\(\int F^{c (a+b x)} (g \cosh (d+\frac {b c x \log (F)}{2+n})+f \sinh (d+\frac {b c x \log (F)}{2+n}))^n \, dx\) [34]

Optimal result

Integrand size = 45, antiderivative size = 143 \[ \int F^{c (a+b x)} \left (g \cosh \left (d+\frac {b c x \log (F)}{2+n}\right )+f \sinh \left (d+\frac {b c x \log (F)}{2+n}\right )\right )^n \, dx=\frac {F^{c (a+b x)} (2+n) \left (g \cosh \left (d+\frac {b c x \log (F)}{2+n}\right )+f \sinh \left (d+\frac {b c x \log (F)}{2+n}\right )\right )^{1+n} \left (f \cosh \left (d+\frac {b c x \log (F)}{2+n}\right )-g \cosh \left (d+\frac {b c x \log (F)}{2+n}\right )-f \sinh \left (d+\frac {b c x \log (F)}{2+n}\right )+g \sinh \left (d+\frac {b c x \log (F)}{2+n}\right )\right )}{b c \left (f^2-g^2\right ) (1+n) \log (F)} \] Output:

F^(c*(b*x+a))*(2+n)*(g*cosh(d+b*c*x*ln(F)/(2+n))+f*sinh(d+b*c*x*ln(F)/(2+n 
)))^(1+n)*(f*cosh(d+b*c*x*ln(F)/(2+n))-g*cosh(d+b*c*x*ln(F)/(2+n))-f*sinh( 
d+b*c*x*ln(F)/(2+n))+g*sinh(d+b*c*x*ln(F)/(2+n)))/b/c/(f^2-g^2)/(1+n)/ln(F 
)
 

Mathematica [A] (warning: unable to verify)

Time = 1.39 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.34 \[ \int F^{c (a+b x)} \left (g \cosh \left (d+\frac {b c x \log (F)}{2+n}\right )+f \sinh \left (d+\frac {b c x \log (F)}{2+n}\right )\right )^n \, dx=\frac {2^{-1-n} e^{-2 d} F^{c \left (a+\frac {b n x}{2+n}\right )} \left (f \left (-1+e^{2 d} F^{\frac {2 b c x}{2+n}}\right )+g+e^{2 d} F^{\frac {2 b c x}{2+n}} g\right )^{-n} \left (e^{-d} F^{-\frac {b c x}{2+n}} \left (f \left (-1+e^{2 d} F^{\frac {2 b c x}{2+n}}\right )+g+e^{2 d} F^{\frac {2 b c x}{2+n}} g\right )\right )^n \left (-f+g+e^{2 d} F^{\frac {2 b c x}{2+n}} (f+g)\right )^{1+n} (2+n)}{b c (f+g) (1+n) \log (F)} \] Input:

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

Output:

(2^(-1 - n)*F^(c*(a + (b*n*x)/(2 + n)))*((f*(-1 + E^(2*d)*F^((2*b*c*x)/(2 
+ n))) + g + E^(2*d)*F^((2*b*c*x)/(2 + n))*g)/(E^d*F^((b*c*x)/(2 + n))))^n 
*(-f + g + E^(2*d)*F^((2*b*c*x)/(2 + n))*(f + g))^(1 + n)*(2 + n))/(b*c*E^ 
(2*d)*(f + g)*(f*(-1 + E^(2*d)*F^((2*b*c*x)/(2 + n))) + g + E^(2*d)*F^((2* 
b*c*x)/(2 + n))*g)^n*(1 + n)*Log[F])
 

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.

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

\(\Big \downarrow \) 7292

\(\displaystyle \int F^{a c+b c x} \left (f \sinh \left (\frac {b c x \log (F)}{n+2}+d\right )+g \cosh \left (\frac {b c x \log (F)}{n+2}+d\right )\right )^ndx\)

\(\Big \downarrow \) 7299

\(\displaystyle \int F^{a c+b c x} \left (f \sinh \left (\frac {b c x \log (F)}{n+2}+d\right )+g \cosh \left (\frac {b c x \log (F)}{n+2}+d\right )\right )^ndx\)

Input:

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

Output:

$Aborted
 

Maple [F]

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

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 493 vs. \(2 (143) = 286\).

Time = 0.10 (sec) , antiderivative size = 493, normalized size of antiderivative = 3.45 \[ \int F^{c (a+b x)} \left (g \cosh \left (d+\frac {b c x \log (F)}{2+n}\right )+f \sinh \left (d+\frac {b c x \log (F)}{2+n}\right )\right )^n \, dx=\frac {{\left ({\left (g n + 2 \, g\right )} \cosh \left ({\left (b c x + a c\right )} \log \left (F\right )\right ) \cosh \left (\frac {b c x \log \left (F\right ) + d n + 2 \, d}{n + 2}\right ) + {\left (f n + 2 \, f\right )} \cosh \left ({\left (b c x + a c\right )} \log \left (F\right )\right ) \sinh \left (\frac {b c x \log \left (F\right ) + d n + 2 \, d}{n + 2}\right ) + {\left ({\left (g n + 2 \, g\right )} \cosh \left (\frac {b c x \log \left (F\right ) + d n + 2 \, d}{n + 2}\right ) + {\left (f n + 2 \, f\right )} \sinh \left (\frac {b c x \log \left (F\right ) + d n + 2 \, d}{n + 2}\right )\right )} \sinh \left ({\left (b c x + a c\right )} \log \left (F\right )\right )\right )} \cosh \left (n \log \left (g \cosh \left (\frac {b c x \log \left (F\right ) + d n + 2 \, d}{n + 2}\right ) + f \sinh \left (\frac {b c x \log \left (F\right ) + d n + 2 \, d}{n + 2}\right )\right )\right ) + {\left ({\left (g n + 2 \, g\right )} \cosh \left ({\left (b c x + a c\right )} \log \left (F\right )\right ) \cosh \left (\frac {b c x \log \left (F\right ) + d n + 2 \, d}{n + 2}\right ) + {\left (f n + 2 \, f\right )} \cosh \left ({\left (b c x + a c\right )} \log \left (F\right )\right ) \sinh \left (\frac {b c x \log \left (F\right ) + d n + 2 \, d}{n + 2}\right ) + {\left ({\left (g n + 2 \, g\right )} \cosh \left (\frac {b c x \log \left (F\right ) + d n + 2 \, d}{n + 2}\right ) + {\left (f n + 2 \, f\right )} \sinh \left (\frac {b c x \log \left (F\right ) + d n + 2 \, d}{n + 2}\right )\right )} \sinh \left ({\left (b c x + a c\right )} \log \left (F\right )\right )\right )} \sinh \left (n \log \left (g \cosh \left (\frac {b c x \log \left (F\right ) + d n + 2 \, d}{n + 2}\right ) + f \sinh \left (\frac {b c x \log \left (F\right ) + d n + 2 \, d}{n + 2}\right )\right )\right )}{{\left (b c f + b c g + {\left (b c f + b c g\right )} n\right )} \cosh \left (\frac {b c x \log \left (F\right ) + d n + 2 \, d}{n + 2}\right ) \log \left (F\right ) + {\left (b c f + b c g + {\left (b c f + b c g\right )} n\right )} \log \left (F\right ) \sinh \left (\frac {b c x \log \left (F\right ) + d n + 2 \, d}{n + 2}\right )} \] Input:

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

Output:

(((g*n + 2*g)*cosh((b*c*x + a*c)*log(F))*cosh((b*c*x*log(F) + d*n + 2*d)/( 
n + 2)) + (f*n + 2*f)*cosh((b*c*x + a*c)*log(F))*sinh((b*c*x*log(F) + d*n 
+ 2*d)/(n + 2)) + ((g*n + 2*g)*cosh((b*c*x*log(F) + d*n + 2*d)/(n + 2)) + 
(f*n + 2*f)*sinh((b*c*x*log(F) + d*n + 2*d)/(n + 2)))*sinh((b*c*x + a*c)*l 
og(F)))*cosh(n*log(g*cosh((b*c*x*log(F) + d*n + 2*d)/(n + 2)) + f*sinh((b* 
c*x*log(F) + d*n + 2*d)/(n + 2)))) + ((g*n + 2*g)*cosh((b*c*x + a*c)*log(F 
))*cosh((b*c*x*log(F) + d*n + 2*d)/(n + 2)) + (f*n + 2*f)*cosh((b*c*x + a* 
c)*log(F))*sinh((b*c*x*log(F) + d*n + 2*d)/(n + 2)) + ((g*n + 2*g)*cosh((b 
*c*x*log(F) + d*n + 2*d)/(n + 2)) + (f*n + 2*f)*sinh((b*c*x*log(F) + d*n + 
 2*d)/(n + 2)))*sinh((b*c*x + a*c)*log(F)))*sinh(n*log(g*cosh((b*c*x*log(F 
) + d*n + 2*d)/(n + 2)) + f*sinh((b*c*x*log(F) + d*n + 2*d)/(n + 2)))))/(( 
b*c*f + b*c*g + (b*c*f + b*c*g)*n)*cosh((b*c*x*log(F) + d*n + 2*d)/(n + 2) 
)*log(F) + (b*c*f + b*c*g + (b*c*f + b*c*g)*n)*log(F)*sinh((b*c*x*log(F) + 
 d*n + 2*d)/(n + 2)))
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1449 vs. \(2 (143) = 286\).

Time = 1.36 (sec) , antiderivative size = 1449, normalized size of antiderivative = 10.13 \[ \int F^{c (a+b x)} \left (g \cosh \left (d+\frac {b c x \log (F)}{2+n}\right )+f \sinh \left (d+\frac {b c x \log (F)}{2+n}\right )\right )^n \, dx=\text {Too large to display} \] Input:

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

Output:

1/2*(F^(a*c)*f*n*e^((2*b*c*x*log(F) - d*n^2 - n^2*log(2) + n^2*log(f*e^(2* 
(b*c*x*log(F) + d*n + 2*d)/(n + 2)) + g*e^(2*(b*c*x*log(F) + d*n + 2*d)/(n 
 + 2)) - f + g) - 2*d*n - 2*n*log(2) + 2*n*log(f*e^(2*(b*c*x*log(F) + d*n 
+ 2*d)/(n + 2)) + g*e^(2*(b*c*x*log(F) + d*n + 2*d)/(n + 2)) - f + g))/(n 
+ 2) + 2*(b*c*x*log(F) + d*n + 2*d)/(n + 2)) + F^(a*c)*g*n*e^((2*b*c*x*log 
(F) - d*n^2 - n^2*log(2) + n^2*log(f*e^(2*(b*c*x*log(F) + d*n + 2*d)/(n + 
2)) + g*e^(2*(b*c*x*log(F) + d*n + 2*d)/(n + 2)) - f + g) - 2*d*n - 2*n*lo 
g(2) + 2*n*log(f*e^(2*(b*c*x*log(F) + d*n + 2*d)/(n + 2)) + g*e^(2*(b*c*x* 
log(F) + d*n + 2*d)/(n + 2)) - f + g))/(n + 2) + 2*(b*c*x*log(F) + d*n + 2 
*d)/(n + 2)) - F^(a*c)*f*n*e^((2*b*c*x*log(F) - d*n^2 - n^2*log(2) + n^2*l 
og(f*e^(2*(b*c*x*log(F) + d*n + 2*d)/(n + 2)) + g*e^(2*(b*c*x*log(F) + d*n 
 + 2*d)/(n + 2)) - f + g) - 2*d*n - 2*n*log(2) + 2*n*log(f*e^(2*(b*c*x*log 
(F) + d*n + 2*d)/(n + 2)) + g*e^(2*(b*c*x*log(F) + d*n + 2*d)/(n + 2)) - f 
 + g))/(n + 2)) + F^(a*c)*g*n*e^((2*b*c*x*log(F) - d*n^2 - n^2*log(2) + n^ 
2*log(f*e^(2*(b*c*x*log(F) + d*n + 2*d)/(n + 2)) + g*e^(2*(b*c*x*log(F) + 
d*n + 2*d)/(n + 2)) - f + g) - 2*d*n - 2*n*log(2) + 2*n*log(f*e^(2*(b*c*x* 
log(F) + d*n + 2*d)/(n + 2)) + g*e^(2*(b*c*x*log(F) + d*n + 2*d)/(n + 2)) 
- f + g))/(n + 2)) + 2*F^(a*c)*f*e^((2*b*c*x*log(F) - d*n^2 - n^2*log(2) + 
 n^2*log(f*e^(2*(b*c*x*log(F) + d*n + 2*d)/(n + 2)) + g*e^(2*(b*c*x*log(F) 
 + d*n + 2*d)/(n + 2)) - f + g) - 2*d*n - 2*n*log(2) + 2*n*log(f*e^(2*(...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 1295, normalized size of antiderivative = 9.06 \[ \int F^{c (a+b x)} \left (g \cosh \left (d+\frac {b c x \log (F)}{2+n}\right )+f \sinh \left (d+\frac {b c x \log (F)}{2+n}\right )\right )^n \, dx =\text {Too large to display} \] Input:

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

Output:

(f**(a*c)*(2*f**(b*c*x)*e**(d*n + 2*d)*(cosh((log(f)*b*c*x + d*n + 2*d)/(n 
 + 2))*g + sinh((log(f)*b*c*x + d*n + 2*d)/(n + 2))*f)**n*2**n*f*g*n**2 + 
6*f**(b*c*x)*e**(d*n + 2*d)*(cosh((log(f)*b*c*x + d*n + 2*d)/(n + 2))*g + 
sinh((log(f)*b*c*x + d*n + 2*d)/(n + 2))*f)**n*2**n*f*g*n + 4*f**(b*c*x)*e 
**(d*n + 2*d)*(cosh((log(f)*b*c*x + d*n + 2*d)/(n + 2))*g + sinh((log(f)*b 
*c*x + d*n + 2*d)/(n + 2))*f)**n*2**n*f*g + 2*f**(b*c*x)*e**(d*n + 2*d)*(c 
osh((log(f)*b*c*x + d*n + 2*d)/(n + 2))*g + sinh((log(f)*b*c*x + d*n + 2*d 
)/(n + 2))*f)**n*2**n*g**2*n**2 + 6*f**(b*c*x)*e**(d*n + 2*d)*(cosh((log(f 
)*b*c*x + d*n + 2*d)/(n + 2))*g + sinh((log(f)*b*c*x + d*n + 2*d)/(n + 2)) 
*f)**n*2**n*g**2*n + 4*f**(b*c*x)*e**(d*n + 2*d)*(cosh((log(f)*b*c*x + d*n 
 + 2*d)/(n + 2))*g + sinh((log(f)*b*c*x + d*n + 2*d)/(n + 2))*f)**n*2**n*g 
**2 + e**((2*log(f)*b*c*x + 2*d*n + 4*d)/(n + 2))*(e**((2*log(f)*b*c*x + 2 
*d*n + 4*d)/(n + 2))*f + e**((2*log(f)*b*c*x + 2*d*n + 4*d)/(n + 2))*g - f 
 + g)**n*f**2*n**2 + 2*e**((2*log(f)*b*c*x + 2*d*n + 4*d)/(n + 2))*(e**((2 
*log(f)*b*c*x + 2*d*n + 4*d)/(n + 2))*f + e**((2*log(f)*b*c*x + 2*d*n + 4* 
d)/(n + 2))*g - f + g)**n*f**2*n - e**((2*log(f)*b*c*x + 2*d*n + 4*d)/(n + 
 2))*(e**((2*log(f)*b*c*x + 2*d*n + 4*d)/(n + 2))*f + e**((2*log(f)*b*c*x 
+ 2*d*n + 4*d)/(n + 2))*g - f + g)**n*g**2*n**2 - 2*e**((2*log(f)*b*c*x + 
2*d*n + 4*d)/(n + 2))*(e**((2*log(f)*b*c*x + 2*d*n + 4*d)/(n + 2))*f + e** 
((2*log(f)*b*c*x + 2*d*n + 4*d)/(n + 2))*g - f + g)**n*g**2*n - (e**((2...