Integrand size = 23, antiderivative size = 77 \[ \int \coth ^7(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx=-\frac {3 a^2 (a+b) \text {csch}^2(c+d x)}{2 d}-\frac {3 a (a+b)^2 \text {csch}^4(c+d x)}{4 d}-\frac {(a+b)^3 \text {csch}^6(c+d x)}{6 d}+\frac {a^3 \log (\sinh (c+d x))}{d} \]
-3/2*a^2*(a+b)*csch(d*x+c)^2/d-3/4*a*(a+b)^2*csch(d*x+c)^4/d-1/6*(a+b)^3*c sch(d*x+c)^6/d+a^3*ln(sinh(d*x+c))/d
Time = 0.85 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.27 \[ \int \coth ^7(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx=-\frac {2 \left (b+a \cosh ^2(c+d x)\right )^3 \left (18 a^2 (a+b) \text {csch}^2(c+d x)+9 a (a+b)^2 \text {csch}^4(c+d x)+2 (a+b)^3 \text {csch}^6(c+d x)-12 a^3 \log (\sinh (c+d x))\right )}{3 d (a+2 b+a \cosh (2 (c+d x)))^3} \]
(-2*(b + a*Cosh[c + d*x]^2)^3*(18*a^2*(a + b)*Csch[c + d*x]^2 + 9*a*(a + b )^2*Csch[c + d*x]^4 + 2*(a + b)^3*Csch[c + d*x]^6 - 12*a^3*Log[Sinh[c + d* x]]))/(3*d*(a + 2*b + a*Cosh[2*(c + d*x)])^3)
Time = 0.32 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.22, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3042, 26, 4626, 353, 49, 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.
| \(\displaystyle \int \coth ^7(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {i \left (a+b \sec (i c+i d x)^2\right )^3}{\tan (i c+i d x)^7}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \int \frac {\left (b \sec (i c+i d x)^2+a\right )^3}{\tan (i c+i d x)^7}dx\) |
| \(\Big \downarrow \) 4626 |
| \(\displaystyle \frac {\int \frac {\cosh (c+d x) \left (a \cosh ^2(c+d x)+b\right )^3}{\left (1-\cosh ^2(c+d x)\right )^4}d\cosh (c+d x)}{d}\) |
| \(\Big \downarrow \) 353 |
| \(\displaystyle \frac {\int \frac {\left (a \cosh ^2(c+d x)+b\right )^3}{\left (1-\cosh ^2(c+d x)\right )^4}d\cosh ^2(c+d x)}{2 d}\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \frac {\int \left (\frac {a^3}{\cosh ^2(c+d x)-1}+\frac {3 (a+b) a^2}{\left (\cosh ^2(c+d x)-1\right )^2}+\frac {3 (a+b)^2 a}{\left (\cosh ^2(c+d x)-1\right )^3}+\frac {(a+b)^3}{\left (\cosh ^2(c+d x)-1\right )^4}\right )d\cosh ^2(c+d x)}{2 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^3 \log \left (1-\cosh ^2(c+d x)\right )+\frac {3 a^2 (a+b)}{1-\cosh ^2(c+d x)}-\frac {3 a (a+b)^2}{2 \left (1-\cosh ^2(c+d x)\right )^2}+\frac {(a+b)^3}{3 \left (1-\cosh ^2(c+d x)\right )^3}}{2 d}\) |
((a + b)^3/(3*(1 - Cosh[c + d*x]^2)^3) - (3*a*(a + b)^2)/(2*(1 - Cosh[c + d*x]^2)^2) + (3*a^2*(a + b))/(1 - Cosh[c + d*x]^2) + a^3*Log[1 - Cosh[c + d*x]^2])/(2*d)
3.2.35.3.1 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[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_Symbol] :> Simp[1/2 Subst[Int[(a + b*x)^p*(c + d*x)^q, x], x, x^2], x] /; FreeQ[ {a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0]
Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*tan[(e_.) + (f_.)*(x_ )]^(m_.), x_Symbol] :> Module[{ff = FreeFactors[Cos[e + f*x], x]}, Simp[-(f *ff^(m + n*p - 1))^(-1) Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*((b + a*(ff* x)^n)^p/x^(m + n*p)), x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, n} , x] && IntegerQ[(m - 1)/2] && IntegerQ[n] && IntegerQ[p]
Leaf count of result is larger than twice the leaf count of optimal. \(148\) vs. \(2(71)=142\).
Time = 173.23 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.94
| method | result | size |
| derivativedivides | \(\frac {a^{3} \left (\ln \left (\sinh \left (d x +c \right )\right )-\frac {\coth \left (d x +c \right )^{2}}{2}-\frac {\coth \left (d x +c \right )^{4}}{4}-\frac {\coth \left (d x +c \right )^{6}}{6}\right )+3 a^{2} b \left (-\frac {\cosh \left (d x +c \right )^{4}}{2 \sinh \left (d x +c \right )^{6}}+\frac {\cosh \left (d x +c \right )^{2}}{2 \sinh \left (d x +c \right )^{6}}-\frac {1}{6 \sinh \left (d x +c \right )^{6}}\right )+3 a \,b^{2} \left (-\frac {\cosh \left (d x +c \right )^{2}}{4 \sinh \left (d x +c \right )^{6}}+\frac {1}{12 \sinh \left (d x +c \right )^{6}}\right )-\frac {b^{3}}{6 \sinh \left (d x +c \right )^{6}}}{d}\) | \(149\) |
| default | \(\frac {a^{3} \left (\ln \left (\sinh \left (d x +c \right )\right )-\frac {\coth \left (d x +c \right )^{2}}{2}-\frac {\coth \left (d x +c \right )^{4}}{4}-\frac {\coth \left (d x +c \right )^{6}}{6}\right )+3 a^{2} b \left (-\frac {\cosh \left (d x +c \right )^{4}}{2 \sinh \left (d x +c \right )^{6}}+\frac {\cosh \left (d x +c \right )^{2}}{2 \sinh \left (d x +c \right )^{6}}-\frac {1}{6 \sinh \left (d x +c \right )^{6}}\right )+3 a \,b^{2} \left (-\frac {\cosh \left (d x +c \right )^{2}}{4 \sinh \left (d x +c \right )^{6}}+\frac {1}{12 \sinh \left (d x +c \right )^{6}}\right )-\frac {b^{3}}{6 \sinh \left (d x +c \right )^{6}}}{d}\) | \(149\) |
| risch | \(-a^{3} x -\frac {2 a^{3} c}{d}-\frac {2 \,{\mathrm e}^{2 d x +2 c} \left (9 a^{3} {\mathrm e}^{8 d x +8 c}+9 a^{2} b \,{\mathrm e}^{8 d x +8 c}-18 a^{3} {\mathrm e}^{6 d x +6 c}+18 a \,b^{2} {\mathrm e}^{6 d x +6 c}+34 a^{3} {\mathrm e}^{4 d x +4 c}+30 a^{2} b \,{\mathrm e}^{4 d x +4 c}+12 a \,b^{2} {\mathrm e}^{4 d x +4 c}+16 \,{\mathrm e}^{4 d x +4 c} b^{3}-18 a^{3} {\mathrm e}^{2 d x +2 c}+18 \,{\mathrm e}^{2 d x +2 c} a \,b^{2}+9 a^{3}+9 a^{2} b \right )}{3 d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{6}}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}-1\right ) a^{3}}{d}\) | \(220\) |
1/d*(a^3*(ln(sinh(d*x+c))-1/2*coth(d*x+c)^2-1/4*coth(d*x+c)^4-1/6*coth(d*x +c)^6)+3*a^2*b*(-1/2/sinh(d*x+c)^6*cosh(d*x+c)^4+1/2/sinh(d*x+c)^6*cosh(d* x+c)^2-1/6/sinh(d*x+c)^6)+3*a*b^2*(-1/4/sinh(d*x+c)^6*cosh(d*x+c)^2+1/12/s inh(d*x+c)^6)-1/6*b^3/sinh(d*x+c)^6)
Leaf count of result is larger than twice the leaf count of optimal. 2632 vs. \(2 (71) = 142\).
Time = 0.28 (sec) , antiderivative size = 2632, normalized size of antiderivative = 34.18 \[ \int \coth ^7(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx=\text {Too large to display} \]
-1/3*(3*a^3*d*x*cosh(d*x + c)^12 + 36*a^3*d*x*cosh(d*x + c)*sinh(d*x + c)^ 11 + 3*a^3*d*x*sinh(d*x + c)^12 - 18*(a^3*d*x - a^3 - a^2*b)*cosh(d*x + c) ^10 + 18*(11*a^3*d*x*cosh(d*x + c)^2 - a^3*d*x + a^3 + a^2*b)*sinh(d*x + c )^10 + 60*(11*a^3*d*x*cosh(d*x + c)^3 - 3*(a^3*d*x - a^3 - a^2*b)*cosh(d*x + c))*sinh(d*x + c)^9 + 9*(5*a^3*d*x - 4*a^3 + 4*a*b^2)*cosh(d*x + c)^8 + 9*(165*a^3*d*x*cosh(d*x + c)^4 + 5*a^3*d*x - 4*a^3 + 4*a*b^2 - 90*(a^3*d* x - a^3 - a^2*b)*cosh(d*x + c)^2)*sinh(d*x + c)^8 + 72*(33*a^3*d*x*cosh(d* x + c)^5 - 30*(a^3*d*x - a^3 - a^2*b)*cosh(d*x + c)^3 + (5*a^3*d*x - 4*a^3 + 4*a*b^2)*cosh(d*x + c))*sinh(d*x + c)^7 - 4*(15*a^3*d*x - 17*a^3 - 15*a ^2*b - 6*a*b^2 - 8*b^3)*cosh(d*x + c)^6 + 4*(693*a^3*d*x*cosh(d*x + c)^6 - 15*a^3*d*x - 945*(a^3*d*x - a^3 - a^2*b)*cosh(d*x + c)^4 + 17*a^3 + 15*a^ 2*b + 6*a*b^2 + 8*b^3 + 63*(5*a^3*d*x - 4*a^3 + 4*a*b^2)*cosh(d*x + c)^2)* sinh(d*x + c)^6 + 24*(99*a^3*d*x*cosh(d*x + c)^7 - 189*(a^3*d*x - a^3 - a^ 2*b)*cosh(d*x + c)^5 + 21*(5*a^3*d*x - 4*a^3 + 4*a*b^2)*cosh(d*x + c)^3 - (15*a^3*d*x - 17*a^3 - 15*a^2*b - 6*a*b^2 - 8*b^3)*cosh(d*x + c))*sinh(d*x + c)^5 + 3*a^3*d*x + 9*(5*a^3*d*x - 4*a^3 + 4*a*b^2)*cosh(d*x + c)^4 + 3* (495*a^3*d*x*cosh(d*x + c)^8 - 1260*(a^3*d*x - a^3 - a^2*b)*cosh(d*x + c)^ 6 + 15*a^3*d*x + 210*(5*a^3*d*x - 4*a^3 + 4*a*b^2)*cosh(d*x + c)^4 - 12*a^ 3 + 12*a*b^2 - 20*(15*a^3*d*x - 17*a^3 - 15*a^2*b - 6*a*b^2 - 8*b^3)*cosh( d*x + c)^2)*sinh(d*x + c)^4 + 4*(165*a^3*d*x*cosh(d*x + c)^9 - 540*(a^3...
Timed out. \[ \int \coth ^7(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 727 vs. \(2 (71) = 142\).
Time = 0.21 (sec) , antiderivative size = 727, normalized size of antiderivative = 9.44 \[ \int \coth ^7(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx=\frac {1}{3} \, a^{3} {\left (3 \, x + \frac {3 \, c}{d} + \frac {3 \, \log \left (e^{\left (-d x - c\right )} + 1\right )}{d} + \frac {3 \, \log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {2 \, {\left (9 \, e^{\left (-2 \, d x - 2 \, c\right )} - 18 \, e^{\left (-4 \, d x - 4 \, c\right )} + 34 \, e^{\left (-6 \, d x - 6 \, c\right )} - 18 \, e^{\left (-8 \, d x - 8 \, c\right )} + 9 \, e^{\left (-10 \, d x - 10 \, c\right )}\right )}}{d {\left (6 \, e^{\left (-2 \, d x - 2 \, c\right )} - 15 \, e^{\left (-4 \, d x - 4 \, c\right )} + 20 \, e^{\left (-6 \, d x - 6 \, c\right )} - 15 \, e^{\left (-8 \, d x - 8 \, c\right )} + 6 \, e^{\left (-10 \, d x - 10 \, c\right )} - e^{\left (-12 \, d x - 12 \, c\right )} - 1\right )}}\right )} + 2 \, a^{2} b {\left (\frac {3 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (6 \, e^{\left (-2 \, d x - 2 \, c\right )} - 15 \, e^{\left (-4 \, d x - 4 \, c\right )} + 20 \, e^{\left (-6 \, d x - 6 \, c\right )} - 15 \, e^{\left (-8 \, d x - 8 \, c\right )} + 6 \, e^{\left (-10 \, d x - 10 \, c\right )} - e^{\left (-12 \, d x - 12 \, c\right )} - 1\right )}} + \frac {10 \, e^{\left (-6 \, d x - 6 \, c\right )}}{d {\left (6 \, e^{\left (-2 \, d x - 2 \, c\right )} - 15 \, e^{\left (-4 \, d x - 4 \, c\right )} + 20 \, e^{\left (-6 \, d x - 6 \, c\right )} - 15 \, e^{\left (-8 \, d x - 8 \, c\right )} + 6 \, e^{\left (-10 \, d x - 10 \, c\right )} - e^{\left (-12 \, d x - 12 \, c\right )} - 1\right )}} + \frac {3 \, e^{\left (-10 \, d x - 10 \, c\right )}}{d {\left (6 \, e^{\left (-2 \, d x - 2 \, c\right )} - 15 \, e^{\left (-4 \, d x - 4 \, c\right )} + 20 \, e^{\left (-6 \, d x - 6 \, c\right )} - 15 \, e^{\left (-8 \, d x - 8 \, c\right )} + 6 \, e^{\left (-10 \, d x - 10 \, c\right )} - e^{\left (-12 \, d x - 12 \, c\right )} - 1\right )}}\right )} + 4 \, a b^{2} {\left (\frac {3 \, e^{\left (-4 \, d x - 4 \, c\right )}}{d {\left (6 \, e^{\left (-2 \, d x - 2 \, c\right )} - 15 \, e^{\left (-4 \, d x - 4 \, c\right )} + 20 \, e^{\left (-6 \, d x - 6 \, c\right )} - 15 \, e^{\left (-8 \, d x - 8 \, c\right )} + 6 \, e^{\left (-10 \, d x - 10 \, c\right )} - e^{\left (-12 \, d x - 12 \, c\right )} - 1\right )}} + \frac {2 \, e^{\left (-6 \, d x - 6 \, c\right )}}{d {\left (6 \, e^{\left (-2 \, d x - 2 \, c\right )} - 15 \, e^{\left (-4 \, d x - 4 \, c\right )} + 20 \, e^{\left (-6 \, d x - 6 \, c\right )} - 15 \, e^{\left (-8 \, d x - 8 \, c\right )} + 6 \, e^{\left (-10 \, d x - 10 \, c\right )} - e^{\left (-12 \, d x - 12 \, c\right )} - 1\right )}} + \frac {3 \, e^{\left (-8 \, d x - 8 \, c\right )}}{d {\left (6 \, e^{\left (-2 \, d x - 2 \, c\right )} - 15 \, e^{\left (-4 \, d x - 4 \, c\right )} + 20 \, e^{\left (-6 \, d x - 6 \, c\right )} - 15 \, e^{\left (-8 \, d x - 8 \, c\right )} + 6 \, e^{\left (-10 \, d x - 10 \, c\right )} - e^{\left (-12 \, d x - 12 \, c\right )} - 1\right )}}\right )} - \frac {32 \, b^{3}}{3 \, d {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{6}} \]
1/3*a^3*(3*x + 3*c/d + 3*log(e^(-d*x - c) + 1)/d + 3*log(e^(-d*x - c) - 1) /d + 2*(9*e^(-2*d*x - 2*c) - 18*e^(-4*d*x - 4*c) + 34*e^(-6*d*x - 6*c) - 1 8*e^(-8*d*x - 8*c) + 9*e^(-10*d*x - 10*c))/(d*(6*e^(-2*d*x - 2*c) - 15*e^( -4*d*x - 4*c) + 20*e^(-6*d*x - 6*c) - 15*e^(-8*d*x - 8*c) + 6*e^(-10*d*x - 10*c) - e^(-12*d*x - 12*c) - 1))) + 2*a^2*b*(3*e^(-2*d*x - 2*c)/(d*(6*e^( -2*d*x - 2*c) - 15*e^(-4*d*x - 4*c) + 20*e^(-6*d*x - 6*c) - 15*e^(-8*d*x - 8*c) + 6*e^(-10*d*x - 10*c) - e^(-12*d*x - 12*c) - 1)) + 10*e^(-6*d*x - 6 *c)/(d*(6*e^(-2*d*x - 2*c) - 15*e^(-4*d*x - 4*c) + 20*e^(-6*d*x - 6*c) - 1 5*e^(-8*d*x - 8*c) + 6*e^(-10*d*x - 10*c) - e^(-12*d*x - 12*c) - 1)) + 3*e ^(-10*d*x - 10*c)/(d*(6*e^(-2*d*x - 2*c) - 15*e^(-4*d*x - 4*c) + 20*e^(-6* d*x - 6*c) - 15*e^(-8*d*x - 8*c) + 6*e^(-10*d*x - 10*c) - e^(-12*d*x - 12* c) - 1))) + 4*a*b^2*(3*e^(-4*d*x - 4*c)/(d*(6*e^(-2*d*x - 2*c) - 15*e^(-4* d*x - 4*c) + 20*e^(-6*d*x - 6*c) - 15*e^(-8*d*x - 8*c) + 6*e^(-10*d*x - 10 *c) - e^(-12*d*x - 12*c) - 1)) + 2*e^(-6*d*x - 6*c)/(d*(6*e^(-2*d*x - 2*c) - 15*e^(-4*d*x - 4*c) + 20*e^(-6*d*x - 6*c) - 15*e^(-8*d*x - 8*c) + 6*e^( -10*d*x - 10*c) - e^(-12*d*x - 12*c) - 1)) + 3*e^(-8*d*x - 8*c)/(d*(6*e^(- 2*d*x - 2*c) - 15*e^(-4*d*x - 4*c) + 20*e^(-6*d*x - 6*c) - 15*e^(-8*d*x - 8*c) + 6*e^(-10*d*x - 10*c) - e^(-12*d*x - 12*c) - 1))) - 32/3*b^3/(d*(e^( d*x + c) - e^(-d*x - c))^6)
Leaf count of result is larger than twice the leaf count of optimal. 242 vs. \(2 (71) = 142\).
Time = 0.44 (sec) , antiderivative size = 242, normalized size of antiderivative = 3.14 \[ \int \coth ^7(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx=-\frac {60 \, {\left (d x + c\right )} a^{3} - 60 \, a^{3} \log \left ({\left | e^{\left (2 \, d x + 2 \, c\right )} - 1 \right |}\right ) + \frac {147 \, a^{3} e^{\left (12 \, d x + 12 \, c\right )} - 522 \, a^{3} e^{\left (10 \, d x + 10 \, c\right )} + 360 \, a^{2} b e^{\left (10 \, d x + 10 \, c\right )} + 1485 \, a^{3} e^{\left (8 \, d x + 8 \, c\right )} + 720 \, a b^{2} e^{\left (8 \, d x + 8 \, c\right )} - 1580 \, a^{3} e^{\left (6 \, d x + 6 \, c\right )} + 1200 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} + 480 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 640 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 1485 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} + 720 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 522 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} + 360 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} + 147 \, a^{3}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{6}}}{60 \, d} \]
-1/60*(60*(d*x + c)*a^3 - 60*a^3*log(abs(e^(2*d*x + 2*c) - 1)) + (147*a^3* e^(12*d*x + 12*c) - 522*a^3*e^(10*d*x + 10*c) + 360*a^2*b*e^(10*d*x + 10*c ) + 1485*a^3*e^(8*d*x + 8*c) + 720*a*b^2*e^(8*d*x + 8*c) - 1580*a^3*e^(6*d *x + 6*c) + 1200*a^2*b*e^(6*d*x + 6*c) + 480*a*b^2*e^(6*d*x + 6*c) + 640*b ^3*e^(6*d*x + 6*c) + 1485*a^3*e^(4*d*x + 4*c) + 720*a*b^2*e^(4*d*x + 4*c) - 522*a^3*e^(2*d*x + 2*c) + 360*a^2*b*e^(2*d*x + 2*c) + 147*a^3)/(e^(2*d*x + 2*c) - 1)^6)/d
Time = 2.32 (sec) , antiderivative size = 411, normalized size of antiderivative = 5.34 \[ \int \coth ^7(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx=\frac {a^3\,\ln \left ({\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}-1\right )}{d}-\frac {32\,\left (a^3+3\,a^2\,b+3\,a\,b^2+b^3\right )}{d\,\left (5\,{\mathrm {e}}^{2\,c+2\,d\,x}-10\,{\mathrm {e}}^{4\,c+4\,d\,x}+10\,{\mathrm {e}}^{6\,c+6\,d\,x}-5\,{\mathrm {e}}^{8\,c+8\,d\,x}+{\mathrm {e}}^{10\,c+10\,d\,x}-1\right )}-\frac {32\,\left (a^3+3\,a^2\,b+3\,a\,b^2+b^3\right )}{3\,d\,\left (15\,{\mathrm {e}}^{4\,c+4\,d\,x}-6\,{\mathrm {e}}^{2\,c+2\,d\,x}-20\,{\mathrm {e}}^{6\,c+6\,d\,x}+15\,{\mathrm {e}}^{8\,c+8\,d\,x}-6\,{\mathrm {e}}^{10\,c+10\,d\,x}+{\mathrm {e}}^{12\,c+12\,d\,x}+1\right )}-\frac {6\,\left (a^3+b\,a^2\right )}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}-\frac {6\,\left (3\,a^3+5\,a^2\,b+2\,a\,b^2\right )}{d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {8\,\left (13\,a^3+30\,a^2\,b+21\,a\,b^2+4\,b^3\right )}{3\,d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}-3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}-1\right )}-\frac {4\,\left (11\,a^3+30\,a^2\,b+27\,a\,b^2+8\,b^3\right )}{d\,\left (6\,{\mathrm {e}}^{4\,c+4\,d\,x}-4\,{\mathrm {e}}^{2\,c+2\,d\,x}-4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1\right )}-a^3\,x \]
(a^3*log(exp(2*c)*exp(2*d*x) - 1))/d - (32*(3*a*b^2 + 3*a^2*b + a^3 + b^3) )/(d*(5*exp(2*c + 2*d*x) - 10*exp(4*c + 4*d*x) + 10*exp(6*c + 6*d*x) - 5*e xp(8*c + 8*d*x) + exp(10*c + 10*d*x) - 1)) - (32*(3*a*b^2 + 3*a^2*b + a^3 + b^3))/(3*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)) - (6*(a^2*b + a^3))/(d*(exp(2*c + 2*d*x) - 1)) - (6*(2*a*b^2 + 5*a^2*b + 3*a^3))/(d*(exp(4*c + 4*d*x) - 2*exp(2*c + 2*d*x) + 1)) - (8*(21*a*b^2 + 3 0*a^2*b + 13*a^3 + 4*b^3))/(3*d*(3*exp(2*c + 2*d*x) - 3*exp(4*c + 4*d*x) + exp(6*c + 6*d*x) - 1)) - (4*(27*a*b^2 + 30*a^2*b + 11*a^3 + 8*b^3))/(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)) - a^3*x