Integrand size = 23, antiderivative size = 171 \[ \int \frac {\coth ^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=-\frac {\coth ^2(c+d x)}{2 a^3 d}+\frac {\log (\cosh (c+d x))}{(a+b)^3 d}+\frac {(a-3 b) \log (\tanh (c+d x))}{a^4 d}+\frac {b^2 \left (6 a^2+8 a b+3 b^2\right ) \log \left (a+b \tanh ^2(c+d x)\right )}{2 a^4 (a+b)^3 d}-\frac {b^2}{4 a^2 (a+b) d \left (a+b \tanh ^2(c+d x)\right )^2}-\frac {b^2 (3 a+2 b)}{2 a^3 (a+b)^2 d \left (a+b \tanh ^2(c+d x)\right )} \]
-1/2*coth(d*x+c)^2/a^3/d+ln(cosh(d*x+c))/(a+b)^3/d+(a-3*b)*ln(tanh(d*x+c)) /a^4/d+1/2*b^2*(6*a^2+8*a*b+3*b^2)*ln(a+b*tanh(d*x+c)^2)/a^4/(a+b)^3/d-1/4 *b^2/a^2/(a+b)/d/(a+b*tanh(d*x+c)^2)^2-1/2*b^2*(3*a+2*b)/a^3/(a+b)^2/d/(a+ b*tanh(d*x+c)^2)
Time = 2.08 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.81 \[ \int \frac {\coth ^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=-\frac {\frac {\coth ^2(c+d x)}{a^3}+\frac {b^4}{2 a^4 (a+b) \left (b+a \coth ^2(c+d x)\right )^2}-\frac {b^3 (4 a+3 b)}{a^4 (a+b)^2 \left (b+a \coth ^2(c+d x)\right )}-\frac {b^2 \left (6 a^2+8 a b+3 b^2\right ) \log \left (b+a \coth ^2(c+d x)\right )}{a^4 (a+b)^3}-\frac {2 \log (\sinh (c+d x))}{(a+b)^3}}{2 d} \]
-1/2*(Coth[c + d*x]^2/a^3 + b^4/(2*a^4*(a + b)*(b + a*Coth[c + d*x]^2)^2) - (b^3*(4*a + 3*b))/(a^4*(a + b)^2*(b + a*Coth[c + d*x]^2)) - (b^2*(6*a^2 + 8*a*b + 3*b^2)*Log[b + a*Coth[c + d*x]^2])/(a^4*(a + b)^3) - (2*Log[Sinh [c + d*x]])/(a + b)^3)/d
Time = 0.42 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.94, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {3042, 26, 4153, 26, 354, 99, 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.
| \(\displaystyle \int \frac {\coth ^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {i}{\tan (i c+i d x)^3 \left (a-b \tan (i c+i d x)^2\right )^3}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \int \frac {1}{\tan (i c+i d x)^3 \left (a-b \tan (i c+i d x)^2\right )^3}dx\) |
| \(\Big \downarrow \) 4153 |
| \(\displaystyle -\frac {i \int \frac {i \coth ^3(c+d x)}{\left (1-\tanh ^2(c+d x)\right ) \left (b \tanh ^2(c+d x)+a\right )^3}d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {\int \frac {\coth ^3(c+d x)}{\left (1-\tanh ^2(c+d x)\right ) \left (b \tanh ^2(c+d x)+a\right )^3}d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 354 |
| \(\displaystyle \frac {\int \frac {\coth ^2(c+d x)}{\left (1-\tanh ^2(c+d x)\right ) \left (b \tanh ^2(c+d x)+a\right )^3}d\tanh ^2(c+d x)}{2 d}\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \frac {\int \left (\frac {\left (6 a^2+8 b a+3 b^2\right ) b^3}{a^4 (a+b)^3 \left (b \tanh ^2(c+d x)+a\right )}+\frac {(3 a+2 b) b^3}{a^3 (a+b)^2 \left (b \tanh ^2(c+d x)+a\right )^2}+\frac {b^3}{a^2 (a+b) \left (b \tanh ^2(c+d x)+a\right )^3}+\frac {\coth ^2(c+d x)}{a^3}+\frac {(a-3 b) \coth (c+d x)}{a^4}-\frac {1}{(a+b)^3 \left (\tanh ^2(c+d x)-1\right )}\right )d\tanh ^2(c+d x)}{2 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {(a-3 b) \log \left (\tanh ^2(c+d x)\right )}{a^4}-\frac {b^2 (3 a+2 b)}{a^3 (a+b)^2 \left (a+b \tanh ^2(c+d x)\right )}-\frac {\coth (c+d x)}{a^3}-\frac {b^2}{2 a^2 (a+b) \left (a+b \tanh ^2(c+d x)\right )^2}+\frac {b^2 \left (6 a^2+8 a b+3 b^2\right ) \log \left (a+b \tanh ^2(c+d x)\right )}{a^4 (a+b)^3}-\frac {\log \left (1-\tanh ^2(c+d x)\right )}{(a+b)^3}}{2 d}\) |
(-(Coth[c + d*x]/a^3) + ((a - 3*b)*Log[Tanh[c + d*x]^2])/a^4 - Log[1 - Tan h[c + d*x]^2]/(a + b)^3 + (b^2*(6*a^2 + 8*a*b + 3*b^2)*Log[a + b*Tanh[c + d*x]^2])/(a^4*(a + b)^3) - b^2/(2*a^2*(a + b)*(a + b*Tanh[c + d*x]^2)^2) - (b^2*(3*a + 2*b))/(a^3*(a + b)^2*(a + b*Tanh[c + d*x]^2)))/(2*d)
3.2.99.3.1 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_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | | (GtQ[m, 0] && GeQ[n, -1]))
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S ymbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(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] && IntegerQ [(m - 1)/2]
Int[((d_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[c*(ff/f) Subst[Int[(d*ff*(x/c))^m*((a + b*(ff*x)^n)^p/(c^2 + f f^2*x^2)), x], x, c*(Tan[e + f*x]/ff)], x]] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && (IGtQ[p, 0] || EqQ[n, 2] || EqQ[n, 4] || (IntegerQ[p] && Ratio nalQ[n]))
Time = 0.69 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.08
| method | result | size |
| derivativedivides | \(-\frac {\frac {\ln \left (\tanh \left (d x +c \right )-1\right )}{2 \left (a +b \right )^{3}}-\frac {b^{3} \left (\frac {\left (6 a^{2}+8 a b +3 b^{2}\right ) \ln \left (a +b \tanh \left (d x +c \right )^{2}\right )}{b}-\frac {a^{2} \left (a^{2}+2 a b +b^{2}\right )}{2 b \left (a +b \tanh \left (d x +c \right )^{2}\right )^{2}}-\frac {a \left (3 a^{2}+5 a b +2 b^{2}\right )}{b \left (a +b \tanh \left (d x +c \right )^{2}\right )}\right )}{2 \left (a +b \right )^{3} a^{4}}+\frac {\ln \left (\tanh \left (d x +c \right )+1\right )}{2 \left (a +b \right )^{3}}+\frac {\left (-a +3 b \right ) \ln \left (\tanh \left (d x +c \right )\right )}{a^{4}}+\frac {1}{2 a^{3} \tanh \left (d x +c \right )^{2}}}{d}\) | \(184\) |
| default | \(-\frac {\frac {\ln \left (\tanh \left (d x +c \right )-1\right )}{2 \left (a +b \right )^{3}}-\frac {b^{3} \left (\frac {\left (6 a^{2}+8 a b +3 b^{2}\right ) \ln \left (a +b \tanh \left (d x +c \right )^{2}\right )}{b}-\frac {a^{2} \left (a^{2}+2 a b +b^{2}\right )}{2 b \left (a +b \tanh \left (d x +c \right )^{2}\right )^{2}}-\frac {a \left (3 a^{2}+5 a b +2 b^{2}\right )}{b \left (a +b \tanh \left (d x +c \right )^{2}\right )}\right )}{2 \left (a +b \right )^{3} a^{4}}+\frac {\ln \left (\tanh \left (d x +c \right )+1\right )}{2 \left (a +b \right )^{3}}+\frac {\left (-a +3 b \right ) \ln \left (\tanh \left (d x +c \right )\right )}{a^{4}}+\frac {1}{2 a^{3} \tanh \left (d x +c \right )^{2}}}{d}\) | \(184\) |
| parallelrisch | \(\frac {48 \left (a^{2}+\frac {4}{3} a b +\frac {1}{2} b^{2}\right ) \left (\frac {\left (a +b \right )^{2} \cosh \left (4 d x +4 c \right )}{4}+\left (a^{2}-b^{2}\right ) \cosh \left (2 d x +2 c \right )+\frac {3 a^{2}}{4}-\frac {a b}{2}+\frac {3 b^{2}}{4}\right ) b^{2} \ln \left (a +b \tanh \left (d x +c \right )^{2}\right )-16 \left (\frac {\left (a +b \right )^{2} \cosh \left (4 d x +4 c \right )}{4}+\left (a^{2}-b^{2}\right ) \cosh \left (2 d x +2 c \right )+\frac {3 a^{2}}{4}-\frac {a b}{2}+\frac {3 b^{2}}{4}\right ) a^{4} \ln \left (1-\tanh \left (d x +c \right )\right )+\left (4 \left (a -3 b \right ) \left (a +b \right )^{5} \ln \left (\tanh \left (d x +c \right )\right )-2 a^{5} \left (a +3 b \right ) \coth \left (d x +c \right )^{2}-4 a^{6} d x -8 a^{5} b d x -4 a^{4} b^{2} d x -22 \operatorname {csch}\left (d x +c \right )^{2} a^{3} b^{3}+51 a^{2} b^{4}+36 a \,b^{5}+9 b^{6}\right ) \cosh \left (4 d x +4 c \right )+\left (16 \left (a -b \right ) \left (a -3 b \right ) \left (a +b \right )^{4} \ln \left (\tanh \left (d x +c \right )\right )+\left (-8 a^{6}-24 a^{5} b -48 a^{4} b^{2}\right ) \coth \left (d x +c \right )^{2}-16 a^{6} d x +16 a^{4} b^{2} d x +\operatorname {csch}\left (d x +c \right )^{2} a^{3} b^{3}-76 a^{2} b^{4}-96 a \,b^{5}-36 b^{6}\right ) \cosh \left (2 d x +2 c \right )+7 \operatorname {csch}\left (d x +c \right )^{2} a^{3} b^{3} \cosh \left (6 d x +6 c \right )+12 \left (a -3 b \right ) \left (a +b \right )^{3} \left (a^{2}-\frac {2}{3} a b +b^{2}\right ) \ln \left (\tanh \left (d x +c \right )\right )-6 a^{5} \left (a +3 b \right ) \coth \left (d x +c \right )^{2}-12 a^{6} d x +8 a^{5} b d x -12 a^{4} b^{2} d x -2 \operatorname {csch}\left (d x +c \right )^{2} a^{3} b^{3}+25 a^{2} b^{4}+60 a \,b^{5}+27 b^{6}}{16 d \left (\frac {\left (a +b \right )^{2} \cosh \left (4 d x +4 c \right )}{4}+\left (a^{2}-b^{2}\right ) \cosh \left (2 d x +2 c \right )+\frac {3 a^{2}}{4}-\frac {a b}{2}+\frac {3 b^{2}}{4}\right ) \left (a +b \right )^{3} a^{4}}\) | \(570\) |
| risch | \(\frac {x}{a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}}-\frac {2 x}{a^{3}}-\frac {2 c}{a^{3} d}+\frac {6 b x}{a^{4}}+\frac {6 b c}{d \,a^{4}}-\frac {12 b^{2} x}{a^{2} \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}-\frac {12 b^{2} c}{a^{2} d \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}-\frac {16 b^{3} x}{a^{3} \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}-\frac {16 b^{3} c}{a^{3} d \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}-\frac {6 b^{4} x}{a^{4} \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}-\frac {6 b^{4} c}{d \,a^{4} \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}-\frac {2 \left (-12 b^{5} {\mathrm e}^{6 d x +6 c}+18 b^{5} {\mathrm e}^{4 d x +4 c}+a^{5}+3 b^{5}+5 a^{4} b +10 a^{3} b^{2}+14 a^{2} b^{3}+11 a \,b^{4}-12 b^{5} {\mathrm e}^{2 d x +2 c}+4 a^{2} b^{3} {\mathrm e}^{4 d x +4 c}-26 a \,b^{4} {\mathrm e}^{6 d x +6 c}+30 a \,b^{4} {\mathrm e}^{4 d x +4 c}-26 a \,b^{4} {\mathrm e}^{2 d x +2 c}+14 a^{2} b^{3} {\mathrm e}^{8 d x +8 c}+11 a \,b^{4} {\mathrm e}^{8 d x +8 c}+3 b^{5} {\mathrm e}^{8 d x +8 c}+4 a^{5} {\mathrm e}^{6 d x +6 c}+a^{5} {\mathrm e}^{8 d x +8 c}+4 \,{\mathrm e}^{2 d x +2 c} a^{5}+6 \,{\mathrm e}^{4 d x +4 c} a^{5}-8 a^{2} b^{3} {\mathrm e}^{6 d x +6 c}+5 \,{\mathrm e}^{8 d x +8 c} a^{4} b +10 \,{\mathrm e}^{8 d x +8 c} a^{3} b^{2}+12 \,{\mathrm e}^{2 d x +2 c} a^{4} b +8 \,{\mathrm e}^{2 d x +2 c} a^{3} b^{2}-8 \,{\mathrm e}^{2 d x +2 c} a^{2} b^{3}+14 \,{\mathrm e}^{4 d x +4 c} a^{4} b +12 \,{\mathrm e}^{4 d x +4 c} a^{3} b^{2}+12 b \,a^{4} {\mathrm e}^{6 d x +6 c}+8 a^{3} b^{2} {\mathrm e}^{6 d x +6 c}\right ) {\mathrm e}^{2 d x +2 c}}{\left (a \,{\mathrm e}^{4 d x +4 c}+b \,{\mathrm e}^{4 d x +4 c}+2 \,{\mathrm e}^{2 d x +2 c} a -2 b \,{\mathrm e}^{2 d x +2 c}+a +b \right )^{2} \left (a +b \right )^{3} a^{3} d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}-1\right )}{a^{3} d}-\frac {3 \ln \left ({\mathrm e}^{2 d x +2 c}-1\right ) b}{d \,a^{4}}+\frac {3 b^{2} \ln \left ({\mathrm e}^{4 d x +4 c}+\frac {2 \left (a -b \right ) {\mathrm e}^{2 d x +2 c}}{a +b}+1\right )}{a^{2} d \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {4 b^{3} \ln \left ({\mathrm e}^{4 d x +4 c}+\frac {2 \left (a -b \right ) {\mathrm e}^{2 d x +2 c}}{a +b}+1\right )}{a^{3} d \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {3 b^{4} \ln \left ({\mathrm e}^{4 d x +4 c}+\frac {2 \left (a -b \right ) {\mathrm e}^{2 d x +2 c}}{a +b}+1\right )}{2 d \,a^{4} \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}\) | \(971\) |
-1/d*(1/2/(a+b)^3*ln(tanh(d*x+c)-1)-1/2*b^3/(a+b)^3/a^4*((6*a^2+8*a*b+3*b^ 2)/b*ln(a+b*tanh(d*x+c)^2)-1/2*a^2*(a^2+2*a*b+b^2)/b/(a+b*tanh(d*x+c)^2)^2 -a*(3*a^2+5*a*b+2*b^2)/b/(a+b*tanh(d*x+c)^2))+1/2/(a+b)^3*ln(tanh(d*x+c)+1 )+(-a+3*b)/a^4*ln(tanh(d*x+c))+1/2/a^3/tanh(d*x+c)^2)
Leaf count of result is larger than twice the leaf count of optimal. 10720 vs. \(2 (163) = 326\).
Time = 0.71 (sec) , antiderivative size = 10720, normalized size of antiderivative = 62.69 \[ \int \frac {\coth ^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\text {Too large to display} \]
\[ \int \frac {\coth ^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\int \frac {\coth ^{3}{\left (c + d x \right )}}{\left (a + b \tanh ^{2}{\left (c + d x \right )}\right )^{3}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 770 vs. \(2 (163) = 326\).
Time = 0.27 (sec) , antiderivative size = 770, normalized size of antiderivative = 4.50 \[ \int \frac {\coth ^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\frac {{\left (6 \, a^{2} b^{2} + 8 \, a b^{3} + 3 \, b^{4}\right )} \log \left (2 \, {\left (a - b\right )} e^{\left (-2 \, d x - 2 \, c\right )} + {\left (a + b\right )} e^{\left (-4 \, d x - 4 \, c\right )} + a + b\right )}{2 \, {\left (a^{7} + 3 \, a^{6} b + 3 \, a^{5} b^{2} + a^{4} b^{3}\right )} d} + \frac {d x + c}{{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} d} - \frac {2 \, {\left ({\left (a^{5} + 5 \, a^{4} b + 10 \, a^{3} b^{2} + 14 \, a^{2} b^{3} + 11 \, a b^{4} + 3 \, b^{5}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + 2 \, {\left (2 \, a^{5} + 6 \, a^{4} b + 4 \, a^{3} b^{2} - 4 \, a^{2} b^{3} - 13 \, a b^{4} - 6 \, b^{5}\right )} e^{\left (-4 \, d x - 4 \, c\right )} + 2 \, {\left (3 \, a^{5} + 7 \, a^{4} b + 6 \, a^{3} b^{2} + 2 \, a^{2} b^{3} + 15 \, a b^{4} + 9 \, b^{5}\right )} e^{\left (-6 \, d x - 6 \, c\right )} + 2 \, {\left (2 \, a^{5} + 6 \, a^{4} b + 4 \, a^{3} b^{2} - 4 \, a^{2} b^{3} - 13 \, a b^{4} - 6 \, b^{5}\right )} e^{\left (-8 \, d x - 8 \, c\right )} + {\left (a^{5} + 5 \, a^{4} b + 10 \, a^{3} b^{2} + 14 \, a^{2} b^{3} + 11 \, a b^{4} + 3 \, b^{5}\right )} e^{\left (-10 \, d x - 10 \, c\right )}\right )}}{{\left (a^{8} + 5 \, a^{7} b + 10 \, a^{6} b^{2} + 10 \, a^{5} b^{3} + 5 \, a^{4} b^{4} + a^{3} b^{5} + 2 \, {\left (a^{8} + a^{7} b - 6 \, a^{6} b^{2} - 14 \, a^{5} b^{3} - 11 \, a^{4} b^{4} - 3 \, a^{3} b^{5}\right )} e^{\left (-2 \, d x - 2 \, c\right )} - {\left (a^{8} + 5 \, a^{7} b - 6 \, a^{6} b^{2} - 38 \, a^{5} b^{3} - 43 \, a^{4} b^{4} - 15 \, a^{3} b^{5}\right )} e^{\left (-4 \, d x - 4 \, c\right )} - 4 \, {\left (a^{8} + a^{7} b + 2 \, a^{6} b^{2} + 10 \, a^{5} b^{3} + 13 \, a^{4} b^{4} + 5 \, a^{3} b^{5}\right )} e^{\left (-6 \, d x - 6 \, c\right )} - {\left (a^{8} + 5 \, a^{7} b - 6 \, a^{6} b^{2} - 38 \, a^{5} b^{3} - 43 \, a^{4} b^{4} - 15 \, a^{3} b^{5}\right )} e^{\left (-8 \, d x - 8 \, c\right )} + 2 \, {\left (a^{8} + a^{7} b - 6 \, a^{6} b^{2} - 14 \, a^{5} b^{3} - 11 \, a^{4} b^{4} - 3 \, a^{3} b^{5}\right )} e^{\left (-10 \, d x - 10 \, c\right )} + {\left (a^{8} + 5 \, a^{7} b + 10 \, a^{6} b^{2} + 10 \, a^{5} b^{3} + 5 \, a^{4} b^{4} + a^{3} b^{5}\right )} e^{\left (-12 \, d x - 12 \, c\right )}\right )} d} + \frac {{\left (a - 3 \, b\right )} \log \left (e^{\left (-d x - c\right )} + 1\right )}{a^{4} d} + \frac {{\left (a - 3 \, b\right )} \log \left (e^{\left (-d x - c\right )} - 1\right )}{a^{4} d} \]
1/2*(6*a^2*b^2 + 8*a*b^3 + 3*b^4)*log(2*(a - b)*e^(-2*d*x - 2*c) + (a + b) *e^(-4*d*x - 4*c) + a + b)/((a^7 + 3*a^6*b + 3*a^5*b^2 + a^4*b^3)*d) + (d* x + c)/((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d) - 2*((a^5 + 5*a^4*b + 10*a^3*b^ 2 + 14*a^2*b^3 + 11*a*b^4 + 3*b^5)*e^(-2*d*x - 2*c) + 2*(2*a^5 + 6*a^4*b + 4*a^3*b^2 - 4*a^2*b^3 - 13*a*b^4 - 6*b^5)*e^(-4*d*x - 4*c) + 2*(3*a^5 + 7 *a^4*b + 6*a^3*b^2 + 2*a^2*b^3 + 15*a*b^4 + 9*b^5)*e^(-6*d*x - 6*c) + 2*(2 *a^5 + 6*a^4*b + 4*a^3*b^2 - 4*a^2*b^3 - 13*a*b^4 - 6*b^5)*e^(-8*d*x - 8*c ) + (a^5 + 5*a^4*b + 10*a^3*b^2 + 14*a^2*b^3 + 11*a*b^4 + 3*b^5)*e^(-10*d* x - 10*c))/((a^8 + 5*a^7*b + 10*a^6*b^2 + 10*a^5*b^3 + 5*a^4*b^4 + a^3*b^5 + 2*(a^8 + a^7*b - 6*a^6*b^2 - 14*a^5*b^3 - 11*a^4*b^4 - 3*a^3*b^5)*e^(-2 *d*x - 2*c) - (a^8 + 5*a^7*b - 6*a^6*b^2 - 38*a^5*b^3 - 43*a^4*b^4 - 15*a^ 3*b^5)*e^(-4*d*x - 4*c) - 4*(a^8 + a^7*b + 2*a^6*b^2 + 10*a^5*b^3 + 13*a^4 *b^4 + 5*a^3*b^5)*e^(-6*d*x - 6*c) - (a^8 + 5*a^7*b - 6*a^6*b^2 - 38*a^5*b ^3 - 43*a^4*b^4 - 15*a^3*b^5)*e^(-8*d*x - 8*c) + 2*(a^8 + a^7*b - 6*a^6*b^ 2 - 14*a^5*b^3 - 11*a^4*b^4 - 3*a^3*b^5)*e^(-10*d*x - 10*c) + (a^8 + 5*a^7 *b + 10*a^6*b^2 + 10*a^5*b^3 + 5*a^4*b^4 + a^3*b^5)*e^(-12*d*x - 12*c))*d) + (a - 3*b)*log(e^(-d*x - c) + 1)/(a^4*d) + (a - 3*b)*log(e^(-d*x - c) - 1)/(a^4*d)
Leaf count of result is larger than twice the leaf count of optimal. 474 vs. \(2 (163) = 326\).
Time = 0.54 (sec) , antiderivative size = 474, normalized size of antiderivative = 2.77 \[ \int \frac {\coth ^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\frac {\frac {{\left (6 \, a^{2} b^{2} + 8 \, a b^{3} + 3 \, b^{4}\right )} \log \left (a e^{\left (4 \, d x + 4 \, c\right )} + b e^{\left (4 \, d x + 4 \, c\right )} + 2 \, a e^{\left (2 \, d x + 2 \, c\right )} - 2 \, b e^{\left (2 \, d x + 2 \, c\right )} + a + b\right )}{a^{7} + 3 \, a^{6} b + 3 \, a^{5} b^{2} + a^{4} b^{3}} - \frac {2 \, {\left (d x + c\right )}}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}} + \frac {2 \, {\left (a - 3 \, b\right )} \log \left ({\left | e^{\left (2 \, d x + 2 \, c\right )} - 1 \right |}\right )}{a^{4}} - \frac {4 \, {\left ({\left (a^{5} + 5 \, a^{4} b + 10 \, a^{3} b^{2} + 14 \, a^{2} b^{3} + 11 \, a b^{4} + 3 \, b^{5}\right )} e^{\left (10 \, d x + 10 \, c\right )} + 2 \, {\left (2 \, a^{5} + 6 \, a^{4} b + 4 \, a^{3} b^{2} - 4 \, a^{2} b^{3} - 13 \, a b^{4} - 6 \, b^{5}\right )} e^{\left (8 \, d x + 8 \, c\right )} + 2 \, {\left (3 \, a^{5} + 7 \, a^{4} b + 6 \, a^{3} b^{2} + 2 \, a^{2} b^{3} + 15 \, a b^{4} + 9 \, b^{5}\right )} e^{\left (6 \, d x + 6 \, c\right )} + 2 \, {\left (2 \, a^{5} + 6 \, a^{4} b + 4 \, a^{3} b^{2} - 4 \, a^{2} b^{3} - 13 \, a b^{4} - 6 \, b^{5}\right )} e^{\left (4 \, d x + 4 \, c\right )} + {\left (a^{5} + 5 \, a^{4} b + 10 \, a^{3} b^{2} + 14 \, a^{2} b^{3} + 11 \, a b^{4} + 3 \, b^{5}\right )} e^{\left (2 \, d x + 2 \, c\right )}\right )}}{{\left (a e^{\left (4 \, d x + 4 \, c\right )} + b e^{\left (4 \, d x + 4 \, c\right )} + 2 \, a e^{\left (2 \, d x + 2 \, c\right )} - 2 \, b e^{\left (2 \, d x + 2 \, c\right )} + a + b\right )}^{2} {\left (a + b\right )}^{3} a^{3} {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{2}}}{2 \, d} \]
1/2*((6*a^2*b^2 + 8*a*b^3 + 3*b^4)*log(a*e^(4*d*x + 4*c) + b*e^(4*d*x + 4* c) + 2*a*e^(2*d*x + 2*c) - 2*b*e^(2*d*x + 2*c) + a + b)/(a^7 + 3*a^6*b + 3 *a^5*b^2 + a^4*b^3) - 2*(d*x + c)/(a^3 + 3*a^2*b + 3*a*b^2 + b^3) + 2*(a - 3*b)*log(abs(e^(2*d*x + 2*c) - 1))/a^4 - 4*((a^5 + 5*a^4*b + 10*a^3*b^2 + 14*a^2*b^3 + 11*a*b^4 + 3*b^5)*e^(10*d*x + 10*c) + 2*(2*a^5 + 6*a^4*b + 4 *a^3*b^2 - 4*a^2*b^3 - 13*a*b^4 - 6*b^5)*e^(8*d*x + 8*c) + 2*(3*a^5 + 7*a^ 4*b + 6*a^3*b^2 + 2*a^2*b^3 + 15*a*b^4 + 9*b^5)*e^(6*d*x + 6*c) + 2*(2*a^5 + 6*a^4*b + 4*a^3*b^2 - 4*a^2*b^3 - 13*a*b^4 - 6*b^5)*e^(4*d*x + 4*c) + ( a^5 + 5*a^4*b + 10*a^3*b^2 + 14*a^2*b^3 + 11*a*b^4 + 3*b^5)*e^(2*d*x + 2*c ))/((a*e^(4*d*x + 4*c) + b*e^(4*d*x + 4*c) + 2*a*e^(2*d*x + 2*c) - 2*b*e^( 2*d*x + 2*c) + a + b)^2*(a + b)^3*a^3*(e^(2*d*x + 2*c) - 1)^2))/d
Timed out. \[ \int \frac {\coth ^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\int \frac {{\mathrm {coth}\left (c+d\,x\right )}^3}{{\left (b\,{\mathrm {tanh}\left (c+d\,x\right )}^2+a\right )}^3} \,d x \]