\(\int e^{\frac {5}{3} (a+b x)} (g \cosh (d+b x)+f \sinh (d+b x))^3 \, dx\) [16]

Optimal result

Integrand size = 31, antiderivative size = 145 \[ \int e^{\frac {5}{3} (a+b x)} (g \cosh (d+b x)+f \sinh (d+b x))^3 \, dx=\frac {3 e^{\frac {5 (a-d)}{3}-\frac {4}{3} (d+b x)} (f-g)^3}{32 b}+\frac {9 e^{\frac {5 (a-d)}{3}+\frac {2}{3} (d+b x)} (f-g)^2 (f+g)}{16 b}-\frac {9 e^{\frac {5 (a-d)}{3}+\frac {8}{3} (d+b x)} (f-g) (f+g)^2}{64 b}+\frac {3 e^{\frac {5 (a-d)}{3}+\frac {14}{3} (d+b x)} (f+g)^3}{112 b} \] Output:

3/32*exp(5/3*a-3*d-4/3*b*x)*(f-g)^3/b+9/16*exp(5/3*a-d+2/3*b*x)*(f-g)^2*(f 
+g)/b-9/64*exp(5/3*a+d+8/3*b*x)*(f-g)*(f+g)^2/b+3/112*exp(5/3*a+3*d+14/3*b 
*x)*(f+g)^3/b
 

Mathematica [A] (verified)

Time = 0.51 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.63 \[ \int e^{\frac {5}{3} (a+b x)} (g \cosh (d+b x)+f \sinh (d+b x))^3 \, dx=\frac {3 e^{\frac {1}{3} (5 a-9 d-4 b x)} \left (14 (f-g)^3+84 e^{2 (d+b x)} (f-g)^2 (f+g)-21 e^{4 (d+b x)} (f-g) (f+g)^2+4 e^{6 (d+b x)} (f+g)^3\right )}{448 b} \] Input:

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

Output:

(3*E^((5*a - 9*d - 4*b*x)/3)*(14*(f - g)^3 + 84*E^(2*(d + b*x))*(f - g)^2* 
(f + g) - 21*E^(4*(d + b*x))*(f - g)*(f + g)^2 + 4*E^(6*(d + b*x))*(f + g) 
^3))/(448*b)
 

Rubi [A] (warning: unable to verify)

Time = 0.29 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.66, 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.

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 802

\(\displaystyle -\frac {3 e^{5 a/3} \int \left (e^{-\frac {5 b x}{3}} (f-g)^3-3 e^{\frac {b x}{3}} (f+g) (f-g)^2+3 e^{\frac {7 b x}{3}} (f+g)^2 (f-g)-e^{\frac {13 b x}{3}} (f+g)^3\right )de^{\frac {b x}{3}}}{8 b}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {3 e^{5 a/3} \left (-\frac {1}{4} e^{-\frac {4 b x}{3}} (f-g)^3-\frac {3}{2} e^{\frac {2 b x}{3}} (f+g) (f-g)^2+\frac {3}{8} e^{\frac {8 b x}{3}} (f+g)^2 (f-g)-\frac {1}{14} e^{\frac {14 b x}{3}} (f+g)^3\right )}{8 b}\)

Input:

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

Output:

(-3*E^((5*a)/3)*(-1/4*(f - g)^3/E^((4*b*x)/3) - (3*E^((2*b*x)/3)*(f - g)^2 
*(f + g))/2 + (3*E^((8*b*x)/3)*(f - g)*(f + g)^2)/8 - (E^((14*b*x)/3)*(f + 
 g)^3)/14))/(8*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. \(317\) vs. \(2(99)=198\).

Time = 0.48 (sec) , antiderivative size = 318, normalized size of antiderivative = 2.19

\[\frac {3 \left (-\frac {3}{8} f^{3}+\frac {3}{8} g^{3}+\frac {3}{8} f \,g^{2}-\frac {3}{8} f^{2} g \right ) \sinh \left (\frac {5 a}{3}+d +\frac {8 b x}{3}\right )}{8 b}+\frac {3 \left (-\frac {3}{8} f^{3}+\frac {3}{8} g^{3}+\frac {3}{8} f \,g^{2}-\frac {3}{8} f^{2} g \right ) \cosh \left (\frac {5 a}{3}+d +\frac {8 b x}{3}\right )}{8 b}-\frac {3 \left (-\frac {1}{8} f^{3}+\frac {1}{8} g^{3}-\frac {3}{8} f \,g^{2}+\frac {3}{8} f^{2} g \right ) \sinh \left (\frac {5 a}{3}-3 d -\frac {4 b x}{3}\right )}{4 b}+\frac {3 \left (\frac {1}{8} f^{3}-\frac {3}{8} f^{2} g +\frac {3}{8} f \,g^{2}-\frac {1}{8} g^{3}\right ) \cosh \left (\frac {5 a}{3}-3 d -\frac {4 b x}{3}\right )}{4 b}+\frac {3 \left (\frac {1}{8} f^{3}+\frac {1}{8} g^{3}+\frac {3}{8} f \,g^{2}+\frac {3}{8} f^{2} g \right ) \sinh \left (\frac {5 a}{3}+3 d +\frac {14 b x}{3}\right )}{14 b}+\frac {3 \left (\frac {1}{8} f^{3}+\frac {1}{8} g^{3}+\frac {3}{8} f \,g^{2}+\frac {3}{8} f^{2} g \right ) \cosh \left (\frac {5 a}{3}+3 d +\frac {14 b x}{3}\right )}{14 b}+\frac {3 \left (\frac {3}{8} f^{3}+\frac {3}{8} g^{3}-\frac {3}{8} f^{2} g -\frac {3}{8} f \,g^{2}\right ) \sinh \left (\frac {5 a}{3}-d +\frac {2 b x}{3}\right )}{2 b}+\frac {3 \left (\frac {3}{8} f^{3}+\frac {3}{8} g^{3}-\frac {3}{8} f^{2} g -\frac {3}{8} f \,g^{2}\right ) \cosh \left (\frac {5 a}{3}-d +\frac {2 b x}{3}\right )}{2 b}\]

Input:

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

Output:

3/8*(-3/8*f^3+3/8*g^3+3/8*f*g^2-3/8*f^2*g)/b*sinh(5/3*a+d+8/3*b*x)+3/8*(-3 
/8*f^3+3/8*g^3+3/8*f*g^2-3/8*f^2*g)*cosh(5/3*a+d+8/3*b*x)/b-3/4*(-1/8*f^3+ 
1/8*g^3-3/8*f*g^2+3/8*f^2*g)/b*sinh(5/3*a-3*d-4/3*b*x)+3/4*(1/8*f^3-3/8*f^ 
2*g+3/8*f*g^2-1/8*g^3)*cosh(5/3*a-3*d-4/3*b*x)/b+3/14*(1/8*f^3+1/8*g^3+3/8 
*f*g^2+3/8*f^2*g)/b*sinh(5/3*a+3*d+14/3*b*x)+3/14*(1/8*f^3+1/8*g^3+3/8*f*g 
^2+3/8*f^2*g)*cosh(5/3*a+3*d+14/3*b*x)/b+3/2*(3/8*f^3+3/8*g^3-3/8*f^2*g-3/ 
8*f*g^2)/b*sinh(5/3*a-d+2/3*b*x)+3/2*(3/8*f^3+3/8*g^3-3/8*f^2*g-3/8*f*g^2) 
*cosh(5/3*a-d+2/3*b*x)/b
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1403 vs. \(2 (99) = 198\).

Time = 0.10 (sec) , antiderivative size = 1403, normalized size of antiderivative = 9.68 \[ \int e^{\frac {5}{3} (a+b x)} (g \cosh (d+b x)+f \sinh (d+b x))^3 \, dx=\text {Too large to display} \] Input:

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

Output:

3/448*(2*(9*f^3 - 15*f^2*g + 27*f*g^2 - 5*g^3)*cosh(1/3*b*x + 1/3*d)^9*cos 
h(-5/3*a + 5/3*d) - 2*((5*f^3 - 27*f^2*g + 15*f*g^2 - 9*g^3)*cosh(-5/3*a + 
 5/3*d) - (5*f^3 - 27*f^2*g + 15*f*g^2 - 9*g^3)*sinh(-5/3*a + 5/3*d))*sinh 
(1/3*b*x + 1/3*d)^9 + 18*((9*f^3 - 15*f^2*g + 27*f*g^2 - 5*g^3)*cosh(1/3*b 
*x + 1/3*d)*cosh(-5/3*a + 5/3*d) - (9*f^3 - 15*f^2*g + 27*f*g^2 - 5*g^3)*c 
osh(1/3*b*x + 1/3*d)*sinh(-5/3*a + 5/3*d))*sinh(1/3*b*x + 1/3*d)^8 - 72*(( 
5*f^3 - 27*f^2*g + 15*f*g^2 - 9*g^3)*cosh(1/3*b*x + 1/3*d)^2*cosh(-5/3*a + 
 5/3*d) - (5*f^3 - 27*f^2*g + 15*f*g^2 - 9*g^3)*cosh(1/3*b*x + 1/3*d)^2*si 
nh(-5/3*a + 5/3*d))*sinh(1/3*b*x + 1/3*d)^7 + 168*((9*f^3 - 15*f^2*g + 27* 
f*g^2 - 5*g^3)*cosh(1/3*b*x + 1/3*d)^3*cosh(-5/3*a + 5/3*d) - (9*f^3 - 15* 
f^2*g + 27*f*g^2 - 5*g^3)*cosh(1/3*b*x + 1/3*d)^3*sinh(-5/3*a + 5/3*d))*si 
nh(1/3*b*x + 1/3*d)^6 - 252*((5*f^3 - 27*f^2*g + 15*f*g^2 - 9*g^3)*cosh(1/ 
3*b*x + 1/3*d)^4*cosh(-5/3*a + 5/3*d) - (5*f^3 - 27*f^2*g + 15*f*g^2 - 9*g 
^3)*cosh(1/3*b*x + 1/3*d)^4*sinh(-5/3*a + 5/3*d))*sinh(1/3*b*x + 1/3*d)^5 
+ 21*(3*f^3 - 5*f^2*g - 3*f*g^2 + 5*g^3)*cosh(1/3*b*x + 1/3*d)^3*cosh(-5/3 
*a + 5/3*d) + 252*((9*f^3 - 15*f^2*g + 27*f*g^2 - 5*g^3)*cosh(1/3*b*x + 1/ 
3*d)^5*cosh(-5/3*a + 5/3*d) - (9*f^3 - 15*f^2*g + 27*f*g^2 - 5*g^3)*cosh(1 
/3*b*x + 1/3*d)^5*sinh(-5/3*a + 5/3*d))*sinh(1/3*b*x + 1/3*d)^4 - 21*(8*(5 
*f^3 - 27*f^2*g + 15*f*g^2 - 9*g^3)*cosh(1/3*b*x + 1/3*d)^6*cosh(-5/3*a + 
5/3*d) + (5*f^3 - 3*f^2*g - 5*f*g^2 + 3*g^3)*cosh(-5/3*a + 5/3*d) - (8*...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 604 vs. \(2 (110) = 220\).

Time = 1.03 (sec) , antiderivative size = 604, normalized size of antiderivative = 4.17 \[ \int e^{\frac {5}{3} (a+b x)} (g \cosh (d+b x)+f \sinh (d+b x))^3 \, dx=\begin {cases} \frac {285 f^{3} e^{\frac {5 a}{3}} e^{\frac {5 b x}{3}} \sinh ^{3}{\left (b x + d \right )}}{448 b} - \frac {27 f^{3} e^{\frac {5 a}{3}} e^{\frac {5 b x}{3}} \sinh ^{2}{\left (b x + d \right )} \cosh {\left (b x + d \right )}}{448 b} - \frac {405 f^{3} e^{\frac {5 a}{3}} e^{\frac {5 b x}{3}} \sinh {\left (b x + d \right )} \cosh ^{2}{\left (b x + d \right )}}{448 b} + \frac {243 f^{3} e^{\frac {5 a}{3}} e^{\frac {5 b x}{3}} \cosh ^{3}{\left (b x + d \right )}}{448 b} - \frac {27 f^{2} g e^{\frac {5 a}{3}} e^{\frac {5 b x}{3}} \sinh ^{3}{\left (b x + d \right )}}{448 b} + \frac {45 f^{2} g e^{\frac {5 a}{3}} e^{\frac {5 b x}{3}} \sinh ^{2}{\left (b x + d \right )} \cosh {\left (b x + d \right )}}{448 b} + \frac {675 f^{2} g e^{\frac {5 a}{3}} e^{\frac {5 b x}{3}} \sinh {\left (b x + d \right )} \cosh ^{2}{\left (b x + d \right )}}{448 b} - \frac {405 f^{2} g e^{\frac {5 a}{3}} e^{\frac {5 b x}{3}} \cosh ^{3}{\left (b x + d \right )}}{448 b} - \frac {405 f g^{2} e^{\frac {5 a}{3}} e^{\frac {5 b x}{3}} \sinh ^{3}{\left (b x + d \right )}}{448 b} + \frac {675 f g^{2} e^{\frac {5 a}{3}} e^{\frac {5 b x}{3}} \sinh ^{2}{\left (b x + d \right )} \cosh {\left (b x + d \right )}}{448 b} + \frac {45 f g^{2} e^{\frac {5 a}{3}} e^{\frac {5 b x}{3}} \sinh {\left (b x + d \right )} \cosh ^{2}{\left (b x + d \right )}}{448 b} - \frac {27 f g^{2} e^{\frac {5 a}{3}} e^{\frac {5 b x}{3}} \cosh ^{3}{\left (b x + d \right )}}{448 b} + \frac {243 g^{3} e^{\frac {5 a}{3}} e^{\frac {5 b x}{3}} \sinh ^{3}{\left (b x + d \right )}}{448 b} - \frac {405 g^{3} e^{\frac {5 a}{3}} e^{\frac {5 b x}{3}} \sinh ^{2}{\left (b x + d \right )} \cosh {\left (b x + d \right )}}{448 b} - \frac {27 g^{3} e^{\frac {5 a}{3}} e^{\frac {5 b x}{3}} \sinh {\left (b x + d \right )} \cosh ^{2}{\left (b x + d \right )}}{448 b} + \frac {285 g^{3} e^{\frac {5 a}{3}} e^{\frac {5 b x}{3}} \cosh ^{3}{\left (b x + d \right )}}{448 b} & \text {for}\: b \neq 0 \\x \left (f \sinh {\left (d \right )} + g \cosh {\left (d \right )}\right )^{3} e^{\frac {5 a}{3}} & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((285*f**3*exp(5*a/3)*exp(5*b*x/3)*sinh(b*x + d)**3/(448*b) - 27* 
f**3*exp(5*a/3)*exp(5*b*x/3)*sinh(b*x + d)**2*cosh(b*x + d)/(448*b) - 405* 
f**3*exp(5*a/3)*exp(5*b*x/3)*sinh(b*x + d)*cosh(b*x + d)**2/(448*b) + 243* 
f**3*exp(5*a/3)*exp(5*b*x/3)*cosh(b*x + d)**3/(448*b) - 27*f**2*g*exp(5*a/ 
3)*exp(5*b*x/3)*sinh(b*x + d)**3/(448*b) + 45*f**2*g*exp(5*a/3)*exp(5*b*x/ 
3)*sinh(b*x + d)**2*cosh(b*x + d)/(448*b) + 675*f**2*g*exp(5*a/3)*exp(5*b* 
x/3)*sinh(b*x + d)*cosh(b*x + d)**2/(448*b) - 405*f**2*g*exp(5*a/3)*exp(5* 
b*x/3)*cosh(b*x + d)**3/(448*b) - 405*f*g**2*exp(5*a/3)*exp(5*b*x/3)*sinh( 
b*x + d)**3/(448*b) + 675*f*g**2*exp(5*a/3)*exp(5*b*x/3)*sinh(b*x + d)**2* 
cosh(b*x + d)/(448*b) + 45*f*g**2*exp(5*a/3)*exp(5*b*x/3)*sinh(b*x + d)*co 
sh(b*x + d)**2/(448*b) - 27*f*g**2*exp(5*a/3)*exp(5*b*x/3)*cosh(b*x + d)** 
3/(448*b) + 243*g**3*exp(5*a/3)*exp(5*b*x/3)*sinh(b*x + d)**3/(448*b) - 40 
5*g**3*exp(5*a/3)*exp(5*b*x/3)*sinh(b*x + d)**2*cosh(b*x + d)/(448*b) - 27 
*g**3*exp(5*a/3)*exp(5*b*x/3)*sinh(b*x + d)*cosh(b*x + d)**2/(448*b) + 285 
*g**3*exp(5*a/3)*exp(5*b*x/3)*cosh(b*x + d)**3/(448*b), Ne(b, 0)), (x*(f*s 
inh(d) + g*cosh(d))**3*exp(5*a/3), True))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 255 vs. \(2 (99) = 198\).

Time = 0.06 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.76 \[ \int e^{\frac {5}{3} (a+b x)} (g \cosh (d+b x)+f \sinh (d+b x))^3 \, dx=\frac {3}{448} \, g^{3} {\left (\frac {{\left (21 \, e^{\left (-2 \, b x - 2 \, d\right )} + 84 \, e^{\left (-4 \, b x - 4 \, d\right )} + 4\right )} e^{\left (\frac {14}{3} \, b x + \frac {5}{3} \, a + 3 \, d\right )}}{b} - \frac {14 \, e^{\left (-\frac {4}{3} \, b x + \frac {5}{3} \, a - 3 \, d\right )}}{b}\right )} - \frac {3}{448} \, f^{3} {\left (\frac {{\left (21 \, e^{\left (-2 \, b x - 2 \, d\right )} - 84 \, e^{\left (-4 \, b x - 4 \, d\right )} - 4\right )} e^{\left (\frac {14}{3} \, b x + \frac {5}{3} \, a + 3 \, d\right )}}{b} - \frac {14 \, e^{\left (-\frac {4}{3} \, b x + \frac {5}{3} \, a - 3 \, d\right )}}{b}\right )} - \frac {9}{448} \, f^{2} g {\left (\frac {{\left (7 \, e^{\left (-2 \, b x - 2 \, d\right )} + 28 \, e^{\left (-4 \, b x - 4 \, d\right )} - 4\right )} e^{\left (\frac {14}{3} \, b x + \frac {5}{3} \, a + 3 \, d\right )}}{b} + \frac {14 \, e^{\left (-\frac {4}{3} \, b x + \frac {5}{3} \, a - 3 \, d\right )}}{b}\right )} + \frac {9}{448} \, f g^{2} {\left (\frac {{\left (7 \, e^{\left (-2 \, b x - 2 \, d\right )} - 28 \, e^{\left (-4 \, b x - 4 \, d\right )} + 4\right )} e^{\left (\frac {14}{3} \, b x + \frac {5}{3} \, a + 3 \, d\right )}}{b} + \frac {14 \, e^{\left (-\frac {4}{3} \, b x + \frac {5}{3} \, a - 3 \, d\right )}}{b}\right )} \] Input:

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

Output:

3/448*g^3*((21*e^(-2*b*x - 2*d) + 84*e^(-4*b*x - 4*d) + 4)*e^(14/3*b*x + 5 
/3*a + 3*d)/b - 14*e^(-4/3*b*x + 5/3*a - 3*d)/b) - 3/448*f^3*((21*e^(-2*b* 
x - 2*d) - 84*e^(-4*b*x - 4*d) - 4)*e^(14/3*b*x + 5/3*a + 3*d)/b - 14*e^(- 
4/3*b*x + 5/3*a - 3*d)/b) - 9/448*f^2*g*((7*e^(-2*b*x - 2*d) + 28*e^(-4*b* 
x - 4*d) - 4)*e^(14/3*b*x + 5/3*a + 3*d)/b + 14*e^(-4/3*b*x + 5/3*a - 3*d) 
/b) + 9/448*f*g^2*((7*e^(-2*b*x - 2*d) - 28*e^(-4*b*x - 4*d) + 4)*e^(14/3* 
b*x + 5/3*a + 3*d)/b + 14*e^(-4/3*b*x + 5/3*a - 3*d)/b)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 265 vs. \(2 (99) = 198\).

Time = 0.12 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.83 \[ \int e^{\frac {5}{3} (a+b x)} (g \cosh (d+b x)+f \sinh (d+b x))^3 \, dx=\frac {3 \, {\left (4 \, f^{3} e^{\left (\frac {14}{3} \, b x + \frac {5}{3} \, a + 6 \, d\right )} + 12 \, f^{2} g e^{\left (\frac {14}{3} \, b x + \frac {5}{3} \, a + 6 \, d\right )} + 12 \, f g^{2} e^{\left (\frac {14}{3} \, b x + \frac {5}{3} \, a + 6 \, d\right )} + 4 \, g^{3} e^{\left (\frac {14}{3} \, b x + \frac {5}{3} \, a + 6 \, d\right )} - 21 \, f^{3} e^{\left (\frac {8}{3} \, b x + \frac {5}{3} \, a + 4 \, d\right )} - 21 \, f^{2} g e^{\left (\frac {8}{3} \, b x + \frac {5}{3} \, a + 4 \, d\right )} + 21 \, f g^{2} e^{\left (\frac {8}{3} \, b x + \frac {5}{3} \, a + 4 \, d\right )} + 21 \, g^{3} e^{\left (\frac {8}{3} \, b x + \frac {5}{3} \, a + 4 \, d\right )} + 84 \, f^{3} e^{\left (\frac {2}{3} \, b x + \frac {5}{3} \, a + 2 \, d\right )} - 84 \, f^{2} g e^{\left (\frac {2}{3} \, b x + \frac {5}{3} \, a + 2 \, d\right )} - 84 \, f g^{2} e^{\left (\frac {2}{3} \, b x + \frac {5}{3} \, a + 2 \, d\right )} + 84 \, g^{3} e^{\left (\frac {2}{3} \, b x + \frac {5}{3} \, a + 2 \, d\right )} + 14 \, {\left (f^{3} e^{\left (\frac {5}{3} \, a\right )} - 3 \, f^{2} g e^{\left (\frac {5}{3} \, a\right )} + 3 \, f g^{2} e^{\left (\frac {5}{3} \, a\right )} - g^{3} e^{\left (\frac {5}{3} \, a\right )}\right )} e^{\left (-\frac {4}{3} \, b x\right )}\right )} e^{\left (-3 \, d\right )}}{448 \, b} \] Input:

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

Output:

3/448*(4*f^3*e^(14/3*b*x + 5/3*a + 6*d) + 12*f^2*g*e^(14/3*b*x + 5/3*a + 6 
*d) + 12*f*g^2*e^(14/3*b*x + 5/3*a + 6*d) + 4*g^3*e^(14/3*b*x + 5/3*a + 6* 
d) - 21*f^3*e^(8/3*b*x + 5/3*a + 4*d) - 21*f^2*g*e^(8/3*b*x + 5/3*a + 4*d) 
 + 21*f*g^2*e^(8/3*b*x + 5/3*a + 4*d) + 21*g^3*e^(8/3*b*x + 5/3*a + 4*d) + 
 84*f^3*e^(2/3*b*x + 5/3*a + 2*d) - 84*f^2*g*e^(2/3*b*x + 5/3*a + 2*d) - 8 
4*f*g^2*e^(2/3*b*x + 5/3*a + 2*d) + 84*g^3*e^(2/3*b*x + 5/3*a + 2*d) + 14* 
(f^3*e^(5/3*a) - 3*f^2*g*e^(5/3*a) + 3*f*g^2*e^(5/3*a) - g^3*e^(5/3*a))*e^ 
(-4/3*b*x))*e^(-3*d)/b
 

Mupad [B] (verification not implemented)

Time = 3.15 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.33 \[ \int e^{\frac {5}{3} (a+b x)} (g \cosh (d+b x)+f \sinh (d+b x))^3 \, dx=\frac {9\,\mathrm {cosh}\left (d+b\,x\right )\,{\mathrm {e}}^{\frac {5\,a}{3}+\frac {5\,b\,x}{3}}\,{\mathrm {sinh}\left (d+b\,x\right )}^2\,\left (-3\,f^3+5\,f^2\,g+75\,f\,g^2-45\,g^3\right )}{448\,b}-\frac {3\,{\mathrm {e}}^{\frac {5\,a}{3}+\frac {5\,b\,x}{3}}\,{\mathrm {sinh}\left (d+b\,x\right )}^3\,\left (-95\,f^3+9\,f^2\,g+135\,f\,g^2-81\,g^3\right )}{448\,b}-\frac {3\,{\mathrm {cosh}\left (d+b\,x\right )}^3\,{\mathrm {e}}^{\frac {5\,a}{3}+\frac {5\,b\,x}{3}}\,\left (-81\,f^3+135\,f^2\,g+9\,f\,g^2-95\,g^3\right )}{448\,b}+\frac {9\,{\mathrm {cosh}\left (d+b\,x\right )}^2\,{\mathrm {e}}^{\frac {5\,a}{3}+\frac {5\,b\,x}{3}}\,\mathrm {sinh}\left (d+b\,x\right )\,\left (-45\,f^3+75\,f^2\,g+5\,f\,g^2-3\,g^3\right )}{448\,b} \] Input:

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

Output:

(9*cosh(d + b*x)*exp((5*a)/3 + (5*b*x)/3)*sinh(d + b*x)^2*(75*f*g^2 + 5*f^ 
2*g - 3*f^3 - 45*g^3))/(448*b) - (3*exp((5*a)/3 + (5*b*x)/3)*sinh(d + b*x) 
^3*(135*f*g^2 + 9*f^2*g - 95*f^3 - 81*g^3))/(448*b) - (3*cosh(d + b*x)^3*e 
xp((5*a)/3 + (5*b*x)/3)*(9*f*g^2 + 135*f^2*g - 81*f^3 - 95*g^3))/(448*b) + 
 (9*cosh(d + b*x)^2*exp((5*a)/3 + (5*b*x)/3)*sinh(d + b*x)*(5*f*g^2 + 75*f 
^2*g - 45*f^3 - 3*g^3))/(448*b)
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.93 \[ \int e^{\frac {5}{3} (a+b x)} (g \cosh (d+b x)+f \sinh (d+b x))^3 \, dx=\frac {3 e^{\frac {5 b x}{3}+\frac {5 a}{3}} \left (81 \cosh \left (b x +d \right )^{3} f^{3}-135 \cosh \left (b x +d \right )^{3} f^{2} g -9 \cosh \left (b x +d \right )^{3} f \,g^{2}+95 \cosh \left (b x +d \right )^{3} g^{3}-135 \cosh \left (b x +d \right )^{2} \sinh \left (b x +d \right ) f^{3}+225 \cosh \left (b x +d \right )^{2} \sinh \left (b x +d \right ) f^{2} g +15 \cosh \left (b x +d \right )^{2} \sinh \left (b x +d \right ) f \,g^{2}-9 \cosh \left (b x +d \right )^{2} \sinh \left (b x +d \right ) g^{3}-9 \cosh \left (b x +d \right ) \sinh \left (b x +d \right )^{2} f^{3}+15 \cosh \left (b x +d \right ) \sinh \left (b x +d \right )^{2} f^{2} g +225 \cosh \left (b x +d \right ) \sinh \left (b x +d \right )^{2} f \,g^{2}-135 \cosh \left (b x +d \right ) \sinh \left (b x +d \right )^{2} g^{3}+95 \sinh \left (b x +d \right )^{3} f^{3}-9 \sinh \left (b x +d \right )^{3} f^{2} g -135 \sinh \left (b x +d \right )^{3} f \,g^{2}+81 \sinh \left (b x +d \right )^{3} g^{3}\right )}{448 b} \] Input:

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

Output:

(3*e**((5*a + 5*b*x)/3)*(81*cosh(b*x + d)**3*f**3 - 135*cosh(b*x + d)**3*f 
**2*g - 9*cosh(b*x + d)**3*f*g**2 + 95*cosh(b*x + d)**3*g**3 - 135*cosh(b* 
x + d)**2*sinh(b*x + d)*f**3 + 225*cosh(b*x + d)**2*sinh(b*x + d)*f**2*g + 
 15*cosh(b*x + d)**2*sinh(b*x + d)*f*g**2 - 9*cosh(b*x + d)**2*sinh(b*x + 
d)*g**3 - 9*cosh(b*x + d)*sinh(b*x + d)**2*f**3 + 15*cosh(b*x + d)*sinh(b* 
x + d)**2*f**2*g + 225*cosh(b*x + d)*sinh(b*x + d)**2*f*g**2 - 135*cosh(b* 
x + d)*sinh(b*x + d)**2*g**3 + 95*sinh(b*x + d)**3*f**3 - 9*sinh(b*x + d)* 
*3*f**2*g - 135*sinh(b*x + d)**3*f*g**2 + 81*sinh(b*x + d)**3*g**3))/(448* 
b)