Integrand size = 29, antiderivative size = 128 \[ \int F^{c (a+b x)} (g \cosh (d+e x)+f \sinh (d+e x))^n \, dx=-\frac {F^{c (a+b x)} \left (1-\frac {e^{2 d+2 e x} (f+g)}{f-g}\right )^{-n} \operatorname {Hypergeometric2F1}\left (-n,\frac {1}{2} \left (-n+\frac {b c \log (F)}{e}\right ),\frac {1}{2} \left (2-n+\frac {b c \log (F)}{e}\right ),\frac {e^{2 d+2 e x} (f+g)}{f-g}\right ) (g \cosh (d+e x)+f \sinh (d+e x))^n}{e n-b c \log (F)} \] Output:
-F^(c*(b*x+a))*hypergeom([-n, -1/2*n+1/2*b*c*ln(F)/e],[1-1/2*n+1/2*b*c*ln( F)/e],exp(2*e*x+2*d)*(f+g)/(f-g))*(g*cosh(e*x+d)+f*sinh(e*x+d))^n/((1-exp( 2*e*x+2*d)*(f+g)/(f-g))^n)/(e*n-b*c*ln(F))
\[ \int F^{c (a+b x)} (g \cosh (d+e x)+f \sinh (d+e x))^n \, dx=\int F^{c (a+b x)} (g \cosh (d+e x)+f \sinh (d+e x))^n \, dx \] Input:
Integrate[F^(c*(a + b*x))*(g*Cosh[d + e*x] + f*Sinh[d + e*x])^n,x]
Output:
Integrate[F^(c*(a + b*x))*(g*Cosh[d + e*x] + f*Sinh[d + e*x])^n, x]
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.
| \(\displaystyle \int F^{c (a+b x)} (f \sinh (d+e x)+g \cosh (d+e x))^n \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int F^{a c+b c x} (f \sinh (d+e x)+g \cosh (d+e x))^ndx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \int F^{a c+b c x} (f \sinh (d+e x)+g \cosh (d+e x))^ndx\) |
Input:
Int[F^(c*(a + b*x))*(g*Cosh[d + e*x] + f*Sinh[d + e*x])^n,x]
Output:
$Aborted
\[\int F^{c \left (b x +a \right )} \left (g \cosh \left (e x +d \right )+f \sinh \left (e x +d \right )\right )^{n}d x\]
Input:
int(F^(c*(b*x+a))*(g*cosh(e*x+d)+f*sinh(e*x+d))^n,x)
Output:
int(F^(c*(b*x+a))*(g*cosh(e*x+d)+f*sinh(e*x+d))^n,x)
\[ \int F^{c (a+b x)} (g \cosh (d+e x)+f \sinh (d+e x))^n \, dx=\int { {\left (g \cosh \left (e x + d\right ) + f \sinh \left (e x + d\right )\right )}^{n} F^{{\left (b x + a\right )} c} \,d x } \] Input:
integrate(F^(c*(b*x+a))*(g*cosh(e*x+d)+f*sinh(e*x+d))^n,x, algorithm="fric as")
Output:
integral((g*cosh(e*x + d) + f*sinh(e*x + d))^n*F^(b*c*x + a*c), x)
\[ \int F^{c (a+b x)} (g \cosh (d+e x)+f \sinh (d+e x))^n \, dx=\int F^{c \left (a + b x\right )} \left (f \sinh {\left (d + e x \right )} + g \cosh {\left (d + e x \right )}\right )^{n}\, dx \] Input:
integrate(F**(c*(b*x+a))*(g*cosh(e*x+d)+f*sinh(e*x+d))**n,x)
Output:
Integral(F**(c*(a + b*x))*(f*sinh(d + e*x) + g*cosh(d + e*x))**n, x)
\[ \int F^{c (a+b x)} (g \cosh (d+e x)+f \sinh (d+e x))^n \, dx=\int { {\left (g \cosh \left (e x + d\right ) + f \sinh \left (e x + d\right )\right )}^{n} F^{{\left (b x + a\right )} c} \,d x } \] Input:
integrate(F^(c*(b*x+a))*(g*cosh(e*x+d)+f*sinh(e*x+d))^n,x, algorithm="maxi ma")
Output:
integrate((g*cosh(e*x + d) + f*sinh(e*x + d))^n*F^((b*x + a)*c), x)
\[ \int F^{c (a+b x)} (g \cosh (d+e x)+f \sinh (d+e x))^n \, dx=\int { {\left (g \cosh \left (e x + d\right ) + f \sinh \left (e x + d\right )\right )}^{n} F^{{\left (b x + a\right )} c} \,d x } \] Input:
integrate(F^(c*(b*x+a))*(g*cosh(e*x+d)+f*sinh(e*x+d))^n,x, algorithm="giac ")
Output:
integrate((g*cosh(e*x + d) + f*sinh(e*x + d))^n*F^((b*x + a)*c), x)
Timed out. \[ \int F^{c (a+b x)} (g \cosh (d+e x)+f \sinh (d+e x))^n \, dx=\int F^{c\,\left (a+b\,x\right )}\,{\left (g\,\mathrm {cosh}\left (d+e\,x\right )+f\,\mathrm {sinh}\left (d+e\,x\right )\right )}^n \,d x \] Input:
int(F^(c*(a + b*x))*(g*cosh(d + e*x) + f*sinh(d + e*x))^n,x)
Output:
int(F^(c*(a + b*x))*(g*cosh(d + e*x) + f*sinh(d + e*x))^n, x)
\[ \int F^{c (a+b x)} (g \cosh (d+e x)+f \sinh (d+e x))^n \, dx=\frac {f^{a c} \left (f^{b c x} \left (\cosh \left (e x +d \right ) g +\sinh \left (e x +d \right ) f \right )^{n} f -\left (\int \frac {f^{b c x} \left (\cosh \left (e x +d \right ) g +\sinh \left (e x +d \right ) f \right )^{n} \cosh \left (e x +d \right )}{\cosh \left (e x +d \right ) \mathrm {log}\left (f \right ) b c f g +\cosh \left (e x +d \right ) e \,g^{2} n +\mathrm {log}\left (f \right ) \sinh \left (e x +d \right ) b c \,f^{2}+\sinh \left (e x +d \right ) e f g n}d x \right ) \mathrm {log}\left (f \right ) b c e \,f^{3} n +\left (\int \frac {f^{b c x} \left (\cosh \left (e x +d \right ) g +\sinh \left (e x +d \right ) f \right )^{n} \cosh \left (e x +d \right )}{\cosh \left (e x +d \right ) \mathrm {log}\left (f \right ) b c f g +\cosh \left (e x +d \right ) e \,g^{2} n +\mathrm {log}\left (f \right ) \sinh \left (e x +d \right ) b c \,f^{2}+\sinh \left (e x +d \right ) e f g n}d x \right ) \mathrm {log}\left (f \right ) b c e f \,g^{2} n -\left (\int \frac {f^{b c x} \left (\cosh \left (e x +d \right ) g +\sinh \left (e x +d \right ) f \right )^{n} \cosh \left (e x +d \right )}{\cosh \left (e x +d \right ) \mathrm {log}\left (f \right ) b c f g +\cosh \left (e x +d \right ) e \,g^{2} n +\mathrm {log}\left (f \right ) \sinh \left (e x +d \right ) b c \,f^{2}+\sinh \left (e x +d \right ) e f g n}d x \right ) e^{2} f^{2} g \,n^{2}+\left (\int \frac {f^{b c x} \left (\cosh \left (e x +d \right ) g +\sinh \left (e x +d \right ) f \right )^{n} \cosh \left (e x +d \right )}{\cosh \left (e x +d \right ) \mathrm {log}\left (f \right ) b c f g +\cosh \left (e x +d \right ) e \,g^{2} n +\mathrm {log}\left (f \right ) \sinh \left (e x +d \right ) b c \,f^{2}+\sinh \left (e x +d \right ) e f g n}d x \right ) e^{2} g^{3} n^{2}\right )}{\mathrm {log}\left (f \right ) b c f +e g n} \] Input:
int(F^(c*(b*x+a))*(g*cosh(e*x+d)+f*sinh(e*x+d))^n,x)
Output:
(f**(a*c)*(f**(b*c*x)*(cosh(d + e*x)*g + sinh(d + e*x)*f)**n*f - int((f**( b*c*x)*(cosh(d + e*x)*g + sinh(d + e*x)*f)**n*cosh(d + e*x))/(cosh(d + e*x )*log(f)*b*c*f*g + cosh(d + e*x)*e*g**2*n + log(f)*sinh(d + e*x)*b*c*f**2 + sinh(d + e*x)*e*f*g*n),x)*log(f)*b*c*e*f**3*n + int((f**(b*c*x)*(cosh(d + e*x)*g + sinh(d + e*x)*f)**n*cosh(d + e*x))/(cosh(d + e*x)*log(f)*b*c*f* g + cosh(d + e*x)*e*g**2*n + log(f)*sinh(d + e*x)*b*c*f**2 + sinh(d + e*x) *e*f*g*n),x)*log(f)*b*c*e*f*g**2*n - int((f**(b*c*x)*(cosh(d + e*x)*g + si nh(d + e*x)*f)**n*cosh(d + e*x))/(cosh(d + e*x)*log(f)*b*c*f*g + cosh(d + e*x)*e*g**2*n + log(f)*sinh(d + e*x)*b*c*f**2 + sinh(d + e*x)*e*f*g*n),x)* e**2*f**2*g*n**2 + int((f**(b*c*x)*(cosh(d + e*x)*g + sinh(d + e*x)*f)**n* cosh(d + e*x))/(cosh(d + e*x)*log(f)*b*c*f*g + cosh(d + e*x)*e*g**2*n + lo g(f)*sinh(d + e*x)*b*c*f**2 + sinh(d + e*x)*e*f*g*n),x)*e**2*g**3*n**2))/( log(f)*b*c*f + e*g*n)