\(\int F^{c (a+b x)} (g \cosh (d-\frac {b c x \log (F)}{2+n})+f \sinh (d-\frac {b c x \log (F)}{2+n}))^n \, dx\) [35]

Optimal result

Integrand size = 47, antiderivative size = 148 \[ \int F^{c (a+b x)} \left (g \cosh \left (d-\frac {b c x \log (F)}{2+n}\right )+f \sinh \left (d-\frac {b c x \log (F)}{2+n}\right )\right )^n \, dx=-\frac {F^{c (a+b x)} (2+n) \left (g \cosh \left (d-\frac {b c x \log (F)}{2+n}\right )+f \sinh \left (d-\frac {b c x \log (F)}{2+n}\right )\right )^{1+n} \left (f \cosh \left (d-\frac {b c x \log (F)}{2+n}\right )+g \cosh \left (d-\frac {b c x \log (F)}{2+n}\right )+f \sinh \left (d-\frac {b c x \log (F)}{2+n}\right )+g \sinh \left (d-\frac {b c x \log (F)}{2+n}\right )\right )}{b c \left (f^2-g^2\right ) (1+n) \log (F)} \] 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] (verified)

Time = 4.19 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.82 \[ \int F^{c (a+b x)} \left (g \cosh \left (d-\frac {b c x \log (F)}{2+n}\right )+f \sinh \left (d-\frac {b c x \log (F)}{2+n}\right )\right )^n \, dx=-\frac {F^{c \left (a+\frac {b n x}{2+n}\right )} \left (F^{\frac {2 b c x}{2+n}} (-f+g)+e^{2 d} (f+g)\right ) \left (\frac {1}{2} e^{-d} F^{\frac {b c x}{2+n}} (-f+g)+\frac {1}{2} e^d F^{-\frac {b c x}{2+n}} (f+g)\right )^n (2+n)}{2 b c (f-g) (1+n) \log (F)} \] 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:

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

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

\(\Big \downarrow \) 7292

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

\(\Big \downarrow \) 7299

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

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. 518 vs. \(2 (157) = 314\).

Time = 0.10 (sec) , antiderivative size = 518, normalized size of antiderivative = 3.50 \[ \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)* 
log(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. 1469 vs. \(2 (157) = 314\).

Time = 1.19 (sec) , antiderivative size = 1469, normalized size of antiderivative = 9.93 \[ \int F^{c (a+b x)} \left (g \cosh \left (d-\frac {b c x \log (F)}{2+n}\right )+f \sinh \left (d-\frac {b c x \log (F)}{2+n}\right )\right )^n \, dx=\text {Too large to display} \] 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*l 
og(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)*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)) - 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*lo 
g(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...
 

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 = 0.29 (sec) , antiderivative size = 1207, normalized size of antiderivative = 8.16 \[ \int F^{c (a+b x)} \left (g \cosh \left (d-\frac {b c x \log (F)}{2+n}\right )+f \sinh \left (d-\frac {b c x \log (F)}{2+n}\right )\right )^n \, dx =\text {Too large to display} \] 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)*(e**((2*log(f)*b*c*x)/(n + 2))*( - e**((2*log(f)*b*c*x)/(n + 2)) 
*f + e**((2*log(f)*b*c*x)/(n + 2))*g + e**(2*d)*f + e**(2*d)*g)**n*f**2*n* 
*2 + 2*e**((2*log(f)*b*c*x)/(n + 2))*( - e**((2*log(f)*b*c*x)/(n + 2))*f + 
 e**((2*log(f)*b*c*x)/(n + 2))*g + e**(2*d)*f + e**(2*d)*g)**n*f**2*n - e* 
*((2*log(f)*b*c*x)/(n + 2))*( - e**((2*log(f)*b*c*x)/(n + 2))*f + e**((2*l 
og(f)*b*c*x)/(n + 2))*g + e**(2*d)*f + e**(2*d)*g)**n*g**2*n**2 - 2*e**((2 
*log(f)*b*c*x)/(n + 2))*( - e**((2*log(f)*b*c*x)/(n + 2))*f + e**((2*log(f 
)*b*c*x)/(n + 2))*g + e**(2*d)*f + e**(2*d)*g)**n*g**2*n - e**(2*d)*( - e* 
*((2*log(f)*b*c*x)/(n + 2))*f + e**((2*log(f)*b*c*x)/(n + 2))*g + e**(2*d) 
*f + e**(2*d)*g)**n*f**2*n**2 - 2*e**(2*d)*( - e**((2*log(f)*b*c*x)/(n + 2 
))*f + e**((2*log(f)*b*c*x)/(n + 2))*g + e**(2*d)*f + e**(2*d)*g)**n*f**2* 
n + 2*e**(2*d)*( - e**((2*log(f)*b*c*x)/(n + 2))*f + e**((2*log(f)*b*c*x)/ 
(n + 2))*g + e**(2*d)*f + e**(2*d)*g)**n*f*g*n + 4*e**(2*d)*( - e**((2*log 
(f)*b*c*x)/(n + 2))*f + e**((2*log(f)*b*c*x)/(n + 2))*g + e**(2*d)*f + e** 
(2*d)*g)**n*f*g + e**(2*d)*( - e**((2*log(f)*b*c*x)/(n + 2))*f + e**((2*lo 
g(f)*b*c*x)/(n + 2))*g + e**(2*d)*f + e**(2*d)*g)**n*g**2*n**2 + 4*e**(2*d 
)*( - e**((2*log(f)*b*c*x)/(n + 2))*f + e**((2*log(f)*b*c*x)/(n + 2))*g + 
e**(2*d)*f + e**(2*d)*g)**n*g**2*n + 4*e**(2*d)*( - e**((2*log(f)*b*c*x)/( 
n + 2))*f + e**((2*log(f)*b*c*x)/(n + 2))*g + e**(2*d)*f + e**(2*d)*g)**n* 
g**2 - 2*f**(b*c*x)*e**(d*n)*(cosh((log(f)*b*c*x - d*n - 2*d)/(n + 2))*...