Integrand size = 12, antiderivative size = 169 \[ \int \frac {1}{(a+b \coth (c+d x))^4} \, dx=\frac {\left (a^4+6 a^2 b^2+b^4\right ) x}{\left (a^2-b^2\right )^4}+\frac {b}{3 \left (a^2-b^2\right ) d (a+b \coth (c+d x))^3}+\frac {a b}{\left (a^2-b^2\right )^2 d (a+b \coth (c+d x))^2}+\frac {b \left (3 a^2+b^2\right )}{\left (a^2-b^2\right )^3 d (a+b \coth (c+d x))}-\frac {4 a b \left (a^2+b^2\right ) \log (b \cosh (c+d x)+a \sinh (c+d x))}{\left (a^2-b^2\right )^4 d} \]
(a^4+6*a^2*b^2+b^4)*x/(a^2-b^2)^4+1/3*b/(a^2-b^2)/d/(a+b*coth(d*x+c))^3+a* b/(a^2-b^2)^2/d/(a+b*coth(d*x+c))^2+b*(3*a^2+b^2)/(a^2-b^2)^3/d/(a+b*coth( d*x+c))-4*a*b*(a^2+b^2)*ln(b*cosh(d*x+c)+a*sinh(d*x+c))/(a^2-b^2)^4/d
Time = 6.24 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.27 \[ \int \frac {1}{(a+b \coth (c+d x))^4} \, dx=-\frac {\log (1-\tanh (c+d x))}{2 (a+b)^4 d}+\frac {\log (1+\tanh (c+d x))}{2 (a-b)^4 d}-\frac {4 a b \left (a^2+b^2\right ) \log (b+a \tanh (c+d x))}{\left (a^2-b^2\right )^4 d}-\frac {b^4}{3 a^3 \left (a^2-b^2\right ) d (b+a \tanh (c+d x))^3}+\frac {b^3 \left (2 a^2-b^2\right )}{a^3 \left (a^2-b^2\right )^2 d (b+a \tanh (c+d x))^2}-\frac {b^2 \left (6 a^4-3 a^2 b^2+b^4\right )}{a^3 \left (a^2-b^2\right )^3 d (b+a \tanh (c+d x))} \]
-1/2*Log[1 - Tanh[c + d*x]]/((a + b)^4*d) + Log[1 + Tanh[c + d*x]]/(2*(a - b)^4*d) - (4*a*b*(a^2 + b^2)*Log[b + a*Tanh[c + d*x]])/((a^2 - b^2)^4*d) - b^4/(3*a^3*(a^2 - b^2)*d*(b + a*Tanh[c + d*x])^3) + (b^3*(2*a^2 - b^2))/ (a^3*(a^2 - b^2)^2*d*(b + a*Tanh[c + d*x])^2) - (b^2*(6*a^4 - 3*a^2*b^2 + b^4))/(a^3*(a^2 - b^2)^3*d*(b + a*Tanh[c + d*x]))
Time = 0.97 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.23, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.917, Rules used = {3042, 3964, 3042, 4012, 3042, 4012, 3042, 4014, 26, 3042, 4013}
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.
| \(\displaystyle \int \frac {1}{(a+b \coth (c+d x))^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\left (a-i b \tan \left (i c+i d x+\frac {\pi }{2}\right )\right )^4}dx\) |
| \(\Big \downarrow \) 3964 |
| \(\displaystyle \frac {\int \frac {a-b \coth (c+d x)}{(a+b \coth (c+d x))^3}dx}{a^2-b^2}+\frac {b}{3 d \left (a^2-b^2\right ) (a+b \coth (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {b}{3 d \left (a^2-b^2\right ) (a+b \coth (c+d x))^3}+\frac {\int \frac {a+i b \tan \left (i c+i d x+\frac {\pi }{2}\right )}{\left (a-i b \tan \left (i c+i d x+\frac {\pi }{2}\right )\right )^3}dx}{a^2-b^2}\) |
| \(\Big \downarrow \) 4012 |
| \(\displaystyle \frac {\frac {\int \frac {a^2-2 b \coth (c+d x) a+b^2}{(a+b \coth (c+d x))^2}dx}{a^2-b^2}+\frac {a b}{d \left (a^2-b^2\right ) (a+b \coth (c+d x))^2}}{a^2-b^2}+\frac {b}{3 d \left (a^2-b^2\right ) (a+b \coth (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {b}{3 d \left (a^2-b^2\right ) (a+b \coth (c+d x))^3}+\frac {\frac {a b}{d \left (a^2-b^2\right ) (a+b \coth (c+d x))^2}+\frac {\int \frac {a^2+2 i b \tan \left (i c+i d x+\frac {\pi }{2}\right ) a+b^2}{\left (a-i b \tan \left (i c+i d x+\frac {\pi }{2}\right )\right )^2}dx}{a^2-b^2}}{a^2-b^2}\) |
| \(\Big \downarrow \) 4012 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {a \left (a^2+3 b^2\right )-b \left (3 a^2+b^2\right ) \coth (c+d x)}{a+b \coth (c+d x)}dx}{a^2-b^2}+\frac {b \left (3 a^2+b^2\right )}{d \left (a^2-b^2\right ) (a+b \coth (c+d x))}}{a^2-b^2}+\frac {a b}{d \left (a^2-b^2\right ) (a+b \coth (c+d x))^2}}{a^2-b^2}+\frac {b}{3 d \left (a^2-b^2\right ) (a+b \coth (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {b}{3 d \left (a^2-b^2\right ) (a+b \coth (c+d x))^3}+\frac {\frac {a b}{d \left (a^2-b^2\right ) (a+b \coth (c+d x))^2}+\frac {\frac {b \left (3 a^2+b^2\right )}{d \left (a^2-b^2\right ) (a+b \coth (c+d x))}+\frac {\int \frac {a \left (a^2+3 b^2\right )+i b \left (3 a^2+b^2\right ) \tan \left (i c+i d x+\frac {\pi }{2}\right )}{a-i b \tan \left (i c+i d x+\frac {\pi }{2}\right )}dx}{a^2-b^2}}{a^2-b^2}}{a^2-b^2}\) |
| \(\Big \downarrow \) 4014 |
| \(\displaystyle \frac {b}{3 d \left (a^2-b^2\right ) (a+b \coth (c+d x))^3}+\frac {\frac {a b}{d \left (a^2-b^2\right ) (a+b \coth (c+d x))^2}+\frac {\frac {b \left (3 a^2+b^2\right )}{d \left (a^2-b^2\right ) (a+b \coth (c+d x))}+\frac {\frac {x \left (a^4+6 a^2 b^2+b^4\right )}{a^2-b^2}-\frac {4 i a b \left (a^2+b^2\right ) \int -\frac {i (b+a \coth (c+d x))}{a+b \coth (c+d x)}dx}{a^2-b^2}}{a^2-b^2}}{a^2-b^2}}{a^2-b^2}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {\frac {\frac {\frac {x \left (a^4+6 a^2 b^2+b^4\right )}{a^2-b^2}-\frac {4 a b \left (a^2+b^2\right ) \int \frac {b+a \coth (c+d x)}{a+b \coth (c+d x)}dx}{a^2-b^2}}{a^2-b^2}+\frac {b \left (3 a^2+b^2\right )}{d \left (a^2-b^2\right ) (a+b \coth (c+d x))}}{a^2-b^2}+\frac {a b}{d \left (a^2-b^2\right ) (a+b \coth (c+d x))^2}}{a^2-b^2}+\frac {b}{3 d \left (a^2-b^2\right ) (a+b \coth (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {b}{3 d \left (a^2-b^2\right ) (a+b \coth (c+d x))^3}+\frac {\frac {a b}{d \left (a^2-b^2\right ) (a+b \coth (c+d x))^2}+\frac {\frac {b \left (3 a^2+b^2\right )}{d \left (a^2-b^2\right ) (a+b \coth (c+d x))}+\frac {\frac {x \left (a^4+6 a^2 b^2+b^4\right )}{a^2-b^2}-\frac {4 a b \left (a^2+b^2\right ) \int \frac {b-i a \tan \left (i c+i d x+\frac {\pi }{2}\right )}{a-i b \tan \left (i c+i d x+\frac {\pi }{2}\right )}dx}{a^2-b^2}}{a^2-b^2}}{a^2-b^2}}{a^2-b^2}\) |
| \(\Big \downarrow \) 4013 |
| \(\displaystyle \frac {b}{3 d \left (a^2-b^2\right ) (a+b \coth (c+d x))^3}+\frac {\frac {a b}{d \left (a^2-b^2\right ) (a+b \coth (c+d x))^2}+\frac {\frac {b \left (3 a^2+b^2\right )}{d \left (a^2-b^2\right ) (a+b \coth (c+d x))}+\frac {\frac {x \left (a^4+6 a^2 b^2+b^4\right )}{a^2-b^2}-\frac {4 a b \left (a^2+b^2\right ) \log (a \sinh (c+d x)+b \cosh (c+d x))}{d \left (a^2-b^2\right )}}{a^2-b^2}}{a^2-b^2}}{a^2-b^2}\) |
b/(3*(a^2 - b^2)*d*(a + b*Coth[c + d*x])^3) + ((a*b)/((a^2 - b^2)*d*(a + b *Coth[c + d*x])^2) + ((b*(3*a^2 + b^2))/((a^2 - b^2)*d*(a + b*Coth[c + d*x ])) + (((a^4 + 6*a^2*b^2 + b^4)*x)/(a^2 - b^2) - (4*a*b*(a^2 + b^2)*Log[b* Cosh[c + d*x] + a*Sinh[c + d*x]])/((a^2 - b^2)*d))/(a^2 - b^2))/(a^2 - b^2 ))/(a^2 - b^2)
3.1.84.3.1 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_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((a + b*Tan[c + d*x])^(n + 1)/(d*(n + 1)*(a^2 + b^2))), x] + Simp[1/(a^2 + b^2) Int[(a - b*Tan[c + d*x])*(a + b*Tan[c + d*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && LtQ[n, -1]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*c - a*d)*((a + b*Tan[e + f*x])^(m + 1)/ (f*(m + 1)*(a^2 + b^2))), x] + Simp[1/(a^2 + b^2) Int[(a + b*Tan[e + f*x] )^(m + 1)*Simp[a*c + b*d - (b*c - a*d)*Tan[e + f*x], x], x], x] /; FreeQ[{a , b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && LtQ[m, -1 ]
Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_) + (b_.)*tan[(e_.) + (f_.)* (x_)]), x_Symbol] :> Simp[(c/(b*f))*Log[RemoveContent[a*Cos[e + f*x] + b*Si n[e + f*x], x]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && EqQ[a*c + b*d, 0]
Int[((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_.) + (b_.)*tan[(e_.) + (f_. )*(x_)]), x_Symbol] :> Simp[(a*c + b*d)*(x/(a^2 + b^2)), x] + Simp[(b*c - a *d)/(a^2 + b^2) Int[(b - a*Tan[e + f*x])/(a + b*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && N eQ[a*c + b*d, 0]
Time = 0.35 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.96
| method | result | size |
| derivativedivides | \(\frac {\frac {\ln \left (\coth \left (d x +c \right )+1\right )}{2 \left (a -b \right )^{4}}+\frac {b}{3 \left (a -b \right ) \left (a +b \right ) \left (a +b \coth \left (d x +c \right )\right )^{3}}+\frac {a b}{\left (a +b \right )^{2} \left (a -b \right )^{2} \left (a +b \coth \left (d x +c \right )\right )^{2}}+\frac {b \left (3 a^{2}+b^{2}\right )}{\left (a +b \right )^{3} \left (a -b \right )^{3} \left (a +b \coth \left (d x +c \right )\right )}-\frac {4 b a \left (a^{2}+b^{2}\right ) \ln \left (a +b \coth \left (d x +c \right )\right )}{\left (a +b \right )^{4} \left (a -b \right )^{4}}-\frac {\ln \left (\coth \left (d x +c \right )-1\right )}{2 \left (a +b \right )^{4}}}{d}\) | \(163\) |
| default | \(\frac {\frac {\ln \left (\coth \left (d x +c \right )+1\right )}{2 \left (a -b \right )^{4}}+\frac {b}{3 \left (a -b \right ) \left (a +b \right ) \left (a +b \coth \left (d x +c \right )\right )^{3}}+\frac {a b}{\left (a +b \right )^{2} \left (a -b \right )^{2} \left (a +b \coth \left (d x +c \right )\right )^{2}}+\frac {b \left (3 a^{2}+b^{2}\right )}{\left (a +b \right )^{3} \left (a -b \right )^{3} \left (a +b \coth \left (d x +c \right )\right )}-\frac {4 b a \left (a^{2}+b^{2}\right ) \ln \left (a +b \coth \left (d x +c \right )\right )}{\left (a +b \right )^{4} \left (a -b \right )^{4}}-\frac {\ln \left (\coth \left (d x +c \right )-1\right )}{2 \left (a +b \right )^{4}}}{d}\) | \(163\) |
| parallelrisch | \(\frac {-4 b \,a^{2} \left (a^{2}+b^{2}\right ) \left (b +a \tanh \left (d x +c \right )\right )^{3} \ln \left (b +a \tanh \left (d x +c \right )\right )+4 b \,a^{2} \left (a^{2}+b^{2}\right ) \left (b +a \tanh \left (d x +c \right )\right )^{3} \ln \left (1-\tanh \left (d x +c \right )\right )+\left (\left (a^{6} d x +b \left (3 d x +2\right ) a^{5}+b^{2} \left (3 d x -2\right ) a^{4}+b^{3} \left (d x -1\right ) a^{3}+a^{2} b^{4}+\frac {a \,b^{5}}{3}-\frac {b^{6}}{3}\right ) a \tanh \left (d x +c \right )^{3}+3 a^{3} b d x \left (a +b \right )^{3} \tanh \left (d x +c \right )^{2}+3 \left (a^{3} d x +b \left (3 d x -\frac {4}{3}\right ) a^{2}+3 \left (d x +\frac {4}{9}\right ) b^{2} a +b^{3} d x \right ) b^{2} a^{2} \tanh \left (d x +c \right )+\left (a^{4} d x +b \left (3 d x -\frac {7}{3}\right ) a^{3}+3 \left (d x +\frac {7}{9}\right ) b^{2} a^{2}+b^{3} \left (d x -\frac {1}{3}\right ) a +\frac {b^{4}}{3}\right ) b^{3}\right ) \left (a +b \right )}{a \left (a -b \right )^{4} \left (a +b \right )^{4} d \left (b +a \tanh \left (d x +c \right )\right )^{3}}\) | \(301\) |
| risch | \(\frac {x}{a^{4}+4 a^{3} b +6 a^{2} b^{2}+4 a \,b^{3}+b^{4}}+\frac {8 b \,a^{3} x}{a^{8}-4 a^{6} b^{2}+6 a^{4} b^{4}-4 a^{2} b^{6}+b^{8}}+\frac {8 b^{3} a x}{a^{8}-4 a^{6} b^{2}+6 a^{4} b^{4}-4 a^{2} b^{6}+b^{8}}+\frac {8 b \,a^{3} c}{d \left (a^{8}-4 a^{6} b^{2}+6 a^{4} b^{4}-4 a^{2} b^{6}+b^{8}\right )}+\frac {8 b^{3} a c}{d \left (a^{8}-4 a^{6} b^{2}+6 a^{4} b^{4}-4 a^{2} b^{6}+b^{8}\right )}-\frac {4 b^{2} \left (9 a^{4} {\mathrm e}^{4 d x +4 c}+12 a^{3} b \,{\mathrm e}^{4 d x +4 c}+3 b^{4} {\mathrm e}^{4 d x +4 c}-18 a^{4} {\mathrm e}^{2 d x +2 c}+6 a^{3} b \,{\mathrm e}^{2 d x +2 c}+15 a^{2} b^{2} {\mathrm e}^{2 d x +2 c}-6 a \,b^{3} {\mathrm e}^{2 d x +2 c}+3 b^{4} {\mathrm e}^{2 d x +2 c}+9 a^{4}-18 a^{3} b +11 a^{2} b^{2}-4 a \,b^{3}+2 b^{4}\right )}{3 \left (a^{4}+4 a^{3} b +6 a^{2} b^{2}+4 a \,b^{3}+b^{4}\right ) \left ({\mathrm e}^{2 d x +2 c} a +b \,{\mathrm e}^{2 d x +2 c}-a +b \right )^{3} d \left (a -b \right )^{3}}-\frac {4 b \,a^{3} \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {a -b}{a +b}\right )}{d \left (a^{8}-4 a^{6} b^{2}+6 a^{4} b^{4}-4 a^{2} b^{6}+b^{8}\right )}-\frac {4 b^{3} a \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {a -b}{a +b}\right )}{d \left (a^{8}-4 a^{6} b^{2}+6 a^{4} b^{4}-4 a^{2} b^{6}+b^{8}\right )}\) | \(551\) |
1/d*(1/2/(a-b)^4*ln(coth(d*x+c)+1)+1/3*b/(a-b)/(a+b)/(a+b*coth(d*x+c))^3+a *b/(a+b)^2/(a-b)^2/(a+b*coth(d*x+c))^2+b*(3*a^2+b^2)/(a+b)^3/(a-b)^3/(a+b* coth(d*x+c))-4*b*a*(a^2+b^2)/(a+b)^4/(a-b)^4*ln(a+b*coth(d*x+c))-1/2/(a+b) ^4*ln(coth(d*x+c)-1))
Leaf count of result is larger than twice the leaf count of optimal. 3698 vs. \(2 (167) = 334\).
Time = 0.32 (sec) , antiderivative size = 3698, normalized size of antiderivative = 21.88 \[ \int \frac {1}{(a+b \coth (c+d x))^4} \, dx=\text {Too large to display} \]
1/3*(3*(a^7 + 7*a^6*b + 21*a^5*b^2 + 35*a^4*b^3 + 35*a^3*b^4 + 21*a^2*b^5 + 7*a*b^6 + b^7)*d*x*cosh(d*x + c)^6 + 18*(a^7 + 7*a^6*b + 21*a^5*b^2 + 35 *a^4*b^3 + 35*a^3*b^4 + 21*a^2*b^5 + 7*a*b^6 + b^7)*d*x*cosh(d*x + c)*sinh (d*x + c)^5 + 3*(a^7 + 7*a^6*b + 21*a^5*b^2 + 35*a^4*b^3 + 35*a^3*b^4 + 21 *a^2*b^5 + 7*a*b^6 + b^7)*d*x*sinh(d*x + c)^6 - 36*a^5*b^2 + 108*a^4*b^3 - 116*a^3*b^4 + 60*a^2*b^5 - 24*a*b^6 + 8*b^7 - 3*(12*a^5*b^2 + 4*a^4*b^3 - 16*a^3*b^4 + 4*a*b^6 - 4*b^7 + 3*(a^7 + 5*a^6*b + 9*a^5*b^2 + 5*a^4*b^3 - 5*a^3*b^4 - 9*a^2*b^5 - 5*a*b^6 - b^7)*d*x)*cosh(d*x + c)^4 - 3*(12*a^5*b ^2 + 4*a^4*b^3 - 16*a^3*b^4 + 4*a*b^6 - 4*b^7 - 15*(a^7 + 7*a^6*b + 21*a^5 *b^2 + 35*a^4*b^3 + 35*a^3*b^4 + 21*a^2*b^5 + 7*a*b^6 + b^7)*d*x*cosh(d*x + c)^2 + 3*(a^7 + 5*a^6*b + 9*a^5*b^2 + 5*a^4*b^3 - 5*a^3*b^4 - 9*a^2*b^5 - 5*a*b^6 - b^7)*d*x)*sinh(d*x + c)^4 + 12*(5*(a^7 + 7*a^6*b + 21*a^5*b^2 + 35*a^4*b^3 + 35*a^3*b^4 + 21*a^2*b^5 + 7*a*b^6 + b^7)*d*x*cosh(d*x + c)^ 3 - (12*a^5*b^2 + 4*a^4*b^3 - 16*a^3*b^4 + 4*a*b^6 - 4*b^7 + 3*(a^7 + 5*a^ 6*b + 9*a^5*b^2 + 5*a^4*b^3 - 5*a^3*b^4 - 9*a^2*b^5 - 5*a*b^6 - b^7)*d*x)* cosh(d*x + c))*sinh(d*x + c)^3 - 3*(a^7 + a^6*b - 3*a^5*b^2 - 3*a^4*b^3 + 3*a^3*b^4 + 3*a^2*b^5 - a*b^6 - b^7)*d*x + 3*(24*a^5*b^2 - 32*a^4*b^3 - 12 *a^3*b^4 + 28*a^2*b^5 - 12*a*b^6 + 4*b^7 + 3*(a^7 + 3*a^6*b + a^5*b^2 - 5* a^4*b^3 - 5*a^3*b^4 + a^2*b^5 + 3*a*b^6 + b^7)*d*x)*cosh(d*x + c)^2 + 3*(2 4*a^5*b^2 - 32*a^4*b^3 - 12*a^3*b^4 + 28*a^2*b^5 - 12*a*b^6 + 4*b^7 + 1...
Exception generated. \[ \int \frac {1}{(a+b \coth (c+d x))^4} \, dx=\text {Exception raised: TypeError} \]
Leaf count of result is larger than twice the leaf count of optimal. 522 vs. \(2 (167) = 334\).
Time = 0.24 (sec) , antiderivative size = 522, normalized size of antiderivative = 3.09 \[ \int \frac {1}{(a+b \coth (c+d x))^4} \, dx=-\frac {4 \, {\left (a^{3} b + a b^{3}\right )} \log \left (-{\left (a - b\right )} e^{\left (-2 \, d x - 2 \, c\right )} + a + b\right )}{{\left (a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}\right )} d} - \frac {4 \, {\left (9 \, a^{4} b^{2} + 18 \, a^{3} b^{3} + 11 \, a^{2} b^{4} + 4 \, a b^{5} + 2 \, b^{6} - 3 \, {\left (6 \, a^{4} b^{2} + 2 \, a^{3} b^{3} - 5 \, a^{2} b^{4} - 2 \, a b^{5} - b^{6}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, {\left (3 \, a^{4} b^{2} - 4 \, a^{3} b^{3} + b^{6}\right )} e^{\left (-4 \, d x - 4 \, c\right )}\right )}}{3 \, {\left (a^{10} + 2 \, a^{9} b - 3 \, a^{8} b^{2} - 8 \, a^{7} b^{3} + 2 \, a^{6} b^{4} + 12 \, a^{5} b^{5} + 2 \, a^{4} b^{6} - 8 \, a^{3} b^{7} - 3 \, a^{2} b^{8} + 2 \, a b^{9} + b^{10} - 3 \, {\left (a^{10} - 5 \, a^{8} b^{2} + 10 \, a^{6} b^{4} - 10 \, a^{4} b^{6} + 5 \, a^{2} b^{8} - b^{10}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, {\left (a^{10} - 2 \, a^{9} b - 3 \, a^{8} b^{2} + 8 \, a^{7} b^{3} + 2 \, a^{6} b^{4} - 12 \, a^{5} b^{5} + 2 \, a^{4} b^{6} + 8 \, a^{3} b^{7} - 3 \, a^{2} b^{8} - 2 \, a b^{9} + b^{10}\right )} e^{\left (-4 \, d x - 4 \, c\right )} - {\left (a^{10} - 4 \, a^{9} b + 3 \, a^{8} b^{2} + 8 \, a^{7} b^{3} - 14 \, a^{6} b^{4} + 14 \, a^{4} b^{6} - 8 \, a^{3} b^{7} - 3 \, a^{2} b^{8} + 4 \, a b^{9} - b^{10}\right )} e^{\left (-6 \, d x - 6 \, c\right )}\right )} d} + \frac {d x + c}{{\left (a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} d} \]
-4*(a^3*b + a*b^3)*log(-(a - b)*e^(-2*d*x - 2*c) + a + b)/((a^8 - 4*a^6*b^ 2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8)*d) - 4/3*(9*a^4*b^2 + 18*a^3*b^3 + 11*a^2 *b^4 + 4*a*b^5 + 2*b^6 - 3*(6*a^4*b^2 + 2*a^3*b^3 - 5*a^2*b^4 - 2*a*b^5 - b^6)*e^(-2*d*x - 2*c) + 3*(3*a^4*b^2 - 4*a^3*b^3 + b^6)*e^(-4*d*x - 4*c))/ ((a^10 + 2*a^9*b - 3*a^8*b^2 - 8*a^7*b^3 + 2*a^6*b^4 + 12*a^5*b^5 + 2*a^4* b^6 - 8*a^3*b^7 - 3*a^2*b^8 + 2*a*b^9 + b^10 - 3*(a^10 - 5*a^8*b^2 + 10*a^ 6*b^4 - 10*a^4*b^6 + 5*a^2*b^8 - b^10)*e^(-2*d*x - 2*c) + 3*(a^10 - 2*a^9* b - 3*a^8*b^2 + 8*a^7*b^3 + 2*a^6*b^4 - 12*a^5*b^5 + 2*a^4*b^6 + 8*a^3*b^7 - 3*a^2*b^8 - 2*a*b^9 + b^10)*e^(-4*d*x - 4*c) - (a^10 - 4*a^9*b + 3*a^8* b^2 + 8*a^7*b^3 - 14*a^6*b^4 + 14*a^4*b^6 - 8*a^3*b^7 - 3*a^2*b^8 + 4*a*b^ 9 - b^10)*e^(-6*d*x - 6*c))*d) + (d*x + c)/((a^4 + 4*a^3*b + 6*a^2*b^2 + 4 *a*b^3 + b^4)*d)
Time = 0.31 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.79 \[ \int \frac {1}{(a+b \coth (c+d x))^4} \, dx=-\frac {\frac {12 \, {\left (a^{3} b + a b^{3}\right )} \log \left ({\left | a e^{\left (2 \, d x + 2 \, c\right )} + b e^{\left (2 \, d x + 2 \, c\right )} - a + b \right |}\right )}{a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}} - \frac {3 \, {\left (d x + c\right )}}{a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}} + \frac {4 \, {\left (3 \, {\left (3 \, a^{4} b^{2} - 2 \, a^{3} b^{3} - 2 \, a^{2} b^{4} + 2 \, a b^{5} - b^{6}\right )} e^{\left (4 \, d x + 4 \, c\right )} - 3 \, {\left (6 \, a^{4} b^{2} - 14 \, a^{3} b^{3} + 11 \, a^{2} b^{4} - 4 \, a b^{5} + b^{6}\right )} e^{\left (2 \, d x + 2 \, c\right )} + \frac {9 \, a^{5} b^{2} - 27 \, a^{4} b^{3} + 29 \, a^{3} b^{4} - 15 \, a^{2} b^{5} + 6 \, a b^{6} - 2 \, b^{7}}{a + b}\right )}}{{\left (a e^{\left (2 \, d x + 2 \, c\right )} + b e^{\left (2 \, d x + 2 \, c\right )} - a + b\right )}^{3} {\left (a + b\right )}^{3} {\left (a - b\right )}^{4}}}{3 \, d} \]
-1/3*(12*(a^3*b + a*b^3)*log(abs(a*e^(2*d*x + 2*c) + b*e^(2*d*x + 2*c) - a + b))/(a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8) - 3*(d*x + c)/(a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4) + 4*(3*(3*a^4*b^2 - 2*a^3*b^3 - 2*a ^2*b^4 + 2*a*b^5 - b^6)*e^(4*d*x + 4*c) - 3*(6*a^4*b^2 - 14*a^3*b^3 + 11*a ^2*b^4 - 4*a*b^5 + b^6)*e^(2*d*x + 2*c) + (9*a^5*b^2 - 27*a^4*b^3 + 29*a^3 *b^4 - 15*a^2*b^5 + 6*a*b^6 - 2*b^7)/(a + b))/((a*e^(2*d*x + 2*c) + b*e^(2 *d*x + 2*c) - a + b)^3*(a + b)^3*(a - b)^4))/d
Time = 2.06 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.83 \[ \int \frac {1}{(a+b \coth (c+d x))^4} \, dx=\frac {x}{{\left (a-b\right )}^4}-\frac {\ln \left (b-a+a\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+b\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}\right )\,\left (4\,a^3\,b+4\,a\,b^3\right )}{d\,a^8-4\,d\,a^6\,b^2+6\,d\,a^4\,b^4-4\,d\,a^2\,b^6+d\,b^8}-\frac {4\,\left (3\,a^2\,b^2-2\,a\,b^3+b^4\right )}{d\,{\left (a+b\right )}^4\,{\left (a-b\right )}^3\,\left (b-a+{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (a+b\right )\right )}-\frac {8\,b^4}{3\,d\,{\left (a+b\right )}^4\,\left (a-b\right )\,\left ({\mathrm {e}}^{6\,c+6\,d\,x}\,{\left (a+b\right )}^3-{\left (a-b\right )}^3+3\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (a+b\right )\,{\left (a-b\right )}^2-3\,{\mathrm {e}}^{4\,c+4\,d\,x}\,{\left (a+b\right )}^2\,\left (a-b\right )\right )}+\frac {4\,\left (2\,a\,b^3-b^4\right )}{d\,{\left (a+b\right )}^4\,{\left (a-b\right )}^2\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}\,{\left (a+b\right )}^2+{\left (a-b\right )}^2-2\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (a+b\right )\,\left (a-b\right )\right )} \]
x/(a - b)^4 - (log(b - a + a*exp(2*c)*exp(2*d*x) + b*exp(2*c)*exp(2*d*x))* (4*a*b^3 + 4*a^3*b))/(a^8*d + b^8*d - 4*a^2*b^6*d + 6*a^4*b^4*d - 4*a^6*b^ 2*d) - (4*(b^4 - 2*a*b^3 + 3*a^2*b^2))/(d*(a + b)^4*(a - b)^3*(b - a + exp (2*c + 2*d*x)*(a + b))) - (8*b^4)/(3*d*(a + b)^4*(a - b)*(exp(6*c + 6*d*x) *(a + b)^3 - (a - b)^3 + 3*exp(2*c + 2*d*x)*(a + b)*(a - b)^2 - 3*exp(4*c + 4*d*x)*(a + b)^2*(a - b))) + (4*(2*a*b^3 - b^4))/(d*(a + b)^4*(a - b)^2* (exp(4*c + 4*d*x)*(a + b)^2 + (a - b)^2 - 2*exp(2*c + 2*d*x)*(a + b)*(a - b)))