\(\int (a+a \cosh (x)+c \sinh (x))^3 \, dx\) [506]

Optimal result

Integrand size = 12, antiderivative size = 105 \[ \int (a+a \cosh (x)+c \sinh (x))^3 \, dx=\frac {1}{2} a \left (5 a^2-3 c^2\right ) x+\frac {1}{6} c \left (15 a^2-4 c^2\right ) \cosh (x)+\frac {1}{6} a \left (15 a^2-4 c^2\right ) \sinh (x)+\frac {5}{6} \left (a c \cosh (x)+a^2 \sinh (x)\right ) (a+a \cosh (x)+c \sinh (x))+\frac {1}{3} (c \cosh (x)+a \sinh (x)) (a+a \cosh (x)+c \sinh (x))^2 \] Output:

1/2*a*(5*a^2-3*c^2)*x+1/6*c*(15*a^2-4*c^2)*cosh(x)+1/6*a*(15*a^2-4*c^2)*si 
nh(x)+5/6*(a*c*cosh(x)+a^2*sinh(x))*(a+a*cosh(x)+c*sinh(x))+1/3*(c*cosh(x) 
+a*sinh(x))*(a+a*cosh(x)+c*sinh(x))^2
 

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.07 \[ \int (a+a \cosh (x)+c \sinh (x))^3 \, dx=\frac {1}{12} \left (30 a^3 x-18 a c^2 x-9 c \left (-5 a^2+c^2\right ) \cosh (x)+18 a^2 c \cosh (2 x)+3 a^2 c \cosh (3 x)+c^3 \cosh (3 x)+45 a^3 \sinh (x)-9 a c^2 \sinh (x)+9 a^3 \sinh (2 x)+9 a c^2 \sinh (2 x)+a^3 \sinh (3 x)+3 a c^2 \sinh (3 x)\right ) \] Input:

Integrate[(a + a*Cosh[x] + c*Sinh[x])^3,x]
 

Output:

(30*a^3*x - 18*a*c^2*x - 9*c*(-5*a^2 + c^2)*Cosh[x] + 18*a^2*c*Cosh[2*x] + 
 3*a^2*c*Cosh[3*x] + c^3*Cosh[3*x] + 45*a^3*Sinh[x] - 9*a*c^2*Sinh[x] + 9* 
a^3*Sinh[2*x] + 9*a*c^2*Sinh[2*x] + a^3*Sinh[3*x] + 3*a*c^2*Sinh[3*x])/12
 

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.10, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {3042, 3599, 3042, 3625, 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.

\(\displaystyle \int (a \cosh (x)+a+c \sinh (x))^3 \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int (a \cos (i x)+a-i c \sin (i x))^3dx\)

\(\Big \downarrow \) 3599

\(\displaystyle \frac {1}{3} \int (\cosh (x) a+a+c \sinh (x)) \left (5 \cosh (x) a^2+5 a^2+5 c \sinh (x) a-2 c^2\right )dx+\frac {1}{3} (a \sinh (x)+c \cosh (x)) (a \cosh (x)+a+c \sinh (x))^2\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{3} (a \sinh (x)+c \cosh (x)) (a \cosh (x)+a+c \sinh (x))^2+\frac {1}{3} \int (\cos (i x) a+a-i c \sin (i x)) \left (5 \cos (i x) a^2+5 a^2-5 i c \sin (i x) a-2 c^2\right )dx\)

\(\Big \downarrow \) 3625

\(\displaystyle \frac {1}{3} \left (\frac {\int \left (3 \left (5 a^2-3 c^2\right ) a^2+\left (15 a^2-4 c^2\right ) \cosh (x) a^2+c \left (15 a^2-4 c^2\right ) \sinh (x) a\right )dx}{2 a}+\frac {5}{2} \left (a^2 \sinh (x)+a c \cosh (x)\right ) (a \cosh (x)+a+c \sinh (x))\right )+\frac {1}{3} (a \sinh (x)+c \cosh (x)) (a \cosh (x)+a+c \sinh (x))^2\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{3} \left (\frac {3 a^2 x \left (5 a^2-3 c^2\right )+a^2 \left (15 a^2-4 c^2\right ) \sinh (x)+a c \left (15 a^2-4 c^2\right ) \cosh (x)}{2 a}+\frac {5}{2} \left (a^2 \sinh (x)+a c \cosh (x)\right ) (a \cosh (x)+a+c \sinh (x))\right )+\frac {1}{3} (a \sinh (x)+c \cosh (x)) (a \cosh (x)+a+c \sinh (x))^2\)

Input:

Int[(a + a*Cosh[x] + c*Sinh[x])^3,x]
 

Output:

((c*Cosh[x] + a*Sinh[x])*(a + a*Cosh[x] + c*Sinh[x])^2)/3 + ((5*(a*c*Cosh[ 
x] + a^2*Sinh[x])*(a + a*Cosh[x] + c*Sinh[x]))/2 + (3*a^2*(5*a^2 - 3*c^2)* 
x + a*c*(15*a^2 - 4*c^2)*Cosh[x] + a^2*(15*a^2 - 4*c^2)*Sinh[x])/(2*a))/3
 

Defintions of rubi rules used

rule 2009

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

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3599

Int[(cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)])^ 
(n_), x_Symbol] :> Simp[(-(c*Cos[d + e*x] - b*Sin[d + e*x]))*((a + b*Cos[d 
+ e*x] + c*Sin[d + e*x])^(n - 1)/(e*n)), x] + Simp[1/n   Int[Simp[n*a^2 + ( 
n - 1)*(b^2 + c^2) + a*b*(2*n - 1)*Cos[d + e*x] + a*c*(2*n - 1)*Sin[d + e*x 
], x]*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n - 2), x], x] /; FreeQ[{a, b, 
 c, d, e}, x] && NeQ[a^2 - b^2 - c^2, 0] && GtQ[n, 1]
 

rule 3625

Int[(cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)])^ 
(n_.)*((A_.) + cos[(d_.) + (e_.)*(x_)]*(B_.) + (C_.)*sin[(d_.) + (e_.)*(x_) 
]), x_Symbol] :> Simp[(B*c - b*C - a*C*Cos[d + e*x] + a*B*Sin[d + e*x])*((a 
 + b*Cos[d + e*x] + c*Sin[d + e*x])^n/(a*e*(n + 1))), x] + Simp[1/(a*(n + 1 
))   Int[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n - 1)*Simp[a*(b*B + c*C)*n 
 + a^2*A*(n + 1) + (n*(a^2*B - B*c^2 + b*c*C) + a*b*A*(n + 1))*Cos[d + e*x] 
 + (n*(b*B*c + a^2*C - b^2*C) + a*c*A*(n + 1))*Sin[d + e*x], x], x], x] /; 
FreeQ[{a, b, c, d, e, A, B, C}, x] && GtQ[n, 0] && NeQ[a^2 - b^2 - c^2, 0]
 

Maple [A] (verified)

Time = 3.59 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.91

method	result	size
parts	\(a^{3} x +c^{3} \left (-\frac {2}{3}+\frac {\sinh \left (x \right )^{2}}{3}\right ) \cosh \left (x \right )+a^{2} c \cosh \left (x \right )^{3}+3 a^{3} \left (\frac {\cosh \left (x \right ) \sinh \left (x \right )}{2}+\frac {x}{2}\right )+\frac {a \left (c \sinh \left (x \right )+a \right )^{3}}{c}+a^{3} \left (\frac {2}{3}+\frac {\cosh \left (x \right )^{2}}{3}\right ) \sinh \left (x \right )+3 a \,c^{2} \left (\frac {\cosh \left (x \right ) \sinh \left (x \right )}{2}-\frac {x}{2}\right )+3 a^{2} c \cosh \left (x \right )\)	\(96\)
default	\(a^{3} x +3 a^{3} \sinh \left (x \right )+3 a^{2} c \cosh \left (x \right )+3 a^{3} \left (\frac {\cosh \left (x \right ) \sinh \left (x \right )}{2}+\frac {x}{2}\right )+3 a^{2} c \cosh \left (x \right )^{2}+3 a \,c^{2} \left (\frac {\cosh \left (x \right ) \sinh \left (x \right )}{2}-\frac {x}{2}\right )+a^{3} \left (\frac {2}{3}+\frac {\cosh \left (x \right )^{2}}{3}\right ) \sinh \left (x \right )+a^{2} c \cosh \left (x \right )^{3}+a \,c^{2} \sinh \left (x \right )^{3}+c^{3} \left (-\frac {2}{3}+\frac {\sinh \left (x \right )^{2}}{3}\right ) \cosh \left (x \right )\)	\(109\)
risch	\(\frac {5 a^{3} x}{2}-\frac {3 a \,c^{2} x}{2}+\frac {{\mathrm e}^{3 x} a^{3}}{24}+\frac {a^{2} c \,{\mathrm e}^{3 x}}{8}+\frac {{\mathrm e}^{3 x} a \,c^{2}}{8}+\frac {{\mathrm e}^{3 x} c^{3}}{24}+\frac {3 \,{\mathrm e}^{2 x} a^{3}}{8}+\frac {3 \,{\mathrm e}^{2 x} a^{2} c}{4}+\frac {3 \,{\mathrm e}^{2 x} a \,c^{2}}{8}+\frac {15 a^{3} {\mathrm e}^{x}}{8}+\frac {15 \,{\mathrm e}^{x} a^{2} c}{8}-\frac {3 \,{\mathrm e}^{x} a \,c^{2}}{8}-\frac {3 \,{\mathrm e}^{x} c^{3}}{8}-\frac {15 \,{\mathrm e}^{-x} a^{3}}{8}+\frac {15 \,{\mathrm e}^{-x} a^{2} c}{8}+\frac {3 \,{\mathrm e}^{-x} a \,c^{2}}{8}-\frac {3 \,{\mathrm e}^{-x} c^{3}}{8}-\frac {3 \,{\mathrm e}^{-2 x} a^{3}}{8}+\frac {3 \,{\mathrm e}^{-2 x} a^{2} c}{4}-\frac {3 \,{\mathrm e}^{-2 x} a \,c^{2}}{8}-\frac {{\mathrm e}^{-3 x} a^{3}}{24}+\frac {{\mathrm e}^{-3 x} a^{2} c}{8}-\frac {{\mathrm e}^{-3 x} a \,c^{2}}{8}+\frac {{\mathrm e}^{-3 x} c^{3}}{24}\)	\(217\)
orering	\(\text {Expression too large to display}\)	\(845\)

Input:

int((a+a*cosh(x)+c*sinh(x))^3,x,method=_RETURNVERBOSE)
 

Output:

a^3*x+c^3*(-2/3+1/3*sinh(x)^2)*cosh(x)+a^2*c*cosh(x)^3+3*a^3*(1/2*cosh(x)* 
sinh(x)+1/2*x)+a*(c*sinh(x)+a)^3/c+a^3*(2/3+1/3*cosh(x)^2)*sinh(x)+3*a*c^2 
*(1/2*cosh(x)*sinh(x)-1/2*x)+3*a^2*c*cosh(x)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.37 \[ \int (a+a \cosh (x)+c \sinh (x))^3 \, dx=\frac {3}{2} \, a^{2} c \cosh \left (x\right )^{2} + \frac {1}{12} \, {\left (3 \, a^{2} c + c^{3}\right )} \cosh \left (x\right )^{3} + \frac {1}{12} \, {\left (a^{3} + 3 \, a c^{2}\right )} \sinh \left (x\right )^{3} + \frac {1}{4} \, {\left (6 \, a^{2} c + {\left (3 \, a^{2} c + c^{3}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} + \frac {1}{2} \, {\left (5 \, a^{3} - 3 \, a c^{2}\right )} x + \frac {3}{4} \, {\left (5 \, a^{2} c - c^{3}\right )} \cosh \left (x\right ) + \frac {1}{4} \, {\left (15 \, a^{3} - 3 \, a c^{2} + {\left (a^{3} + 3 \, a c^{2}\right )} \cosh \left (x\right )^{2} + 6 \, {\left (a^{3} + a c^{2}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right ) \] Input:

integrate((a+a*cosh(x)+c*sinh(x))^3,x, algorithm="fricas")
 

Output:

3/2*a^2*c*cosh(x)^2 + 1/12*(3*a^2*c + c^3)*cosh(x)^3 + 1/12*(a^3 + 3*a*c^2 
)*sinh(x)^3 + 1/4*(6*a^2*c + (3*a^2*c + c^3)*cosh(x))*sinh(x)^2 + 1/2*(5*a 
^3 - 3*a*c^2)*x + 3/4*(5*a^2*c - c^3)*cosh(x) + 1/4*(15*a^3 - 3*a*c^2 + (a 
^3 + 3*a*c^2)*cosh(x)^2 + 6*(a^3 + a*c^2)*cosh(x))*sinh(x)
 

Sympy [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.80 \[ \int (a+a \cosh (x)+c \sinh (x))^3 \, dx=- \frac {3 a^{3} x \sinh ^{2}{\left (x \right )}}{2} + \frac {3 a^{3} x \cosh ^{2}{\left (x \right )}}{2} + a^{3} x - \frac {2 a^{3} \sinh ^{3}{\left (x \right )}}{3} + a^{3} \sinh {\left (x \right )} \cosh ^{2}{\left (x \right )} + \frac {3 a^{3} \sinh {\left (x \right )} \cosh {\left (x \right )}}{2} + 3 a^{3} \sinh {\left (x \right )} + 3 a^{2} c \sinh ^{2}{\left (x \right )} + a^{2} c \cosh ^{3}{\left (x \right )} + 3 a^{2} c \cosh {\left (x \right )} + \frac {3 a c^{2} x \sinh ^{2}{\left (x \right )}}{2} - \frac {3 a c^{2} x \cosh ^{2}{\left (x \right )}}{2} + a c^{2} \sinh ^{3}{\left (x \right )} + \frac {3 a c^{2} \sinh {\left (x \right )} \cosh {\left (x \right )}}{2} + c^{3} \sinh ^{2}{\left (x \right )} \cosh {\left (x \right )} - \frac {2 c^{3} \cosh ^{3}{\left (x \right )}}{3} \] Input:

integrate((a+a*cosh(x)+c*sinh(x))**3,x)
 

Output:

-3*a**3*x*sinh(x)**2/2 + 3*a**3*x*cosh(x)**2/2 + a**3*x - 2*a**3*sinh(x)** 
3/3 + a**3*sinh(x)*cosh(x)**2 + 3*a**3*sinh(x)*cosh(x)/2 + 3*a**3*sinh(x) 
+ 3*a**2*c*sinh(x)**2 + a**2*c*cosh(x)**3 + 3*a**2*c*cosh(x) + 3*a*c**2*x* 
sinh(x)**2/2 - 3*a*c**2*x*cosh(x)**2/2 + a*c**2*sinh(x)**3 + 3*a*c**2*sinh 
(x)*cosh(x)/2 + c**3*sinh(x)**2*cosh(x) - 2*c**3*cosh(x)**3/3
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.30 \[ \int (a+a \cosh (x)+c \sinh (x))^3 \, dx=a^{2} c \cosh \left (x\right )^{3} + a c^{2} \sinh \left (x\right )^{3} + a^{3} x + \frac {1}{24} \, c^{3} {\left (e^{\left (3 \, x\right )} - 9 \, e^{\left (-x\right )} + e^{\left (-3 \, x\right )} - 9 \, e^{x}\right )} + \frac {1}{24} \, a^{3} {\left (e^{\left (3 \, x\right )} - 9 \, e^{\left (-x\right )} - e^{\left (-3 \, x\right )} + 9 \, e^{x}\right )} + 3 \, {\left (c \cosh \left (x\right ) + a \sinh \left (x\right )\right )} a^{2} + \frac {3}{8} \, {\left (8 \, a c \cosh \left (x\right )^{2} + a^{2} {\left (4 \, x + e^{\left (2 \, x\right )} - e^{\left (-2 \, x\right )}\right )} - c^{2} {\left (4 \, x - e^{\left (2 \, x\right )} + e^{\left (-2 \, x\right )}\right )}\right )} a \] Input:

integrate((a+a*cosh(x)+c*sinh(x))^3,x, algorithm="maxima")
 

Output:

a^2*c*cosh(x)^3 + a*c^2*sinh(x)^3 + a^3*x + 1/24*c^3*(e^(3*x) - 9*e^(-x) + 
 e^(-3*x) - 9*e^x) + 1/24*a^3*(e^(3*x) - 9*e^(-x) - e^(-3*x) + 9*e^x) + 3* 
(c*cosh(x) + a*sinh(x))*a^2 + 3/8*(8*a*c*cosh(x)^2 + a^2*(4*x + e^(2*x) - 
e^(-2*x)) - c^2*(4*x - e^(2*x) + e^(-2*x)))*a
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.77 \[ \int (a+a \cosh (x)+c \sinh (x))^3 \, dx=\frac {1}{24} \, a^{3} e^{\left (3 \, x\right )} + \frac {1}{8} \, a^{2} c e^{\left (3 \, x\right )} + \frac {1}{8} \, a c^{2} e^{\left (3 \, x\right )} + \frac {1}{24} \, c^{3} e^{\left (3 \, x\right )} + \frac {3}{8} \, a^{3} e^{\left (2 \, x\right )} + \frac {3}{4} \, a^{2} c e^{\left (2 \, x\right )} + \frac {3}{8} \, a c^{2} e^{\left (2 \, x\right )} + \frac {15}{8} \, a^{3} e^{x} + \frac {15}{8} \, a^{2} c e^{x} - \frac {3}{8} \, a c^{2} e^{x} - \frac {3}{8} \, c^{3} e^{x} + \frac {1}{2} \, {\left (5 \, a^{3} - 3 \, a c^{2}\right )} x - \frac {1}{24} \, {\left (a^{3} - 3 \, a^{2} c + 3 \, a c^{2} - c^{3} + 9 \, {\left (5 \, a^{3} - 5 \, a^{2} c - a c^{2} + c^{3}\right )} e^{\left (2 \, x\right )} + 9 \, {\left (a^{3} - 2 \, a^{2} c + a c^{2}\right )} e^{x}\right )} e^{\left (-3 \, x\right )} \] Input:

integrate((a+a*cosh(x)+c*sinh(x))^3,x, algorithm="giac")
 

Output:

1/24*a^3*e^(3*x) + 1/8*a^2*c*e^(3*x) + 1/8*a*c^2*e^(3*x) + 1/24*c^3*e^(3*x 
) + 3/8*a^3*e^(2*x) + 3/4*a^2*c*e^(2*x) + 3/8*a*c^2*e^(2*x) + 15/8*a^3*e^x 
 + 15/8*a^2*c*e^x - 3/8*a*c^2*e^x - 3/8*c^3*e^x + 1/2*(5*a^3 - 3*a*c^2)*x 
- 1/24*(a^3 - 3*a^2*c + 3*a*c^2 - c^3 + 9*(5*a^3 - 5*a^2*c - a*c^2 + c^3)* 
e^(2*x) + 9*(a^3 - 2*a^2*c + a*c^2)*e^x)*e^(-3*x)
 

Mupad [B] (verification not implemented)

Time = 0.88 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.25 \[ \int (a+a \cosh (x)+c \sinh (x))^3 \, dx=3\,a^3\,\mathrm {sinh}\left (x\right )+a^3\,x+{\mathrm {cosh}\left (x\right )}^3\,\left (a^2\,c-\frac {2\,c^3}{3}\right )+{\mathrm {sinh}\left (x\right )}^3\,\left (a\,c^2-\frac {2\,a^3}{3}\right )+a^3\,{\mathrm {cosh}\left (x\right )}^2\,\mathrm {sinh}\left (x\right )+c^3\,\mathrm {cosh}\left (x\right )\,{\mathrm {sinh}\left (x\right )}^2+3\,a^2\,c\,\mathrm {cosh}\left (x\right )+3\,a^2\,c\,{\mathrm {cosh}\left (x\right )}^2+\frac {3\,a\,\mathrm {cosh}\left (x\right )\,\mathrm {sinh}\left (x\right )\,\left (a^2+c^2\right )}{2}+\frac {3\,a\,x\,{\mathrm {cosh}\left (x\right )}^2\,\left (a^2-c^2\right )}{2}-\frac {3\,a\,x\,{\mathrm {sinh}\left (x\right )}^2\,\left (a^2-c^2\right )}{2} \] Input:

int((a + a*cosh(x) + c*sinh(x))^3,x)
 

Output:

3*a^3*sinh(x) + a^3*x + cosh(x)^3*(a^2*c - (2*c^3)/3) + sinh(x)^3*(a*c^2 - 
 (2*a^3)/3) + a^3*cosh(x)^2*sinh(x) + c^3*cosh(x)*sinh(x)^2 + 3*a^2*c*cosh 
(x) + 3*a^2*c*cosh(x)^2 + (3*a*cosh(x)*sinh(x)*(a^2 + c^2))/2 + (3*a*x*cos 
h(x)^2*(a^2 - c^2))/2 - (3*a*x*sinh(x)^2*(a^2 - c^2))/2
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.41 \[ \int (a+a \cosh (x)+c \sinh (x))^3 \, dx=\cosh \left (x \right )^{3} a^{2} c -\frac {2 \cosh \left (x \right )^{3} c^{3}}{3}+\cosh \left (x \right )^{2} \sinh \left (x \right ) a^{3}+\frac {3 \cosh \left (x \right )^{2} a^{3} x}{2}+3 \cosh \left (x \right )^{2} a^{2} c -\frac {3 \cosh \left (x \right )^{2} a \,c^{2} x}{2}+\cosh \left (x \right ) \sinh \left (x \right )^{2} c^{3}+\frac {3 \cosh \left (x \right ) \sinh \left (x \right ) a^{3}}{2}+\frac {3 \cosh \left (x \right ) \sinh \left (x \right ) a \,c^{2}}{2}+3 \cosh \left (x \right ) a^{2} c -\frac {2 \sinh \left (x \right )^{3} a^{3}}{3}+\sinh \left (x \right )^{3} a \,c^{2}-\frac {3 \sinh \left (x \right )^{2} a^{3} x}{2}+\frac {3 \sinh \left (x \right )^{2} a \,c^{2} x}{2}+3 \sinh \left (x \right ) a^{3}+a^{3} x \] Input:

int((a+a*cosh(x)+c*sinh(x))^3,x)
 

Output:

(6*cosh(x)**3*a**2*c - 4*cosh(x)**3*c**3 + 6*cosh(x)**2*sinh(x)*a**3 + 9*c 
osh(x)**2*a**3*x + 18*cosh(x)**2*a**2*c - 9*cosh(x)**2*a*c**2*x + 6*cosh(x 
)*sinh(x)**2*c**3 + 9*cosh(x)*sinh(x)*a**3 + 9*cosh(x)*sinh(x)*a*c**2 + 18 
*cosh(x)*a**2*c - 4*sinh(x)**3*a**3 + 6*sinh(x)**3*a*c**2 - 9*sinh(x)**2*a 
**3*x + 9*sinh(x)**2*a*c**2*x + 18*sinh(x)*a**3 + 6*a**3*x)/6