Integrand size = 23, antiderivative size = 131 \[ \int \text {csch}^6(c+d x) \left (a+b \sinh ^3(c+d x)\right )^3 \, dx=3 a b^2 x+\frac {3 a^2 b \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {b^3 \cosh (c+d x)}{d}+\frac {b^3 \cosh ^3(c+d x)}{3 d}-\frac {a^3 \coth (c+d x)}{d}+\frac {2 a^3 \coth ^3(c+d x)}{3 d}-\frac {a^3 \coth ^5(c+d x)}{5 d}-\frac {3 a^2 b \coth (c+d x) \text {csch}(c+d x)}{2 d} \]
3*a*b^2*x+3/2*a^2*b*arctanh(cosh(d*x+c))/d-b^3*cosh(d*x+c)/d+1/3*b^3*cosh( d*x+c)^3/d-a^3*coth(d*x+c)/d+2/3*a^3*coth(d*x+c)^3/d-1/5*a^3*coth(d*x+c)^5 /d-3/2*a^2*b*coth(d*x+c)*csch(d*x+c)/d
Time = 3.88 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.82 \[ \int \text {csch}^6(c+d x) \left (a+b \sinh ^3(c+d x)\right )^3 \, dx=\frac {-360 b^3 \cosh (c+d x)+40 b^3 \cosh (3 (c+d x))+\frac {1}{2} a \left (-256 a^2 \coth \left (\frac {1}{2} (c+d x)\right )-360 a b \text {csch}^2\left (\frac {1}{2} (c+d x)\right )+19 a^2 \text {csch}^4\left (\frac {1}{2} (c+d x)\right ) \sinh (c+d x)-3 a^2 \text {csch}^6\left (\frac {1}{2} (c+d x)\right ) \sinh (c+d x)+8 \left (180 b \left (2 b (c+d x)+a \log \left (\cosh \left (\frac {1}{2} (c+d x)\right )\right )-a \log \left (\sinh \left (\frac {1}{2} (c+d x)\right )\right )\right )-45 a b \text {sech}^2\left (\frac {1}{2} (c+d x)\right )-38 a^2 \text {csch}^3(c+d x) \sinh ^4\left (\frac {1}{2} (c+d x)\right )-24 a^2 \text {csch}^5(c+d x) \sinh ^6\left (\frac {1}{2} (c+d x)\right )-32 a^2 \tanh \left (\frac {1}{2} (c+d x)\right )\right )\right )}{480 d} \]
(-360*b^3*Cosh[c + d*x] + 40*b^3*Cosh[3*(c + d*x)] + (a*(-256*a^2*Coth[(c + d*x)/2] - 360*a*b*Csch[(c + d*x)/2]^2 + 19*a^2*Csch[(c + d*x)/2]^4*Sinh[ c + d*x] - 3*a^2*Csch[(c + d*x)/2]^6*Sinh[c + d*x] + 8*(180*b*(2*b*(c + d* x) + a*Log[Cosh[(c + d*x)/2]] - a*Log[Sinh[(c + d*x)/2]]) - 45*a*b*Sech[(c + d*x)/2]^2 - 38*a^2*Csch[c + d*x]^3*Sinh[(c + d*x)/2]^4 - 24*a^2*Csch[c + d*x]^5*Sinh[(c + d*x)/2]^6 - 32*a^2*Tanh[(c + d*x)/2])))/2)/(480*d)
Time = 0.36 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3042, 25, 3699, 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.
| \(\displaystyle \int \text {csch}^6(c+d x) \left (a+b \sinh ^3(c+d x)\right )^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {\left (a+i b \sin (i c+i d x)^3\right )^3}{\sin (i c+i d x)^6}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {\left (i b \sin (i c+i d x)^3+a\right )^3}{\sin (i c+i d x)^6}dx\) |
| \(\Big \downarrow \) 3699 |
| \(\displaystyle -\int \left (-a^3 \text {csch}^6(c+d x)-3 a^2 b \text {csch}^3(c+d x)-b^3 \sinh ^3(c+d x)-3 a b^2\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a^3 \coth ^5(c+d x)}{5 d}+\frac {2 a^3 \coth ^3(c+d x)}{3 d}-\frac {a^3 \coth (c+d x)}{d}+\frac {3 a^2 b \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {3 a^2 b \coth (c+d x) \text {csch}(c+d x)}{2 d}+3 a b^2 x+\frac {b^3 \cosh ^3(c+d x)}{3 d}-\frac {b^3 \cosh (c+d x)}{d}\) |
3*a*b^2*x + (3*a^2*b*ArcTanh[Cosh[c + d*x]])/(2*d) - (b^3*Cosh[c + d*x])/d + (b^3*Cosh[c + d*x]^3)/(3*d) - (a^3*Coth[c + d*x])/d + (2*a^3*Coth[c + d *x]^3)/(3*d) - (a^3*Coth[c + d*x]^5)/(5*d) - (3*a^2*b*Coth[c + d*x]*Csch[c + d*x])/(2*d)
3.2.69.3.1 Defintions of rubi rules used
Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^(n_ ))^(p_.), x_Symbol] :> Int[ExpandTrig[sin[e + f*x]^m*(a + b*sin[e + f*x]^n) ^p, x], x] /; FreeQ[{a, b, e, f}, x] && IntegersQ[m, p] && (EqQ[n, 4] || Gt Q[p, 0] || (EqQ[p, -1] && IntegerQ[n]))
Time = 0.66 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.76
| method | result | size |
| derivativedivides | \(\frac {a^{3} \left (-\frac {8}{15}-\frac {\operatorname {csch}\left (d x +c \right )^{4}}{5}+\frac {4 \operatorname {csch}\left (d x +c \right )^{2}}{15}\right ) \coth \left (d x +c \right )+3 a^{2} b \left (-\frac {\operatorname {csch}\left (d x +c \right ) \coth \left (d x +c \right )}{2}+\operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )\right )+3 a \,b^{2} \left (d x +c \right )+b^{3} \left (-\frac {2}{3}+\frac {\sinh \left (d x +c \right )^{2}}{3}\right ) \cosh \left (d x +c \right )}{d}\) | \(99\) |
| default | \(\frac {a^{3} \left (-\frac {8}{15}-\frac {\operatorname {csch}\left (d x +c \right )^{4}}{5}+\frac {4 \operatorname {csch}\left (d x +c \right )^{2}}{15}\right ) \coth \left (d x +c \right )+3 a^{2} b \left (-\frac {\operatorname {csch}\left (d x +c \right ) \coth \left (d x +c \right )}{2}+\operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )\right )+3 a \,b^{2} \left (d x +c \right )+b^{3} \left (-\frac {2}{3}+\frac {\sinh \left (d x +c \right )^{2}}{3}\right ) \cosh \left (d x +c \right )}{d}\) | \(99\) |
| parallelrisch | \(\frac {-144 \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2} b -\left (\cosh \left (d x +c \right )-\frac {\cosh \left (3 d x +3 c \right )}{2}+\frac {\cosh \left (5 d x +5 c \right )}{10}\right ) a^{3} \operatorname {sech}\left (\frac {d x}{2}+\frac {c}{2}\right )^{5} \operatorname {csch}\left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+36 b \,a^{2} \left (\operatorname {sech}\left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-5\right ) \operatorname {csch}\left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+108 a^{2} b \coth \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+288 x a \,b^{2} d -72 b^{3} \cosh \left (d x +c \right )+8 b^{3} \cosh \left (3 d x +3 c \right )-64 b^{3}}{96 d}\) | \(164\) |
| risch | \(3 a \,b^{2} x +\frac {{\mathrm e}^{3 d x +3 c} b^{3}}{24 d}-\frac {3 \,{\mathrm e}^{d x +c} b^{3}}{8 d}-\frac {3 \,{\mathrm e}^{-d x -c} b^{3}}{8 d}+\frac {{\mathrm e}^{-3 d x -3 c} b^{3}}{24 d}-\frac {a^{2} \left (45 b \,{\mathrm e}^{9 d x +9 c}-90 b \,{\mathrm e}^{7 d x +7 c}+160 \,{\mathrm e}^{4 d x +4 c} a +90 b \,{\mathrm e}^{3 d x +3 c}-80 a \,{\mathrm e}^{2 d x +2 c}-45 \,{\mathrm e}^{d x +c} b +16 a \right )}{15 d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{5}}-\frac {3 a^{2} \ln \left ({\mathrm e}^{d x +c}-1\right ) b}{2 d}+\frac {3 a^{2} \ln \left ({\mathrm e}^{d x +c}+1\right ) b}{2 d}\) | \(204\) |
1/d*(a^3*(-8/15-1/5*csch(d*x+c)^4+4/15*csch(d*x+c)^2)*coth(d*x+c)+3*a^2*b* (-1/2*csch(d*x+c)*coth(d*x+c)+arctanh(exp(d*x+c)))+3*a*b^2*(d*x+c)+b^3*(-2 /3+1/3*sinh(d*x+c)^2)*cosh(d*x+c))
Leaf count of result is larger than twice the leaf count of optimal. 4629 vs. \(2 (121) = 242\).
Time = 0.32 (sec) , antiderivative size = 4629, normalized size of antiderivative = 35.34 \[ \int \text {csch}^6(c+d x) \left (a+b \sinh ^3(c+d x)\right )^3 \, dx=\text {Too large to display} \]
1/120*(5*b^3*cosh(d*x + c)^16 + 80*b^3*cosh(d*x + c)*sinh(d*x + c)^15 + 5* b^3*sinh(d*x + c)^16 + 360*a*b^2*d*x*cosh(d*x + c)^13 - 70*b^3*cosh(d*x + c)^14 - 1800*a*b^2*d*x*cosh(d*x + c)^11 + 10*(60*b^3*cosh(d*x + c)^2 - 7*b ^3)*sinh(d*x + c)^14 + 3600*a*b^2*d*x*cosh(d*x + c)^9 + 20*(140*b^3*cosh(d *x + c)^3 + 18*a*b^2*d*x - 49*b^3*cosh(d*x + c))*sinh(d*x + c)^13 - 10*(36 *a^2*b - 23*b^3)*cosh(d*x + c)^12 + 10*(910*b^3*cosh(d*x + c)^4 + 468*a*b^ 2*d*x*cosh(d*x + c) - 637*b^3*cosh(d*x + c)^2 - 36*a^2*b + 23*b^3)*sinh(d* x + c)^12 + 40*(546*b^3*cosh(d*x + c)^5 + 702*a*b^2*d*x*cosh(d*x + c)^2 - 637*b^3*cosh(d*x + c)^3 - 45*a*b^2*d*x - 3*(36*a^2*b - 23*b^3)*cosh(d*x + c))*sinh(d*x + c)^11 + 90*(8*a^2*b - 3*b^3)*cosh(d*x + c)^10 + 10*(4004*b^ 3*cosh(d*x + c)^6 + 10296*a*b^2*d*x*cosh(d*x + c)^3 - 7007*b^3*cosh(d*x + c)^4 - 1980*a*b^2*d*x*cosh(d*x + c) + 72*a^2*b - 27*b^3 - 66*(36*a^2*b - 2 3*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^10 + 20*(2860*b^3*cosh(d*x + c)^7 + 12870*a*b^2*d*x*cosh(d*x + c)^4 - 7007*b^3*cosh(d*x + c)^5 - 4950*a*b^2*d* x*cosh(d*x + c)^2 + 180*a*b^2*d*x - 110*(36*a^2*b - 23*b^3)*cosh(d*x + c)^ 3 + 45*(8*a^2*b - 3*b^3)*cosh(d*x + c))*sinh(d*x + c)^9 + 30*(2145*b^3*cos h(d*x + c)^8 + 15444*a*b^2*d*x*cosh(d*x + c)^5 - 7007*b^3*cosh(d*x + c)^6 - 9900*a*b^2*d*x*cosh(d*x + c)^3 + 1080*a*b^2*d*x*cosh(d*x + c) - 165*(36* a^2*b - 23*b^3)*cosh(d*x + c)^4 + 135*(8*a^2*b - 3*b^3)*cosh(d*x + c)^2)*s inh(d*x + c)^8 - 80*(45*a*b^2*d*x + 16*a^3)*cosh(d*x + c)^7 + 80*(715*b...
Timed out. \[ \int \text {csch}^6(c+d x) \left (a+b \sinh ^3(c+d x)\right )^3 \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 365 vs. \(2 (121) = 242\).
Time = 0.22 (sec) , antiderivative size = 365, normalized size of antiderivative = 2.79 \[ \int \text {csch}^6(c+d x) \left (a+b \sinh ^3(c+d x)\right )^3 \, dx=3 \, a b^{2} x + \frac {1}{24} \, b^{3} {\left (\frac {e^{\left (3 \, d x + 3 \, c\right )}}{d} - \frac {9 \, e^{\left (d x + c\right )}}{d} - \frac {9 \, e^{\left (-d x - c\right )}}{d} + \frac {e^{\left (-3 \, d x - 3 \, c\right )}}{d}\right )} + \frac {3}{2} \, a^{2} b {\left (\frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {2 \, {\left (e^{\left (-d x - c\right )} + e^{\left (-3 \, d x - 3 \, c\right )}\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )}}\right )} - \frac {16}{15} \, a^{3} {\left (\frac {5 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} - 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} - 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} - 1\right )}} - \frac {10 \, e^{\left (-4 \, d x - 4 \, c\right )}}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} - 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} - 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} - 1\right )}} - \frac {1}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} - 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} - 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} - 1\right )}}\right )} \]
3*a*b^2*x + 1/24*b^3*(e^(3*d*x + 3*c)/d - 9*e^(d*x + c)/d - 9*e^(-d*x - c) /d + e^(-3*d*x - 3*c)/d) + 3/2*a^2*b*(log(e^(-d*x - c) + 1)/d - log(e^(-d* x - c) - 1)/d + 2*(e^(-d*x - c) + e^(-3*d*x - 3*c))/(d*(2*e^(-2*d*x - 2*c) - e^(-4*d*x - 4*c) - 1))) - 16/15*a^3*(5*e^(-2*d*x - 2*c)/(d*(5*e^(-2*d*x - 2*c) - 10*e^(-4*d*x - 4*c) + 10*e^(-6*d*x - 6*c) - 5*e^(-8*d*x - 8*c) + e^(-10*d*x - 10*c) - 1)) - 10*e^(-4*d*x - 4*c)/(d*(5*e^(-2*d*x - 2*c) - 1 0*e^(-4*d*x - 4*c) + 10*e^(-6*d*x - 6*c) - 5*e^(-8*d*x - 8*c) + e^(-10*d*x - 10*c) - 1)) - 1/(d*(5*e^(-2*d*x - 2*c) - 10*e^(-4*d*x - 4*c) + 10*e^(-6 *d*x - 6*c) - 5*e^(-8*d*x - 8*c) + e^(-10*d*x - 10*c) - 1)))
Leaf count of result is larger than twice the leaf count of optimal. 270 vs. \(2 (121) = 242\).
Time = 0.47 (sec) , antiderivative size = 270, normalized size of antiderivative = 2.06 \[ \int \text {csch}^6(c+d x) \left (a+b \sinh ^3(c+d x)\right )^3 \, dx=\frac {360 \, {\left (d x + c\right )} a b^{2} + 5 \, b^{3} e^{\left (3 \, d x + 3 \, c\right )} - 45 \, b^{3} e^{\left (d x + c\right )} + 180 \, a^{2} b \log \left (e^{\left (d x + c\right )} + 1\right ) - 180 \, a^{2} b \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right ) - \frac {{\left (475 \, b^{3} e^{\left (8 \, d x + 8 \, c\right )} + 1280 \, a^{3} e^{\left (7 \, d x + 7 \, c\right )} - 640 \, a^{3} e^{\left (5 \, d x + 5 \, c\right )} + 128 \, a^{3} e^{\left (3 \, d x + 3 \, c\right )} - 70 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 5 \, b^{3} + 45 \, {\left (8 \, a^{2} b + b^{3}\right )} e^{\left (12 \, d x + 12 \, c\right )} - 10 \, {\left (72 \, a^{2} b + 23 \, b^{3}\right )} e^{\left (10 \, d x + 10 \, c\right )} + 20 \, {\left (36 \, a^{2} b - 25 \, b^{3}\right )} e^{\left (6 \, d x + 6 \, c\right )} - 5 \, {\left (72 \, a^{2} b - 55 \, b^{3}\right )} e^{\left (4 \, d x + 4 \, c\right )}\right )} e^{\left (-3 \, d x - 3 \, c\right )}}{{\left (e^{\left (d x + c\right )} + 1\right )}^{5} {\left (e^{\left (d x + c\right )} - 1\right )}^{5}}}{120 \, d} \]
1/120*(360*(d*x + c)*a*b^2 + 5*b^3*e^(3*d*x + 3*c) - 45*b^3*e^(d*x + c) + 180*a^2*b*log(e^(d*x + c) + 1) - 180*a^2*b*log(abs(e^(d*x + c) - 1)) - (47 5*b^3*e^(8*d*x + 8*c) + 1280*a^3*e^(7*d*x + 7*c) - 640*a^3*e^(5*d*x + 5*c) + 128*a^3*e^(3*d*x + 3*c) - 70*b^3*e^(2*d*x + 2*c) + 5*b^3 + 45*(8*a^2*b + b^3)*e^(12*d*x + 12*c) - 10*(72*a^2*b + 23*b^3)*e^(10*d*x + 10*c) + 20*( 36*a^2*b - 25*b^3)*e^(6*d*x + 6*c) - 5*(72*a^2*b - 55*b^3)*e^(4*d*x + 4*c) )*e^(-3*d*x - 3*c)/((e^(d*x + c) + 1)^5*(e^(d*x + c) - 1)^5))/d
Time = 1.56 (sec) , antiderivative size = 432, normalized size of antiderivative = 3.30 \[ \int \text {csch}^6(c+d x) \left (a+b \sinh ^3(c+d x)\right )^3 \, dx=\frac {b^3\,{\mathrm {e}}^{-3\,c-3\,d\,x}}{24\,d}-\frac {3\,b^3\,{\mathrm {e}}^{c+d\,x}}{8\,d}-\frac {3\,b^3\,{\mathrm {e}}^{-c-d\,x}}{8\,d}-\frac {\frac {32\,a^3\,{\mathrm {e}}^{4\,c+4\,d\,x}}{5\,d}+\frac {36\,a^2\,b\,{\mathrm {e}}^{3\,c+3\,d\,x}}{5\,d}-\frac {36\,a^2\,b\,{\mathrm {e}}^{5\,c+5\,d\,x}}{5\,d}+\frac {12\,a^2\,b\,{\mathrm {e}}^{7\,c+7\,d\,x}}{5\,d}-\frac {12\,a^2\,b\,{\mathrm {e}}^{c+d\,x}}{5\,d}}{5\,{\mathrm {e}}^{2\,c+2\,d\,x}-10\,{\mathrm {e}}^{4\,c+4\,d\,x}+10\,{\mathrm {e}}^{6\,c+6\,d\,x}-5\,{\mathrm {e}}^{8\,c+8\,d\,x}+{\mathrm {e}}^{10\,c+10\,d\,x}-1}+\frac {b^3\,{\mathrm {e}}^{3\,c+3\,d\,x}}{24\,d}-\frac {64\,a^3}{15\,d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}-3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}-1\right )}-\frac {16\,a^3}{5\,d\,\left (6\,{\mathrm {e}}^{4\,c+4\,d\,x}-4\,{\mathrm {e}}^{2\,c+2\,d\,x}-4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1\right )}+\frac {3\,\mathrm {atan}\left (\frac {a^2\,b\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {-d^2}}{d\,\sqrt {a^4\,b^2}}\right )\,\sqrt {a^4\,b^2}}{\sqrt {-d^2}}+3\,a\,b^2\,x-\frac {3\,a^2\,b\,{\mathrm {e}}^{c+d\,x}}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}-\frac {18\,a^2\,b\,{\mathrm {e}}^{c+d\,x}}{5\,d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )} \]
(b^3*exp(- 3*c - 3*d*x))/(24*d) - (3*b^3*exp(c + d*x))/(8*d) - (3*b^3*exp( - c - d*x))/(8*d) - ((32*a^3*exp(4*c + 4*d*x))/(5*d) + (36*a^2*b*exp(3*c + 3*d*x))/(5*d) - (36*a^2*b*exp(5*c + 5*d*x))/(5*d) + (12*a^2*b*exp(7*c + 7 *d*x))/(5*d) - (12*a^2*b*exp(c + d*x))/(5*d))/(5*exp(2*c + 2*d*x) - 10*exp (4*c + 4*d*x) + 10*exp(6*c + 6*d*x) - 5*exp(8*c + 8*d*x) + exp(10*c + 10*d *x) - 1) + (b^3*exp(3*c + 3*d*x))/(24*d) - (64*a^3)/(15*d*(3*exp(2*c + 2*d *x) - 3*exp(4*c + 4*d*x) + exp(6*c + 6*d*x) - 1)) - (16*a^3)/(5*d*(6*exp(4 *c + 4*d*x) - 4*exp(2*c + 2*d*x) - 4*exp(6*c + 6*d*x) + exp(8*c + 8*d*x) + 1)) + (3*atan((a^2*b*exp(d*x)*exp(c)*(-d^2)^(1/2))/(d*(a^4*b^2)^(1/2)))*( a^4*b^2)^(1/2))/(-d^2)^(1/2) + 3*a*b^2*x - (3*a^2*b*exp(c + d*x))/(d*(exp( 2*c + 2*d*x) - 1)) - (18*a^2*b*exp(c + d*x))/(5*d*(exp(4*c + 4*d*x) - 2*ex p(2*c + 2*d*x) + 1))