(f+g)/(f-g 
))*(g*cosh(b*x+a)+f*sinh(b*x+a))^n/b/((1-exp(2*b*x+2*a)*(f+g)/(f-g))^n)/(1 
-n)
 

Mathematica [F]

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

Integrate[E^(a + b*x)*(g*Cosh[a + b*x] + f*Sinh[a + b*x])^n,x]
 

Output:

Integrate[E^(a + b*x)*(g*Cosh[a + b*x] + f*Sinh[a + b*x])^n, 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.

\(\displaystyle \int e^{a+b x} (f \sinh (a+b x)+g \cosh (a+b x))^n \, dx\)

\(\Big \downarrow \) 2720

\(\displaystyle \frac {\int 2^{-n} \left (e^{-a-b x} \left (1+e^{2 a+2 b x}\right ) g-e^{-a-b x} \left (1-e^{2 a+2 b x}\right ) f\right )^nde^{a+b x}}{b}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2^{-n} \int \left (e^{-a-b x} \left (1+e^{2 a+2 b x}\right ) g-e^{-a-b x} \left (1-e^{2 a+2 b x}\right ) f\right )^nde^{a+b x}}{b}\)

\(\Big \downarrow \) 7299

\(\displaystyle \frac {2^{-n} \int \left (e^{-a-b x} \left (1+e^{2 a+2 b x}\right ) g-e^{-a-b x} \left (1-e^{2 a+2 b x}\right ) f\right )^nde^{a+b x}}{b}\)

Input:

Int[E^(a + b*x)*(g*Cosh[a + b*x] + f*Sinh[a + b*x])^n,x]
 

Output:

$Aborted
 

Maple [F]

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

Input:

int(exp(b*x+a)*(g*cosh(b*x+a)+f*sinh(b*x+a))^n,x)
 

Output:

int(exp(b*x+a)*(g*cosh(b*x+a)+f*sinh(b*x+a))^n,x)
 

Fricas [F]

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

integrate(exp(b*x+a)*(g*cosh(b*x+a)+f*sinh(b*x+a))^n,x, algorithm="fricas" 
)
                                                                                    
                                                                                    
 

Output:

integral((g*cosh(b*x + a) + f*sinh(b*x + a))^n*e^(b*x + a), x)
 

Sympy [F]

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

integrate(exp(b*x+a)*(g*cosh(b*x+a)+f*sinh(b*x+a))**n,x)
 

Output:

exp(a)*Integral((f*sinh(a + b*x) + g*cosh(a + b*x))**n*exp(b*x), x)
 

Maxima [F]

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

integrate(exp(b*x+a)*(g*cosh(b*x+a)+f*sinh(b*x+a))^n,x, algorithm="maxima" 
)
 

Output:

integrate((g*cosh(b*x + a) + f*sinh(b*x + a))^n*e^(b*x + a), x)
 

Giac [F]

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

integrate(exp(b*x+a)*(g*cosh(b*x+a)+f*sinh(b*x+a))^n,x, algorithm="giac")
 

Output:

integrate((g*cosh(b*x + a) + f*sinh(b*x + a))^n*e^(b*x + a), x)
 

Mupad [F(-1)]

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

int(exp(a + b*x)*(g*cosh(a + b*x) + f*sinh(a + b*x))^n,x)
 

Output:

int(exp(a + b*x)*(g*cosh(a + b*x) + f*sinh(a + b*x))^n, x)
 

Reduce [F]

\[ \int e^{a+b x} (g \cosh (a+b x)+f \sinh (a+b x))^n \, dx =\text {Too large to display} \] Input:

int(exp(b*x+a)*(g*cosh(b*x+a)+f*sinh(b*x+a))^n,x)
 

Output:

(e**a*(e**(a*n + b*n*x + b*x)*(cosh(a + b*x)*g + sinh(a + b*x)*f)**n*2**n* 
g*n + e**(a*n + b*n*x + b*x)*(cosh(a + b*x)*g + sinh(a + b*x)*f)**n*2**n*g 
 + e**(b*x)*(e**(2*a + 2*b*x)*f + e**(2*a + 2*b*x)*g - f + g)**n*f*n - e** 
(b*x)*(e**(2*a + 2*b*x)*f + e**(2*a + 2*b*x)*g - f + g)**n*g*n - 2*e**(a*n 
 + b*n*x)*int((e**(b*x)*(e**(2*a + 2*b*x)*f + e**(2*a + 2*b*x)*g - f + g)* 
*n)/(e**(a*n + 2*a + b*n*x + 2*b*x)*f*n + e**(a*n + 2*a + b*n*x + 2*b*x)*f 
 + e**(a*n + 2*a + b*n*x + 2*b*x)*g*n + e**(a*n + 2*a + b*n*x + 2*b*x)*g - 
 e**(a*n + b*n*x)*f*n - e**(a*n + b*n*x)*f + e**(a*n + b*n*x)*g*n + e**(a* 
n + b*n*x)*g),x)*b*f**2*n**3 - 2*e**(a*n + b*n*x)*int((e**(b*x)*(e**(2*a + 
 2*b*x)*f + e**(2*a + 2*b*x)*g - f + g)**n)/(e**(a*n + 2*a + b*n*x + 2*b*x 
)*f*n + e**(a*n + 2*a + b*n*x + 2*b*x)*f + e**(a*n + 2*a + b*n*x + 2*b*x)* 
g*n + e**(a*n + 2*a + b*n*x + 2*b*x)*g - e**(a*n + b*n*x)*f*n - e**(a*n + 
b*n*x)*f + e**(a*n + b*n*x)*g*n + e**(a*n + b*n*x)*g),x)*b*f**2*n**2 + 2*e 
**(a*n + b*n*x)*int((e**(b*x)*(e**(2*a + 2*b*x)*f + e**(2*a + 2*b*x)*g - f 
 + g)**n)/(e**(a*n + 2*a + b*n*x + 2*b*x)*f*n + e**(a*n + 2*a + b*n*x + 2* 
b*x)*f + e**(a*n + 2*a + b*n*x + 2*b*x)*g*n + e**(a*n + 2*a + b*n*x + 2*b* 
x)*g - e**(a*n + b*n*x)*f*n - e**(a*n + b*n*x)*f + e**(a*n + b*n*x)*g*n + 
e**(a*n + b*n*x)*g),x)*b*f*g*n**3 - 2*e**(a*n + b*n*x)*int((e**(b*x)*(e**( 
2*a + 2*b*x)*f + e**(2*a + 2*b*x)*g - f + g)**n)/(e**(a*n + 2*a + b*n*x + 
2*b*x)*f*n + e**(a*n + 2*a + b*n*x + 2*b*x)*f + e**(a*n + 2*a + b*n*x +...