Integrand size = 23, antiderivative size = 147 \[ \int \text {csch}^{12}(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx=b^3 x+\frac {a \left (a^2+3 a b+3 b^2\right ) \coth (c+d x)}{d}-\frac {a \left (5 a^2+9 a b+3 b^2\right ) \coth ^3(c+d x)}{3 d}+\frac {a^2 (10 a+9 b) \coth ^5(c+d x)}{5 d}-\frac {a^2 (10 a+3 b) \coth ^7(c+d x)}{7 d}+\frac {5 a^3 \coth ^9(c+d x)}{9 d}-\frac {a^3 \coth ^{11}(c+d x)}{11 d} \]
b^3*x+a*(a^2+3*a*b+3*b^2)*coth(d*x+c)/d-1/3*a*(5*a^2+9*a*b+3*b^2)*coth(d*x +c)^3/d+1/5*a^2*(10*a+9*b)*coth(d*x+c)^5/d-1/7*a^2*(10*a+3*b)*coth(d*x+c)^ 7/d+5/9*a^3*coth(d*x+c)^9/d-1/11*a^3*coth(d*x+c)^11/d
Time = 6.29 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.63 \[ \int \text {csch}^{12}(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx=\frac {b^3 (c+d x)}{d}+\frac {2 \left (640 a^3 \cosh (c+d x)+2376 a^2 b \cosh (c+d x)+3465 a b^2 \cosh (c+d x)\right ) \text {csch}(c+d x)}{3465 d}+\frac {\left (-640 a^3 \cosh (c+d x)-2376 a^2 b \cosh (c+d x)-3465 a b^2 \cosh (c+d x)\right ) \text {csch}^3(c+d x)}{3465 d}+\frac {2 \left (80 a^3 \cosh (c+d x)+297 a^2 b \cosh (c+d x)\right ) \text {csch}^5(c+d x)}{1155 d}+\frac {\left (-80 a^3 \cosh (c+d x)-297 a^2 b \cosh (c+d x)\right ) \text {csch}^7(c+d x)}{693 d}+\frac {10 a^3 \coth (c+d x) \text {csch}^8(c+d x)}{99 d}-\frac {a^3 \coth (c+d x) \text {csch}^{10}(c+d x)}{11 d} \]
(b^3*(c + d*x))/d + (2*(640*a^3*Cosh[c + d*x] + 2376*a^2*b*Cosh[c + d*x] + 3465*a*b^2*Cosh[c + d*x])*Csch[c + d*x])/(3465*d) + ((-640*a^3*Cosh[c + d *x] - 2376*a^2*b*Cosh[c + d*x] - 3465*a*b^2*Cosh[c + d*x])*Csch[c + d*x]^3 )/(3465*d) + (2*(80*a^3*Cosh[c + d*x] + 297*a^2*b*Cosh[c + d*x])*Csch[c + d*x]^5)/(1155*d) + ((-80*a^3*Cosh[c + d*x] - 297*a^2*b*Cosh[c + d*x])*Csch [c + d*x]^7)/(693*d) + (10*a^3*Coth[c + d*x]*Csch[c + d*x]^8)/(99*d) - (a^ 3*Coth[c + d*x]*Csch[c + d*x]^10)/(11*d)
Time = 0.37 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.95, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3042, 3696, 1584, 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.
| \(\displaystyle \int \text {csch}^{12}(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \sin (i c+i d x)^4\right )^3}{\sin (i c+i d x)^{12}}dx\) |
| \(\Big \downarrow \) 3696 |
| \(\displaystyle \frac {\int \frac {\coth ^{12}(c+d x) \left ((a+b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )^3}{1-\tanh ^2(c+d x)}d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 1584 |
| \(\displaystyle \frac {\int \left (a^3 \coth ^{12}(c+d x)-5 a^3 \coth ^{10}(c+d x)+a^2 (10 a+3 b) \coth ^8(c+d x)-a^2 (10 a+9 b) \coth ^6(c+d x)+a \left (5 a^2+9 b a+3 b^2\right ) \coth ^4(c+d x)-a \left (a^2+3 b a+3 b^2\right ) \coth ^2(c+d x)-\frac {b^3}{\tanh ^2(c+d x)-1}\right )d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {1}{11} a^3 \coth ^{11}(c+d x)+\frac {5}{9} a^3 \coth ^9(c+d x)-\frac {1}{3} a \left (5 a^2+9 a b+3 b^2\right ) \coth ^3(c+d x)+a \left (a^2+3 a b+3 b^2\right ) \coth (c+d x)-\frac {1}{7} a^2 (10 a+3 b) \coth ^7(c+d x)+\frac {1}{5} a^2 (10 a+9 b) \coth ^5(c+d x)+b^3 \text {arctanh}(\tanh (c+d x))}{d}\) |
(b^3*ArcTanh[Tanh[c + d*x]] + a*(a^2 + 3*a*b + 3*b^2)*Coth[c + d*x] - (a*( 5*a^2 + 9*a*b + 3*b^2)*Coth[c + d*x]^3)/3 + (a^2*(10*a + 9*b)*Coth[c + d*x ]^5)/5 - (a^2*(10*a + 3*b)*Coth[c + d*x]^7)/7 + (5*a^3*Coth[c + d*x]^9)/9 - (a^3*Coth[c + d*x]^11)/11)/d
3.3.24.3.1 Defintions of rubi rules used
Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + ( c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q* (a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && NeQ[ b^2 - 4*a*c, 0] && IGtQ[p, 0] && IGtQ[q, -2]
Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^( p_.), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[ff^(m + 1 )/f Subst[Int[x^m*((a + 2*a*ff^2*x^2 + (a + b)*ff^4*x^4)^p/(1 + ff^2*x^2) ^(m/2 + 2*p + 1)), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] & & IntegerQ[m/2] && IntegerQ[p]
Time = 1.16 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.99
| method | result | size |
| derivativedivides | \(\frac {a^{3} \left (\frac {256}{693}-\frac {\operatorname {csch}\left (d x +c \right )^{10}}{11}+\frac {10 \operatorname {csch}\left (d x +c \right )^{8}}{99}-\frac {80 \operatorname {csch}\left (d x +c \right )^{6}}{693}+\frac {32 \operatorname {csch}\left (d x +c \right )^{4}}{231}-\frac {128 \operatorname {csch}\left (d x +c \right )^{2}}{693}\right ) \coth \left (d x +c \right )+3 a^{2} b \left (\frac {16}{35}-\frac {\operatorname {csch}\left (d x +c \right )^{6}}{7}+\frac {6 \operatorname {csch}\left (d x +c \right )^{4}}{35}-\frac {8 \operatorname {csch}\left (d x +c \right )^{2}}{35}\right ) \coth \left (d x +c \right )+3 a \,b^{2} \left (\frac {2}{3}-\frac {\operatorname {csch}\left (d x +c \right )^{2}}{3}\right ) \coth \left (d x +c \right )+b^{3} \left (d x +c \right )}{d}\) | \(145\) |
| default | \(\frac {a^{3} \left (\frac {256}{693}-\frac {\operatorname {csch}\left (d x +c \right )^{10}}{11}+\frac {10 \operatorname {csch}\left (d x +c \right )^{8}}{99}-\frac {80 \operatorname {csch}\left (d x +c \right )^{6}}{693}+\frac {32 \operatorname {csch}\left (d x +c \right )^{4}}{231}-\frac {128 \operatorname {csch}\left (d x +c \right )^{2}}{693}\right ) \coth \left (d x +c \right )+3 a^{2} b \left (\frac {16}{35}-\frac {\operatorname {csch}\left (d x +c \right )^{6}}{7}+\frac {6 \operatorname {csch}\left (d x +c \right )^{4}}{35}-\frac {8 \operatorname {csch}\left (d x +c \right )^{2}}{35}\right ) \coth \left (d x +c \right )+3 a \,b^{2} \left (\frac {2}{3}-\frac {\operatorname {csch}\left (d x +c \right )^{2}}{3}\right ) \coth \left (d x +c \right )+b^{3} \left (d x +c \right )}{d}\) | \(145\) |
| parallelrisch | \(\frac {-\left (\cosh \left (9 d x +9 c \right )-\frac {\cosh \left (11 d x +11 c \right )}{11}+42 \cosh \left (d x +c \right )-30 \cosh \left (3 d x +3 c \right )+15 \cosh \left (5 d x +5 c \right )-5 \cosh \left (7 d x +7 c \right )\right ) \operatorname {sech}\left (\frac {d x}{2}+\frac {c}{2}\right )^{11} a^{3} \operatorname {csch}\left (\frac {d x}{2}+\frac {c}{2}\right )^{11}-3024 \left (\cosh \left (d x +c \right )-\frac {3 \cosh \left (3 d x +3 c \right )}{5}+\frac {\cosh \left (5 d x +5 c \right )}{5}-\frac {\cosh \left (7 d x +7 c \right )}{35}\right ) \operatorname {sech}\left (\frac {d x}{2}+\frac {c}{2}\right )^{7} a^{2} b \operatorname {csch}\left (\frac {d x}{2}+\frac {c}{2}\right )^{7}-96768 \operatorname {sech}\left (\frac {d x}{2}+\frac {c}{2}\right )^{3} a \,b^{2} \left (\cosh \left (d x +c \right )-\frac {\cosh \left (3 d x +3 c \right )}{3}\right ) \operatorname {csch}\left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+516096 x \,b^{3} d}{516096 d}\) | \(217\) |
| risch | \(b^{3} x -\frac {4 a \left (10395 b^{2} {\mathrm e}^{18 d x +18 c}-86625 b^{2} {\mathrm e}^{16 d x +16 c}+83160 a b \,{\mathrm e}^{14 d x +14 c}+318780 b^{2} {\mathrm e}^{14 d x +14 c}-382536 a b \,{\mathrm e}^{12 d x +12 c}-679140 b^{2} {\mathrm e}^{12 d x +12 c}+295680 a^{2} {\mathrm e}^{10 d x +10 c}+715176 a b \,{\mathrm e}^{10 d x +10 c}+921690 b^{2} {\mathrm e}^{10 d x +10 c}-211200 \,{\mathrm e}^{8 d x +8 c} a^{2}-700920 \,{\mathrm e}^{8 d x +8 c} a b -824670 b^{2} {\mathrm e}^{8 d x +8 c}+105600 a^{2} {\mathrm e}^{6 d x +6 c}+392040 \,{\mathrm e}^{6 d x +6 c} a b +485100 b^{2} {\mathrm e}^{6 d x +6 c}-35200 \,{\mathrm e}^{4 d x +4 c} a^{2}-130680 \,{\mathrm e}^{4 d x +4 c} a b -180180 b^{2} {\mathrm e}^{4 d x +4 c}+7040 \,{\mathrm e}^{2 d x +2 c} a^{2}+26136 \,{\mathrm e}^{2 d x +2 c} b a +38115 b^{2} {\mathrm e}^{2 d x +2 c}-640 a^{2}-2376 a b -3465 b^{2}\right )}{3465 d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{11}}\) | \(328\) |
1/d*(a^3*(256/693-1/11*csch(d*x+c)^10+10/99*csch(d*x+c)^8-80/693*csch(d*x+ c)^6+32/231*csch(d*x+c)^4-128/693*csch(d*x+c)^2)*coth(d*x+c)+3*a^2*b*(16/3 5-1/7*csch(d*x+c)^6+6/35*csch(d*x+c)^4-8/35*csch(d*x+c)^2)*coth(d*x+c)+3*a *b^2*(2/3-1/3*csch(d*x+c)^2)*coth(d*x+c)+b^3*(d*x+c))
Leaf count of result is larger than twice the leaf count of optimal. 1607 vs. \(2 (137) = 274\).
Time = 0.28 (sec) , antiderivative size = 1607, normalized size of antiderivative = 10.93 \[ \int \text {csch}^{12}(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx=\text {Too large to display} \]
1/3465*(2*(640*a^3 + 2376*a^2*b + 3465*a*b^2)*cosh(d*x + c)^11 + 22*(640*a ^3 + 2376*a^2*b + 3465*a*b^2)*cosh(d*x + c)*sinh(d*x + c)^10 + (3465*b^3*d *x - 1280*a^3 - 4752*a^2*b - 6930*a*b^2)*sinh(d*x + c)^11 - 22*(640*a^3 + 2376*a^2*b + 3465*a*b^2)*cosh(d*x + c)^9 - 11*(3465*b^3*d*x - 1280*a^3 - 4 752*a^2*b - 6930*a*b^2 - 5*(3465*b^3*d*x - 1280*a^3 - 4752*a^2*b - 6930*a* b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^9 + 66*(5*(640*a^3 + 2376*a^2*b + 3465 *a*b^2)*cosh(d*x + c)^3 - 3*(640*a^3 + 2376*a^2*b + 3465*a*b^2)*cosh(d*x + c))*sinh(d*x + c)^8 + 110*(640*a^3 + 2376*a^2*b + 3087*a*b^2)*cosh(d*x + c)^7 + 11*(17325*b^3*d*x + 30*(3465*b^3*d*x - 1280*a^3 - 4752*a^2*b - 6930 *a*b^2)*cosh(d*x + c)^4 - 6400*a^3 - 23760*a^2*b - 34650*a*b^2 - 36*(3465* b^3*d*x - 1280*a^3 - 4752*a^2*b - 6930*a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^7 + 154*(6*(640*a^3 + 2376*a^2*b + 3465*a*b^2)*cosh(d*x + c)^5 - 12*(64 0*a^3 + 2376*a^2*b + 3465*a*b^2)*cosh(d*x + c)^3 + 5*(640*a^3 + 2376*a^2*b + 3087*a*b^2)*cosh(d*x + c))*sinh(d*x + c)^6 - 330*(640*a^3 + 2376*a^2*b + 2415*a*b^2)*cosh(d*x + c)^5 + 33*(14*(3465*b^3*d*x - 1280*a^3 - 4752*a^2 *b - 6930*a*b^2)*cosh(d*x + c)^6 - 17325*b^3*d*x - 42*(3465*b^3*d*x - 1280 *a^3 - 4752*a^2*b - 6930*a*b^2)*cosh(d*x + c)^4 + 6400*a^3 + 23760*a^2*b + 34650*a*b^2 + 35*(3465*b^3*d*x - 1280*a^3 - 4752*a^2*b - 6930*a*b^2)*cosh (d*x + c)^2)*sinh(d*x + c)^5 + 22*(30*(640*a^3 + 2376*a^2*b + 3465*a*b^2)* cosh(d*x + c)^7 - 126*(640*a^3 + 2376*a^2*b + 3465*a*b^2)*cosh(d*x + c)...
Timed out. \[ \int \text {csch}^{12}(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 1291 vs. \(2 (137) = 274\).
Time = 0.20 (sec) , antiderivative size = 1291, normalized size of antiderivative = 8.78 \[ \int \text {csch}^{12}(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx=\text {Too large to display} \]
b^3*x + 512/693*a^3*(11*e^(-2*d*x - 2*c)/(d*(11*e^(-2*d*x - 2*c) - 55*e^(- 4*d*x - 4*c) + 165*e^(-6*d*x - 6*c) - 330*e^(-8*d*x - 8*c) + 462*e^(-10*d* x - 10*c) - 462*e^(-12*d*x - 12*c) + 330*e^(-14*d*x - 14*c) - 165*e^(-16*d *x - 16*c) + 55*e^(-18*d*x - 18*c) - 11*e^(-20*d*x - 20*c) + e^(-22*d*x - 22*c) - 1)) - 55*e^(-4*d*x - 4*c)/(d*(11*e^(-2*d*x - 2*c) - 55*e^(-4*d*x - 4*c) + 165*e^(-6*d*x - 6*c) - 330*e^(-8*d*x - 8*c) + 462*e^(-10*d*x - 10* c) - 462*e^(-12*d*x - 12*c) + 330*e^(-14*d*x - 14*c) - 165*e^(-16*d*x - 16 *c) + 55*e^(-18*d*x - 18*c) - 11*e^(-20*d*x - 20*c) + e^(-22*d*x - 22*c) - 1)) + 165*e^(-6*d*x - 6*c)/(d*(11*e^(-2*d*x - 2*c) - 55*e^(-4*d*x - 4*c) + 165*e^(-6*d*x - 6*c) - 330*e^(-8*d*x - 8*c) + 462*e^(-10*d*x - 10*c) - 4 62*e^(-12*d*x - 12*c) + 330*e^(-14*d*x - 14*c) - 165*e^(-16*d*x - 16*c) + 55*e^(-18*d*x - 18*c) - 11*e^(-20*d*x - 20*c) + e^(-22*d*x - 22*c) - 1)) - 330*e^(-8*d*x - 8*c)/(d*(11*e^(-2*d*x - 2*c) - 55*e^(-4*d*x - 4*c) + 165* e^(-6*d*x - 6*c) - 330*e^(-8*d*x - 8*c) + 462*e^(-10*d*x - 10*c) - 462*e^( -12*d*x - 12*c) + 330*e^(-14*d*x - 14*c) - 165*e^(-16*d*x - 16*c) + 55*e^( -18*d*x - 18*c) - 11*e^(-20*d*x - 20*c) + e^(-22*d*x - 22*c) - 1)) + 462*e ^(-10*d*x - 10*c)/(d*(11*e^(-2*d*x - 2*c) - 55*e^(-4*d*x - 4*c) + 165*e^(- 6*d*x - 6*c) - 330*e^(-8*d*x - 8*c) + 462*e^(-10*d*x - 10*c) - 462*e^(-12* d*x - 12*c) + 330*e^(-14*d*x - 14*c) - 165*e^(-16*d*x - 16*c) + 55*e^(-18* d*x - 18*c) - 11*e^(-20*d*x - 20*c) + e^(-22*d*x - 22*c) - 1)) - 1/(d*(...
Leaf count of result is larger than twice the leaf count of optimal. 359 vs. \(2 (137) = 274\).
Time = 0.51 (sec) , antiderivative size = 359, normalized size of antiderivative = 2.44 \[ \int \text {csch}^{12}(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx=\frac {3465 \, {\left (d x + c\right )} b^{3} - \frac {4 \, {\left (10395 \, a b^{2} e^{\left (18 \, d x + 18 \, c\right )} - 86625 \, a b^{2} e^{\left (16 \, d x + 16 \, c\right )} + 83160 \, a^{2} b e^{\left (14 \, d x + 14 \, c\right )} + 318780 \, a b^{2} e^{\left (14 \, d x + 14 \, c\right )} - 382536 \, a^{2} b e^{\left (12 \, d x + 12 \, c\right )} - 679140 \, a b^{2} e^{\left (12 \, d x + 12 \, c\right )} + 295680 \, a^{3} e^{\left (10 \, d x + 10 \, c\right )} + 715176 \, a^{2} b e^{\left (10 \, d x + 10 \, c\right )} + 921690 \, a b^{2} e^{\left (10 \, d x + 10 \, c\right )} - 211200 \, a^{3} e^{\left (8 \, d x + 8 \, c\right )} - 700920 \, a^{2} b e^{\left (8 \, d x + 8 \, c\right )} - 824670 \, a b^{2} e^{\left (8 \, d x + 8 \, c\right )} + 105600 \, a^{3} e^{\left (6 \, d x + 6 \, c\right )} + 392040 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} + 485100 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} - 35200 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} - 130680 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} - 180180 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 7040 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} + 26136 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} + 38115 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 640 \, a^{3} - 2376 \, a^{2} b - 3465 \, a b^{2}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{11}}}{3465 \, d} \]
1/3465*(3465*(d*x + c)*b^3 - 4*(10395*a*b^2*e^(18*d*x + 18*c) - 86625*a*b^ 2*e^(16*d*x + 16*c) + 83160*a^2*b*e^(14*d*x + 14*c) + 318780*a*b^2*e^(14*d *x + 14*c) - 382536*a^2*b*e^(12*d*x + 12*c) - 679140*a*b^2*e^(12*d*x + 12* c) + 295680*a^3*e^(10*d*x + 10*c) + 715176*a^2*b*e^(10*d*x + 10*c) + 92169 0*a*b^2*e^(10*d*x + 10*c) - 211200*a^3*e^(8*d*x + 8*c) - 700920*a^2*b*e^(8 *d*x + 8*c) - 824670*a*b^2*e^(8*d*x + 8*c) + 105600*a^3*e^(6*d*x + 6*c) + 392040*a^2*b*e^(6*d*x + 6*c) + 485100*a*b^2*e^(6*d*x + 6*c) - 35200*a^3*e^ (4*d*x + 4*c) - 130680*a^2*b*e^(4*d*x + 4*c) - 180180*a*b^2*e^(4*d*x + 4*c ) + 7040*a^3*e^(2*d*x + 2*c) + 26136*a^2*b*e^(2*d*x + 2*c) + 38115*a*b^2*e ^(2*d*x + 2*c) - 640*a^3 - 2376*a^2*b - 3465*a*b^2)/(e^(2*d*x + 2*c) - 1)^ 11)/d
Time = 1.66 (sec) , antiderivative size = 1955, normalized size of antiderivative = 13.30 \[ \int \text {csch}^{12}(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx=\text {Too large to display} \]
((64*a*b^2)/(165*d) - (32*exp(2*c + 2*d*x)*(7*a*b^2 + 4*a^2*b))/(55*d) + ( 128*exp(4*c + 4*d*x)*(7*a*b^2 + 8*a^2*b))/(55*d) + (64*exp(8*c + 8*d*x)*(7 *a*b^2 + 8*a^2*b))/(11*d) - (224*exp(10*c + 10*d*x)*(7*a*b^2 + 4*a^2*b))/( 55*d) - (32*exp(6*c + 6*d*x)*(105*a*b^2 + 144*a^2*b + 128*a^3))/(99*d) + ( 1792*a*b^2*exp(12*c + 12*d*x))/(165*d) - (96*a*b^2*exp(14*c + 14*d*x))/(55 *d))/(9*exp(2*c + 2*d*x) - 36*exp(4*c + 4*d*x) + 84*exp(6*c + 6*d*x) - 126 *exp(8*c + 8*d*x) + 126*exp(10*c + 10*d*x) - 84*exp(12*c + 12*d*x) + 36*ex p(14*c + 14*d*x) - 9*exp(16*c + 16*d*x) + exp(18*c + 18*d*x) - 1) - ((4*(1 05*a*b^2 + 144*a^2*b + 128*a^3))/(693*d) - (32*exp(2*c + 2*d*x)*(7*a*b^2 + 8*a^2*b))/(77*d) + (8*exp(4*c + 4*d*x)*(7*a*b^2 + 4*a^2*b))/(11*d) - (128 *a*b^2*exp(6*c + 6*d*x))/(33*d) + (12*a*b^2*exp(8*c + 8*d*x))/(11*d))/(15* exp(4*c + 4*d*x) - 6*exp(2*c + 2*d*x) - 20*exp(6*c + 6*d*x) + 15*exp(8*c + 8*d*x) - 6*exp(10*c + 10*d*x) + exp(12*c + 12*d*x) + 1) - ((4*(7*a*b^2 + 4*a^2*b))/(55*d) - (32*exp(2*c + 2*d*x)*(7*a*b^2 + 8*a^2*b))/(55*d) - (32* exp(6*c + 6*d*x)*(7*a*b^2 + 8*a^2*b))/(11*d) + (28*exp(8*c + 8*d*x)*(7*a*b ^2 + 4*a^2*b))/(11*d) + (4*exp(4*c + 4*d*x)*(105*a*b^2 + 144*a^2*b + 128*a ^3))/(33*d) - (448*a*b^2*exp(10*c + 10*d*x))/(55*d) + (84*a*b^2*exp(12*c + 12*d*x))/(55*d))/(28*exp(4*c + 4*d*x) - 8*exp(2*c + 2*d*x) - 56*exp(6*c + 6*d*x) + 70*exp(8*c + 8*d*x) - 56*exp(10*c + 10*d*x) + 28*exp(12*c + 12*d *x) - 8*exp(14*c + 14*d*x) + exp(16*c + 16*d*x) + 1) - ((4*(7*a*b^2 + 4...