\(\int e^{a+b x} (g \cosh (d+b x)+f \sinh (d+b x))^4 \, dx\) [1]

Optimal result

Integrand size = 27, antiderivative size = 130 \[ \int e^{a+b x} (g \cosh (d+b x)+f \sinh (d+b x))^4 \, dx=-\frac {e^{a-4 d-3 b x} (f-g)^4}{48 b}+\frac {e^{a-2 d-b x} (f-g)^3 (f+g)}{4 b}-\frac {e^{a+2 d+3 b x} (f-g) (f+g)^3}{12 b}+\frac {e^{a+4 d+5 b x} (f+g)^4}{80 b}+\frac {3 e^{a+b x} \left (f^2-g^2\right )^2}{8 b} \] Output:

-1/48*exp(-3*b*x+a-4*d)*(f-g)^4/b+1/4*exp(-b*x+a-2*d)*(f-g)^3*(f+g)/b-1/12 
*exp(3*b*x+a+2*d)*(f-g)*(f+g)^3/b+1/80*exp(5*b*x+a+4*d)*(f+g)^4/b+3/8*exp( 
b*x+a)*(f^2-g^2)^2/b
 

Mathematica [A] (verified)

Time = 2.59 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.91 \[ \int e^{a+b x} (g \cosh (d+b x)+f \sinh (d+b x))^4 \, dx=\frac {e^{a-d} \left (-\frac {1}{3} e^{-3 (d+b x)} (f-g)^4+4 e^{-d-b x} (f-g)^3 (f+g)-\frac {4}{3} e^{3 (d+b x)} (f-g) (f+g)^3+\frac {1}{5} e^{5 (d+b x)} (f+g)^4+6 e^{d+b x} \left (f^2-g^2\right )^2\right )}{16 b} \] Input:

Integrate[E^(a + b*x)*(g*Cosh[d + b*x] + f*Sinh[d + b*x])^4,x]
 

Output:

(E^(a - d)*(-1/3*(f - g)^4/E^(3*(d + b*x)) + 4*E^(-d - b*x)*(f - g)^3*(f + 
 g) - (4*E^(3*(d + b*x))*(f - g)*(f + g)^3)/3 + (E^(5*(d + b*x))*(f + g)^4 
)/5 + 6*E^(d + b*x)*(f^2 - g^2)^2))/(16*b)
 

Rubi [A] (warning: unable to verify)

Time = 0.42 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.76, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {2720, 27, 244, 2009}

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.

\(\displaystyle \int e^{a+b x} (f \sinh (b x+d)+g \cosh (b x+d))^4 \, dx\)

\(\Big \downarrow \) 2720

\(\displaystyle \frac {\int \frac {1}{16} e^{a-4 b x} \left (f-g-e^{2 b x} (f+g)\right )^4de^{b x}}{b}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {e^a \int e^{-4 b x} \left (f-g-e^{2 b x} (f+g)\right )^4de^{b x}}{16 b}\)

\(\Big \downarrow \) 244

\(\displaystyle \frac {e^a \int \left (e^{-4 b x} (f-g)^4-4 e^{-2 b x} (f+g) (f-g)^3-4 e^{2 b x} (f+g)^3 (f-g)+e^{4 b x} (f+g)^4+6 \left (f^2-g^2\right )^2\right )de^{b x}}{16 b}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {e^a \left (6 e^{b x} \left (f^2-g^2\right )^2-\frac {1}{3} e^{-3 b x} (f-g)^4+4 e^{-b x} (f+g) (f-g)^3-\frac {4}{3} e^{3 b x} (f+g)^3 (f-g)+\frac {1}{5} e^{5 b x} (f+g)^4\right )}{16 b}\)

Input:

Int[E^(a + b*x)*(g*Cosh[d + b*x] + f*Sinh[d + b*x])^4,x]
 

Output:

(E^a*(-1/3*(f - g)^4/E^(3*b*x) + (4*(f - g)^3*(f + g))/E^(b*x) - (4*E^(3*b 
*x)*(f - g)*(f + g)^3)/3 + (E^(5*b*x)*(f + g)^4)/5 + 6*E^(b*x)*(f^2 - g^2) 
^2))/(16*b)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 244

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand 
Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p 
, 0]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2720

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] 
   Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct 
ionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ 
[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) 
*(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(394\) vs. \(2(115)=230\).

Time = 0.18 (sec) , antiderivative size = 395, normalized size of antiderivative = 3.04

\[\frac {\left (-\frac {3}{4} f^{2} g^{2}+\frac {3}{8} g^{4}+\frac {3}{8} f^{4}\right ) \sinh \left (b x +a \right )}{b}+\frac {\left (-\frac {3}{4} f^{2} g^{2}+\frac {3}{8} g^{4}+\frac {3}{8} f^{4}\right ) \cosh \left (b x +a \right )}{b}+\frac {\left (\frac {1}{4} g^{4}-\frac {1}{4} f^{4}-\frac {1}{2} g \,f^{3}+\frac {1}{2} g^{3} f \right ) \sinh \left (3 b x +a +2 d \right )}{3 b}+\frac {\left (\frac {1}{4} g^{4}-\frac {1}{4} f^{4}-\frac {1}{2} g \,f^{3}+\frac {1}{2} g^{3} f \right ) \cosh \left (3 b x +a +2 d \right )}{3 b}-\frac {\left (\frac {1}{4} g^{4}-\frac {1}{4} f^{4}+\frac {1}{2} g \,f^{3}-\frac {1}{2} g^{3} f \right ) \sinh \left (-b x +a -2 d \right )}{b}+\frac {\left (\frac {1}{4} f^{4}-\frac {1}{2} g \,f^{3}+\frac {1}{2} g^{3} f -\frac {1}{4} g^{4}\right ) \cosh \left (-b x +a -2 d \right )}{b}-\frac {\left (\frac {3}{8} f^{2} g^{2}+\frac {1}{16} g^{4}+\frac {1}{16} f^{4}-\frac {1}{4} g \,f^{3}-\frac {1}{4} g^{3} f \right ) \sinh \left (-3 b x +a -4 d \right )}{3 b}+\frac {\left (-\frac {1}{16} f^{4}+\frac {1}{4} g \,f^{3}-\frac {3}{8} f^{2} g^{2}+\frac {1}{4} g^{3} f -\frac {1}{16} g^{4}\right ) \cosh \left (-3 b x +a -4 d \right )}{3 b}+\frac {\left (\frac {3}{8} f^{2} g^{2}+\frac {1}{16} g^{4}+\frac {1}{16} f^{4}+\frac {1}{4} g \,f^{3}+\frac {1}{4} g^{3} f \right ) \sinh \left (5 b x +a +4 d \right )}{5 b}+\frac {\left (\frac {3}{8} f^{2} g^{2}+\frac {1}{16} g^{4}+\frac {1}{16} f^{4}+\frac {1}{4} g \,f^{3}+\frac {1}{4} g^{3} f \right ) \cosh \left (5 b x +a +4 d \right )}{5 b}\]

Input:

int(exp(b*x+a)*(g*cosh(b*x+d)+f*sinh(b*x+d))^4,x)
 

Output:

(-3/4*f^2*g^2+3/8*g^4+3/8*f^4)/b*sinh(b*x+a)+(-3/4*f^2*g^2+3/8*g^4+3/8*f^4 
)*cosh(b*x+a)/b+1/3*(1/4*g^4-1/4*f^4-1/2*g*f^3+1/2*g^3*f)/b*sinh(3*b*x+a+2 
*d)+1/3*(1/4*g^4-1/4*f^4-1/2*g*f^3+1/2*g^3*f)*cosh(3*b*x+a+2*d)/b-(1/4*g^4 
-1/4*f^4+1/2*g*f^3-1/2*g^3*f)/b*sinh(-b*x+a-2*d)+(1/4*f^4-1/2*g*f^3+1/2*g^ 
3*f-1/4*g^4)*cosh(-b*x+a-2*d)/b-1/3*(3/8*f^2*g^2+1/16*g^4+1/16*f^4-1/4*g*f 
^3-1/4*g^3*f)/b*sinh(-3*b*x+a-4*d)+1/3*(-1/16*f^4+1/4*g*f^3-3/8*f^2*g^2+1/ 
4*g^3*f-1/16*g^4)*cosh(-3*b*x+a-4*d)/b+1/5*(3/8*f^2*g^2+1/16*g^4+1/16*f^4+ 
1/4*g*f^3+1/4*g^3*f)/b*sinh(5*b*x+a+4*d)+1/5*(3/8*f^2*g^2+1/16*g^4+1/16*f^ 
4+1/4*g*f^3+1/4*g^3*f)*cosh(5*b*x+a+4*d)/b
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 698 vs. \(2 (115) = 230\).

Time = 0.09 (sec) , antiderivative size = 698, normalized size of antiderivative = 5.37 \[ \int e^{a+b x} (g \cosh (d+b x)+f \sinh (d+b x))^4 \, dx =\text {Too large to display} \] Input:

integrate(exp(b*x+a)*(g*cosh(b*x+d)+f*sinh(b*x+d))^4,x, algorithm="fricas" 
)
 

Output:

-1/120*((f^4 - 16*f^3*g + 6*f^2*g^2 - 16*f*g^3 + g^4)*cosh(b*x + d)^4*cosh 
(-a + d) + ((f^4 - 16*f^3*g + 6*f^2*g^2 - 16*f*g^3 + g^4)*cosh(-a + d) - ( 
f^4 - 16*f^3*g + 6*f^2*g^2 - 16*f*g^3 + g^4)*sinh(-a + d))*sinh(b*x + d)^4 
 - 20*(f^4 - 4*f^3*g + 4*f*g^3 - g^4)*cosh(b*x + d)^2*cosh(-a + d) - 16*(( 
f^4 - f^3*g + 6*f^2*g^2 - f*g^3 + g^4)*cosh(b*x + d)*cosh(-a + d) - (f^4 - 
 f^3*g + 6*f^2*g^2 - f*g^3 + g^4)*cosh(b*x + d)*sinh(-a + d))*sinh(b*x + d 
)^3 + 2*(3*(f^4 - 16*f^3*g + 6*f^2*g^2 - 16*f*g^3 + g^4)*cosh(b*x + d)^2*c 
osh(-a + d) - 10*(f^4 - 4*f^3*g + 4*f*g^3 - g^4)*cosh(-a + d) + (10*f^4 - 
40*f^3*g + 40*f*g^3 - 10*g^4 - 3*(f^4 - 16*f^3*g + 6*f^2*g^2 - 16*f*g^3 + 
g^4)*cosh(b*x + d)^2)*sinh(-a + d))*sinh(b*x + d)^2 - 45*(f^4 - 2*f^2*g^2 
+ g^4)*cosh(-a + d) - 16*((f^4 - f^3*g + 6*f^2*g^2 - f*g^3 + g^4)*cosh(b*x 
 + d)^3*cosh(-a + d) - 5*(f^4 - f^3*g + f*g^3 - g^4)*cosh(b*x + d)*cosh(-a 
 + d) - ((f^4 - f^3*g + 6*f^2*g^2 - f*g^3 + g^4)*cosh(b*x + d)^3 - 5*(f^4 
- f^3*g + f*g^3 - g^4)*cosh(b*x + d))*sinh(-a + d))*sinh(b*x + d) - ((f^4 
- 16*f^3*g + 6*f^2*g^2 - 16*f*g^3 + g^4)*cosh(b*x + d)^4 - 45*f^4 + 90*f^2 
*g^2 - 45*g^4 - 20*(f^4 - 4*f^3*g + 4*f*g^3 - g^4)*cosh(b*x + d)^2)*sinh(- 
a + d))/(b*cosh(b*x + d) - b*sinh(b*x + d))
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 794 vs. \(2 (105) = 210\).

Time = 2.49 (sec) , antiderivative size = 794, normalized size of antiderivative = 6.11 \[ \int e^{a+b x} (g \cosh (d+b x)+f \sinh (d+b x))^4 \, dx =\text {Too large to display} \] Input:

integrate(exp(b*x+a)*(g*cosh(b*x+d)+f*sinh(b*x+d))**4,x)
 

Output:

Piecewise((f**4*exp(a)*exp(b*x)*sinh(b*x + d)**4/(5*b) + 4*f**4*exp(a)*exp 
(b*x)*sinh(b*x + d)**3*cosh(b*x + d)/(5*b) - 4*f**4*exp(a)*exp(b*x)*sinh(b 
*x + d)**2*cosh(b*x + d)**2/(5*b) - 8*f**4*exp(a)*exp(b*x)*sinh(b*x + d)*c 
osh(b*x + d)**3/(15*b) + 8*f**4*exp(a)*exp(b*x)*cosh(b*x + d)**4/(15*b) + 
4*f**3*g*exp(a)*exp(b*x)*sinh(b*x + d)**4/(5*b) - 4*f**3*g*exp(a)*exp(b*x) 
*sinh(b*x + d)**3*cosh(b*x + d)/(5*b) + 4*f**3*g*exp(a)*exp(b*x)*sinh(b*x 
+ d)**2*cosh(b*x + d)**2/(5*b) + 8*f**3*g*exp(a)*exp(b*x)*sinh(b*x + d)*co 
sh(b*x + d)**3/(15*b) - 8*f**3*g*exp(a)*exp(b*x)*cosh(b*x + d)**4/(15*b) - 
 4*f**2*g**2*exp(a)*exp(b*x)*sinh(b*x + d)**4/(5*b) + 4*f**2*g**2*exp(a)*e 
xp(b*x)*sinh(b*x + d)**3*cosh(b*x + d)/(5*b) + 6*f**2*g**2*exp(a)*exp(b*x) 
*sinh(b*x + d)**2*cosh(b*x + d)**2/(5*b) + 4*f**2*g**2*exp(a)*exp(b*x)*sin 
h(b*x + d)*cosh(b*x + d)**3/(5*b) - 4*f**2*g**2*exp(a)*exp(b*x)*cosh(b*x + 
 d)**4/(5*b) - 8*f*g**3*exp(a)*exp(b*x)*sinh(b*x + d)**4/(15*b) + 8*f*g**3 
*exp(a)*exp(b*x)*sinh(b*x + d)**3*cosh(b*x + d)/(15*b) + 4*f*g**3*exp(a)*e 
xp(b*x)*sinh(b*x + d)**2*cosh(b*x + d)**2/(5*b) - 4*f*g**3*exp(a)*exp(b*x) 
*sinh(b*x + d)*cosh(b*x + d)**3/(5*b) + 4*f*g**3*exp(a)*exp(b*x)*cosh(b*x 
+ d)**4/(5*b) + 8*g**4*exp(a)*exp(b*x)*sinh(b*x + d)**4/(15*b) - 8*g**4*ex 
p(a)*exp(b*x)*sinh(b*x + d)**3*cosh(b*x + d)/(15*b) - 4*g**4*exp(a)*exp(b* 
x)*sinh(b*x + d)**2*cosh(b*x + d)**2/(5*b) + 4*g**4*exp(a)*exp(b*x)*sinh(b 
*x + d)*cosh(b*x + d)**3/(5*b) + g**4*exp(a)*exp(b*x)*cosh(b*x + d)**4/...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 374 vs. \(2 (115) = 230\).

Time = 0.06 (sec) , antiderivative size = 374, normalized size of antiderivative = 2.88 \[ \int e^{a+b x} (g \cosh (d+b x)+f \sinh (d+b x))^4 \, dx=\frac {1}{60} \, f g^{3} {\left (\frac {5 \, {\left (6 \, e^{\left (2 \, b x + 4 \, a + 2 \, d\right )} + e^{\left (4 \, a\right )}\right )} e^{\left (-3 \, b x - 3 \, a - 4 \, d\right )}}{b} + \frac {{\left (3 \, e^{\left (5 \, b x + 5 \, a + 4 \, d\right )} + 10 \, e^{\left (3 \, b x + 5 \, a + 2 \, d\right )}\right )} e^{\left (-4 \, a\right )}}{b}\right )} - \frac {1}{60} \, f^{3} g {\left (\frac {5 \, {\left (6 \, e^{\left (2 \, b x + 4 \, a + 2 \, d\right )} - e^{\left (4 \, a\right )}\right )} e^{\left (-3 \, b x - 3 \, a - 4 \, d\right )}}{b} - \frac {{\left (3 \, e^{\left (5 \, b x + 5 \, a + 4 \, d\right )} - 10 \, e^{\left (3 \, b x + 5 \, a + 2 \, d\right )}\right )} e^{\left (-4 \, a\right )}}{b}\right )} + \frac {1}{40} \, f^{2} g^{2} {\left (\frac {3 \, {\left (e^{\left (5 \, b x + 5 \, a + 4 \, d\right )} - 10 \, e^{\left (b x + 5 \, a\right )}\right )} e^{\left (-4 \, a\right )}}{b} - \frac {5 \, e^{\left (-3 \, b x + a - 4 \, d\right )}}{b}\right )} + \frac {1}{240} \, g^{4} {\left (\frac {3 \, e^{\left (5 \, b x + a + 4 \, d\right )}}{b} + \frac {20 \, e^{\left (3 \, b x + a + 2 \, d\right )}}{b} + \frac {90 \, e^{\left (b x + a\right )}}{b} - \frac {60 \, e^{\left (-b x + a - 2 \, d\right )}}{b} - \frac {5 \, e^{\left (-3 \, b x + a - 4 \, d\right )}}{b}\right )} + \frac {1}{240} \, f^{4} {\left (\frac {3 \, e^{\left (5 \, b x + a + 4 \, d\right )}}{b} - \frac {20 \, e^{\left (3 \, b x + a + 2 \, d\right )}}{b} + \frac {90 \, e^{\left (b x + a\right )}}{b} + \frac {60 \, e^{\left (-b x + a - 2 \, d\right )}}{b} - \frac {5 \, e^{\left (-3 \, b x + a - 4 \, d\right )}}{b}\right )} \] Input:

integrate(exp(b*x+a)*(g*cosh(b*x+d)+f*sinh(b*x+d))^4,x, algorithm="maxima" 
)
 

Output:

1/60*f*g^3*(5*(6*e^(2*b*x + 4*a + 2*d) + e^(4*a))*e^(-3*b*x - 3*a - 4*d)/b 
 + (3*e^(5*b*x + 5*a + 4*d) + 10*e^(3*b*x + 5*a + 2*d))*e^(-4*a)/b) - 1/60 
*f^3*g*(5*(6*e^(2*b*x + 4*a + 2*d) - e^(4*a))*e^(-3*b*x - 3*a - 4*d)/b - ( 
3*e^(5*b*x + 5*a + 4*d) - 10*e^(3*b*x + 5*a + 2*d))*e^(-4*a)/b) + 1/40*f^2 
*g^2*(3*(e^(5*b*x + 5*a + 4*d) - 10*e^(b*x + 5*a))*e^(-4*a)/b - 5*e^(-3*b* 
x + a - 4*d)/b) + 1/240*g^4*(3*e^(5*b*x + a + 4*d)/b + 20*e^(3*b*x + a + 2 
*d)/b + 90*e^(b*x + a)/b - 60*e^(-b*x + a - 2*d)/b - 5*e^(-3*b*x + a - 4*d 
)/b) + 1/240*f^4*(3*e^(5*b*x + a + 4*d)/b - 20*e^(3*b*x + a + 2*d)/b + 90* 
e^(b*x + a)/b + 60*e^(-b*x + a - 2*d)/b - 5*e^(-3*b*x + a - 4*d)/b)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 307 vs. \(2 (115) = 230\).

Time = 0.14 (sec) , antiderivative size = 307, normalized size of antiderivative = 2.36 \[ \int e^{a+b x} (g \cosh (d+b x)+f \sinh (d+b x))^4 \, dx=\frac {{\left (3 \, f^{4} e^{\left (5 \, b x + a + 8 \, d\right )} + 12 \, f^{3} g e^{\left (5 \, b x + a + 8 \, d\right )} + 18 \, f^{2} g^{2} e^{\left (5 \, b x + a + 8 \, d\right )} + 12 \, f g^{3} e^{\left (5 \, b x + a + 8 \, d\right )} + 3 \, g^{4} e^{\left (5 \, b x + a + 8 \, d\right )} - 20 \, f^{4} e^{\left (3 \, b x + a + 6 \, d\right )} - 40 \, f^{3} g e^{\left (3 \, b x + a + 6 \, d\right )} + 40 \, f g^{3} e^{\left (3 \, b x + a + 6 \, d\right )} + 20 \, g^{4} e^{\left (3 \, b x + a + 6 \, d\right )} + 90 \, f^{4} e^{\left (b x + a + 4 \, d\right )} - 180 \, f^{2} g^{2} e^{\left (b x + a + 4 \, d\right )} + 90 \, g^{4} e^{\left (b x + a + 4 \, d\right )} + 5 \, {\left (12 \, f^{4} e^{\left (2 \, b x + a + 2 \, d\right )} - 24 \, f^{3} g e^{\left (2 \, b x + a + 2 \, d\right )} + 24 \, f g^{3} e^{\left (2 \, b x + a + 2 \, d\right )} - 12 \, g^{4} e^{\left (2 \, b x + a + 2 \, d\right )} - f^{4} e^{a} + 4 \, f^{3} g e^{a} - 6 \, f^{2} g^{2} e^{a} + 4 \, f g^{3} e^{a} - g^{4} e^{a}\right )} e^{\left (-3 \, b x\right )}\right )} e^{\left (-4 \, d\right )}}{240 \, b} \] Input:

integrate(exp(b*x+a)*(g*cosh(b*x+d)+f*sinh(b*x+d))^4,x, algorithm="giac")
 

Output:

1/240*(3*f^4*e^(5*b*x + a + 8*d) + 12*f^3*g*e^(5*b*x + a + 8*d) + 18*f^2*g 
^2*e^(5*b*x + a + 8*d) + 12*f*g^3*e^(5*b*x + a + 8*d) + 3*g^4*e^(5*b*x + a 
 + 8*d) - 20*f^4*e^(3*b*x + a + 6*d) - 40*f^3*g*e^(3*b*x + a + 6*d) + 40*f 
*g^3*e^(3*b*x + a + 6*d) + 20*g^4*e^(3*b*x + a + 6*d) + 90*f^4*e^(b*x + a 
+ 4*d) - 180*f^2*g^2*e^(b*x + a + 4*d) + 90*g^4*e^(b*x + a + 4*d) + 5*(12* 
f^4*e^(2*b*x + a + 2*d) - 24*f^3*g*e^(2*b*x + a + 2*d) + 24*f*g^3*e^(2*b*x 
 + a + 2*d) - 12*g^4*e^(2*b*x + a + 2*d) - f^4*e^a + 4*f^3*g*e^a - 6*f^2*g 
^2*e^a + 4*f*g^3*e^a - g^4*e^a)*e^(-3*b*x))*e^(-4*d)/b
 

Mupad [B] (verification not implemented)

Time = 0.79 (sec) , antiderivative size = 271, normalized size of antiderivative = 2.08 \[ \int e^{a+b x} (g \cosh (d+b x)+f \sinh (d+b x))^4 \, dx=\frac {{\mathrm {cosh}\left (d+b\,x\right )}^4\,{\mathrm {e}}^{a+b\,x}\,\left (8\,f^4-8\,f^3\,g-12\,f^2\,g^2+12\,f\,g^3+3\,g^4\right )}{15\,b}+\frac {{\mathrm {e}}^{a+b\,x}\,{\mathrm {sinh}\left (d+b\,x\right )}^4\,\left (3\,f^4+12\,f^3\,g-12\,f^2\,g^2-8\,f\,g^3+8\,g^4\right )}{15\,b}+\frac {4\,\mathrm {cosh}\left (d+b\,x\right )\,{\mathrm {e}}^{a+b\,x}\,{\mathrm {sinh}\left (d+b\,x\right )}^3\,\left (3\,f^4-3\,f^3\,g+3\,f^2\,g^2+2\,f\,g^3-2\,g^4\right )}{15\,b}+\frac {4\,{\mathrm {cosh}\left (d+b\,x\right )}^3\,{\mathrm {e}}^{a+b\,x}\,\mathrm {sinh}\left (d+b\,x\right )\,\left (-2\,f^4+2\,f^3\,g+3\,f^2\,g^2-3\,f\,g^3+3\,g^4\right )}{15\,b}+\frac {2\,{\mathrm {cosh}\left (d+b\,x\right )}^2\,{\mathrm {e}}^{a+b\,x}\,{\mathrm {sinh}\left (d+b\,x\right )}^2\,\left (-2\,f^4+2\,f^3\,g+3\,f^2\,g^2+2\,f\,g^3-2\,g^4\right )}{5\,b} \] Input:

int(exp(a + b*x)*(g*cosh(d + b*x) + f*sinh(d + b*x))^4,x)
 

Output:

(cosh(d + b*x)^4*exp(a + b*x)*(12*f*g^3 - 8*f^3*g + 8*f^4 + 3*g^4 - 12*f^2 
*g^2))/(15*b) + (exp(a + b*x)*sinh(d + b*x)^4*(12*f^3*g - 8*f*g^3 + 3*f^4 
+ 8*g^4 - 12*f^2*g^2))/(15*b) + (4*cosh(d + b*x)*exp(a + b*x)*sinh(d + b*x 
)^3*(2*f*g^3 - 3*f^3*g + 3*f^4 - 2*g^4 + 3*f^2*g^2))/(15*b) + (4*cosh(d + 
b*x)^3*exp(a + b*x)*sinh(d + b*x)*(2*f^3*g - 3*f*g^3 - 2*f^4 + 3*g^4 + 3*f 
^2*g^2))/(15*b) + (2*cosh(d + b*x)^2*exp(a + b*x)*sinh(d + b*x)^2*(2*f*g^3 
 + 2*f^3*g - 2*f^4 - 2*g^4 + 3*f^2*g^2))/(5*b)
 

Reduce [B] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 463, normalized size of antiderivative = 3.56 \[ \int e^{a+b x} (g \cosh (d+b x)+f \sinh (d+b x))^4 \, dx=\frac {e^{b x +a} \left (8 \cosh \left (b x +d \right )^{3} \sinh \left (b x +d \right ) f^{3} g +12 \cosh \left (b x +d \right )^{3} \sinh \left (b x +d \right ) f^{2} g^{2}-12 \cosh \left (b x +d \right )^{3} \sinh \left (b x +d \right ) f \,g^{3}+12 \cosh \left (b x +d \right )^{2} \sinh \left (b x +d \right )^{2} f^{3} g +18 \cosh \left (b x +d \right )^{2} \sinh \left (b x +d \right )^{2} f^{2} g^{2}+12 \cosh \left (b x +d \right )^{2} \sinh \left (b x +d \right )^{2} f \,g^{3}-12 \cosh \left (b x +d \right ) \sinh \left (b x +d \right )^{3} f^{3} g +12 \cosh \left (b x +d \right ) \sinh \left (b x +d \right )^{3} f^{2} g^{2}+8 \cosh \left (b x +d \right ) \sinh \left (b x +d \right )^{3} f \,g^{3}+8 \cosh \left (b x +d \right )^{4} f^{4}+3 \cosh \left (b x +d \right )^{4} g^{4}-8 \sinh \left (b x +d \right )^{4} f \,g^{3}-12 \cosh \left (b x +d \right )^{2} \sinh \left (b x +d \right )^{2} f^{4}+12 \sinh \left (b x +d \right )^{4} f^{3} g +12 \cosh \left (b x +d \right )^{3} \sinh \left (b x +d \right ) g^{4}-12 \sinh \left (b x +d \right )^{4} f^{2} g^{2}-8 \cosh \left (b x +d \right )^{4} f^{3} g +12 \cosh \left (b x +d \right )^{4} f \,g^{3}-12 \cosh \left (b x +d \right )^{2} \sinh \left (b x +d \right )^{2} g^{4}+12 \cosh \left (b x +d \right ) \sinh \left (b x +d \right )^{3} f^{4}-8 \cosh \left (b x +d \right ) \sinh \left (b x +d \right )^{3} g^{4}-12 \cosh \left (b x +d \right )^{4} f^{2} g^{2}-8 \cosh \left (b x +d \right )^{3} \sinh \left (b x +d \right ) f^{4}+3 \sinh \left (b x +d \right )^{4} f^{4}+8 \sinh \left (b x +d \right )^{4} g^{4}\right )}{15 b} \] Input:

int(exp(b*x+a)*(g*cosh(b*x+d)+f*sinh(b*x+d))^4,x)
 

Output:

(e**(a + b*x)*(8*cosh(b*x + d)**4*f**4 - 8*cosh(b*x + d)**4*f**3*g - 12*co 
sh(b*x + d)**4*f**2*g**2 + 12*cosh(b*x + d)**4*f*g**3 + 3*cosh(b*x + d)**4 
*g**4 - 8*cosh(b*x + d)**3*sinh(b*x + d)*f**4 + 8*cosh(b*x + d)**3*sinh(b* 
x + d)*f**3*g + 12*cosh(b*x + d)**3*sinh(b*x + d)*f**2*g**2 - 12*cosh(b*x 
+ d)**3*sinh(b*x + d)*f*g**3 + 12*cosh(b*x + d)**3*sinh(b*x + d)*g**4 - 12 
*cosh(b*x + d)**2*sinh(b*x + d)**2*f**4 + 12*cosh(b*x + d)**2*sinh(b*x + d 
)**2*f**3*g + 18*cosh(b*x + d)**2*sinh(b*x + d)**2*f**2*g**2 + 12*cosh(b*x 
 + d)**2*sinh(b*x + d)**2*f*g**3 - 12*cosh(b*x + d)**2*sinh(b*x + d)**2*g* 
*4 + 12*cosh(b*x + d)*sinh(b*x + d)**3*f**4 - 12*cosh(b*x + d)*sinh(b*x + 
d)**3*f**3*g + 12*cosh(b*x + d)*sinh(b*x + d)**3*f**2*g**2 + 8*cosh(b*x + 
d)*sinh(b*x + d)**3*f*g**3 - 8*cosh(b*x + d)*sinh(b*x + d)**3*g**4 + 3*sin 
h(b*x + d)**4*f**4 + 12*sinh(b*x + d)**4*f**3*g - 12*sinh(b*x + d)**4*f**2 
*g**2 - 8*sinh(b*x + d)**4*f*g**3 + 8*sinh(b*x + d)**4*g**4))/(15*b)