\(\int e^{\frac {5}{3} (a+b x)} (g \cosh (d+b x)+f \sinh (d+b x))^4 \, dx\) [15]

Optimal result

Integrand size = 31, antiderivative size = 187 \[ \int e^{\frac {5}{3} (a+b x)} (g \cosh (d+b x)+f \sinh (d+b x))^4 \, dx=-\frac {3 e^{\frac {5 (a-d)}{3}-\frac {7}{3} (d+b x)} (f-g)^4}{112 b}+\frac {3 e^{\frac {5 (a-d)}{3}+\frac {1}{3} (-d-b x)} (f-g)^3 (f+g)}{4 b}-\frac {3 e^{\frac {5 (a-d)}{3}+\frac {11}{3} (d+b x)} (f-g) (f+g)^3}{44 b}+\frac {3 e^{\frac {5 (a-d)}{3}+\frac {17}{3} (d+b x)} (f+g)^4}{272 b}+\frac {9 e^{\frac {5 (a-d)}{3}+\frac {5}{3} (d+b x)} \left (f^2-g^2\right )^2}{40 b} \] Output:

-3/112*exp(5/3*a-4*d-7/3*b*x)*(f-g)^4/b+3/4*exp(5/3*a-2*d-1/3*b*x)*(f-g)^3 
*(f+g)/b-3/44*exp(5/3*a+2*d+11/3*b*x)*(f-g)*(f+g)^3/b+3/272*exp(5/3*a+4*d+ 
17/3*b*x)*(f+g)^4/b+9/40*exp(5/3*b*x+5/3*a)*(f^2-g^2)^2/b
 

Mathematica [A] (verified)

Time = 3.18 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.74 \[ \int e^{\frac {5}{3} (a+b x)} (g \cosh (d+b x)+f \sinh (d+b x))^4 \, dx=\frac {3 e^{\frac {5 (a-d)}{3}} \left (-\frac {1}{7} e^{-\frac {7}{3} (d+b x)} (f-g)^4+4 e^{\frac {1}{3} (-d-b x)} (f-g)^3 (f+g)-\frac {4}{11} e^{\frac {11}{3} (d+b x)} (f-g) (f+g)^3+\frac {1}{17} e^{\frac {17}{3} (d+b x)} (f+g)^4+\frac {6}{5} e^{\frac {5}{3} (d+b x)} \left (f^2-g^2\right )^2\right )}{16 b} \] Input:

Integrate[E^((5*(a + b*x))/3)*(g*Cosh[d + b*x] + f*Sinh[d + b*x])^4,x]
 

Output:

(3*E^((5*(a - d))/3)*(-1/7*(f - g)^4/E^((7*(d + b*x))/3) + 4*E^((-d - b*x) 
/3)*(f - g)^3*(f + g) - (4*E^((11*(d + b*x))/3)*(f - g)*(f + g)^3)/11 + (E 
^((17*(d + b*x))/3)*(f + g)^4)/17 + (6*E^((5*(d + b*x))/3)*(f^2 - g^2)^2)/ 
5))/(16*b)
 

Rubi [A] (warning: unable to verify)

Time = 0.31 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.62, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {2720, 27, 802, 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.

\(\displaystyle \int e^{\frac {5}{3} (a+b x)} (f \sinh (b x+d)+g \cosh (b x+d))^4 \, dx\)

\(\Big \downarrow \) 2720

\(\displaystyle \frac {3 \int \frac {1}{16} e^{\frac {5 a}{3}-\frac {8 b x}{3}} \left (f-g-e^{2 b x} (f+g)\right )^4de^{\frac {b x}{3}}}{b}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {3 e^{5 a/3} \int e^{-\frac {8 b x}{3}} \left (f-g-e^{2 b x} (f+g)\right )^4de^{\frac {b x}{3}}}{16 b}\)

\(\Big \downarrow \) 802

\(\displaystyle \frac {3 e^{5 a/3} \int \left (e^{-\frac {8 b x}{3}} (f-g)^4-4 e^{-\frac {2 b x}{3}} (f+g) (f-g)^3-4 e^{\frac {10 b x}{3}} (f+g)^3 (f-g)+e^{\frac {16 b x}{3}} (f+g)^4+6 e^{\frac {4 b x}{3}} \left (f^2-g^2\right )^2\right )de^{\frac {b x}{3}}}{16 b}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {3 e^{5 a/3} \left (\frac {6}{5} e^{\frac {5 b x}{3}} \left (f^2-g^2\right )^2-\frac {1}{7} e^{-\frac {7 b x}{3}} (f-g)^4+4 e^{-\frac {b x}{3}} (f+g) (f-g)^3-\frac {4}{11} e^{\frac {11 b x}{3}} (f+g)^3 (f-g)+\frac {1}{17} e^{\frac {17 b x}{3}} (f+g)^4\right )}{16 b}\)

Input:

Int[E^((5*(a + b*x))/3)*(g*Cosh[d + b*x] + f*Sinh[d + b*x])^4,x]
 

Output:

(3*E^((5*a)/3)*(-1/7*(f - g)^4/E^((7*b*x)/3) + (4*(f - g)^3*(f + g))/E^((b 
*x)/3) - (4*E^((11*b*x)/3)*(f - g)*(f + g)^3)/11 + (E^((17*b*x)/3)*(f + g) 
^4)/17 + (6*E^((5*b*x)/3)*(f^2 - g^2)^2)/5))/(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 802

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[Exp 
andIntegrand[(c*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, n}, 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. \(419\) vs. \(2(126)=252\).

Time = 0.20 (sec) , antiderivative size = 420, normalized size of antiderivative = 2.25

\[\frac {3 \left (-\frac {3}{4} f^{2} g^{2}+\frac {3}{8} g^{4}+\frac {3}{8} f^{4}\right ) \sinh \left (\frac {5 b x}{3}+\frac {5 a}{3}\right )}{5 b}+\frac {3 \left (-\frac {3}{4} f^{2} g^{2}+\frac {3}{8} g^{4}+\frac {3}{8} f^{4}\right ) \cosh \left (\frac {5 b x}{3}+\frac {5 a}{3}\right )}{5 b}+\frac {3 \left (\frac {1}{4} g^{4}-\frac {1}{4} f^{4}-\frac {1}{2} g \,f^{3}+\frac {1}{2} g^{3} f \right ) \sinh \left (\frac {5 a}{3}+2 d +\frac {11 b x}{3}\right )}{11 b}+\frac {3 \left (\frac {1}{4} g^{4}-\frac {1}{4} f^{4}-\frac {1}{2} g \,f^{3}+\frac {1}{2} g^{3} f \right ) \cosh \left (\frac {5 a}{3}+2 d +\frac {11 b x}{3}\right )}{11 b}-\frac {3 \left (\frac {1}{4} g^{4}-\frac {1}{4} f^{4}+\frac {1}{2} g \,f^{3}-\frac {1}{2} g^{3} f \right ) \sinh \left (\frac {5 a}{3}-2 d -\frac {b x}{3}\right )}{b}+\frac {3 \left (\frac {1}{4} f^{4}-\frac {1}{2} g \,f^{3}+\frac {1}{2} g^{3} f -\frac {1}{4} g^{4}\right ) \cosh \left (\frac {5 a}{3}-2 d -\frac {b x}{3}\right )}{b}-\frac {3 \left (\frac {3}{8} f^{2} g^{2}+\frac {1}{16} g^{4}+\frac {1}{16} f^{4}-\frac {1}{4} g \,f^{3}-\frac {1}{4} g^{3} f \right ) \sinh \left (\frac {5 a}{3}-4 d -\frac {7 b x}{3}\right )}{7 b}+\frac {3 \left (-\frac {1}{16} f^{4}+\frac {1}{4} g \,f^{3}-\frac {3}{8} f^{2} g^{2}+\frac {1}{4} g^{3} f -\frac {1}{16} g^{4}\right ) \cosh \left (\frac {5 a}{3}-4 d -\frac {7 b x}{3}\right )}{7 b}+\frac {3 \left (\frac {3}{8} f^{2} g^{2}+\frac {1}{16} g^{4}+\frac {1}{16} f^{4}+\frac {1}{4} g \,f^{3}+\frac {1}{4} g^{3} f \right ) \sinh \left (\frac {5 a}{3}+4 d +\frac {17 b x}{3}\right )}{17 b}+\frac {3 \left (\frac {3}{8} f^{2} g^{2}+\frac {1}{16} g^{4}+\frac {1}{16} f^{4}+\frac {1}{4} g \,f^{3}+\frac {1}{4} g^{3} f \right ) \cosh \left (\frac {5 a}{3}+4 d +\frac {17 b x}{3}\right )}{17 b}\]

Input:

int(exp(5/3*b*x+5/3*a)*(g*cosh(b*x+d)+f*sinh(b*x+d))^4,x)
 

Output:

3/5*(-3/4*f^2*g^2+3/8*g^4+3/8*f^4)/b*sinh(5/3*b*x+5/3*a)+3/5*(-3/4*f^2*g^2 
+3/8*g^4+3/8*f^4)*cosh(5/3*b*x+5/3*a)/b+3/11*(1/4*g^4-1/4*f^4-1/2*g*f^3+1/ 
2*g^3*f)/b*sinh(5/3*a+2*d+11/3*b*x)+3/11*(1/4*g^4-1/4*f^4-1/2*g*f^3+1/2*g^ 
3*f)*cosh(5/3*a+2*d+11/3*b*x)/b-3*(1/4*g^4-1/4*f^4+1/2*g*f^3-1/2*g^3*f)/b* 
sinh(5/3*a-2*d-1/3*b*x)+3*(1/4*f^4-1/2*g*f^3+1/2*g^3*f-1/4*g^4)*cosh(5/3*a 
-2*d-1/3*b*x)/b-3/7*(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(5/3*a-4*d-7/3*b*x)+3/7*(-1/16*f^4+1/4*g*f^3-3/8*f^2*g^2+1/4*g^3*f-1/1 
6*g^4)*cosh(5/3*a-4*d-7/3*b*x)/b+3/17*(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(5/3*a+4*d+17/3*b*x)+3/17*(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(5/3*a+4*d+17/3*b*x)/b
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2210 vs. \(2 (126) = 252\).

Time = 0.11 (sec) , antiderivative size = 2210, normalized size of antiderivative = 11.82 \[ \int e^{\frac {5}{3} (a+b x)} (g \cosh (d+b x)+f \sinh (d+b x))^4 \, dx=\text {Too large to display} \] Input:

integrate(exp(5/3*b*x+5/3*a)*(g*cosh(b*x+d)+f*sinh(b*x+d))^4,x, algorithm= 
"fricas")
 

Output:

-3/52360*(55*(5*f^4 - 48*f^3*g + 30*f^2*g^2 - 48*f*g^3 + 5*g^4)*cosh(1/3*b 
*x + 1/3*d)^12*cosh(-5/3*a + 5/3*d) + 55*((5*f^4 - 48*f^3*g + 30*f^2*g^2 - 
 48*f*g^3 + 5*g^4)*cosh(-5/3*a + 5/3*d) - (5*f^4 - 48*f^3*g + 30*f^2*g^2 - 
 48*f*g^3 + 5*g^4)*sinh(-5/3*a + 5/3*d))*sinh(1/3*b*x + 1/3*d)^12 - 2640*( 
(3*f^4 - 5*f^3*g + 18*f^2*g^2 - 5*f*g^3 + 3*g^4)*cosh(1/3*b*x + 1/3*d)*cos 
h(-5/3*a + 5/3*d) - (3*f^4 - 5*f^3*g + 18*f^2*g^2 - 5*f*g^3 + 3*g^4)*cosh( 
1/3*b*x + 1/3*d)*sinh(-5/3*a + 5/3*d))*sinh(1/3*b*x + 1/3*d)^11 + 3630*((5 
*f^4 - 48*f^3*g + 30*f^2*g^2 - 48*f*g^3 + 5*g^4)*cosh(1/3*b*x + 1/3*d)^2*c 
osh(-5/3*a + 5/3*d) - (5*f^4 - 48*f^3*g + 30*f^2*g^2 - 48*f*g^3 + 5*g^4)*c 
osh(1/3*b*x + 1/3*d)^2*sinh(-5/3*a + 5/3*d))*sinh(1/3*b*x + 1/3*d)^10 - 48 
400*((3*f^4 - 5*f^3*g + 18*f^2*g^2 - 5*f*g^3 + 3*g^4)*cosh(1/3*b*x + 1/3*d 
)^3*cosh(-5/3*a + 5/3*d) - (3*f^4 - 5*f^3*g + 18*f^2*g^2 - 5*f*g^3 + 3*g^4 
)*cosh(1/3*b*x + 1/3*d)^3*sinh(-5/3*a + 5/3*d))*sinh(1/3*b*x + 1/3*d)^9 + 
27225*((5*f^4 - 48*f^3*g + 30*f^2*g^2 - 48*f*g^3 + 5*g^4)*cosh(1/3*b*x + 1 
/3*d)^4*cosh(-5/3*a + 5/3*d) - (5*f^4 - 48*f^3*g + 30*f^2*g^2 - 48*f*g^3 + 
 5*g^4)*cosh(1/3*b*x + 1/3*d)^4*sinh(-5/3*a + 5/3*d))*sinh(1/3*b*x + 1/3*d 
)^8 - 2380*(5*f^4 - 12*f^3*g + 12*f*g^3 - 5*g^4)*cosh(1/3*b*x + 1/3*d)^6*c 
osh(-5/3*a + 5/3*d) - 174240*((3*f^4 - 5*f^3*g + 18*f^2*g^2 - 5*f*g^3 + 3* 
g^4)*cosh(1/3*b*x + 1/3*d)^5*cosh(-5/3*a + 5/3*d) - (3*f^4 - 5*f^3*g + 18* 
f^2*g^2 - 5*f*g^3 + 3*g^4)*cosh(1/3*b*x + 1/3*d)^5*sinh(-5/3*a + 5/3*d)...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 971 vs. \(2 (139) = 278\).

Time = 2.51 (sec) , antiderivative size = 971, normalized size of antiderivative = 5.19 \[ \int e^{\frac {5}{3} (a+b x)} (g \cosh (d+b x)+f \sinh (d+b x))^4 \, dx=\text {Too large to display} \] Input:

integrate(exp(5/3*b*x+5/3*a)*(g*cosh(b*x+d)+f*sinh(b*x+d))**4,x)
 

Output:

Piecewise((-3093*f**4*exp(5*a/3)*exp(5*b*x/3)*sinh(b*x + d)**4/(6545*b) + 
2340*f**4*exp(5*a/3)*exp(5*b*x/3)*sinh(b*x + d)**3*cosh(b*x + d)/(1309*b) 
- 324*f**4*exp(5*a/3)*exp(5*b*x/3)*sinh(b*x + d)**2*cosh(b*x + d)**2/(595* 
b) - 1944*f**4*exp(5*a/3)*exp(5*b*x/3)*sinh(b*x + d)*cosh(b*x + d)**3/(130 
9*b) + 5832*f**4*exp(5*a/3)*exp(5*b*x/3)*cosh(b*x + d)**4/(6545*b) + 2340* 
f**3*g*exp(5*a/3)*exp(5*b*x/3)*sinh(b*x + d)**4/(1309*b) - 3900*f**3*g*exp 
(5*a/3)*exp(5*b*x/3)*sinh(b*x + d)**3*cosh(b*x + d)/(1309*b) + 108*f**3*g* 
exp(5*a/3)*exp(5*b*x/3)*sinh(b*x + d)**2*cosh(b*x + d)**2/(119*b) + 3240*f 
**3*g*exp(5*a/3)*exp(5*b*x/3)*sinh(b*x + d)*cosh(b*x + d)**3/(1309*b) - 19 
44*f**3*g*exp(5*a/3)*exp(5*b*x/3)*cosh(b*x + d)**4/(1309*b) - 324*f**2*g** 
2*exp(5*a/3)*exp(5*b*x/3)*sinh(b*x + d)**4/(595*b) + 108*f**2*g**2*exp(5*a 
/3)*exp(5*b*x/3)*sinh(b*x + d)**3*cosh(b*x + d)/(119*b) + 198*f**2*g**2*ex 
p(5*a/3)*exp(5*b*x/3)*sinh(b*x + d)**2*cosh(b*x + d)**2/(595*b) + 108*f**2 
*g**2*exp(5*a/3)*exp(5*b*x/3)*sinh(b*x + d)*cosh(b*x + d)**3/(119*b) - 324 
*f**2*g**2*exp(5*a/3)*exp(5*b*x/3)*cosh(b*x + d)**4/(595*b) - 1944*f*g**3* 
exp(5*a/3)*exp(5*b*x/3)*sinh(b*x + d)**4/(1309*b) + 3240*f*g**3*exp(5*a/3) 
*exp(5*b*x/3)*sinh(b*x + d)**3*cosh(b*x + d)/(1309*b) + 108*f*g**3*exp(5*a 
/3)*exp(5*b*x/3)*sinh(b*x + d)**2*cosh(b*x + d)**2/(119*b) - 3900*f*g**3*e 
xp(5*a/3)*exp(5*b*x/3)*sinh(b*x + d)*cosh(b*x + d)**3/(1309*b) + 2340*f*g* 
*3*exp(5*a/3)*exp(5*b*x/3)*cosh(b*x + d)**4/(1309*b) + 5832*g**4*exp(5*...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 365 vs. \(2 (126) = 252\).

Time = 0.07 (sec) , antiderivative size = 365, normalized size of antiderivative = 1.95 \[ \int e^{\frac {5}{3} (a+b x)} (g \cosh (d+b x)+f \sinh (d+b x))^4 \, dx=\frac {3}{104720} \, g^{4} {\left (\frac {7 \, {\left (340 \, e^{\left (-2 \, b x - 2 \, d\right )} + 1122 \, e^{\left (-4 \, b x - 4 \, d\right )} + 55\right )} e^{\left (\frac {17}{3} \, b x + \frac {5}{3} \, a + 4 \, d\right )}}{b} - \frac {935 \, {\left (28 \, e^{\left (-\frac {1}{3} \, b x - \frac {1}{3} \, d\right )} + e^{\left (-\frac {7}{3} \, b x - \frac {7}{3} \, d\right )}\right )} e^{\left (\frac {5}{3} \, a - \frac {5}{3} \, d\right )}}{b}\right )} - \frac {3}{104720} \, f^{4} {\left (\frac {7 \, {\left (340 \, e^{\left (-2 \, b x - 2 \, d\right )} - 1122 \, e^{\left (-4 \, b x - 4 \, d\right )} - 55\right )} e^{\left (\frac {17}{3} \, b x + \frac {5}{3} \, a + 4 \, d\right )}}{b} - \frac {935 \, {\left (28 \, e^{\left (-\frac {1}{3} \, b x - \frac {1}{3} \, d\right )} - e^{\left (-\frac {7}{3} \, b x - \frac {7}{3} \, d\right )}\right )} e^{\left (\frac {5}{3} \, a - \frac {5}{3} \, d\right )}}{b}\right )} + \frac {3}{5236} \, f g^{3} {\left (\frac {7 \, {\left (34 \, e^{\left (-2 \, b x - 2 \, d\right )} + 11\right )} e^{\left (\frac {17}{3} \, b x + \frac {5}{3} \, a + 4 \, d\right )}}{b} + \frac {187 \, {\left (14 \, e^{\left (-\frac {1}{3} \, b x - \frac {1}{3} \, d\right )} + e^{\left (-\frac {7}{3} \, b x - \frac {7}{3} \, d\right )}\right )} e^{\left (\frac {5}{3} \, a - \frac {5}{3} \, d\right )}}{b}\right )} - \frac {3}{5236} \, f^{3} g {\left (\frac {7 \, {\left (34 \, e^{\left (-2 \, b x - 2 \, d\right )} - 11\right )} e^{\left (\frac {17}{3} \, b x + \frac {5}{3} \, a + 4 \, d\right )}}{b} + \frac {187 \, {\left (14 \, e^{\left (-\frac {1}{3} \, b x - \frac {1}{3} \, d\right )} - e^{\left (-\frac {7}{3} \, b x - \frac {7}{3} \, d\right )}\right )} e^{\left (\frac {5}{3} \, a - \frac {5}{3} \, d\right )}}{b}\right )} - \frac {9}{4760} \, f^{2} g^{2} {\left (\frac {7 \, {\left (34 \, e^{\left (-4 \, b x - 4 \, d\right )} - 5\right )} e^{\left (\frac {17}{3} \, b x + \frac {5}{3} \, a + 4 \, d\right )}}{b} + \frac {85 \, e^{\left (-\frac {7}{3} \, b x + \frac {5}{3} \, a - 4 \, d\right )}}{b}\right )} \] Input:

integrate(exp(5/3*b*x+5/3*a)*(g*cosh(b*x+d)+f*sinh(b*x+d))^4,x, algorithm= 
"maxima")
 

Output:

3/104720*g^4*(7*(340*e^(-2*b*x - 2*d) + 1122*e^(-4*b*x - 4*d) + 55)*e^(17/ 
3*b*x + 5/3*a + 4*d)/b - 935*(28*e^(-1/3*b*x - 1/3*d) + e^(-7/3*b*x - 7/3* 
d))*e^(5/3*a - 5/3*d)/b) - 3/104720*f^4*(7*(340*e^(-2*b*x - 2*d) - 1122*e^ 
(-4*b*x - 4*d) - 55)*e^(17/3*b*x + 5/3*a + 4*d)/b - 935*(28*e^(-1/3*b*x - 
1/3*d) - e^(-7/3*b*x - 7/3*d))*e^(5/3*a - 5/3*d)/b) + 3/5236*f*g^3*(7*(34* 
e^(-2*b*x - 2*d) + 11)*e^(17/3*b*x + 5/3*a + 4*d)/b + 187*(14*e^(-1/3*b*x 
- 1/3*d) + e^(-7/3*b*x - 7/3*d))*e^(5/3*a - 5/3*d)/b) - 3/5236*f^3*g*(7*(3 
4*e^(-2*b*x - 2*d) - 11)*e^(17/3*b*x + 5/3*a + 4*d)/b + 187*(14*e^(-1/3*b* 
x - 1/3*d) - e^(-7/3*b*x - 7/3*d))*e^(5/3*a - 5/3*d)/b) - 9/4760*f^2*g^2*( 
7*(34*e^(-4*b*x - 4*d) - 5)*e^(17/3*b*x + 5/3*a + 4*d)/b + 85*e^(-7/3*b*x 
+ 5/3*a - 4*d)/b)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 352 vs. \(2 (126) = 252\).

Time = 0.14 (sec) , antiderivative size = 352, normalized size of antiderivative = 1.88 \[ \int e^{\frac {5}{3} (a+b x)} (g \cosh (d+b x)+f \sinh (d+b x))^4 \, dx=\frac {3 \, {\left (385 \, f^{4} e^{\left (\frac {17}{3} \, b x + \frac {5}{3} \, a + 8 \, d\right )} + 1540 \, f^{3} g e^{\left (\frac {17}{3} \, b x + \frac {5}{3} \, a + 8 \, d\right )} + 2310 \, f^{2} g^{2} e^{\left (\frac {17}{3} \, b x + \frac {5}{3} \, a + 8 \, d\right )} + 1540 \, f g^{3} e^{\left (\frac {17}{3} \, b x + \frac {5}{3} \, a + 8 \, d\right )} + 385 \, g^{4} e^{\left (\frac {17}{3} \, b x + \frac {5}{3} \, a + 8 \, d\right )} - 2380 \, f^{4} e^{\left (\frac {11}{3} \, b x + \frac {5}{3} \, a + 6 \, d\right )} - 4760 \, f^{3} g e^{\left (\frac {11}{3} \, b x + \frac {5}{3} \, a + 6 \, d\right )} + 4760 \, f g^{3} e^{\left (\frac {11}{3} \, b x + \frac {5}{3} \, a + 6 \, d\right )} + 2380 \, g^{4} e^{\left (\frac {11}{3} \, b x + \frac {5}{3} \, a + 6 \, d\right )} + 7854 \, f^{4} e^{\left (\frac {5}{3} \, b x + \frac {5}{3} \, a + 4 \, d\right )} - 15708 \, f^{2} g^{2} e^{\left (\frac {5}{3} \, b x + \frac {5}{3} \, a + 4 \, d\right )} + 7854 \, g^{4} e^{\left (\frac {5}{3} \, b x + \frac {5}{3} \, a + 4 \, d\right )} + 935 \, {\left (28 \, f^{4} e^{\left (2 \, b x + \frac {5}{3} \, a + 2 \, d\right )} - 56 \, f^{3} g e^{\left (2 \, b x + \frac {5}{3} \, a + 2 \, d\right )} + 56 \, f g^{3} e^{\left (2 \, b x + \frac {5}{3} \, a + 2 \, d\right )} - 28 \, g^{4} e^{\left (2 \, b x + \frac {5}{3} \, a + 2 \, d\right )} - f^{4} e^{\left (\frac {5}{3} \, a\right )} + 4 \, f^{3} g e^{\left (\frac {5}{3} \, a\right )} - 6 \, f^{2} g^{2} e^{\left (\frac {5}{3} \, a\right )} + 4 \, f g^{3} e^{\left (\frac {5}{3} \, a\right )} - g^{4} e^{\left (\frac {5}{3} \, a\right )}\right )} e^{\left (-\frac {7}{3} \, b x\right )}\right )} e^{\left (-4 \, d\right )}}{104720 \, b} \] Input:

integrate(exp(5/3*b*x+5/3*a)*(g*cosh(b*x+d)+f*sinh(b*x+d))^4,x, algorithm= 
"giac")
 

Output:

3/104720*(385*f^4*e^(17/3*b*x + 5/3*a + 8*d) + 1540*f^3*g*e^(17/3*b*x + 5/ 
3*a + 8*d) + 2310*f^2*g^2*e^(17/3*b*x + 5/3*a + 8*d) + 1540*f*g^3*e^(17/3* 
b*x + 5/3*a + 8*d) + 385*g^4*e^(17/3*b*x + 5/3*a + 8*d) - 2380*f^4*e^(11/3 
*b*x + 5/3*a + 6*d) - 4760*f^3*g*e^(11/3*b*x + 5/3*a + 6*d) + 4760*f*g^3*e 
^(11/3*b*x + 5/3*a + 6*d) + 2380*g^4*e^(11/3*b*x + 5/3*a + 6*d) + 7854*f^4 
*e^(5/3*b*x + 5/3*a + 4*d) - 15708*f^2*g^2*e^(5/3*b*x + 5/3*a + 4*d) + 785 
4*g^4*e^(5/3*b*x + 5/3*a + 4*d) + 935*(28*f^4*e^(2*b*x + 5/3*a + 2*d) - 56 
*f^3*g*e^(2*b*x + 5/3*a + 2*d) + 56*f*g^3*e^(2*b*x + 5/3*a + 2*d) - 28*g^4 
*e^(2*b*x + 5/3*a + 2*d) - f^4*e^(5/3*a) + 4*f^3*g*e^(5/3*a) - 6*f^2*g^2*e 
^(5/3*a) + 4*f*g^3*e^(5/3*a) - g^4*e^(5/3*a))*e^(-7/3*b*x))*e^(-4*d)/b
 

Mupad [B] (verification not implemented)

Time = 3.45 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.53 \[ \int e^{\frac {5}{3} (a+b x)} (g \cosh (d+b x)+f \sinh (d+b x))^4 \, dx=\frac {18\,{\mathrm {cosh}\left (d+b\,x\right )}^2\,{\mathrm {e}}^{\frac {5\,a}{3}+\frac {5\,b\,x}{3}}\,{\mathrm {sinh}\left (d+b\,x\right )}^2\,\left (-18\,f^4+30\,f^3\,g+11\,f^2\,g^2+30\,f\,g^3-18\,g^4\right )}{595\,b}-\frac {3\,{\mathrm {e}}^{\frac {5\,a}{3}+\frac {5\,b\,x}{3}}\,{\mathrm {sinh}\left (d+b\,x\right )}^4\,\left (1031\,f^4-3900\,f^3\,g+1188\,f^2\,g^2+3240\,f\,g^3-1944\,g^4\right )}{6545\,b}-\frac {3\,{\mathrm {cosh}\left (d+b\,x\right )}^4\,{\mathrm {e}}^{\frac {5\,a}{3}+\frac {5\,b\,x}{3}}\,\left (-1944\,f^4+3240\,f^3\,g+1188\,f^2\,g^2-3900\,f\,g^3+1031\,g^4\right )}{6545\,b}+\frac {12\,\mathrm {cosh}\left (d+b\,x\right )\,{\mathrm {e}}^{\frac {5\,a}{3}+\frac {5\,b\,x}{3}}\,{\mathrm {sinh}\left (d+b\,x\right )}^3\,\left (195\,f^4-325\,f^3\,g+99\,f^2\,g^2+270\,f\,g^3-162\,g^4\right )}{1309\,b}+\frac {12\,{\mathrm {cosh}\left (d+b\,x\right )}^3\,{\mathrm {e}}^{\frac {5\,a}{3}+\frac {5\,b\,x}{3}}\,\mathrm {sinh}\left (d+b\,x\right )\,\left (-162\,f^4+270\,f^3\,g+99\,f^2\,g^2-325\,f\,g^3+195\,g^4\right )}{1309\,b} \] Input:

int(exp((5*a)/3 + (5*b*x)/3)*(g*cosh(d + b*x) + f*sinh(d + b*x))^4,x)
 

Output:

(18*cosh(d + b*x)^2*exp((5*a)/3 + (5*b*x)/3)*sinh(d + b*x)^2*(30*f*g^3 + 3 
0*f^3*g - 18*f^4 - 18*g^4 + 11*f^2*g^2))/(595*b) - (3*exp((5*a)/3 + (5*b*x 
)/3)*sinh(d + b*x)^4*(3240*f*g^3 - 3900*f^3*g + 1031*f^4 - 1944*g^4 + 1188 
*f^2*g^2))/(6545*b) - (3*cosh(d + b*x)^4*exp((5*a)/3 + (5*b*x)/3)*(3240*f^ 
3*g - 3900*f*g^3 - 1944*f^4 + 1031*g^4 + 1188*f^2*g^2))/(6545*b) + (12*cos 
h(d + b*x)*exp((5*a)/3 + (5*b*x)/3)*sinh(d + b*x)^3*(270*f*g^3 - 325*f^3*g 
 + 195*f^4 - 162*g^4 + 99*f^2*g^2))/(1309*b) + (12*cosh(d + b*x)^3*exp((5* 
a)/3 + (5*b*x)/3)*sinh(d + b*x)*(270*f^3*g - 325*f*g^3 - 162*f^4 + 195*g^4 
 + 99*f^2*g^2))/(1309*b)
 

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 466, normalized size of antiderivative = 2.49 \[ \int e^{\frac {5}{3} (a+b x)} (g \cosh (d+b x)+f \sinh (d+b x))^4 \, dx=\frac {3 e^{\frac {5 b x}{3}+\frac {5 a}{3}} \left (5400 \cosh \left (b x +d \right )^{3} \sinh \left (b x +d \right ) f^{3} g +1980 \cosh \left (b x +d \right )^{3} \sinh \left (b x +d \right ) f^{2} g^{2}-6500 \cosh \left (b x +d \right )^{3} \sinh \left (b x +d \right ) f \,g^{3}+1980 \cosh \left (b x +d \right )^{2} \sinh \left (b x +d \right )^{2} f^{3} g +726 \cosh \left (b x +d \right )^{2} \sinh \left (b x +d \right )^{2} f^{2} g^{2}+1980 \cosh \left (b x +d \right )^{2} \sinh \left (b x +d \right )^{2} f \,g^{3}-6500 \cosh \left (b x +d \right ) \sinh \left (b x +d \right )^{3} f^{3} g +1980 \cosh \left (b x +d \right ) \sinh \left (b x +d \right )^{3} f^{2} g^{2}+5400 \cosh \left (b x +d \right ) \sinh \left (b x +d \right )^{3} f \,g^{3}+1944 \cosh \left (b x +d \right )^{4} f^{4}-1031 \cosh \left (b x +d \right )^{4} g^{4}-3240 \sinh \left (b x +d \right )^{4} f \,g^{3}-1188 \cosh \left (b x +d \right )^{2} \sinh \left (b x +d \right )^{2} f^{4}+3900 \sinh \left (b x +d \right )^{4} f^{3} g +3900 \cosh \left (b x +d \right )^{3} \sinh \left (b x +d \right ) g^{4}-1188 \sinh \left (b x +d \right )^{4} f^{2} g^{2}-3240 \cosh \left (b x +d \right )^{4} f^{3} g +3900 \cosh \left (b x +d \right )^{4} f \,g^{3}-1188 \cosh \left (b x +d \right )^{2} \sinh \left (b x +d \right )^{2} g^{4}+3900 \cosh \left (b x +d \right ) \sinh \left (b x +d \right )^{3} f^{4}-3240 \cosh \left (b x +d \right ) \sinh \left (b x +d \right )^{3} g^{4}-1188 \cosh \left (b x +d \right )^{4} f^{2} g^{2}-3240 \cosh \left (b x +d \right )^{3} \sinh \left (b x +d \right ) f^{4}-1031 \sinh \left (b x +d \right )^{4} f^{4}+1944 \sinh \left (b x +d \right )^{4} g^{4}\right )}{6545 b} \] Input:

int(exp(5/3*b*x+5/3*a)*(g*cosh(b*x+d)+f*sinh(b*x+d))^4,x)
 

Output:

(3*e**((5*a + 5*b*x)/3)*(1944*cosh(b*x + d)**4*f**4 - 3240*cosh(b*x + d)** 
4*f**3*g - 1188*cosh(b*x + d)**4*f**2*g**2 + 3900*cosh(b*x + d)**4*f*g**3 
- 1031*cosh(b*x + d)**4*g**4 - 3240*cosh(b*x + d)**3*sinh(b*x + d)*f**4 + 
5400*cosh(b*x + d)**3*sinh(b*x + d)*f**3*g + 1980*cosh(b*x + d)**3*sinh(b* 
x + d)*f**2*g**2 - 6500*cosh(b*x + d)**3*sinh(b*x + d)*f*g**3 + 3900*cosh( 
b*x + d)**3*sinh(b*x + d)*g**4 - 1188*cosh(b*x + d)**2*sinh(b*x + d)**2*f* 
*4 + 1980*cosh(b*x + d)**2*sinh(b*x + d)**2*f**3*g + 726*cosh(b*x + d)**2* 
sinh(b*x + d)**2*f**2*g**2 + 1980*cosh(b*x + d)**2*sinh(b*x + d)**2*f*g**3 
 - 1188*cosh(b*x + d)**2*sinh(b*x + d)**2*g**4 + 3900*cosh(b*x + d)*sinh(b 
*x + d)**3*f**4 - 6500*cosh(b*x + d)*sinh(b*x + d)**3*f**3*g + 1980*cosh(b 
*x + d)*sinh(b*x + d)**3*f**2*g**2 + 5400*cosh(b*x + d)*sinh(b*x + d)**3*f 
*g**3 - 3240*cosh(b*x + d)*sinh(b*x + d)**3*g**4 - 1031*sinh(b*x + d)**4*f 
**4 + 3900*sinh(b*x + d)**4*f**3*g - 1188*sinh(b*x + d)**4*f**2*g**2 - 324 
0*sinh(b*x + d)**4*f*g**3 + 1944*sinh(b*x + d)**4*g**4))/(6545*b)