\(\int e^{2 (a+b x)} (g \cosh (d+b x)+f \sinh (d+b x))^4 \, dx\) [8]

Optimal result

Integrand size = 29, antiderivative size = 136 \[ \int e^{2 (a+b x)} (g \cosh (d+b x)+f \sinh (d+b x))^4 \, dx=-\frac {e^{2 (a-2 d)-2 b x} (f-g)^4}{32 b}-\frac {e^{2 (a+d)+4 b x} (f-g) (f+g)^3}{16 b}+\frac {e^{2 (a+2 d)+6 b x} (f+g)^4}{96 b}+\frac {3 e^{2 a+2 b x} \left (f^2-g^2\right )^2}{16 b}-\frac {1}{4} e^{2 a-2 d} (f-g)^3 (f+g) x \] Output:

-1/32*exp(-2*b*x+2*a-4*d)*(f-g)^4/b-1/16*exp(4*b*x+2*a+2*d)*(f-g)*(f+g)^3/ 
b+1/96*exp(6*b*x+2*a+4*d)*(f+g)^4/b+3/16*exp(2*b*x+2*a)*(f^2-g^2)^2/b-1/4* 
exp(2*a-2*d)*(f-g)^3*(f+g)*x
 

Mathematica [A] (verified)

Time = 2.73 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.83 \[ \int e^{2 (a+b x)} (g \cosh (d+b x)+f \sinh (d+b x))^4 \, dx=\frac {e^{2 a-2 d} \left (-e^{-2 (d+b x)} (f-g)^4-2 e^{4 (d+b x)} (f-g) (f+g)^3+\frac {1}{3} e^{6 (d+b x)} (f+g)^4+6 e^{2 (d+b x)} \left (f^2-g^2\right )^2-8 (f-g)^3 (f+g) (d+b x)\right )}{32 b} \] Input:

Integrate[E^(2*(a + b*x))*(g*Cosh[d + b*x] + f*Sinh[d + b*x])^4,x]
 

Output:

(E^(2*a - 2*d)*(-((f - g)^4/E^(2*(d + b*x))) - 2*E^(4*(d + b*x))*(f - g)*( 
f + g)^3 + (E^(6*(d + b*x))*(f + g)^4)/3 + 6*E^(2*(d + b*x))*(f^2 - g^2)^2 
 - 8*(f - g)^3*(f + g)*(d + b*x)))/(32*b)
 

Rubi [A] (warning: unable to verify)

Time = 0.34 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.73, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {2720, 27, 243, 49, 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.

\(\displaystyle \int e^{2 (a+b x)} (f \sinh (b x+d)+g \cosh (b x+d))^4 \, dx\)

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 243

\(\displaystyle \frac {e^{2 a} \int e^{-2 b x} \left (f-g-e^{2 b x} (f+g)\right )^4de^{2 b x}}{32 b}\)

\(\Big \downarrow \) 49

\(\displaystyle \frac {e^{2 a} \int \left (e^{-2 b x} (f-g)^4-4 e^{-b x} (f+g) (f-g)^3-4 e^{2 b x} (f+g)^3 (f-g)+e^{2 b x} (f+g)^4+6 \left (f^2-g^2\right )^2\right )de^{2 b x}}{32 b}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {e^{2 a} \left (6 e^{2 b x} \left (f^2-g^2\right )^2-e^{-b x} (f-g)^4-2 e^{2 b x} (f+g)^3 (f-g)+\frac {1}{3} e^{3 b x} (f+g)^4-4 (f+g) (f-g)^3 \log \left (e^{2 b x}\right )\right )}{32 b}\)

Input:

Int[E^(2*(a + b*x))*(g*Cosh[d + b*x] + f*Sinh[d + b*x])^4,x]
 

Output:

(E^(2*a)*(-((f - g)^4/E^(b*x)) - 2*E^(2*b*x)*(f - g)*(f + g)^3 + (E^(3*b*x 
)*(f + g)^4)/3 + 6*E^(2*b*x)*(f^2 - g^2)^2 - 4*(f - g)^3*(f + g)*Log[E^(2* 
b*x)]))/(32*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 49

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int 
[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] 
&& IGtQ[m, 0] && IGtQ[m + n + 2, 0]
 

rule 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

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. \(403\) vs. \(2(120)=240\).

Time = 0.19 (sec) , antiderivative size = 404, normalized size of antiderivative = 2.97

\[\frac {\left (-\frac {3}{4} f^{2} g^{2}+\frac {3}{8} g^{4}+\frac {3}{8} f^{4}\right ) \sinh \left (2 b x +2 a \right )}{2 b}+\frac {\left (-\frac {3}{4} f^{2} g^{2}+\frac {3}{8} g^{4}+\frac {3}{8} f^{4}\right ) \cosh \left (2 b x +2 a \right )}{2 b}+\frac {\left (\frac {1}{4} g^{4}-\frac {1}{4} f^{4}-\frac {1}{2} g \,f^{3}+\frac {1}{2} g^{3} f \right ) \sinh \left (4 b x +2 a +2 d \right )}{4 b}+\frac {\left (\frac {1}{4} g^{4}-\frac {1}{4} f^{4}-\frac {1}{2} g \,f^{3}+\frac {1}{2} g^{3} f \right ) \cosh \left (4 b x +2 a +2 d \right )}{4 b}+\frac {\left (-f^{4}+2 g \,f^{3}-2 g^{3} f +g^{4}\right ) \cosh \left (2 a -2 d \right ) x}{4}+\frac {\left (-f^{4}+2 g \,f^{3}-2 g^{3} f +g^{4}\right ) \sinh \left (2 a -2 d \right ) x}{4}-\frac {\left (\frac {3}{8} f^{2} g^{2}+\frac {1}{16} g^{4}+\frac {1}{16} f^{4}-\frac {1}{4} g \,f^{3}-\frac {1}{4} g^{3} f \right ) \sinh \left (-2 b x +2 a -4 d \right )}{2 b}+\frac {\left (-\frac {1}{16} f^{4}+\frac {1}{4} g \,f^{3}-\frac {3}{8} f^{2} g^{2}+\frac {1}{4} g^{3} f -\frac {1}{16} g^{4}\right ) \cosh \left (-2 b x +2 a -4 d \right )}{2 b}+\frac {\left (\frac {3}{8} f^{2} g^{2}+\frac {1}{16} g^{4}+\frac {1}{16} f^{4}+\frac {1}{4} g \,f^{3}+\frac {1}{4} g^{3} f \right ) \sinh \left (6 b x +2 a +4 d \right )}{6 b}+\frac {\left (\frac {3}{8} f^{2} g^{2}+\frac {1}{16} g^{4}+\frac {1}{16} f^{4}+\frac {1}{4} g \,f^{3}+\frac {1}{4} g^{3} f \right ) \cosh \left (6 b x +2 a +4 d \right )}{6 b}\]

Input:

int(exp(2*b*x+2*a)*(g*cosh(b*x+d)+f*sinh(b*x+d))^4,x)
 

Output:

1/2*(-3/4*f^2*g^2+3/8*g^4+3/8*f^4)/b*sinh(2*b*x+2*a)+1/2*(-3/4*f^2*g^2+3/8 
*g^4+3/8*f^4)*cosh(2*b*x+2*a)/b+1/4*(1/4*g^4-1/4*f^4-1/2*g*f^3+1/2*g^3*f)/ 
b*sinh(4*b*x+2*a+2*d)+1/4*(1/4*g^4-1/4*f^4-1/2*g*f^3+1/2*g^3*f)*cosh(4*b*x 
+2*a+2*d)/b+1/4*(-f^4+2*f^3*g-2*f*g^3+g^4)*cosh(2*a-2*d)*x+1/4*(-f^4+2*f^3 
*g-2*f*g^3+g^4)*sinh(2*a-2*d)*x-1/2*(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(-2*b*x+2*a-4*d)+1/2*(-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(-2*b*x+2*a-4*d)/b+1/6*(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(6*b*x+2*a+4*d)+1/6*(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(6*b*x+2*a+4*d)/b
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 920 vs. \(2 (120) = 240\).

Time = 0.09 (sec) , antiderivative size = 920, normalized size of antiderivative = 6.76 \[ \int e^{2 (a+b x)} (g \cosh (d+b x)+f \sinh (d+b x))^4 \, dx =\text {Too large to display} \] Input:

integrate(exp(2*b*x+2*a)*(g*cosh(b*x+d)+f*sinh(b*x+d))^4,x, algorithm="fri 
cas")
 

Output:

-1/48*((f^4 - 8*f^3*g + 6*f^2*g^2 - 8*f*g^3 + g^4)*cosh(b*x + d)^4*cosh(-2 
*a + 2*d) + ((f^4 - 8*f^3*g + 6*f^2*g^2 - 8*f*g^3 + g^4)*cosh(-2*a + 2*d) 
- (f^4 - 8*f^3*g + 6*f^2*g^2 - 8*f*g^3 + g^4)*sinh(-2*a + 2*d))*sinh(b*x + 
 d)^4 + 3*(f^4 + 2*f^3*g - 2*f*g^3 - g^4 + 4*(b*f^4 - 2*b*f^3*g + 2*b*f*g^ 
3 - b*g^4)*x)*cosh(b*x + d)^2*cosh(-2*a + 2*d) - 8*((f^4 - 2*f^3*g + 6*f^2 
*g^2 - 2*f*g^3 + g^4)*cosh(b*x + d)*cosh(-2*a + 2*d) - (f^4 - 2*f^3*g + 6* 
f^2*g^2 - 2*f*g^3 + g^4)*cosh(b*x + d)*sinh(-2*a + 2*d))*sinh(b*x + d)^3 + 
 3*(2*(f^4 - 8*f^3*g + 6*f^2*g^2 - 8*f*g^3 + g^4)*cosh(b*x + d)^2*cosh(-2* 
a + 2*d) + (f^4 + 2*f^3*g - 2*f*g^3 - g^4 + 4*(b*f^4 - 2*b*f^3*g + 2*b*f*g 
^3 - b*g^4)*x)*cosh(-2*a + 2*d) - (f^4 + 2*f^3*g - 2*f*g^3 - g^4 + 2*(f^4 
- 8*f^3*g + 6*f^2*g^2 - 8*f*g^3 + g^4)*cosh(b*x + d)^2 + 4*(b*f^4 - 2*b*f^ 
3*g + 2*b*f*g^3 - b*g^4)*x)*sinh(-2*a + 2*d))*sinh(b*x + d)^2 - 9*(f^4 - 2 
*f^2*g^2 + g^4)*cosh(-2*a + 2*d) - 2*(4*(f^4 - 2*f^3*g + 6*f^2*g^2 - 2*f*g 
^3 + g^4)*cosh(b*x + d)^3*cosh(-2*a + 2*d) - 3*(f^4 + 2*f^3*g - 2*f*g^3 - 
g^4 - 4*(b*f^4 - 2*b*f^3*g + 2*b*f*g^3 - b*g^4)*x)*cosh(b*x + d)*cosh(-2*a 
 + 2*d) - (4*(f^4 - 2*f^3*g + 6*f^2*g^2 - 2*f*g^3 + g^4)*cosh(b*x + d)^3 - 
 3*(f^4 + 2*f^3*g - 2*f*g^3 - g^4 - 4*(b*f^4 - 2*b*f^3*g + 2*b*f*g^3 - b*g 
^4)*x)*cosh(b*x + d))*sinh(-2*a + 2*d))*sinh(b*x + d) - ((f^4 - 8*f^3*g + 
6*f^2*g^2 - 8*f*g^3 + g^4)*cosh(b*x + d)^4 - 9*f^4 + 18*f^2*g^2 - 9*g^4 + 
3*(f^4 + 2*f^3*g - 2*f*g^3 - g^4 + 4*(b*f^4 - 2*b*f^3*g + 2*b*f*g^3 - b...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1246 vs. \(2 (112) = 224\).

Time = 2.46 (sec) , antiderivative size = 1246, normalized size of antiderivative = 9.16 \[ \int e^{2 (a+b x)} (g \cosh (d+b x)+f \sinh (d+b x))^4 \, dx=\text {Too large to display} \] Input:

integrate(exp(2*b*x+2*a)*(g*cosh(b*x+d)+f*sinh(b*x+d))**4,x)
 

Output:

Piecewise((f**4*x*exp(2*a)*exp(2*b*x)*sinh(b*x + d)**4/4 - f**4*x*exp(2*a) 
*exp(2*b*x)*sinh(b*x + d)**3*cosh(b*x + d)/2 + f**4*x*exp(2*a)*exp(2*b*x)* 
sinh(b*x + d)*cosh(b*x + d)**3/2 - f**4*x*exp(2*a)*exp(2*b*x)*cosh(b*x + d 
)**4/4 - f**3*g*x*exp(2*a)*exp(2*b*x)*sinh(b*x + d)**4/2 + f**3*g*x*exp(2* 
a)*exp(2*b*x)*sinh(b*x + d)**3*cosh(b*x + d) - f**3*g*x*exp(2*a)*exp(2*b*x 
)*sinh(b*x + d)*cosh(b*x + d)**3 + f**3*g*x*exp(2*a)*exp(2*b*x)*cosh(b*x + 
 d)**4/2 + f*g**3*x*exp(2*a)*exp(2*b*x)*sinh(b*x + d)**4/2 - f*g**3*x*exp( 
2*a)*exp(2*b*x)*sinh(b*x + d)**3*cosh(b*x + d) + f*g**3*x*exp(2*a)*exp(2*b 
*x)*sinh(b*x + d)*cosh(b*x + d)**3 - f*g**3*x*exp(2*a)*exp(2*b*x)*cosh(b*x 
 + d)**4/2 - g**4*x*exp(2*a)*exp(2*b*x)*sinh(b*x + d)**4/4 + g**4*x*exp(2* 
a)*exp(2*b*x)*sinh(b*x + d)**3*cosh(b*x + d)/2 - g**4*x*exp(2*a)*exp(2*b*x 
)*sinh(b*x + d)*cosh(b*x + d)**3/2 + g**4*x*exp(2*a)*exp(2*b*x)*cosh(b*x + 
 d)**4/4 + f**4*exp(2*a)*exp(2*b*x)*sinh(b*x + d)**4/(48*b) + 17*f**4*exp( 
2*a)*exp(2*b*x)*sinh(b*x + d)**3*cosh(b*x + d)/(24*b) - f**4*exp(2*a)*exp( 
2*b*x)*sinh(b*x + d)**2*cosh(b*x + d)**2/(2*b) - 3*f**4*exp(2*a)*exp(2*b*x 
)*sinh(b*x + d)*cosh(b*x + d)**3/(8*b) + 5*f**4*exp(2*a)*exp(2*b*x)*cosh(b 
*x + d)**4/(16*b) + 7*f**3*g*exp(2*a)*exp(2*b*x)*sinh(b*x + d)**4/(12*b) - 
 2*f**3*g*exp(2*a)*exp(2*b*x)*sinh(b*x + d)**3*cosh(b*x + d)/(3*b) + f**3* 
g*exp(2*a)*exp(2*b*x)*sinh(b*x + d)**2*cosh(b*x + d)**2/b - f**3*g*exp(2*a 
)*exp(2*b*x)*cosh(b*x + d)**4/(4*b) - 5*f**2*g**2*exp(2*a)*exp(2*b*x)*s...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 360 vs. \(2 (120) = 240\).

Time = 0.07 (sec) , antiderivative size = 360, normalized size of antiderivative = 2.65 \[ \int e^{2 (a+b x)} (g \cosh (d+b x)+f \sinh (d+b x))^4 \, dx=\frac {1}{96} \, g^{4} {\left (\frac {{\left (6 \, e^{\left (-2 \, b x - 2 \, d\right )} + 18 \, e^{\left (-4 \, b x - 4 \, d\right )} + 1\right )} e^{\left (6 \, b x + 2 \, a + 4 \, d\right )}}{b} + \frac {24 \, {\left (b x + d\right )} e^{\left (2 \, a - 2 \, d\right )}}{b} - \frac {3 \, e^{\left (-2 \, b x + 2 \, a - 4 \, d\right )}}{b}\right )} - \frac {1}{96} \, f^{4} {\left (\frac {{\left (6 \, e^{\left (-2 \, b x - 2 \, d\right )} - 18 \, e^{\left (-4 \, b x - 4 \, d\right )} - 1\right )} e^{\left (6 \, b x + 2 \, a + 4 \, d\right )}}{b} + \frac {24 \, {\left (b x + d\right )} e^{\left (2 \, a - 2 \, d\right )}}{b} + \frac {3 \, e^{\left (-2 \, b x + 2 \, a - 4 \, d\right )}}{b}\right )} + \frac {1}{24} \, f g^{3} {\left (\frac {{\left (3 \, e^{\left (-2 \, b x - 2 \, d\right )} + 1\right )} e^{\left (6 \, b x + 2 \, a + 4 \, d\right )}}{b} - \frac {12 \, {\left (b x + d\right )} e^{\left (2 \, a - 2 \, d\right )}}{b} + \frac {3 \, e^{\left (-2 \, b x + 2 \, a - 4 \, d\right )}}{b}\right )} - \frac {1}{24} \, f^{3} g {\left (\frac {{\left (3 \, e^{\left (-2 \, b x - 2 \, d\right )} - 1\right )} e^{\left (6 \, b x + 2 \, a + 4 \, d\right )}}{b} - \frac {12 \, {\left (b x + d\right )} e^{\left (2 \, a - 2 \, d\right )}}{b} - \frac {3 \, e^{\left (-2 \, b x + 2 \, a - 4 \, d\right )}}{b}\right )} - \frac {1}{16} \, f^{2} g^{2} {\left (\frac {{\left (6 \, e^{\left (-4 \, b x - 4 \, d\right )} - 1\right )} e^{\left (6 \, b x + 2 \, a + 4 \, d\right )}}{b} + \frac {3 \, e^{\left (-2 \, b x + 2 \, a - 4 \, d\right )}}{b}\right )} \] Input:

integrate(exp(2*b*x+2*a)*(g*cosh(b*x+d)+f*sinh(b*x+d))^4,x, algorithm="max 
ima")
 

Output:

1/96*g^4*((6*e^(-2*b*x - 2*d) + 18*e^(-4*b*x - 4*d) + 1)*e^(6*b*x + 2*a + 
4*d)/b + 24*(b*x + d)*e^(2*a - 2*d)/b - 3*e^(-2*b*x + 2*a - 4*d)/b) - 1/96 
*f^4*((6*e^(-2*b*x - 2*d) - 18*e^(-4*b*x - 4*d) - 1)*e^(6*b*x + 2*a + 4*d) 
/b + 24*(b*x + d)*e^(2*a - 2*d)/b + 3*e^(-2*b*x + 2*a - 4*d)/b) + 1/24*f*g 
^3*((3*e^(-2*b*x - 2*d) + 1)*e^(6*b*x + 2*a + 4*d)/b - 12*(b*x + d)*e^(2*a 
 - 2*d)/b + 3*e^(-2*b*x + 2*a - 4*d)/b) - 1/24*f^3*g*((3*e^(-2*b*x - 2*d) 
- 1)*e^(6*b*x + 2*a + 4*d)/b - 12*(b*x + d)*e^(2*a - 2*d)/b - 3*e^(-2*b*x 
+ 2*a - 4*d)/b) - 1/16*f^2*g^2*((6*e^(-4*b*x - 4*d) - 1)*e^(6*b*x + 2*a + 
4*d)/b + 3*e^(-2*b*x + 2*a - 4*d)/b)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 399 vs. \(2 (120) = 240\).

Time = 0.12 (sec) , antiderivative size = 399, normalized size of antiderivative = 2.93 \[ \int e^{2 (a+b x)} (g \cosh (d+b x)+f \sinh (d+b x))^4 \, dx=\frac {{\left (f^{4} e^{\left (6 \, b x + 2 \, a + 6 \, d\right )} + 4 \, f^{3} g e^{\left (6 \, b x + 2 \, a + 6 \, d\right )} + 6 \, f^{2} g^{2} e^{\left (6 \, b x + 2 \, a + 6 \, d\right )} + 4 \, f g^{3} e^{\left (6 \, b x + 2 \, a + 6 \, d\right )} + g^{4} e^{\left (6 \, b x + 2 \, a + 6 \, d\right )} - 6 \, f^{4} e^{\left (4 \, b x + 2 \, a + 4 \, d\right )} - 12 \, f^{3} g e^{\left (4 \, b x + 2 \, a + 4 \, d\right )} + 12 \, f g^{3} e^{\left (4 \, b x + 2 \, a + 4 \, d\right )} + 6 \, g^{4} e^{\left (4 \, b x + 2 \, a + 4 \, d\right )} + 18 \, f^{4} e^{\left (2 \, b x + 2 \, a + 2 \, d\right )} - 36 \, f^{2} g^{2} e^{\left (2 \, b x + 2 \, a + 2 \, d\right )} + 18 \, g^{4} e^{\left (2 \, b x + 2 \, a + 2 \, d\right )} - 24 \, {\left (f^{4} e^{\left (2 \, a\right )} - 2 \, f^{3} g e^{\left (2 \, a\right )} + 2 \, f g^{3} e^{\left (2 \, a\right )} - g^{4} e^{\left (2 \, a\right )}\right )} {\left (b x + d\right )} + 3 \, {\left (4 \, f^{4} e^{\left (2 \, b x + 2 \, a + 2 \, d\right )} - 8 \, f^{3} g e^{\left (2 \, b x + 2 \, a + 2 \, d\right )} + 8 \, f g^{3} e^{\left (2 \, b x + 2 \, a + 2 \, d\right )} - 4 \, g^{4} e^{\left (2 \, b x + 2 \, a + 2 \, d\right )} - f^{4} e^{\left (2 \, a\right )} + 4 \, f^{3} g e^{\left (2 \, a\right )} - 6 \, f^{2} g^{2} e^{\left (2 \, a\right )} + 4 \, f g^{3} e^{\left (2 \, a\right )} - g^{4} e^{\left (2 \, a\right )}\right )} e^{\left (-2 \, b x - 2 \, d\right )}\right )} e^{\left (-2 \, d\right )}}{96 \, b} \] Input:

integrate(exp(2*b*x+2*a)*(g*cosh(b*x+d)+f*sinh(b*x+d))^4,x, algorithm="gia 
c")
 

Output:

1/96*(f^4*e^(6*b*x + 2*a + 6*d) + 4*f^3*g*e^(6*b*x + 2*a + 6*d) + 6*f^2*g^ 
2*e^(6*b*x + 2*a + 6*d) + 4*f*g^3*e^(6*b*x + 2*a + 6*d) + g^4*e^(6*b*x + 2 
*a + 6*d) - 6*f^4*e^(4*b*x + 2*a + 4*d) - 12*f^3*g*e^(4*b*x + 2*a + 4*d) + 
 12*f*g^3*e^(4*b*x + 2*a + 4*d) + 6*g^4*e^(4*b*x + 2*a + 4*d) + 18*f^4*e^( 
2*b*x + 2*a + 2*d) - 36*f^2*g^2*e^(2*b*x + 2*a + 2*d) + 18*g^4*e^(2*b*x + 
2*a + 2*d) - 24*(f^4*e^(2*a) - 2*f^3*g*e^(2*a) + 2*f*g^3*e^(2*a) - g^4*e^( 
2*a))*(b*x + d) + 3*(4*f^4*e^(2*b*x + 2*a + 2*d) - 8*f^3*g*e^(2*b*x + 2*a 
+ 2*d) + 8*f*g^3*e^(2*b*x + 2*a + 2*d) - 4*g^4*e^(2*b*x + 2*a + 2*d) - f^4 
*e^(2*a) + 4*f^3*g*e^(2*a) - 6*f^2*g^2*e^(2*a) + 4*f*g^3*e^(2*a) - g^4*e^( 
2*a))*e^(-2*b*x - 2*d))*e^(-2*d)/b
 

Mupad [B] (verification not implemented)

Time = 0.80 (sec) , antiderivative size = 347, normalized size of antiderivative = 2.55 \[ \int e^{2 (a+b x)} (g \cosh (d+b x)+f \sinh (d+b x))^4 \, dx=\frac {{\mathrm {cosh}\left (d+b\,x\right )}^4\,{\mathrm {e}}^{2\,a+2\,b\,x}\,\left (\frac {f^4}{3}+\frac {f^3\,g}{3}-f^2\,g^2+\frac {f\,g^3}{3}+\frac {g^4}{3}\right )}{b}+\frac {{\mathrm {cosh}\left (d+b\,x\right )}^2\,{\mathrm {e}}^{2\,a+2\,b\,x}\,{\mathrm {sinh}\left (d+b\,x\right )}^2\,\left (-f^4+2\,f^3\,g+2\,f\,g^3-g^4\right )}{2\,b}-\frac {x\,{\mathrm {cosh}\left (d+b\,x\right )}^4\,{\mathrm {e}}^{2\,a+2\,b\,x}\,\left (f+g\right )\,{\left (f-g\right )}^3}{4}+\frac {x\,{\mathrm {e}}^{2\,a+2\,b\,x}\,{\mathrm {sinh}\left (d+b\,x\right )}^4\,\left (f+g\right )\,{\left (f-g\right )}^3}{4}-\frac {{\mathrm {cosh}\left (d+b\,x\right )}^3\,{\mathrm {e}}^{2\,a+2\,b\,x}\,\mathrm {sinh}\left (d+b\,x\right )\,\left (\frac {5\,f^4}{12}+\frac {7\,f^3\,g}{6}-2\,f^2\,g^2+\frac {f\,g^3}{6}-\frac {g^4}{12}\right )}{b}+\frac {\mathrm {cosh}\left (d+b\,x\right )\,{\mathrm {e}}^{2\,a+2\,b\,x}\,{\mathrm {sinh}\left (d+b\,x\right )}^3\,\left (\frac {3\,f^4}{4}+\frac {f^3\,g}{2}-\frac {f\,g^3}{2}+\frac {g^4}{4}\right )}{b}-\frac {x\,\mathrm {cosh}\left (d+b\,x\right )\,{\mathrm {e}}^{2\,a+2\,b\,x}\,{\mathrm {sinh}\left (d+b\,x\right )}^3\,\left (f+g\right )\,{\left (f-g\right )}^3}{2}+\frac {x\,{\mathrm {cosh}\left (d+b\,x\right )}^3\,{\mathrm {e}}^{2\,a+2\,b\,x}\,\mathrm {sinh}\left (d+b\,x\right )\,\left (f+g\right )\,{\left (f-g\right )}^3}{2} \] Input:

int(exp(2*a + 2*b*x)*(g*cosh(d + b*x) + f*sinh(d + b*x))^4,x)
 

Output:

(cosh(d + b*x)^4*exp(2*a + 2*b*x)*((f*g^3)/3 + (f^3*g)/3 + f^4/3 + g^4/3 - 
 f^2*g^2))/b + (cosh(d + b*x)^2*exp(2*a + 2*b*x)*sinh(d + b*x)^2*(2*f*g^3 
+ 2*f^3*g - f^4 - g^4))/(2*b) - (x*cosh(d + b*x)^4*exp(2*a + 2*b*x)*(f + g 
)*(f - g)^3)/4 + (x*exp(2*a + 2*b*x)*sinh(d + b*x)^4*(f + g)*(f - g)^3)/4 
- (cosh(d + b*x)^3*exp(2*a + 2*b*x)*sinh(d + b*x)*((f*g^3)/6 + (7*f^3*g)/6 
 + (5*f^4)/12 - g^4/12 - 2*f^2*g^2))/b + (cosh(d + b*x)*exp(2*a + 2*b*x)*s 
inh(d + b*x)^3*((f^3*g)/2 - (f*g^3)/2 + (3*f^4)/4 + g^4/4))/b - (x*cosh(d 
+ b*x)*exp(2*a + 2*b*x)*sinh(d + b*x)^3*(f + g)*(f - g)^3)/2 + (x*cosh(d + 
 b*x)^3*exp(2*a + 2*b*x)*sinh(d + b*x)*(f + g)*(f - g)^3)/2
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 622, normalized size of antiderivative = 4.57 \[ \int e^{2 (a+b x)} (g \cosh (d+b x)+f \sinh (d+b x))^4 \, dx =\text {Too large to display} \] Input:

int(exp(2*b*x+2*a)*(g*cosh(b*x+d)+f*sinh(b*x+d))^4,x)
 

Output:

(e**(2*a + 2*b*x)*( - 6*cosh(b*x + d)**4*b*f**4*x + 12*cosh(b*x + d)**4*b* 
f**3*g*x - 12*cosh(b*x + d)**4*b*f*g**3*x + 6*cosh(b*x + d)**4*b*g**4*x + 
3*cosh(b*x + d)**4*f**4 - 6*cosh(b*x + d)**4*f**3*g + 6*cosh(b*x + d)**4*f 
*g**3 + 9*cosh(b*x + d)**4*g**4 + 12*cosh(b*x + d)**3*sinh(b*x + d)*b*f**4 
*x - 24*cosh(b*x + d)**3*sinh(b*x + d)*b*f**3*g*x + 24*cosh(b*x + d)**3*si 
nh(b*x + d)*b*f*g**3*x - 12*cosh(b*x + d)**3*sinh(b*x + d)*b*g**4*x - 12*c 
osh(b*x + d)**2*sinh(b*x + d)**2*f**4 + 24*cosh(b*x + d)**2*sinh(b*x + d)* 
*2*f**3*g + 24*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*b*f**4*x + 2 
4*cosh(b*x + d)*sinh(b*x + d)**3*b*f**3*g*x - 24*cosh(b*x + d)*sinh(b*x + 
d)**3*b*f*g**3*x + 12*cosh(b*x + d)*sinh(b*x + d)**3*b*g**4*x + 8*cosh(b*x 
 + d)*sinh(b*x + d)**3*f**4 - 16*cosh(b*x + d)*sinh(b*x + d)**3*f**3*g + 4 
8*cosh(b*x + d)*sinh(b*x + d)**3*f**2*g**2 - 16*cosh(b*x + d)*sinh(b*x + d 
)**3*f*g**3 + 8*cosh(b*x + d)*sinh(b*x + d)**3*g**4 + 6*sinh(b*x + d)**4*b 
*f**4*x - 12*sinh(b*x + d)**4*b*f**3*g*x + 12*sinh(b*x + d)**4*b*f*g**3*x 
- 6*sinh(b*x + d)**4*b*g**4*x + 5*sinh(b*x + d)**4*f**4 + 14*sinh(b*x + d) 
**4*f**3*g - 24*sinh(b*x + d)**4*f**2*g**2 + 2*sinh(b*x + d)**4*f*g**3 - s 
inh(b*x + d)**4*g**4))/(24*b)