\(\int F^{c (a+b x)} (g \cosh (d+e x))^q (f \sinh (d+e x))^p \, dx\) [167]

Optimal result

Integrand size = 30, antiderivative size = 160 \[ \int F^{c (a+b x)} (g \cosh (d+e x))^q (f \sinh (d+e x))^p \, dx=-\frac {2^{-1+q} \left (e^{2 d+2 e x}\right )^{\frac {1}{2} \left (p+q-\frac {b c \log (F)}{e}\right )} \left (1-e^{2 d+2 e x}\right ) \left (1+e^{2 d+2 e x}\right )^{-q} F^{c (a+b x)} \operatorname {AppellF1}\left (1+p,\frac {1}{2} \left (2+p+q-\frac {b c \log (F)}{e}\right ),-q,2+p,1-e^{2 d+2 e x},\frac {1}{2} \left (1-e^{2 d+2 e x}\right )\right ) (g \cosh (d+e x))^q (f \sinh (d+e x))^p}{e (1+p)} \] Output:

-2^(-1+q)*exp(2*e*x+2*d)^(1/2*p+1/2*q-1/2*b*c*ln(F)/e)*(1-exp(2*e*x+2*d))* 
F^(c*(b*x+a))*AppellF1(p+1,1+1/2*p+1/2*q-1/2*b*c*ln(F)/e,-q,2+p,1-exp(2*e* 
x+2*d),1/2-1/2*exp(2*e*x+2*d))*(g*cosh(e*x+d))^q*(f*sinh(e*x+d))^p/e/((1+e 
xp(2*e*x+2*d))^q)/(p+1)
 

Mathematica [F]

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

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

Output:

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

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

\(\Big \downarrow \) 7271

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

\(\Big \downarrow \) 7271

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

\(\Big \downarrow \) 7292

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

\(\Big \downarrow \) 7299

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

Input:

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

Output:

$Aborted
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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