\(\int e^{a+b x} (g \cosh (d+b x)+f \sinh (d+b x))^3 \, dx\) [2]

Optimal result

Integrand size = 27, antiderivative size = 97 \[ \int e^{a+b x} (g \cosh (d+b x)+f \sinh (d+b x))^3 \, dx=\frac {e^{a-3 d-2 b x} (f-g)^3}{16 b}-\frac {3 e^{a+d+2 b x} (f-g) (f+g)^2}{16 b}+\frac {e^{a+3 d+4 b x} (f+g)^3}{32 b}+\frac {3}{8} e^{a-d} (f-g)^2 (f+g) x \] Output:

1/16*exp(-2*b*x+a-3*d)*(f-g)^3/b-3/16*exp(2*b*x+a+d)*(f-g)*(f+g)^2/b+1/32* 
exp(4*b*x+a+3*d)*(f+g)^3/b+3/8*exp(a-d)*(f-g)^2*(f+g)*x
 

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.91 \[ \int e^{a+b x} (g \cosh (d+b x)+f \sinh (d+b x))^3 \, dx=\frac {e^{a-d} \left (e^{-2 (d+b x)} (f-g)^3-3 e^{2 (d+b x)} (f-g) (f+g)^2+\frac {1}{2} e^{4 (d+b x)} (f+g)^3+6 (f-g)^2 (f+g) (d+b x)\right )}{16 b} \] Input:

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

Output:

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

Rubi [A] (warning: unable to verify)

Time = 0.29 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.80, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, 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.

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 49

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

-1/16*(E^a*(-((f - g)^3/E^(b*x)) + 3*E^(2*b*x)*(f - g)*(f + g)^2 - (E^(2*b 
*x)*(f + g)^3)/2 - 3*(f - g)^2*(f + g)*Log[E^(2*b*x)]))/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. \(281\) vs. \(2(85)=170\).

Time = 0.33 (sec) , antiderivative size = 282, normalized size of antiderivative = 2.91

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

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 562 vs. \(2 (85) = 170\).

Time = 0.09 (sec) , antiderivative size = 562, normalized size of antiderivative = 5.79 \[ \int e^{a+b x} (g \cosh (d+b x)+f \sinh (d+b x))^3 \, dx=\frac {{\left (3 \, f^{3} - 3 \, f^{2} g + 9 \, f g^{2} - g^{3}\right )} \cosh \left (b x + d\right )^{3} \cosh \left (-a + d\right ) - {\left ({\left (f^{3} - 9 \, f^{2} g + 3 \, f g^{2} - 3 \, g^{3}\right )} \cosh \left (-a + d\right ) - {\left (f^{3} - 9 \, f^{2} g + 3 \, f g^{2} - 3 \, g^{3}\right )} \sinh \left (-a + d\right )\right )} \sinh \left (b x + d\right )^{3} - 6 \, {\left (f^{3} + f^{2} g - f g^{2} - g^{3} - 2 \, {\left (b f^{3} - b f^{2} g - b f g^{2} + b g^{3}\right )} x\right )} \cosh \left (b x + d\right ) \cosh \left (-a + d\right ) + 3 \, {\left ({\left (3 \, f^{3} - 3 \, f^{2} g + 9 \, f g^{2} - g^{3}\right )} \cosh \left (b x + d\right ) \cosh \left (-a + d\right ) - {\left (3 \, f^{3} - 3 \, f^{2} g + 9 \, f g^{2} - g^{3}\right )} \cosh \left (b x + d\right ) \sinh \left (-a + d\right )\right )} \sinh \left (b x + d\right )^{2} - 3 \, {\left ({\left (f^{3} - 9 \, f^{2} g + 3 \, f g^{2} - 3 \, g^{3}\right )} \cosh \left (b x + d\right )^{2} \cosh \left (-a + d\right ) + 2 \, {\left (f^{3} + f^{2} g - f g^{2} - g^{3} + 2 \, {\left (b f^{3} - b f^{2} g - b f g^{2} + b g^{3}\right )} x\right )} \cosh \left (-a + d\right ) - {\left (2 \, f^{3} + 2 \, f^{2} g - 2 \, f g^{2} - 2 \, g^{3} + {\left (f^{3} - 9 \, f^{2} g + 3 \, f g^{2} - 3 \, g^{3}\right )} \cosh \left (b x + d\right )^{2} + 4 \, {\left (b f^{3} - b f^{2} g - b f g^{2} + b g^{3}\right )} x\right )} \sinh \left (-a + d\right )\right )} \sinh \left (b x + d\right ) - {\left ({\left (3 \, f^{3} - 3 \, f^{2} g + 9 \, f g^{2} - g^{3}\right )} \cosh \left (b x + d\right )^{3} - 6 \, {\left (f^{3} + f^{2} g - f g^{2} - g^{3} - 2 \, {\left (b f^{3} - b f^{2} g - b f g^{2} + b g^{3}\right )} x\right )} \cosh \left (b x + d\right )\right )} \sinh \left (-a + d\right )}{32 \, {\left (b \cosh \left (b x + d\right ) - b \sinh \left (b x + d\right )\right )}} \] Input:

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

Output:

1/32*((3*f^3 - 3*f^2*g + 9*f*g^2 - g^3)*cosh(b*x + d)^3*cosh(-a + d) - ((f 
^3 - 9*f^2*g + 3*f*g^2 - 3*g^3)*cosh(-a + d) - (f^3 - 9*f^2*g + 3*f*g^2 - 
3*g^3)*sinh(-a + d))*sinh(b*x + d)^3 - 6*(f^3 + f^2*g - f*g^2 - g^3 - 2*(b 
*f^3 - b*f^2*g - b*f*g^2 + b*g^3)*x)*cosh(b*x + d)*cosh(-a + d) + 3*((3*f^ 
3 - 3*f^2*g + 9*f*g^2 - g^3)*cosh(b*x + d)*cosh(-a + d) - (3*f^3 - 3*f^2*g 
 + 9*f*g^2 - g^3)*cosh(b*x + d)*sinh(-a + d))*sinh(b*x + d)^2 - 3*((f^3 - 
9*f^2*g + 3*f*g^2 - 3*g^3)*cosh(b*x + d)^2*cosh(-a + d) + 2*(f^3 + f^2*g - 
 f*g^2 - g^3 + 2*(b*f^3 - b*f^2*g - b*f*g^2 + b*g^3)*x)*cosh(-a + d) - (2* 
f^3 + 2*f^2*g - 2*f*g^2 - 2*g^3 + (f^3 - 9*f^2*g + 3*f*g^2 - 3*g^3)*cosh(b 
*x + d)^2 + 4*(b*f^3 - b*f^2*g - b*f*g^2 + b*g^3)*x)*sinh(-a + d))*sinh(b* 
x + d) - ((3*f^3 - 3*f^2*g + 9*f*g^2 - g^3)*cosh(b*x + d)^3 - 6*(f^3 + f^2 
*g - f*g^2 - g^3 - 2*(b*f^3 - b*f^2*g - b*f*g^2 + b*g^3)*x)*cosh(b*x + d)) 
*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. 860 vs. \(2 (82) = 164\).

Time = 1.08 (sec) , antiderivative size = 860, normalized size of antiderivative = 8.87 \[ \int e^{a+b x} (g \cosh (d+b x)+f \sinh (d+b x))^3 \, dx =\text {Too large to display} \] Input:

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

Output:

Piecewise((3*f**3*x*exp(a)*exp(b*x)*sinh(b*x + d)**3/8 - 3*f**3*x*exp(a)*e 
xp(b*x)*sinh(b*x + d)**2*cosh(b*x + d)/8 - 3*f**3*x*exp(a)*exp(b*x)*sinh(b 
*x + d)*cosh(b*x + d)**2/8 + 3*f**3*x*exp(a)*exp(b*x)*cosh(b*x + d)**3/8 - 
 3*f**2*g*x*exp(a)*exp(b*x)*sinh(b*x + d)**3/8 + 3*f**2*g*x*exp(a)*exp(b*x 
)*sinh(b*x + d)**2*cosh(b*x + d)/8 + 3*f**2*g*x*exp(a)*exp(b*x)*sinh(b*x + 
 d)*cosh(b*x + d)**2/8 - 3*f**2*g*x*exp(a)*exp(b*x)*cosh(b*x + d)**3/8 - 3 
*f*g**2*x*exp(a)*exp(b*x)*sinh(b*x + d)**3/8 + 3*f*g**2*x*exp(a)*exp(b*x)* 
sinh(b*x + d)**2*cosh(b*x + d)/8 + 3*f*g**2*x*exp(a)*exp(b*x)*sinh(b*x + d 
)*cosh(b*x + d)**2/8 - 3*f*g**2*x*exp(a)*exp(b*x)*cosh(b*x + d)**3/8 + 3*g 
**3*x*exp(a)*exp(b*x)*sinh(b*x + d)**3/8 - 3*g**3*x*exp(a)*exp(b*x)*sinh(b 
*x + d)**2*cosh(b*x + d)/8 - 3*g**3*x*exp(a)*exp(b*x)*sinh(b*x + d)*cosh(b 
*x + d)**2/8 + 3*g**3*x*exp(a)*exp(b*x)*cosh(b*x + d)**3/8 - f**3*exp(a)*e 
xp(b*x)*sinh(b*x + d)**3/(8*b) + 3*f**3*exp(a)*exp(b*x)*sinh(b*x + d)**2*c 
osh(b*x + d)/(4*b) - 3*f**3*exp(a)*exp(b*x)*cosh(b*x + d)**3/(8*b) + 9*f** 
2*g*exp(a)*exp(b*x)*sinh(b*x + d)**3/(8*b) - 3*f**2*g*exp(a)*exp(b*x)*sinh 
(b*x + d)**2*cosh(b*x + d)/(4*b) + 3*f**2*g*exp(a)*exp(b*x)*cosh(b*x + d)* 
*3/(8*b) - 3*f*g**2*exp(a)*exp(b*x)*sinh(b*x + d)**3/(8*b) + 3*f*g**2*exp( 
a)*exp(b*x)*sinh(b*x + d)**2*cosh(b*x + d)/(4*b) + 3*f*g**2*exp(a)*exp(b*x 
)*cosh(b*x + d)**3/(8*b) - 5*g**3*exp(a)*exp(b*x)*sinh(b*x + d)**3/(8*b) + 
 g**3*exp(a)*exp(b*x)*sinh(b*x + d)**2*cosh(b*x + d)/(4*b) + g**3*exp(a...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 271 vs. \(2 (85) = 170\).

Time = 0.05 (sec) , antiderivative size = 271, normalized size of antiderivative = 2.79 \[ \int e^{a+b x} (g \cosh (d+b x)+f \sinh (d+b x))^3 \, dx=\frac {3}{32} \, f g^{2} {\left (\frac {{\left (e^{\left (4 \, b x + 4 \, a + 3 \, d\right )} + 2 \, e^{\left (2 \, b x + 4 \, a + d\right )}\right )} e^{\left (-3 \, a\right )}}{b} - \frac {4 \, {\left (b x + a\right )} e^{\left (a - d\right )}}{b} + \frac {2 \, e^{\left (-2 \, b x + a - 3 \, d\right )}}{b}\right )} + \frac {3}{32} \, f^{2} g {\left (\frac {{\left (e^{\left (4 \, b x + 4 \, a + 3 \, d\right )} - 2 \, e^{\left (2 \, b x + 4 \, a + d\right )}\right )} e^{\left (-3 \, a\right )}}{b} - \frac {4 \, {\left (b x + a\right )} e^{\left (a - d\right )}}{b} - \frac {2 \, e^{\left (-2 \, b x + a - 3 \, d\right )}}{b}\right )} + \frac {1}{32} \, g^{3} {\left (\frac {12 \, {\left (b x + a\right )} e^{\left (a - d\right )}}{b} + \frac {e^{\left (4 \, b x + a + 3 \, d\right )}}{b} + \frac {6 \, e^{\left (2 \, b x + a + d\right )}}{b} - \frac {2 \, e^{\left (-2 \, b x + a - 3 \, d\right )}}{b}\right )} + \frac {1}{32} \, f^{3} {\left (\frac {12 \, {\left (b x + a\right )} e^{\left (a - d\right )}}{b} + \frac {e^{\left (4 \, b x + a + 3 \, d\right )}}{b} - \frac {6 \, e^{\left (2 \, b x + a + d\right )}}{b} + \frac {2 \, e^{\left (-2 \, b x + a - 3 \, d\right )}}{b}\right )} \] Input:

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

Output:

3/32*f*g^2*((e^(4*b*x + 4*a + 3*d) + 2*e^(2*b*x + 4*a + d))*e^(-3*a)/b - 4 
*(b*x + a)*e^(a - d)/b + 2*e^(-2*b*x + a - 3*d)/b) + 3/32*f^2*g*((e^(4*b*x 
 + 4*a + 3*d) - 2*e^(2*b*x + 4*a + d))*e^(-3*a)/b - 4*(b*x + a)*e^(a - d)/ 
b - 2*e^(-2*b*x + a - 3*d)/b) + 1/32*g^3*(12*(b*x + a)*e^(a - d)/b + e^(4* 
b*x + a + 3*d)/b + 6*e^(2*b*x + a + d)/b - 2*e^(-2*b*x + a - 3*d)/b) + 1/3 
2*f^3*(12*(b*x + a)*e^(a - d)/b + e^(4*b*x + a + 3*d)/b - 6*e^(2*b*x + a + 
 d)/b + 2*e^(-2*b*x + a - 3*d)/b)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 280 vs. \(2 (85) = 170\).

Time = 0.13 (sec) , antiderivative size = 280, normalized size of antiderivative = 2.89 \[ \int e^{a+b x} (g \cosh (d+b x)+f \sinh (d+b x))^3 \, dx=\frac {{\left (f^{3} e^{\left (4 \, b x + a + 6 \, d\right )} + 3 \, f^{2} g e^{\left (4 \, b x + a + 6 \, d\right )} + 3 \, f g^{2} e^{\left (4 \, b x + a + 6 \, d\right )} + g^{3} e^{\left (4 \, b x + a + 6 \, d\right )} - 6 \, f^{3} e^{\left (2 \, b x + a + 4 \, d\right )} - 6 \, f^{2} g e^{\left (2 \, b x + a + 4 \, d\right )} + 6 \, f g^{2} e^{\left (2 \, b x + a + 4 \, d\right )} + 6 \, g^{3} e^{\left (2 \, b x + a + 4 \, d\right )} + 12 \, {\left (f^{3} e^{\left (a + 2 \, d\right )} - f^{2} g e^{\left (a + 2 \, d\right )} - f g^{2} e^{\left (a + 2 \, d\right )} + g^{3} e^{\left (a + 2 \, d\right )}\right )} b x - 2 \, {\left (3 \, f^{3} e^{\left (2 \, b x + a + 2 \, d\right )} - 3 \, f^{2} g e^{\left (2 \, b x + a + 2 \, d\right )} - 3 \, f g^{2} e^{\left (2 \, b x + a + 2 \, d\right )} + 3 \, g^{3} e^{\left (2 \, b x + a + 2 \, d\right )} - f^{3} e^{a} + 3 \, f^{2} g e^{a} - 3 \, f g^{2} e^{a} + g^{3} e^{a}\right )} e^{\left (-2 \, b x\right )}\right )} e^{\left (-3 \, d\right )}}{32 \, b} \] Input:

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

Output:

1/32*(f^3*e^(4*b*x + a + 6*d) + 3*f^2*g*e^(4*b*x + a + 6*d) + 3*f*g^2*e^(4 
*b*x + a + 6*d) + g^3*e^(4*b*x + a + 6*d) - 6*f^3*e^(2*b*x + a + 4*d) - 6* 
f^2*g*e^(2*b*x + a + 4*d) + 6*f*g^2*e^(2*b*x + a + 4*d) + 6*g^3*e^(2*b*x + 
 a + 4*d) + 12*(f^3*e^(a + 2*d) - f^2*g*e^(a + 2*d) - f*g^2*e^(a + 2*d) + 
g^3*e^(a + 2*d))*b*x - 2*(3*f^3*e^(2*b*x + a + 2*d) - 3*f^2*g*e^(2*b*x + a 
 + 2*d) - 3*f*g^2*e^(2*b*x + a + 2*d) + 3*g^3*e^(2*b*x + a + 2*d) - f^3*e^ 
a + 3*f^2*g*e^a - 3*f*g^2*e^a + g^3*e^a)*e^(-2*b*x))*e^(-3*d)/b
 

Mupad [B] (verification not implemented)

Time = 0.44 (sec) , antiderivative size = 257, normalized size of antiderivative = 2.65 \[ \int e^{a+b x} (g \cosh (d+b x)+f \sinh (d+b x))^3 \, dx=\frac {{\mathrm {cosh}\left (d+b\,x\right )}^3\,{\mathrm {e}}^{a+b\,x}\,\left (-\frac {f^3}{4}-\frac {3\,f^2\,g}{4}+\frac {3\,f\,g^2}{4}+\frac {g^3}{4}\right )}{b}+\frac {3\,x\,{\mathrm {cosh}\left (d+b\,x\right )}^3\,{\mathrm {e}}^{a+b\,x}\,\left (f+g\right )\,{\left (f-g\right )}^2}{8}+\frac {3\,x\,{\mathrm {e}}^{a+b\,x}\,{\mathrm {sinh}\left (d+b\,x\right )}^3\,\left (f+g\right )\,{\left (f-g\right )}^2}{8}+\frac {\mathrm {cosh}\left (d+b\,x\right )\,{\mathrm {e}}^{a+b\,x}\,{\mathrm {sinh}\left (d+b\,x\right )}^2\,\left (\frac {5\,f^3}{8}+\frac {3\,f^2\,g}{8}+\frac {3\,f\,g^2}{8}-\frac {3\,g^3}{8}\right )}{b}-\frac {{\mathrm {cosh}\left (d+b\,x\right )}^2\,{\mathrm {e}}^{a+b\,x}\,\mathrm {sinh}\left (d+b\,x\right )\,\left (\frac {f^3}{8}-\frac {9\,f^2\,g}{8}+\frac {3\,f\,g^2}{8}-\frac {3\,g^3}{8}\right )}{b}-\frac {3\,x\,\mathrm {cosh}\left (d+b\,x\right )\,{\mathrm {e}}^{a+b\,x}\,{\mathrm {sinh}\left (d+b\,x\right )}^2\,\left (f+g\right )\,{\left (f-g\right )}^2}{8}-\frac {3\,x\,{\mathrm {cosh}\left (d+b\,x\right )}^2\,{\mathrm {e}}^{a+b\,x}\,\mathrm {sinh}\left (d+b\,x\right )\,\left (f+g\right )\,{\left (f-g\right )}^2}{8} \] Input:

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

Output:

(cosh(d + b*x)^3*exp(a + b*x)*((3*f*g^2)/4 - (3*f^2*g)/4 - f^3/4 + g^3/4)) 
/b + (3*x*cosh(d + b*x)^3*exp(a + b*x)*(f + g)*(f - g)^2)/8 + (3*x*exp(a + 
 b*x)*sinh(d + b*x)^3*(f + g)*(f - g)^2)/8 + (cosh(d + b*x)*exp(a + b*x)*s 
inh(d + b*x)^2*((3*f*g^2)/8 + (3*f^2*g)/8 + (5*f^3)/8 - (3*g^3)/8))/b - (c 
osh(d + b*x)^2*exp(a + b*x)*sinh(d + b*x)*((3*f*g^2)/8 - (9*f^2*g)/8 + f^3 
/8 - (3*g^3)/8))/b - (3*x*cosh(d + b*x)*exp(a + b*x)*sinh(d + b*x)^2*(f + 
g)*(f - g)^2)/8 - (3*x*cosh(d + b*x)^2*exp(a + b*x)*sinh(d + b*x)*(f + g)* 
(f - g)^2)/8
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 495, normalized size of antiderivative = 5.10 \[ \int e^{a+b x} (g \cosh (d+b x)+f \sinh (d+b x))^3 \, dx=\frac {e^{b x +a} \left (3 \cosh \left (b x +d \right )^{3} b \,f^{3} x +3 \cosh \left (b x +d \right )^{3} b \,g^{3} x -6 \cosh \left (b x +d \right ) \sinh \left (b x +d \right )^{2} f^{2} g +6 \cosh \left (b x +d \right ) \sinh \left (b x +d \right )^{2} f \,g^{2}+3 \sinh \left (b x +d \right )^{3} b \,f^{3} x +3 \sinh \left (b x +d \right )^{3} b \,g^{3} x -3 \cosh \left (b x +d \right )^{3} b \,f^{2} g x -3 \cosh \left (b x +d \right )^{3} b f \,g^{2} x -3 \cosh \left (b x +d \right )^{2} \sinh \left (b x +d \right ) b \,f^{3} x -3 \cosh \left (b x +d \right )^{2} \sinh \left (b x +d \right ) b \,g^{3} x -3 \cosh \left (b x +d \right ) \sinh \left (b x +d \right )^{2} b \,f^{3} x -3 \cosh \left (b x +d \right ) \sinh \left (b x +d \right )^{2} b \,g^{3} x -3 \sinh \left (b x +d \right )^{3} b \,f^{2} g x -3 \sinh \left (b x +d \right )^{3} b f \,g^{2} x -\sinh \left (b x +d \right )^{3} f^{3}-3 \cosh \left (b x +d \right )^{3} f^{3}+5 \cosh \left (b x +d \right )^{3} g^{3}+3 \sinh \left (b x +d \right )^{3} g^{3}+3 \cosh \left (b x +d \right )^{2} \sinh \left (b x +d \right ) b \,f^{2} g x +3 \cosh \left (b x +d \right )^{2} \sinh \left (b x +d \right ) b f \,g^{2} x +3 \cosh \left (b x +d \right ) \sinh \left (b x +d \right )^{2} b \,f^{2} g x +3 \cosh \left (b x +d \right ) \sinh \left (b x +d \right )^{2} b f \,g^{2} x +3 \cosh \left (b x +d \right )^{3} f^{2} g +3 \cosh \left (b x +d \right )^{3} f \,g^{2}+6 \cosh \left (b x +d \right ) \sinh \left (b x +d \right )^{2} f^{3}-6 \cosh \left (b x +d \right ) \sinh \left (b x +d \right )^{2} g^{3}+9 \sinh \left (b x +d \right )^{3} f^{2} g -3 \sinh \left (b x +d \right )^{3} f \,g^{2}\right )}{8 b} \] Input:

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

Output:

(e**(a + b*x)*(3*cosh(b*x + d)**3*b*f**3*x - 3*cosh(b*x + d)**3*b*f**2*g*x 
 - 3*cosh(b*x + d)**3*b*f*g**2*x + 3*cosh(b*x + d)**3*b*g**3*x - 3*cosh(b* 
x + d)**3*f**3 + 3*cosh(b*x + d)**3*f**2*g + 3*cosh(b*x + d)**3*f*g**2 + 5 
*cosh(b*x + d)**3*g**3 - 3*cosh(b*x + d)**2*sinh(b*x + d)*b*f**3*x + 3*cos 
h(b*x + d)**2*sinh(b*x + d)*b*f**2*g*x + 3*cosh(b*x + d)**2*sinh(b*x + d)* 
b*f*g**2*x - 3*cosh(b*x + d)**2*sinh(b*x + d)*b*g**3*x - 3*cosh(b*x + d)*s 
inh(b*x + d)**2*b*f**3*x + 3*cosh(b*x + d)*sinh(b*x + d)**2*b*f**2*g*x + 3 
*cosh(b*x + d)*sinh(b*x + d)**2*b*f*g**2*x - 3*cosh(b*x + d)*sinh(b*x + d) 
**2*b*g**3*x + 6*cosh(b*x + d)*sinh(b*x + d)**2*f**3 - 6*cosh(b*x + d)*sin 
h(b*x + d)**2*f**2*g + 6*cosh(b*x + d)*sinh(b*x + d)**2*f*g**2 - 6*cosh(b* 
x + d)*sinh(b*x + d)**2*g**3 + 3*sinh(b*x + d)**3*b*f**3*x - 3*sinh(b*x + 
d)**3*b*f**2*g*x - 3*sinh(b*x + d)**3*b*f*g**2*x + 3*sinh(b*x + d)**3*b*g* 
*3*x - sinh(b*x + d)**3*f**3 + 9*sinh(b*x + d)**3*f**2*g - 3*sinh(b*x + d) 
**3*f*g**2 + 3*sinh(b*x + d)**3*g**3))/(8*b)