Integrand size = 23, antiderivative size = 71 \[ \int \text {csch}^2(c+d x) \left (a+b \tanh ^3(c+d x)\right )^3 \, dx=-\frac {a^3 \coth (c+d x)}{d}+\frac {3 a^2 b \tanh ^2(c+d x)}{2 d}+\frac {3 a b^2 \tanh ^5(c+d x)}{5 d}+\frac {b^3 \tanh ^8(c+d x)}{8 d} \]
-a^3*coth(d*x+c)/d+3/2*a^2*b*tanh(d*x+c)^2/d+3/5*a*b^2*tanh(d*x+c)^5/d+1/8 *b^3*tanh(d*x+c)^8/d
Time = 2.39 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.59 \[ \int \text {csch}^2(c+d x) \left (a+b \tanh ^3(c+d x)\right )^3 \, dx=\frac {-40 a^3 \coth (c+d x)+b \left (-20 b^2 \text {sech}^6(c+d x)+5 b^2 \text {sech}^8(c+d x)+24 a b \tanh (c+d x)+6 b \text {sech}^4(c+d x) (5 b+4 a \tanh (c+d x))-4 \text {sech}^2(c+d x) \left (15 a^2+5 b^2+12 a b \tanh (c+d x)\right )\right )}{40 d} \]
(-40*a^3*Coth[c + d*x] + b*(-20*b^2*Sech[c + d*x]^6 + 5*b^2*Sech[c + d*x]^ 8 + 24*a*b*Tanh[c + d*x] + 6*b*Sech[c + d*x]^4*(5*b + 4*a*Tanh[c + d*x]) - 4*Sech[c + d*x]^2*(15*a^2 + 5*b^2 + 12*a*b*Tanh[c + d*x])))/(40*d)
Time = 0.27 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.89, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3042, 25, 4146, 802, 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.
| \(\displaystyle \int \text {csch}^2(c+d x) \left (a+b \tanh ^3(c+d x)\right )^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {\left (a+i b \tan (i c+i d x)^3\right )^3}{\sin (i c+i d x)^2}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {\left (i b \tan (i c+i d x)^3+a\right )^3}{\sin (i c+i d x)^2}dx\) |
| \(\Big \downarrow \) 4146 |
| \(\displaystyle \frac {\int \coth ^2(c+d x) \left (b \tanh ^3(c+d x)+a\right )^3d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 802 |
| \(\displaystyle \frac {\int \left (b^3 \tanh ^7(c+d x)+3 a b^2 \tanh ^4(c+d x)+3 a^2 b \tanh (c+d x)+a^3 \coth ^2(c+d x)\right )d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-a^3 \coth (c+d x)+\frac {3}{2} a^2 b \tanh ^2(c+d x)+\frac {3}{5} a b^2 \tanh ^5(c+d x)+\frac {1}{8} b^3 \tanh ^8(c+d x)}{d}\) |
(-(a^3*Coth[c + d*x]) + (3*a^2*b*Tanh[c + d*x]^2)/2 + (3*a*b^2*Tanh[c + d* x]^5)/5 + (b^3*Tanh[c + d*x]^8)/8)/d
3.1.70.3.1 Defintions of rubi rules used
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[Exp andIntegrand[(c*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]
Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_ )])^(n_))^(p_.), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Sim p[c*(ff^(m + 1)/f) Subst[Int[x^m*((a + b*(ff*x)^n)^p/(c^2 + ff^2*x^2)^(m/ 2 + 1)), x], x, c*(Tan[e + f*x]/ff)], x]] /; FreeQ[{a, b, c, e, f, n, p}, x ] && IntegerQ[m/2]
Leaf count of result is larger than twice the leaf count of optimal. \(170\) vs. \(2(65)=130\).
Time = 20.42 (sec) , antiderivative size = 171, normalized size of antiderivative = 2.41
| method | result | size |
| derivativedivides | \(\frac {-a^{3} \coth \left (d x +c \right )-\frac {3 a^{2} b}{2 \cosh \left (d x +c \right )^{2}}+3 a \,b^{2} \left (-\frac {\sinh \left (d x +c \right )^{3}}{2 \cosh \left (d x +c \right )^{5}}-\frac {3 \sinh \left (d x +c \right )}{8 \cosh \left (d x +c \right )^{5}}+\frac {3 \left (\frac {8}{15}+\frac {\operatorname {sech}\left (d x +c \right )^{4}}{5}+\frac {4 \operatorname {sech}\left (d x +c \right )^{2}}{15}\right ) \tanh \left (d x +c \right )}{8}\right )+b^{3} \left (-\frac {\sinh \left (d x +c \right )^{6}}{2 \cosh \left (d x +c \right )^{8}}-\frac {3 \sinh \left (d x +c \right )^{4}}{4 \cosh \left (d x +c \right )^{8}}-\frac {\sinh \left (d x +c \right )^{2}}{2 \cosh \left (d x +c \right )^{8}}-\frac {1}{8 \cosh \left (d x +c \right )^{8}}\right )}{d}\) | \(171\) |
| default | \(\frac {-a^{3} \coth \left (d x +c \right )-\frac {3 a^{2} b}{2 \cosh \left (d x +c \right )^{2}}+3 a \,b^{2} \left (-\frac {\sinh \left (d x +c \right )^{3}}{2 \cosh \left (d x +c \right )^{5}}-\frac {3 \sinh \left (d x +c \right )}{8 \cosh \left (d x +c \right )^{5}}+\frac {3 \left (\frac {8}{15}+\frac {\operatorname {sech}\left (d x +c \right )^{4}}{5}+\frac {4 \operatorname {sech}\left (d x +c \right )^{2}}{15}\right ) \tanh \left (d x +c \right )}{8}\right )+b^{3} \left (-\frac {\sinh \left (d x +c \right )^{6}}{2 \cosh \left (d x +c \right )^{8}}-\frac {3 \sinh \left (d x +c \right )^{4}}{4 \cosh \left (d x +c \right )^{8}}-\frac {\sinh \left (d x +c \right )^{2}}{2 \cosh \left (d x +c \right )^{8}}-\frac {1}{8 \cosh \left (d x +c \right )^{8}}\right )}{d}\) | \(171\) |
| risch | \(-\frac {2 \left (-3 a \,b^{2}+5 a^{3} {\mathrm e}^{16 d x +16 c}+5 b^{3} {\mathrm e}^{16 d x +16 c}+35 b^{3} {\mathrm e}^{12 d x +12 c}+5 a^{3}+5 \,{\mathrm e}^{4 d x +4 c} b^{3}+40 a^{3} {\mathrm e}^{14 d x +14 c}-5 b^{3} {\mathrm e}^{14 d x +14 c}-6 \,{\mathrm e}^{2 d x +2 c} a \,b^{2}-15 a^{2} b \,{\mathrm e}^{2 d x +2 c}-30 a \,b^{2} {\mathrm e}^{4 d x +4 c}-75 a^{2} b \,{\mathrm e}^{4 d x +4 c}-135 a^{2} b \,{\mathrm e}^{6 d x +6 c}-54 a \,b^{2} {\mathrm e}^{6 d x +6 c}+140 a^{3} {\mathrm e}^{12 d x +12 c}-75 a^{2} b \,{\mathrm e}^{8 d x +8 c}-12 a \,b^{2} {\mathrm e}^{8 d x +8 c}+15 a^{2} b \,{\mathrm e}^{16 d x +16 c}+15 a \,b^{2} {\mathrm e}^{16 d x +16 c}+75 a^{2} b \,{\mathrm e}^{14 d x +14 c}+30 a \,b^{2} {\mathrm e}^{14 d x +14 c}+75 a^{2} b \,{\mathrm e}^{10 d x +10 c}-35 b^{3} {\mathrm e}^{10 d x +10 c}+280 a^{3} {\mathrm e}^{10 d x +10 c}+280 a^{3} {\mathrm e}^{6 d x +6 c}-35 \,{\mathrm e}^{6 d x +6 c} b^{3}+140 a^{3} {\mathrm e}^{4 d x +4 c}+30 a \,b^{2} {\mathrm e}^{12 d x +12 c}+135 a^{2} b \,{\mathrm e}^{12 d x +12 c}+40 a^{3} {\mathrm e}^{2 d x +2 c}-5 \,{\mathrm e}^{2 d x +2 c} b^{3}+35 b^{3} {\mathrm e}^{8 d x +8 c}+350 a^{3} {\mathrm e}^{8 d x +8 c}+30 a \,b^{2} {\mathrm e}^{10 d x +10 c}\right )}{5 d \left ({\mathrm e}^{2 d x +2 c}-1\right ) \left ({\mathrm e}^{2 d x +2 c}+1\right )^{8}}\) | \(508\) |
1/d*(-a^3*coth(d*x+c)-3/2*a^2*b/cosh(d*x+c)^2+3*a*b^2*(-1/2*sinh(d*x+c)^3/ cosh(d*x+c)^5-3/8*sinh(d*x+c)/cosh(d*x+c)^5+3/8*(8/15+1/5*sech(d*x+c)^4+4/ 15*sech(d*x+c)^2)*tanh(d*x+c))+b^3*(-1/2*sinh(d*x+c)^6/cosh(d*x+c)^8-3/4*s inh(d*x+c)^4/cosh(d*x+c)^8-1/2*sinh(d*x+c)^2/cosh(d*x+c)^8-1/8/cosh(d*x+c) ^8))
Leaf count of result is larger than twice the leaf count of optimal. 1192 vs. \(2 (65) = 130\).
Time = 0.27 (sec) , antiderivative size = 1192, normalized size of antiderivative = 16.79 \[ \int \text {csch}^2(c+d x) \left (a+b \tanh ^3(c+d x)\right )^3 \, dx=\text {Too large to display} \]
-2/5*((10*a^3 + 15*a^2*b + 12*a*b^2 + 5*b^3)*cosh(d*x + c)^8 + 8*(15*a^2*b + 18*a*b^2 + 5*b^3)*cosh(d*x + c)*sinh(d*x + c)^7 + (10*a^3 + 15*a^2*b + 12*a*b^2 + 5*b^3)*sinh(d*x + c)^8 + 2*(40*a^3 + 30*a^2*b + 12*a*b^2 - 5*b^ 3)*cosh(d*x + c)^6 + 2*(40*a^3 + 30*a^2*b + 12*a*b^2 - 5*b^3 + 14*(10*a^3 + 15*a^2*b + 12*a*b^2 + 5*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 4*(14*(1 5*a^2*b + 18*a*b^2 + 5*b^3)*cosh(d*x + c)^3 + 27*(5*a^2*b + 2*a*b^2)*cosh( d*x + c))*sinh(d*x + c)^5 + 20*(14*a^3 + 3*a^2*b + 2*b^3)*cosh(d*x + c)^4 + 10*(7*(10*a^3 + 15*a^2*b + 12*a*b^2 + 5*b^3)*cosh(d*x + c)^4 + 28*a^3 + 6*a^2*b + 4*b^3 + 3*(40*a^3 + 30*a^2*b + 12*a*b^2 - 5*b^3)*cosh(d*x + c)^2 )*sinh(d*x + c)^4 + 8*(7*(15*a^2*b + 18*a*b^2 + 5*b^3)*cosh(d*x + c)^5 + 4 5*(5*a^2*b + 2*a*b^2)*cosh(d*x + c)^3 + 15*(7*a^2*b + 2*a*b^2 + b^3)*cosh( d*x + c))*sinh(d*x + c)^3 + 350*a^3 - 75*a^2*b - 12*a*b^2 + 35*b^3 + 2*(28 0*a^3 - 30*a^2*b - 12*a*b^2 - 35*b^3)*cosh(d*x + c)^2 + 2*(14*(10*a^3 + 15 *a^2*b + 12*a*b^2 + 5*b^3)*cosh(d*x + c)^6 + 15*(40*a^3 + 30*a^2*b + 12*a* b^2 - 5*b^3)*cosh(d*x + c)^4 + 280*a^3 - 30*a^2*b - 12*a*b^2 - 35*b^3 + 60 *(14*a^3 + 3*a^2*b + 2*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*(2*(15*a^ 2*b + 18*a*b^2 + 5*b^3)*cosh(d*x + c)^7 + 27*(5*a^2*b + 2*a*b^2)*cosh(d*x + c)^5 + 30*(7*a^2*b + 2*a*b^2 + b^3)*cosh(d*x + c)^3 + 21*(5*a^2*b + 2*a* b^2)*cosh(d*x + c))*sinh(d*x + c))/(d*cosh(d*x + c)^10 + 10*d*cosh(d*x + c )*sinh(d*x + c)^9 + d*sinh(d*x + c)^10 + 6*d*cosh(d*x + c)^8 + 3*(15*d*...
\[ \int \text {csch}^2(c+d x) \left (a+b \tanh ^3(c+d x)\right )^3 \, dx=\int \left (a + b \tanh ^{3}{\left (c + d x \right )}\right )^{3} \operatorname {csch}^{2}{\left (c + d x \right )}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 679 vs. \(2 (65) = 130\).
Time = 0.21 (sec) , antiderivative size = 679, normalized size of antiderivative = 9.56 \[ \int \text {csch}^2(c+d x) \left (a+b \tanh ^3(c+d x)\right )^3 \, dx=-2 \, b^{3} {\left (\frac {e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (8 \, e^{\left (-2 \, d x - 2 \, c\right )} + 28 \, e^{\left (-4 \, d x - 4 \, c\right )} + 56 \, e^{\left (-6 \, d x - 6 \, c\right )} + 70 \, e^{\left (-8 \, d x - 8 \, c\right )} + 56 \, e^{\left (-10 \, d x - 10 \, c\right )} + 28 \, e^{\left (-12 \, d x - 12 \, c\right )} + 8 \, e^{\left (-14 \, d x - 14 \, c\right )} + e^{\left (-16 \, d x - 16 \, c\right )} + 1\right )}} + \frac {7 \, e^{\left (-6 \, d x - 6 \, c\right )}}{d {\left (8 \, e^{\left (-2 \, d x - 2 \, c\right )} + 28 \, e^{\left (-4 \, d x - 4 \, c\right )} + 56 \, e^{\left (-6 \, d x - 6 \, c\right )} + 70 \, e^{\left (-8 \, d x - 8 \, c\right )} + 56 \, e^{\left (-10 \, d x - 10 \, c\right )} + 28 \, e^{\left (-12 \, d x - 12 \, c\right )} + 8 \, e^{\left (-14 \, d x - 14 \, c\right )} + e^{\left (-16 \, d x - 16 \, c\right )} + 1\right )}} + \frac {7 \, e^{\left (-10 \, d x - 10 \, c\right )}}{d {\left (8 \, e^{\left (-2 \, d x - 2 \, c\right )} + 28 \, e^{\left (-4 \, d x - 4 \, c\right )} + 56 \, e^{\left (-6 \, d x - 6 \, c\right )} + 70 \, e^{\left (-8 \, d x - 8 \, c\right )} + 56 \, e^{\left (-10 \, d x - 10 \, c\right )} + 28 \, e^{\left (-12 \, d x - 12 \, c\right )} + 8 \, e^{\left (-14 \, d x - 14 \, c\right )} + e^{\left (-16 \, d x - 16 \, c\right )} + 1\right )}} + \frac {e^{\left (-14 \, d x - 14 \, c\right )}}{d {\left (8 \, e^{\left (-2 \, d x - 2 \, c\right )} + 28 \, e^{\left (-4 \, d x - 4 \, c\right )} + 56 \, e^{\left (-6 \, d x - 6 \, c\right )} + 70 \, e^{\left (-8 \, d x - 8 \, c\right )} + 56 \, e^{\left (-10 \, d x - 10 \, c\right )} + 28 \, e^{\left (-12 \, d x - 12 \, c\right )} + 8 \, e^{\left (-14 \, d x - 14 \, c\right )} + e^{\left (-16 \, d x - 16 \, c\right )} + 1\right )}}\right )} + \frac {6}{5} \, a b^{2} {\left (\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 {5 \, e^{\left (-8 \, d x - 8 \, 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 )} + \frac {2 \, a^{3}}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} - 1\right )}} - \frac {6 \, a^{2} b}{d {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{2}} \]
-2*b^3*(e^(-2*d*x - 2*c)/(d*(8*e^(-2*d*x - 2*c) + 28*e^(-4*d*x - 4*c) + 56 *e^(-6*d*x - 6*c) + 70*e^(-8*d*x - 8*c) + 56*e^(-10*d*x - 10*c) + 28*e^(-1 2*d*x - 12*c) + 8*e^(-14*d*x - 14*c) + e^(-16*d*x - 16*c) + 1)) + 7*e^(-6* d*x - 6*c)/(d*(8*e^(-2*d*x - 2*c) + 28*e^(-4*d*x - 4*c) + 56*e^(-6*d*x - 6 *c) + 70*e^(-8*d*x - 8*c) + 56*e^(-10*d*x - 10*c) + 28*e^(-12*d*x - 12*c) + 8*e^(-14*d*x - 14*c) + e^(-16*d*x - 16*c) + 1)) + 7*e^(-10*d*x - 10*c)/( d*(8*e^(-2*d*x - 2*c) + 28*e^(-4*d*x - 4*c) + 56*e^(-6*d*x - 6*c) + 70*e^( -8*d*x - 8*c) + 56*e^(-10*d*x - 10*c) + 28*e^(-12*d*x - 12*c) + 8*e^(-14*d *x - 14*c) + e^(-16*d*x - 16*c) + 1)) + e^(-14*d*x - 14*c)/(d*(8*e^(-2*d*x - 2*c) + 28*e^(-4*d*x - 4*c) + 56*e^(-6*d*x - 6*c) + 70*e^(-8*d*x - 8*c) + 56*e^(-10*d*x - 10*c) + 28*e^(-12*d*x - 12*c) + 8*e^(-14*d*x - 14*c) + e ^(-16*d*x - 16*c) + 1))) + 6/5*a*b^2*(10*e^(-4*d*x - 4*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)) + 5*e^(-8*d*x - 8*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)) + 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))) + 2*a^3/(d*(e^( -2*d*x - 2*c) - 1)) - 6*a^2*b/(d*(e^(d*x + c) + e^(-d*x - c))^2)
Leaf count of result is larger than twice the leaf count of optimal. 311 vs. \(2 (65) = 130\).
Time = 0.55 (sec) , antiderivative size = 311, normalized size of antiderivative = 4.38 \[ \int \text {csch}^2(c+d x) \left (a+b \tanh ^3(c+d x)\right )^3 \, dx=-\frac {2 \, {\left (\frac {5 \, a^{3}}{e^{\left (2 \, d x + 2 \, c\right )} - 1} + \frac {15 \, a^{2} b e^{\left (14 \, d x + 14 \, c\right )} + 15 \, a b^{2} e^{\left (14 \, d x + 14 \, c\right )} + 5 \, b^{3} e^{\left (14 \, d x + 14 \, c\right )} + 90 \, a^{2} b e^{\left (12 \, d x + 12 \, c\right )} + 45 \, a b^{2} e^{\left (12 \, d x + 12 \, c\right )} + 225 \, a^{2} b e^{\left (10 \, d x + 10 \, c\right )} + 75 \, a b^{2} e^{\left (10 \, d x + 10 \, c\right )} + 35 \, b^{3} e^{\left (10 \, d x + 10 \, c\right )} + 300 \, a^{2} b e^{\left (8 \, d x + 8 \, c\right )} + 105 \, a b^{2} e^{\left (8 \, d x + 8 \, c\right )} + 225 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} + 93 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 35 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 90 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} + 39 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 15 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} + 9 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 5 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 3 \, a b^{2}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{8}}\right )}}{5 \, d} \]
-2/5*(5*a^3/(e^(2*d*x + 2*c) - 1) + (15*a^2*b*e^(14*d*x + 14*c) + 15*a*b^2 *e^(14*d*x + 14*c) + 5*b^3*e^(14*d*x + 14*c) + 90*a^2*b*e^(12*d*x + 12*c) + 45*a*b^2*e^(12*d*x + 12*c) + 225*a^2*b*e^(10*d*x + 10*c) + 75*a*b^2*e^(1 0*d*x + 10*c) + 35*b^3*e^(10*d*x + 10*c) + 300*a^2*b*e^(8*d*x + 8*c) + 105 *a*b^2*e^(8*d*x + 8*c) + 225*a^2*b*e^(6*d*x + 6*c) + 93*a*b^2*e^(6*d*x + 6 *c) + 35*b^3*e^(6*d*x + 6*c) + 90*a^2*b*e^(4*d*x + 4*c) + 39*a*b^2*e^(4*d* x + 4*c) + 15*a^2*b*e^(2*d*x + 2*c) + 9*a*b^2*e^(2*d*x + 2*c) + 5*b^3*e^(2 *d*x + 2*c) + 3*a*b^2)/(e^(2*d*x + 2*c) + 1)^8)/d
Time = 2.09 (sec) , antiderivative size = 1515, normalized size of antiderivative = 21.34 \[ \int \text {csch}^2(c+d x) \left (a+b \tanh ^3(c+d x)\right )^3 \, dx=\text {Too large to display} \]
((3*a*b^2 - 15*a^2*b + 7*b^3)/(28*d) - (exp(2*c + 2*d*x)*(3*a*b^2 + 3*a^2* b + b^3))/(4*d))/(2*exp(2*c + 2*d*x) + exp(4*c + 4*d*x) + 1) - ((9*a*b^2 + 15*a^2*b - 35*b^3)/(140*d) + (exp(6*c + 6*d*x)*(3*a*b^2 + 3*a^2*b + b^3)) /(4*d) + (3*exp(2*c + 2*d*x)*(9*a^2*b - 3*a*b^2 + 7*b^3))/(28*d) - (3*exp( 4*c + 4*d*x)*(3*a*b^2 - 15*a^2*b + 7*b^3))/(28*d))/(4*exp(2*c + 2*d*x) + 6 *exp(4*c + 4*d*x) + 4*exp(6*c + 6*d*x) + exp(8*c + 8*d*x) + 1) - ((exp(10* c + 10*d*x)*(3*a*b^2 + 3*a^2*b + b^3))/(4*d) - (3*a*b^2 + 9*a^2*b + 7*b^3) /(28*d) + (5*exp(6*c + 6*d*x)*(9*a^2*b - 3*a*b^2 + 7*b^3))/(14*d) - (5*exp (8*c + 8*d*x)*(3*a*b^2 - 15*a^2*b + 7*b^3))/(28*d) + (exp(2*c + 2*d*x)*(9* a*b^2 - 15*a^2*b + 35*b^3))/(28*d) + (exp(4*c + 4*d*x)*(9*a*b^2 + 15*a^2*b - 35*b^3))/(14*d))/(6*exp(2*c + 2*d*x) + 15*exp(4*c + 4*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) - ((9*a^2*b - 3*a*b^2 + 7*b^3)/(28*d) + (exp(4*c + 4*d*x)*(3*a*b^2 + 3*a^2*b + b^3))/(4*d) - (exp(2*c + 2*d*x)*(3*a*b^2 - 15*a^2*b + 7*b^3))/ (14*d))/(3*exp(2*c + 2*d*x) + 3*exp(4*c + 4*d*x) + exp(6*c + 6*d*x) + 1) - ((9*a*b^2 - 15*a^2*b + 35*b^3)/(140*d) + (exp(8*c + 8*d*x)*(3*a*b^2 + 3*a ^2*b + b^3))/(4*d) + (3*exp(4*c + 4*d*x)*(9*a^2*b - 3*a*b^2 + 7*b^3))/(14* d) - (exp(6*c + 6*d*x)*(3*a*b^2 - 15*a^2*b + 7*b^3))/(7*d) + (exp(2*c + 2* d*x)*(9*a*b^2 + 15*a^2*b - 35*b^3))/(35*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*...