Integrand size = 21, antiderivative size = 201 \[ \int \text {csch}(c+d x) \left (a+b \sinh ^3(c+d x)\right )^3 \, dx=-\frac {3}{2} a^2 b x+\frac {35 b^3 x}{128}-\frac {a^3 \text {arctanh}(\cosh (c+d x))}{d}+\frac {3 a b^2 \cosh (c+d x)}{d}-\frac {2 a b^2 \cosh ^3(c+d x)}{d}+\frac {3 a b^2 \cosh ^5(c+d x)}{5 d}+\frac {3 a^2 b \cosh (c+d x) \sinh (c+d x)}{2 d}-\frac {35 b^3 \cosh (c+d x) \sinh (c+d x)}{128 d}+\frac {35 b^3 \cosh (c+d x) \sinh ^3(c+d x)}{192 d}-\frac {7 b^3 \cosh (c+d x) \sinh ^5(c+d x)}{48 d}+\frac {b^3 \cosh (c+d x) \sinh ^7(c+d x)}{8 d} \]
-3/2*a^2*b*x+35/128*b^3*x-a^3*arctanh(cosh(d*x+c))/d+3*a*b^2*cosh(d*x+c)/d -2*a*b^2*cosh(d*x+c)^3/d+3/5*a*b^2*cosh(d*x+c)^5/d+3/2*a^2*b*cosh(d*x+c)*s inh(d*x+c)/d-35/128*b^3*cosh(d*x+c)*sinh(d*x+c)/d+35/192*b^3*cosh(d*x+c)*s inh(d*x+c)^3/d-7/48*b^3*cosh(d*x+c)*sinh(d*x+c)^5/d+1/8*b^3*cosh(d*x+c)*si nh(d*x+c)^7/d
Time = 5.79 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.87 \[ \int \text {csch}(c+d x) \left (a+b \sinh ^3(c+d x)\right )^3 \, dx=\frac {-23040 a^2 b c+4200 b^3 c-23040 a^2 b d x+4200 b^3 d x+28800 a b^2 \cosh (c+d x)-4800 a b^2 \cosh (3 (c+d x))+576 a b^2 \cosh (5 (c+d x))-15360 a^3 \log \left (\cosh \left (\frac {1}{2} (c+d x)\right )\right )+15360 a^3 \log \left (\sinh \left (\frac {1}{2} (c+d x)\right )\right )+11520 a^2 b \sinh (2 (c+d x))-3360 b^3 \sinh (2 (c+d x))+840 b^3 \sinh (4 (c+d x))-160 b^3 \sinh (6 (c+d x))+15 b^3 \sinh (8 (c+d x))}{15360 d} \]
(-23040*a^2*b*c + 4200*b^3*c - 23040*a^2*b*d*x + 4200*b^3*d*x + 28800*a*b^ 2*Cosh[c + d*x] - 4800*a*b^2*Cosh[3*(c + d*x)] + 576*a*b^2*Cosh[5*(c + d*x )] - 15360*a^3*Log[Cosh[(c + d*x)/2]] + 15360*a^3*Log[Sinh[(c + d*x)/2]] + 11520*a^2*b*Sinh[2*(c + d*x)] - 3360*b^3*Sinh[2*(c + d*x)] + 840*b^3*Sinh [4*(c + d*x)] - 160*b^3*Sinh[6*(c + d*x)] + 15*b^3*Sinh[8*(c + d*x)])/(153 60*d)
Result contains complex when optimal does not.
Time = 0.42 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.13, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3042, 26, 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.
| \(\displaystyle \int \text {csch}(c+d x) \left (a+b \sinh ^3(c+d x)\right )^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {i \left (a+i b \sin (i c+i d x)^3\right )^3}{\sin (i c+i d x)}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \int \frac {\left (i b \sin (i c+i d x)^3+a\right )^3}{\sin (i c+i d x)}dx\) |
| \(\Big \downarrow \) 3699 |
| \(\displaystyle i \int \left (-i b^3 \sinh ^8(c+d x)-3 i a b^2 \sinh ^5(c+d x)-3 i a^2 b \sinh ^2(c+d x)-i a^3 \text {csch}(c+d x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle i \left (\frac {i a^3 \text {arctanh}(\cosh (c+d x))}{d}-\frac {3 i a^2 b \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {3}{2} i a^2 b x-\frac {3 i a b^2 \cosh ^5(c+d x)}{5 d}+\frac {2 i a b^2 \cosh ^3(c+d x)}{d}-\frac {3 i a b^2 \cosh (c+d x)}{d}-\frac {i b^3 \sinh ^7(c+d x) \cosh (c+d x)}{8 d}+\frac {7 i b^3 \sinh ^5(c+d x) \cosh (c+d x)}{48 d}-\frac {35 i b^3 \sinh ^3(c+d x) \cosh (c+d x)}{192 d}+\frac {35 i b^3 \sinh (c+d x) \cosh (c+d x)}{128 d}-\frac {35}{128} i b^3 x\right )\) |
I*(((3*I)/2)*a^2*b*x - ((35*I)/128)*b^3*x + (I*a^3*ArcTanh[Cosh[c + d*x]]) /d - ((3*I)*a*b^2*Cosh[c + d*x])/d + ((2*I)*a*b^2*Cosh[c + d*x]^3)/d - ((( 3*I)/5)*a*b^2*Cosh[c + d*x]^5)/d - (((3*I)/2)*a^2*b*Cosh[c + d*x]*Sinh[c + d*x])/d + (((35*I)/128)*b^3*Cosh[c + d*x]*Sinh[c + d*x])/d - (((35*I)/192 )*b^3*Cosh[c + d*x]*Sinh[c + d*x]^3)/d + (((7*I)/48)*b^3*Cosh[c + d*x]*Sin h[c + d*x]^5)/d - ((I/8)*b^3*Cosh[c + d*x]*Sinh[c + d*x]^7)/d)
3.2.64.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
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 = 1.92 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.69
| method | result | size |
| derivativedivides | \(\frac {-2 a^{3} \operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )+3 a^{2} b \left (\frac {\sinh \left (d x +c \right ) \cosh \left (d x +c \right )}{2}-\frac {d x}{2}-\frac {c}{2}\right )+3 a \,b^{2} \left (\frac {8}{15}+\frac {\sinh \left (d x +c \right )^{4}}{5}-\frac {4 \sinh \left (d x +c \right )^{2}}{15}\right ) \cosh \left (d x +c \right )+b^{3} \left (\left (\frac {\sinh \left (d x +c \right )^{7}}{8}-\frac {7 \sinh \left (d x +c \right )^{5}}{48}+\frac {35 \sinh \left (d x +c \right )^{3}}{192}-\frac {35 \sinh \left (d x +c \right )}{128}\right ) \cosh \left (d x +c \right )+\frac {35 d x}{128}+\frac {35 c}{128}\right )}{d}\) | \(138\) |
| default | \(\frac {-2 a^{3} \operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )+3 a^{2} b \left (\frac {\sinh \left (d x +c \right ) \cosh \left (d x +c \right )}{2}-\frac {d x}{2}-\frac {c}{2}\right )+3 a \,b^{2} \left (\frac {8}{15}+\frac {\sinh \left (d x +c \right )^{4}}{5}-\frac {4 \sinh \left (d x +c \right )^{2}}{15}\right ) \cosh \left (d x +c \right )+b^{3} \left (\left (\frac {\sinh \left (d x +c \right )^{7}}{8}-\frac {7 \sinh \left (d x +c \right )^{5}}{48}+\frac {35 \sinh \left (d x +c \right )^{3}}{192}-\frac {35 \sinh \left (d x +c \right )}{128}\right ) \cosh \left (d x +c \right )+\frac {35 d x}{128}+\frac {35 c}{128}\right )}{d}\) | \(138\) |
| parallelrisch | \(\frac {-23040 x \,a^{2} b d +4200 x \,b^{3} d +576 a \,b^{2} \cosh \left (5 d x +5 c \right )-4800 a \,b^{2} \cosh \left (3 d x +3 c \right )+11520 a^{2} b \sinh \left (2 d x +2 c \right )-3360 b^{3} \sinh \left (2 d x +2 c \right )-160 b^{3} \sinh \left (6 d x +6 c \right )+15 b^{3} \sinh \left (8 d x +8 c \right )+840 b^{3} \sinh \left (4 d x +4 c \right )+28800 a \,b^{2} \cosh \left (d x +c \right )+15360 a^{3} \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+24576 a \,b^{2}}{15360 d}\) | \(156\) |
| risch | \(-\frac {3 a^{2} b x}{2}+\frac {35 b^{3} x}{128}+\frac {b^{3} {\mathrm e}^{8 d x +8 c}}{2048 d}-\frac {b^{3} {\mathrm e}^{6 d x +6 c}}{192 d}+\frac {3 b^{2} {\mathrm e}^{5 d x +5 c} a}{160 d}+\frac {7 \,{\mathrm e}^{4 d x +4 c} b^{3}}{256 d}-\frac {5 \,{\mathrm e}^{3 d x +3 c} a \,b^{2}}{32 d}+\frac {3 \,{\mathrm e}^{2 d x +2 c} a^{2} b}{8 d}-\frac {7 \,{\mathrm e}^{2 d x +2 c} b^{3}}{64 d}+\frac {15 \,{\mathrm e}^{d x +c} a \,b^{2}}{16 d}+\frac {15 \,{\mathrm e}^{-d x -c} a \,b^{2}}{16 d}-\frac {3 \,{\mathrm e}^{-2 d x -2 c} a^{2} b}{8 d}+\frac {7 \,{\mathrm e}^{-2 d x -2 c} b^{3}}{64 d}-\frac {5 \,{\mathrm e}^{-3 d x -3 c} a \,b^{2}}{32 d}-\frac {7 \,{\mathrm e}^{-4 d x -4 c} b^{3}}{256 d}+\frac {3 b^{2} {\mathrm e}^{-5 d x -5 c} a}{160 d}+\frac {b^{3} {\mathrm e}^{-6 d x -6 c}}{192 d}-\frac {b^{3} {\mathrm e}^{-8 d x -8 c}}{2048 d}+\frac {a^{3} \ln \left ({\mathrm e}^{d x +c}-1\right )}{d}-\frac {a^{3} \ln \left ({\mathrm e}^{d x +c}+1\right )}{d}\) | \(325\) |
1/d*(-2*a^3*arctanh(exp(d*x+c))+3*a^2*b*(1/2*sinh(d*x+c)*cosh(d*x+c)-1/2*d *x-1/2*c)+3*a*b^2*(8/15+1/5*sinh(d*x+c)^4-4/15*sinh(d*x+c)^2)*cosh(d*x+c)+ b^3*((1/8*sinh(d*x+c)^7-7/48*sinh(d*x+c)^5+35/192*sinh(d*x+c)^3-35/128*sin h(d*x+c))*cosh(d*x+c)+35/128*d*x+35/128*c))
Leaf count of result is larger than twice the leaf count of optimal. 2609 vs. \(2 (185) = 370\).
Time = 0.30 (sec) , antiderivative size = 2609, normalized size of antiderivative = 12.98 \[ \int \text {csch}(c+d x) \left (a+b \sinh ^3(c+d x)\right )^3 \, dx=\text {Too large to display} \]
1/30720*(15*b^3*cosh(d*x + c)^16 + 240*b^3*cosh(d*x + c)*sinh(d*x + c)^15 + 15*b^3*sinh(d*x + c)^16 - 160*b^3*cosh(d*x + c)^14 + 576*a*b^2*cosh(d*x + c)^13 + 840*b^3*cosh(d*x + c)^12 + 40*(45*b^3*cosh(d*x + c)^2 - 4*b^3)*s inh(d*x + c)^14 - 4800*a*b^2*cosh(d*x + c)^11 + 16*(525*b^3*cosh(d*x + c)^ 3 - 140*b^3*cosh(d*x + c) + 36*a*b^2)*sinh(d*x + c)^13 + 4*(6825*b^3*cosh( d*x + c)^4 - 3640*b^3*cosh(d*x + c)^2 + 1872*a*b^2*cosh(d*x + c) + 210*b^3 )*sinh(d*x + c)^12 + 28800*a*b^2*cosh(d*x + c)^9 + 16*(4095*b^3*cosh(d*x + c)^5 - 3640*b^3*cosh(d*x + c)^3 + 2808*a*b^2*cosh(d*x + c)^2 + 630*b^3*co sh(d*x + c) - 300*a*b^2)*sinh(d*x + c)^11 - 240*(192*a^2*b - 35*b^3)*d*x*c osh(d*x + c)^8 + 480*(24*a^2*b - 7*b^3)*cosh(d*x + c)^10 + 8*(15015*b^3*co sh(d*x + c)^6 - 20020*b^3*cosh(d*x + c)^4 + 20592*a*b^2*cosh(d*x + c)^3 + 6930*b^3*cosh(d*x + c)^2 - 6600*a*b^2*cosh(d*x + c) + 1440*a^2*b - 420*b^3 )*sinh(d*x + c)^10 + 28800*a*b^2*cosh(d*x + c)^7 + 80*(2145*b^3*cosh(d*x + c)^7 - 4004*b^3*cosh(d*x + c)^5 + 5148*a*b^2*cosh(d*x + c)^4 + 2310*b^3*c osh(d*x + c)^3 - 3300*a*b^2*cosh(d*x + c)^2 + 360*a*b^2 + 60*(24*a^2*b - 7 *b^3)*cosh(d*x + c))*sinh(d*x + c)^9 + 6*(32175*b^3*cosh(d*x + c)^8 - 8008 0*b^3*cosh(d*x + c)^6 + 123552*a*b^2*cosh(d*x + c)^5 + 69300*b^3*cosh(d*x + c)^4 - 132000*a*b^2*cosh(d*x + c)^3 + 43200*a*b^2*cosh(d*x + c) - 40*(19 2*a^2*b - 35*b^3)*d*x + 3600*(24*a^2*b - 7*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^8 - 4800*a*b^2*cosh(d*x + c)^5 + 48*(3575*b^3*cosh(d*x + c)^9 - 11...
Timed out. \[ \int \text {csch}(c+d x) \left (a+b \sinh ^3(c+d x)\right )^3 \, dx=\text {Timed out} \]
Time = 0.21 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.28 \[ \int \text {csch}(c+d x) \left (a+b \sinh ^3(c+d x)\right )^3 \, dx=-\frac {3}{8} \, a^{2} b {\left (4 \, x - \frac {e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac {e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right )} - \frac {1}{6144} \, b^{3} {\left (\frac {{\left (32 \, e^{\left (-2 \, d x - 2 \, c\right )} - 168 \, e^{\left (-4 \, d x - 4 \, c\right )} + 672 \, e^{\left (-6 \, d x - 6 \, c\right )} - 3\right )} e^{\left (8 \, d x + 8 \, c\right )}}{d} - \frac {1680 \, {\left (d x + c\right )}}{d} - \frac {672 \, e^{\left (-2 \, d x - 2 \, c\right )} - 168 \, e^{\left (-4 \, d x - 4 \, c\right )} + 32 \, e^{\left (-6 \, d x - 6 \, c\right )} - 3 \, e^{\left (-8 \, d x - 8 \, c\right )}}{d}\right )} + \frac {1}{160} \, a b^{2} {\left (\frac {3 \, e^{\left (5 \, d x + 5 \, c\right )}}{d} - \frac {25 \, e^{\left (3 \, d x + 3 \, c\right )}}{d} + \frac {150 \, e^{\left (d x + c\right )}}{d} + \frac {150 \, e^{\left (-d x - c\right )}}{d} - \frac {25 \, e^{\left (-3 \, d x - 3 \, c\right )}}{d} + \frac {3 \, e^{\left (-5 \, d x - 5 \, c\right )}}{d}\right )} + \frac {a^{3} \log \left (\tanh \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{d} \]
-3/8*a^2*b*(4*x - e^(2*d*x + 2*c)/d + e^(-2*d*x - 2*c)/d) - 1/6144*b^3*((3 2*e^(-2*d*x - 2*c) - 168*e^(-4*d*x - 4*c) + 672*e^(-6*d*x - 6*c) - 3)*e^(8 *d*x + 8*c)/d - 1680*(d*x + c)/d - (672*e^(-2*d*x - 2*c) - 168*e^(-4*d*x - 4*c) + 32*e^(-6*d*x - 6*c) - 3*e^(-8*d*x - 8*c))/d) + 1/160*a*b^2*(3*e^(5 *d*x + 5*c)/d - 25*e^(3*d*x + 3*c)/d + 150*e^(d*x + c)/d + 150*e^(-d*x - c )/d - 25*e^(-3*d*x - 3*c)/d + 3*e^(-5*d*x - 5*c)/d) + a^3*log(tanh(1/2*d*x + 1/2*c))/d
Time = 0.40 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.39 \[ \int \text {csch}(c+d x) \left (a+b \sinh ^3(c+d x)\right )^3 \, dx=\frac {15 \, b^{3} e^{\left (8 \, d x + 8 \, c\right )} - 160 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 576 \, a b^{2} e^{\left (5 \, d x + 5 \, c\right )} + 840 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} - 4800 \, a b^{2} e^{\left (3 \, d x + 3 \, c\right )} + 11520 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} - 3360 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 28800 \, a b^{2} e^{\left (d x + c\right )} - 30720 \, a^{3} \log \left (e^{\left (d x + c\right )} + 1\right ) + 30720 \, a^{3} \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right ) - 240 \, {\left (192 \, a^{2} b - 35 \, b^{3}\right )} {\left (d x + c\right )} + {\left (28800 \, a b^{2} e^{\left (7 \, d x + 7 \, c\right )} - 4800 \, a b^{2} e^{\left (5 \, d x + 5 \, c\right )} - 840 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 576 \, a b^{2} e^{\left (3 \, d x + 3 \, c\right )} + 160 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} - 15 \, b^{3} - 480 \, {\left (24 \, a^{2} b - 7 \, b^{3}\right )} e^{\left (6 \, d x + 6 \, c\right )}\right )} e^{\left (-8 \, d x - 8 \, c\right )}}{30720 \, d} \]
1/30720*(15*b^3*e^(8*d*x + 8*c) - 160*b^3*e^(6*d*x + 6*c) + 576*a*b^2*e^(5 *d*x + 5*c) + 840*b^3*e^(4*d*x + 4*c) - 4800*a*b^2*e^(3*d*x + 3*c) + 11520 *a^2*b*e^(2*d*x + 2*c) - 3360*b^3*e^(2*d*x + 2*c) + 28800*a*b^2*e^(d*x + c ) - 30720*a^3*log(e^(d*x + c) + 1) + 30720*a^3*log(abs(e^(d*x + c) - 1)) - 240*(192*a^2*b - 35*b^3)*(d*x + c) + (28800*a*b^2*e^(7*d*x + 7*c) - 4800* a*b^2*e^(5*d*x + 5*c) - 840*b^3*e^(4*d*x + 4*c) + 576*a*b^2*e^(3*d*x + 3*c ) + 160*b^3*e^(2*d*x + 2*c) - 15*b^3 - 480*(24*a^2*b - 7*b^3)*e^(6*d*x + 6 *c))*e^(-8*d*x - 8*c))/d
Time = 0.54 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.57 \[ \int \text {csch}(c+d x) \left (a+b \sinh ^3(c+d x)\right )^3 \, dx=\frac {7\,b^3\,{\mathrm {e}}^{4\,c+4\,d\,x}}{256\,d}-\frac {2\,\mathrm {atan}\left (\frac {a^3\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {-d^2}}{d\,\sqrt {a^6}}\right )\,\sqrt {a^6}}{\sqrt {-d^2}}-\frac {7\,b^3\,{\mathrm {e}}^{-4\,c-4\,d\,x}}{256\,d}-x\,\left (\frac {3\,a^2\,b}{2}-\frac {35\,b^3}{128}\right )+\frac {b^3\,{\mathrm {e}}^{-6\,c-6\,d\,x}}{192\,d}-\frac {b^3\,{\mathrm {e}}^{6\,c+6\,d\,x}}{192\,d}-\frac {b^3\,{\mathrm {e}}^{-8\,c-8\,d\,x}}{2048\,d}+\frac {b^3\,{\mathrm {e}}^{8\,c+8\,d\,x}}{2048\,d}-\frac {{\mathrm {e}}^{-2\,c-2\,d\,x}\,\left (24\,a^2\,b-7\,b^3\right )}{64\,d}+\frac {{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (24\,a^2\,b-7\,b^3\right )}{64\,d}+\frac {15\,a\,b^2\,{\mathrm {e}}^{-c-d\,x}}{16\,d}-\frac {5\,a\,b^2\,{\mathrm {e}}^{-3\,c-3\,d\,x}}{32\,d}-\frac {5\,a\,b^2\,{\mathrm {e}}^{3\,c+3\,d\,x}}{32\,d}+\frac {3\,a\,b^2\,{\mathrm {e}}^{-5\,c-5\,d\,x}}{160\,d}+\frac {3\,a\,b^2\,{\mathrm {e}}^{5\,c+5\,d\,x}}{160\,d}+\frac {15\,a\,b^2\,{\mathrm {e}}^{c+d\,x}}{16\,d} \]
(7*b^3*exp(4*c + 4*d*x))/(256*d) - (2*atan((a^3*exp(d*x)*exp(c)*(-d^2)^(1/ 2))/(d*(a^6)^(1/2)))*(a^6)^(1/2))/(-d^2)^(1/2) - (7*b^3*exp(- 4*c - 4*d*x) )/(256*d) - x*((3*a^2*b)/2 - (35*b^3)/128) + (b^3*exp(- 6*c - 6*d*x))/(192 *d) - (b^3*exp(6*c + 6*d*x))/(192*d) - (b^3*exp(- 8*c - 8*d*x))/(2048*d) + (b^3*exp(8*c + 8*d*x))/(2048*d) - (exp(- 2*c - 2*d*x)*(24*a^2*b - 7*b^3)) /(64*d) + (exp(2*c + 2*d*x)*(24*a^2*b - 7*b^3))/(64*d) + (15*a*b^2*exp(- c - d*x))/(16*d) - (5*a*b^2*exp(- 3*c - 3*d*x))/(32*d) - (5*a*b^2*exp(3*c + 3*d*x))/(32*d) + (3*a*b^2*exp(- 5*c - 5*d*x))/(160*d) + (3*a*b^2*exp(5*c + 5*d*x))/(160*d) + (15*a*b^2*exp(c + d*x))/(16*d)