\(\int \coth ^3(c+b x) \sinh (a+b x) \, dx\) [43]

Optimal result

Integrand size = 15, antiderivative size = 73 \[ \int \coth ^3(c+b x) \sinh (a+b x) \, dx=-\frac {\cosh (a-c) \text {csch}(c+b x)}{b}-\frac {3 \text {arctanh}(\cosh (c+b x)) \sinh (a-c)}{2 b}-\frac {\coth (c+b x) \text {csch}(c+b x) \sinh (a-c)}{2 b}+\frac {\sinh (a+b x)}{b} \] Output:

-cosh(a-c)*csch(b*x+c)/b-3/2*arctanh(cosh(b*x+c))*sinh(a-c)/b-1/2*coth(b*x 
+c)*csch(b*x+c)*sinh(a-c)/b+sinh(b*x+a)/b
 

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.96 \[ \int \coth ^3(c+b x) \sinh (a+b x) \, dx=\frac {-12 \text {arctanh}\left (\cosh (c)+\sinh (c) \tanh \left (\frac {b x}{2}\right )\right ) \sinh (a-c)+\text {csch}^2(c+b x) (2 \sinh (a-2 c-b x)-5 \sinh (a+b x)+\sinh (a+2 c+3 b x))}{4 b} \] Input:

Integrate[Coth[c + b*x]^3*Sinh[a + b*x],x]
 

Output:

(-12*ArcTanh[Cosh[c] + Sinh[c]*Tanh[(b*x)/2]]*Sinh[a - c] + Csch[c + b*x]^ 
2*(2*Sinh[a - 2*c - b*x] - 5*Sinh[a + b*x] + Sinh[a + 2*c + 3*b*x]))/(4*b)
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.78 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.29, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.133, Rules used = {6156, 3042, 26, 3091, 26, 3042, 26, 4257, 6155, 3042, 3086, 24, 6156, 3042, 26, 3117, 4257}

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.

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

\(\Big \downarrow \) 6156

\(\displaystyle \int \cosh (a+b x) \coth ^2(c+b x)dx+\sinh (a-c) \int \coth ^2(c+b x) \text {csch}(c+b x)dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \cosh (a+b x) \coth ^2(c+b x)dx+\sinh (a-c) \int -i \sec \left (i c+i b x-\frac {\pi }{2}\right ) \tan \left (i c+i b x-\frac {\pi }{2}\right )^2dx\)

\(\Big \downarrow \) 26

\(\displaystyle \int \cosh (a+b x) \coth ^2(c+b x)dx-i \sinh (a-c) \int \sec \left (\frac {1}{2} (2 i c-\pi )+i b x\right ) \tan \left (\frac {1}{2} (2 i c-\pi )+i b x\right )^2dx\)

\(\Big \downarrow \) 3091

\(\displaystyle \int \cosh (a+b x) \coth ^2(c+b x)dx-i \sinh (a-c) \left (-\frac {1}{2} \int -i \text {csch}(c+b x)dx-\frac {i \coth (b x+c) \text {csch}(b x+c)}{2 b}\right )\)

\(\Big \downarrow \) 26

\(\displaystyle \int \cosh (a+b x) \coth ^2(c+b x)dx-i \sinh (a-c) \left (\frac {1}{2} i \int \text {csch}(c+b x)dx-\frac {i \coth (b x+c) \text {csch}(b x+c)}{2 b}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \cosh (a+b x) \coth ^2(c+b x)dx-i \sinh (a-c) \left (\frac {1}{2} i \int i \csc (i c+i b x)dx-\frac {i \coth (b x+c) \text {csch}(b x+c)}{2 b}\right )\)

\(\Big \downarrow \) 26

\(\displaystyle \int \cosh (a+b x) \coth ^2(c+b x)dx-i \sinh (a-c) \left (-\frac {1}{2} \int \csc (i c+i b x)dx-\frac {i \coth (b x+c) \text {csch}(b x+c)}{2 b}\right )\)

\(\Big \downarrow \) 4257

\(\displaystyle \int \cosh (a+b x) \coth ^2(c+b x)dx-i \sinh (a-c) \left (-\frac {i \text {arctanh}(\cosh (b x+c))}{2 b}-\frac {i \coth (b x+c) \text {csch}(b x+c)}{2 b}\right )\)

\(\Big \downarrow \) 6155

\(\displaystyle \int \coth (c+b x) \sinh (a+b x)dx+\cosh (a-c) \int \coth (c+b x) \text {csch}(c+b x)dx-i \sinh (a-c) \left (-\frac {i \text {arctanh}(\cosh (b x+c))}{2 b}-\frac {i \coth (b x+c) \text {csch}(b x+c)}{2 b}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \coth (c+b x) \sinh (a+b x)dx+\cosh (a-c) \int \sec \left (i c+i b x-\frac {\pi }{2}\right ) \tan \left (i c+i b x-\frac {\pi }{2}\right )dx-i \sinh (a-c) \left (-\frac {i \text {arctanh}(\cosh (b x+c))}{2 b}-\frac {i \coth (b x+c) \text {csch}(b x+c)}{2 b}\right )\)

\(\Big \downarrow \) 3086

\(\displaystyle -\frac {i \cosh (a-c) \int 1d(-i \text {csch}(c+b x))}{b}+\int \coth (c+b x) \sinh (a+b x)dx-i \sinh (a-c) \left (-\frac {i \text {arctanh}(\cosh (b x+c))}{2 b}-\frac {i \coth (b x+c) \text {csch}(b x+c)}{2 b}\right )\)

\(\Big \downarrow \) 24

\(\displaystyle \int \coth (c+b x) \sinh (a+b x)dx-i \sinh (a-c) \left (-\frac {i \text {arctanh}(\cosh (b x+c))}{2 b}-\frac {i \coth (b x+c) \text {csch}(b x+c)}{2 b}\right )-\frac {\cosh (a-c) \text {csch}(b x+c)}{b}\)

\(\Big \downarrow \) 6156

\(\displaystyle \sinh (a-c) \int \text {csch}(c+b x)dx+\int \cosh (a+b x)dx-i \sinh (a-c) \left (-\frac {i \text {arctanh}(\cosh (b x+c))}{2 b}-\frac {i \coth (b x+c) \text {csch}(b x+c)}{2 b}\right )-\frac {\cosh (a-c) \text {csch}(b x+c)}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \sinh (a-c) \int i \csc (i c+i b x)dx+\int \sin \left (i a+i b x+\frac {\pi }{2}\right )dx-i \sinh (a-c) \left (-\frac {i \text {arctanh}(\cosh (b x+c))}{2 b}-\frac {i \coth (b x+c) \text {csch}(b x+c)}{2 b}\right )-\frac {\cosh (a-c) \text {csch}(b x+c)}{b}\)

\(\Big \downarrow \) 26

\(\displaystyle i \sinh (a-c) \int \csc (i c+i b x)dx+\int \sin \left (i a+i b x+\frac {\pi }{2}\right )dx-i \sinh (a-c) \left (-\frac {i \text {arctanh}(\cosh (b x+c))}{2 b}-\frac {i \coth (b x+c) \text {csch}(b x+c)}{2 b}\right )-\frac {\cosh (a-c) \text {csch}(b x+c)}{b}\)

\(\Big \downarrow \) 3117

\(\displaystyle i \sinh (a-c) \int \csc (i c+i b x)dx-i \sinh (a-c) \left (-\frac {i \text {arctanh}(\cosh (b x+c))}{2 b}-\frac {i \coth (b x+c) \text {csch}(b x+c)}{2 b}\right )-\frac {\cosh (a-c) \text {csch}(b x+c)}{b}+\frac {\sinh (a+b x)}{b}\)

\(\Big \downarrow \) 4257

\(\displaystyle -\frac {\sinh (a-c) \text {arctanh}(\cosh (b x+c))}{b}-i \sinh (a-c) \left (-\frac {i \text {arctanh}(\cosh (b x+c))}{2 b}-\frac {i \coth (b x+c) \text {csch}(b x+c)}{2 b}\right )-\frac {\cosh (a-c) \text {csch}(b x+c)}{b}+\frac {\sinh (a+b x)}{b}\)

Input:

Int[Coth[c + b*x]^3*Sinh[a + b*x],x]
 

Output:

-((Cosh[a - c]*Csch[c + b*x])/b) - (ArcTanh[Cosh[c + b*x]]*Sinh[a - c])/b 
- I*(((-1/2*I)*ArcTanh[Cosh[c + b*x]])/b - ((I/2)*Coth[c + b*x]*Csch[c + b 
*x])/b)*Sinh[a - c] + Sinh[a + b*x]/b
 

Defintions of rubi rules used

rule 24

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
 

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 3042

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

rule 3086

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( 
n_.), x_Symbol] :> Simp[a/f   Subst[Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2 
), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2 
] &&  !(IntegerQ[m/2] && LtQ[0, m, n + 1])
 

rule 3091

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( 
n_), x_Symbol] :> Simp[b*(a*Sec[e + f*x])^m*((b*Tan[e + f*x])^(n - 1)/(f*(m 
 + n - 1))), x] - Simp[b^2*((n - 1)/(m + n - 1))   Int[(a*Sec[e + f*x])^m*( 
b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] & 
& NeQ[m + n - 1, 0] && IntegersQ[2*m, 2*n]
 

rule 3117

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; 
 FreeQ[{c, d}, x]
 

rule 4257

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] 
 /; FreeQ[{c, d}, x]
 

rule 6155

Int[Cosh[v_]*Coth[w_]^(n_.), x_Symbol] :> Int[Sinh[v]*Coth[w]^(n - 1), x] + 
 Simp[Cosh[v - w]   Int[Csch[w]*Coth[w]^(n - 1), x], x] /; GtQ[n, 0] && NeQ 
[w, v] && FreeQ[v - w, x]
 

rule 6156

Int[Coth[w_]^(n_.)*Sinh[v_], x_Symbol] :> Int[Cosh[v]*Coth[w]^(n - 1), x] + 
 Simp[Sinh[v - w]   Int[Csch[w]*Coth[w]^(n - 1), x], x] /; GtQ[n, 0] && NeQ 
[w, v] && FreeQ[v - w, x]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(229\) vs. \(2(69)=138\).

Time = 0.18 (sec) , antiderivative size = 230, normalized size of antiderivative = 3.15

method	result	size
risch	\(\frac {{\mathrm e}^{b x +a}}{2 b}-\frac {{\mathrm e}^{-b x -a}}{2 b}+\frac {{\mathrm e}^{b x +a} \left (-3 \,{\mathrm e}^{2 b x +4 a +2 c}-{\mathrm e}^{2 b x +2 a +4 c}+{\mathrm e}^{4 a}+3 \,{\mathrm e}^{2 a +2 c}\right )}{2 b \left (-{\mathrm e}^{2 b x +2 a +2 c}+{\mathrm e}^{2 a}\right )^{2}}+\frac {3 \ln \left ({\mathrm e}^{b x +a}-{\mathrm e}^{a -c}\right ) {\mathrm e}^{-a -c} {\mathrm e}^{2 a}}{4 b}-\frac {3 \ln \left ({\mathrm e}^{b x +a}-{\mathrm e}^{a -c}\right ) {\mathrm e}^{-a -c} {\mathrm e}^{2 c}}{4 b}-\frac {3 \ln \left ({\mathrm e}^{b x +a}+{\mathrm e}^{a -c}\right ) {\mathrm e}^{-a -c} {\mathrm e}^{2 a}}{4 b}+\frac {3 \ln \left ({\mathrm e}^{b x +a}+{\mathrm e}^{a -c}\right ) {\mathrm e}^{-a -c} {\mathrm e}^{2 c}}{4 b}\)	\(230\)

Input:

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

Output:

1/2/b*exp(b*x+a)-1/2/b*exp(-b*x-a)+1/2*exp(b*x+a)*(-3*exp(2*b*x+4*a+2*c)-e 
xp(2*b*x+2*a+4*c)+exp(4*a)+3*exp(2*a+2*c))/b/(-exp(2*b*x+2*a+2*c)+exp(2*a) 
)^2+3/4*ln(exp(b*x+a)-exp(a-c))/b*exp(-a-c)*exp(2*a)-3/4*ln(exp(b*x+a)-exp 
(a-c))/b*exp(-a-c)*exp(2*c)-3/4*ln(exp(b*x+a)+exp(a-c))/b*exp(-a-c)*exp(2* 
a)+3/4*ln(exp(b*x+a)+exp(a-c))/b*exp(-a-c)*exp(2*c)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2372 vs. \(2 (69) = 138\).

Time = 0.11 (sec) , antiderivative size = 2372, normalized size of antiderivative = 32.49 \[ \int \coth ^3(c+b x) \sinh (a+b x) \, dx=\text {Too large to display} \] Input:

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

Output:

1/4*(2*cosh(b*x + c)^6*cosh(-a + c)^2 + 2*(cosh(-a + c)^2 - 2*cosh(-a + c) 
*sinh(-a + c) + sinh(-a + c)^2)*sinh(b*x + c)^6 + 12*(cosh(b*x + c)*cosh(- 
a + c)^2 - 2*cosh(b*x + c)*cosh(-a + c)*sinh(-a + c) + cosh(b*x + c)*sinh( 
-a + c)^2)*sinh(b*x + c)^5 - 2*(5*cosh(-a + c)^2 + 2)*cosh(b*x + c)^4 + 2* 
(15*cosh(b*x + c)^2*cosh(-a + c)^2 + 5*(3*cosh(b*x + c)^2 - 1)*sinh(-a + c 
)^2 - 5*cosh(-a + c)^2 - 10*(3*cosh(b*x + c)^2*cosh(-a + c) - cosh(-a + c) 
)*sinh(-a + c) - 2)*sinh(b*x + c)^4 + 8*(5*cosh(b*x + c)^3*cosh(-a + c)^2 
+ 5*(cosh(b*x + c)^3 - cosh(b*x + c))*sinh(-a + c)^2 - (5*cosh(-a + c)^2 + 
 2)*cosh(b*x + c) - 10*(cosh(b*x + c)^3*cosh(-a + c) - cosh(b*x + c)*cosh( 
-a + c))*sinh(-a + c))*sinh(b*x + c)^3 + 2*(2*cosh(-a + c)^2 + 5)*cosh(b*x 
 + c)^2 + 2*(15*cosh(b*x + c)^4*cosh(-a + c)^2 - 6*(5*cosh(-a + c)^2 + 2)* 
cosh(b*x + c)^2 + (15*cosh(b*x + c)^4 - 30*cosh(b*x + c)^2 + 2)*sinh(-a + 
c)^2 + 2*cosh(-a + c)^2 - 2*(15*cosh(b*x + c)^4*cosh(-a + c) - 30*cosh(b*x 
 + c)^2*cosh(-a + c) + 2*cosh(-a + c))*sinh(-a + c) + 5)*sinh(b*x + c)^2 + 
 2*(cosh(b*x + c)^6 - 5*cosh(b*x + c)^4 + 2*cosh(b*x + c)^2)*sinh(-a + c)^ 
2 - 3*((cosh(-a + c)^2 - 1)*cosh(b*x + c)^5 + (cosh(-a + c)^2 - 2*cosh(-a 
+ c)*sinh(-a + c) + sinh(-a + c)^2 - 1)*sinh(b*x + c)^5 - 5*(2*cosh(b*x + 
c)*cosh(-a + c)*sinh(-a + c) - cosh(b*x + c)*sinh(-a + c)^2 - (cosh(-a + c 
)^2 - 1)*cosh(b*x + c))*sinh(b*x + c)^4 - 2*(cosh(-a + c)^2 - 1)*cosh(b*x 
+ c)^3 + 2*(5*(cosh(-a + c)^2 - 1)*cosh(b*x + c)^2 + (5*cosh(b*x + c)^2...
 

Sympy [F]

\[ \int \coth ^3(c+b x) \sinh (a+b x) \, dx=\int \sinh {\left (a + b x \right )} \coth ^{3}{\left (b x + c \right )}\, dx \] Input:

integrate(coth(b*x+c)**3*sinh(b*x+a),x)
 

Output:

Integral(sinh(a + b*x)*coth(b*x + c)**3, x)
                                                                                    
                                                                                    
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 186 vs. \(2 (69) = 138\).

Time = 0.04 (sec) , antiderivative size = 186, normalized size of antiderivative = 2.55 \[ \int \coth ^3(c+b x) \sinh (a+b x) \, dx=-\frac {3 \, {\left (e^{\left (2 \, a\right )} - e^{\left (2 \, c\right )}\right )} e^{\left (-a - c\right )} \log \left (e^{\left (-b x\right )} + e^{c}\right )}{4 \, b} + \frac {3 \, {\left (e^{\left (2 \, a\right )} - e^{\left (2 \, c\right )}\right )} e^{\left (-a - c\right )} \log \left (e^{\left (-b x\right )} - e^{c}\right )}{4 \, b} - \frac {e^{\left (-b x - a\right )}}{2 \, b} - \frac {{\left (5 \, e^{\left (2 \, a + 2 \, c\right )} + e^{\left (4 \, c\right )}\right )} e^{\left (-2 \, b x - 2 \, a\right )} - {\left (2 \, e^{\left (4 \, a\right )} + 3 \, e^{\left (2 \, a + 2 \, c\right )}\right )} e^{\left (-4 \, b x - 4 \, a\right )} - e^{\left (4 \, c\right )}}{2 \, b {\left (e^{\left (-b x - a + 4 \, c\right )} - 2 \, e^{\left (-3 \, b x - a + 2 \, c\right )} + e^{\left (-5 \, b x - a\right )}\right )}} \] Input:

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

Output:

-3/4*(e^(2*a) - e^(2*c))*e^(-a - c)*log(e^(-b*x) + e^c)/b + 3/4*(e^(2*a) - 
 e^(2*c))*e^(-a - c)*log(e^(-b*x) - e^c)/b - 1/2*e^(-b*x - a)/b - 1/2*((5* 
e^(2*a + 2*c) + e^(4*c))*e^(-2*b*x - 2*a) - (2*e^(4*a) + 3*e^(2*a + 2*c))* 
e^(-4*b*x - 4*a) - e^(4*c))/(b*(e^(-b*x - a + 4*c) - 2*e^(-3*b*x - a + 2*c 
) + e^(-5*b*x - a)))
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 169 vs. \(2 (69) = 138\).

Time = 0.12 (sec) , antiderivative size = 169, normalized size of antiderivative = 2.32 \[ \int \coth ^3(c+b x) \sinh (a+b x) \, dx=-\frac {3 \, {\left (e^{\left (2 \, a + c\right )} - e^{\left (3 \, c\right )}\right )} e^{\left (-a - 2 \, c\right )} \log \left (e^{\left (b x + a + c\right )} + e^{a}\right ) - 3 \, {\left (e^{\left (2 \, a + c\right )} - e^{\left (3 \, c\right )}\right )} e^{\left (-a - 2 \, c\right )} \log \left ({\left | e^{\left (b x + a + c\right )} - e^{a} \right |}\right ) + \frac {2 \, {\left (3 \, e^{\left (3 \, b x + 5 \, a + 2 \, c\right )} + e^{\left (3 \, b x + 3 \, a + 4 \, c\right )} - e^{\left (b x + 5 \, a\right )} - 3 \, e^{\left (b x + 3 \, a + 2 \, c\right )}\right )}}{{\left (e^{\left (2 \, b x + 2 \, a + 2 \, c\right )} - e^{\left (2 \, a\right )}\right )}^{2}} - 2 \, e^{\left (b x + a\right )} + 2 \, e^{\left (-b x - a\right )}}{4 \, b} \] Input:

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

Output:

-1/4*(3*(e^(2*a + c) - e^(3*c))*e^(-a - 2*c)*log(e^(b*x + a + c) + e^a) - 
3*(e^(2*a + c) - e^(3*c))*e^(-a - 2*c)*log(abs(e^(b*x + a + c) - e^a)) + 2 
*(3*e^(3*b*x + 5*a + 2*c) + e^(3*b*x + 3*a + 4*c) - e^(b*x + 5*a) - 3*e^(b 
*x + 3*a + 2*c))/(e^(2*b*x + 2*a + 2*c) - e^(2*a))^2 - 2*e^(b*x + a) + 2*e 
^(-b*x - a))/b
 

Mupad [F(-1)]

Timed out. \[ \int \coth ^3(c+b x) \sinh (a+b x) \, dx=\int {\mathrm {coth}\left (c+b\,x\right )}^3\,\mathrm {sinh}\left (a+b\,x\right ) \,d x \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 180, normalized size of antiderivative = 2.47 \[ \int \coth ^3(c+b x) \sinh (a+b x) \, dx=\frac {-2 e^{b x +a +c} \cosh \left (b x +a \right ) \coth \left (b x +c \right )-2 e^{b x +a +c} \coth \left (b x +c \right )^{2} \sinh \left (b x +a \right )+3 e^{2 b x +2 a +c}+3 e^{b x +2 a} \mathrm {log}\left (e^{b x +c}-1\right )-3 e^{b x +2 a} \mathrm {log}\left (e^{b x +c}+1\right )+2 e^{b x +a +c} \sinh \left (b x +a \right )-3 e^{b x +2 c} \mathrm {log}\left (e^{b x +c}-1\right )+3 e^{b x +2 c} \mathrm {log}\left (e^{b x +c}+1\right )-3 e^{c}}{4 e^{b x +a +c} b} \] Input:

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

Output:

( - 2*e**(a + b*x + c)*cosh(a + b*x)*coth(b*x + c) - 2*e**(a + b*x + c)*co 
th(b*x + c)**2*sinh(a + b*x) + 3*e**(2*a + 2*b*x + c) + 3*e**(2*a + b*x)*l 
og(e**(b*x + c) - 1) - 3*e**(2*a + b*x)*log(e**(b*x + c) + 1) + 2*e**(a + 
b*x + c)*sinh(a + b*x) - 3*e**(b*x + 2*c)*log(e**(b*x + c) - 1) + 3*e**(b* 
x + 2*c)*log(e**(b*x + c) + 1) - 3*e**c)/(4*e**(a + b*x + c)*b)