Integrand size = 13, antiderivative size = 170 \[ \int \frac {1}{x^3 \coth ^{-1}(\tanh (a+b x))^3} \, dx=-\frac {3 b^2}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3 \coth ^{-1}(\tanh (a+b x))^2}+\frac {2 b}{x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \coth ^{-1}(\tanh (a+b x))^2}+\frac {1}{2 x^2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))^2}+\frac {6 b^2}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^4 \coth ^{-1}(\tanh (a+b x))}-\frac {6 b^2 \log (x)}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^5}+\frac {6 b^2 \log \left (\coth ^{-1}(\tanh (a+b x))\right )}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^5} \]
-3*b^2/(b*x-arccoth(tanh(b*x+a)))^3/arccoth(tanh(b*x+a))^2+2*b/x/(b*x-arcc oth(tanh(b*x+a)))^2/arccoth(tanh(b*x+a))^2+1/2/x^2/(b*x-arccoth(tanh(b*x+a )))/arccoth(tanh(b*x+a))^2+6*b^2/(b*x-arccoth(tanh(b*x+a)))^4/arccoth(tanh (b*x+a))-6*b^2*ln(x)/(b*x-arccoth(tanh(b*x+a)))^5+6*b^2*ln(arccoth(tanh(b* x+a)))/(b*x-arccoth(tanh(b*x+a)))^5
Time = 0.03 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.63 \[ \int \frac {1}{x^3 \coth ^{-1}(\tanh (a+b x))^3} \, dx=\frac {-b^4 x^4+8 b^3 x^3 \coth ^{-1}(\tanh (a+b x))-8 b x \coth ^{-1}(\tanh (a+b x))^3+\coth ^{-1}(\tanh (a+b x))^4-12 b^2 x^2 \coth ^{-1}(\tanh (a+b x))^2 \left (\log (x)-\log \left (\coth ^{-1}(\tanh (a+b x))\right )\right )}{2 x^2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^5 \coth ^{-1}(\tanh (a+b x))^2} \]
(-(b^4*x^4) + 8*b^3*x^3*ArcCoth[Tanh[a + b*x]] - 8*b*x*ArcCoth[Tanh[a + b* x]]^3 + ArcCoth[Tanh[a + b*x]]^4 - 12*b^2*x^2*ArcCoth[Tanh[a + b*x]]^2*(Lo g[x] - Log[ArcCoth[Tanh[a + b*x]]]))/(2*x^2*(b*x - ArcCoth[Tanh[a + b*x]]) ^5*ArcCoth[Tanh[a + b*x]]^2)
Time = 0.57 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.36, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.615, Rules used = {2602, 2602, 2594, 2594, 2591, 14, 2588, 14}
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.
\(\displaystyle \int \frac {1}{x^3 \coth ^{-1}(\tanh (a+b x))^3} \, dx\) |
\(\Big \downarrow \) 2602 |
\(\displaystyle \frac {2 b \int \frac {1}{x^2 \coth ^{-1}(\tanh (a+b x))^3}dx}{b x-\coth ^{-1}(\tanh (a+b x))}+\frac {1}{2 x^2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))^2}\) |
\(\Big \downarrow \) 2602 |
\(\displaystyle \frac {2 b \left (\frac {3 b \int \frac {1}{x \coth ^{-1}(\tanh (a+b x))^3}dx}{b x-\coth ^{-1}(\tanh (a+b x))}+\frac {1}{x \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))^2}\right )}{b x-\coth ^{-1}(\tanh (a+b x))}+\frac {1}{2 x^2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))^2}\) |
\(\Big \downarrow \) 2594 |
\(\displaystyle \frac {2 b \left (\frac {3 b \left (-\frac {\int \frac {1}{x \coth ^{-1}(\tanh (a+b x))^2}dx}{b x-\coth ^{-1}(\tanh (a+b x))}-\frac {1}{2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))^2}\right )}{b x-\coth ^{-1}(\tanh (a+b x))}+\frac {1}{x \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))^2}\right )}{b x-\coth ^{-1}(\tanh (a+b x))}+\frac {1}{2 x^2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))^2}\) |
\(\Big \downarrow \) 2594 |
\(\displaystyle \frac {2 b \left (\frac {3 b \left (-\frac {-\frac {\int \frac {1}{x \coth ^{-1}(\tanh (a+b x))}dx}{b x-\coth ^{-1}(\tanh (a+b x))}-\frac {1}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))}}{b x-\coth ^{-1}(\tanh (a+b x))}-\frac {1}{2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))^2}\right )}{b x-\coth ^{-1}(\tanh (a+b x))}+\frac {1}{x \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))^2}\right )}{b x-\coth ^{-1}(\tanh (a+b x))}+\frac {1}{2 x^2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))^2}\) |
\(\Big \downarrow \) 2591 |
\(\displaystyle \frac {2 b \left (\frac {3 b \left (-\frac {-\frac {\frac {b \int \frac {1}{\coth ^{-1}(\tanh (a+b x))}dx}{b x-\coth ^{-1}(\tanh (a+b x))}-\frac {\int \frac {1}{x}dx}{b x-\coth ^{-1}(\tanh (a+b x))}}{b x-\coth ^{-1}(\tanh (a+b x))}-\frac {1}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))}}{b x-\coth ^{-1}(\tanh (a+b x))}-\frac {1}{2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))^2}\right )}{b x-\coth ^{-1}(\tanh (a+b x))}+\frac {1}{x \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))^2}\right )}{b x-\coth ^{-1}(\tanh (a+b x))}+\frac {1}{2 x^2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))^2}\) |
\(\Big \downarrow \) 14 |
\(\displaystyle \frac {2 b \left (\frac {3 b \left (-\frac {-\frac {\frac {b \int \frac {1}{\coth ^{-1}(\tanh (a+b x))}dx}{b x-\coth ^{-1}(\tanh (a+b x))}-\frac {\log (x)}{b x-\coth ^{-1}(\tanh (a+b x))}}{b x-\coth ^{-1}(\tanh (a+b x))}-\frac {1}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))}}{b x-\coth ^{-1}(\tanh (a+b x))}-\frac {1}{2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))^2}\right )}{b x-\coth ^{-1}(\tanh (a+b x))}+\frac {1}{x \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))^2}\right )}{b x-\coth ^{-1}(\tanh (a+b x))}+\frac {1}{2 x^2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))^2}\) |
\(\Big \downarrow \) 2588 |
\(\displaystyle \frac {2 b \left (\frac {3 b \left (-\frac {-\frac {\frac {\int \frac {1}{\coth ^{-1}(\tanh (a+b x))}d\coth ^{-1}(\tanh (a+b x))}{b x-\coth ^{-1}(\tanh (a+b x))}-\frac {\log (x)}{b x-\coth ^{-1}(\tanh (a+b x))}}{b x-\coth ^{-1}(\tanh (a+b x))}-\frac {1}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))}}{b x-\coth ^{-1}(\tanh (a+b x))}-\frac {1}{2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))^2}\right )}{b x-\coth ^{-1}(\tanh (a+b x))}+\frac {1}{x \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))^2}\right )}{b x-\coth ^{-1}(\tanh (a+b x))}+\frac {1}{2 x^2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))^2}\) |
\(\Big \downarrow \) 14 |
\(\displaystyle \frac {1}{2 x^2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))^2}+\frac {2 b \left (\frac {1}{x \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))^2}+\frac {3 b \left (-\frac {1}{2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))^2}-\frac {-\frac {1}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))}-\frac {\frac {\log \left (\coth ^{-1}(\tanh (a+b x))\right )}{b x-\coth ^{-1}(\tanh (a+b x))}-\frac {\log (x)}{b x-\coth ^{-1}(\tanh (a+b x))}}{b x-\coth ^{-1}(\tanh (a+b x))}}{b x-\coth ^{-1}(\tanh (a+b x))}\right )}{b x-\coth ^{-1}(\tanh (a+b x))}\right )}{b x-\coth ^{-1}(\tanh (a+b x))}\) |
1/(2*x^2*(b*x - ArcCoth[Tanh[a + b*x]])*ArcCoth[Tanh[a + b*x]]^2) + (2*b*( 1/(x*(b*x - ArcCoth[Tanh[a + b*x]])*ArcCoth[Tanh[a + b*x]]^2) + (3*b*(-1/2 *1/((b*x - ArcCoth[Tanh[a + b*x]])*ArcCoth[Tanh[a + b*x]]^2) - (-(1/((b*x - ArcCoth[Tanh[a + b*x]])*ArcCoth[Tanh[a + b*x]])) - (-(Log[x]/(b*x - ArcC oth[Tanh[a + b*x]])) + Log[ArcCoth[Tanh[a + b*x]]]/(b*x - ArcCoth[Tanh[a + b*x]]))/(b*x - ArcCoth[Tanh[a + b*x]]))/(b*x - ArcCoth[Tanh[a + b*x]])))/ (b*x - ArcCoth[Tanh[a + b*x]])))/(b*x - ArcCoth[Tanh[a + b*x]])
3.2.83.3.1 Defintions of rubi rules used
Int[(u_)^(m_.), x_Symbol] :> With[{c = Simplify[D[u, x]]}, Simp[1/c Subst [Int[x^m, x], x, u], x]] /; FreeQ[m, x] && PiecewiseLinearQ[u, x]
Int[1/((u_)*(v_)), x_Symbol] :> With[{a = Simplify[D[u, x]], b = Simplify[D [v, x]]}, Simp[b/(b*u - a*v) Int[1/v, x], x] - Simp[a/(b*u - a*v) Int[1 /u, x], x] /; NeQ[b*u - a*v, 0]] /; PiecewiseLinearQ[u, v, x]
Int[(v_)^(n_)/(u_), x_Symbol] :> With[{a = Simplify[D[u, x]], b = Simplify[ D[v, x]]}, Simp[v^(n + 1)/((n + 1)*(b*u - a*v)), x] - Simp[a*((n + 1)/((n + 1)*(b*u - a*v))) Int[v^(n + 1)/u, x], x] /; NeQ[b*u - a*v, 0]] /; Piecew iseLinearQ[u, v, x] && LtQ[n, -1]
Int[(u_)^(m_)*(v_)^(n_), x_Symbol] :> With[{a = Simplify[D[u, x]], b = Simp lify[D[v, x]]}, Simp[(-u^(m + 1))*(v^(n + 1)/((m + 1)*(b*u - a*v))), x] + S imp[b*((m + n + 2)/((m + 1)*(b*u - a*v))) Int[u^(m + 1)*v^n, x], x] /; Ne Q[b*u - a*v, 0]] /; PiecewiseLinearQ[u, v, x] && NeQ[m + n + 2, 0] && LtQ[m , -1]
Timed out.
\[\int \frac {1}{x^{3} \operatorname {arccoth}\left (\tanh \left (b x +a \right )\right )^{3}}d x\]
Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 461, normalized size of antiderivative = 2.71 \[ \int \frac {1}{x^3 \coth ^{-1}(\tanh (a+b x))^3} \, dx=\frac {4 \, {\left (192 \, a b^{3} x^{3} + 288 \, a^{2} b^{2} x^{2} + 64 \, a^{3} b x - \pi ^{4} - 8 i \, \pi ^{3} {\left (b x - a\right )} - 16 \, a^{4} - 24 \, \pi ^{2} {\left (3 \, b^{2} x^{2} + 2 \, a b x - a^{2}\right )} + 32 i \, \pi {\left (3 \, b^{3} x^{3} + 9 \, a b^{2} x^{2} + 3 \, a^{2} b x - a^{3}\right )} - 48 \, {\left (4 \, b^{4} x^{4} + 8 \, a b^{3} x^{3} - \pi ^{2} b^{2} x^{2} + 4 \, a^{2} b^{2} x^{2} + 4 i \, \pi {\left (b^{3} x^{3} + a b^{2} x^{2}\right )}\right )} \log \left (i \, \pi + 2 \, b x + 2 \, a\right ) + 48 \, {\left (4 \, b^{4} x^{4} + 8 \, a b^{3} x^{3} - \pi ^{2} b^{2} x^{2} + 4 \, a^{2} b^{2} x^{2} + 4 i \, \pi {\left (b^{3} x^{3} + a b^{2} x^{2}\right )}\right )} \log \left (x\right )\right )}}{128 \, a^{5} b^{2} x^{4} + 256 \, a^{6} b x^{3} - i \, \pi ^{7} x^{2} + 128 \, a^{7} x^{2} - 2 \, \pi ^{6} {\left (2 \, b x^{3} + 7 \, a x^{2}\right )} + 4 i \, \pi ^{5} {\left (b^{2} x^{4} + 12 \, a b x^{3} + 21 \, a^{2} x^{2}\right )} + 40 \, \pi ^{4} {\left (a b^{2} x^{4} + 6 \, a^{2} b x^{3} + 7 \, a^{3} x^{2}\right )} - 80 i \, \pi ^{3} {\left (2 \, a^{2} b^{2} x^{4} + 8 \, a^{3} b x^{3} + 7 \, a^{4} x^{2}\right )} - 32 \, \pi ^{2} {\left (10 \, a^{3} b^{2} x^{4} + 30 \, a^{4} b x^{3} + 21 \, a^{5} x^{2}\right )} + 64 i \, \pi {\left (5 \, a^{4} b^{2} x^{4} + 12 \, a^{5} b x^{3} + 7 \, a^{6} x^{2}\right )}} \]
4*(192*a*b^3*x^3 + 288*a^2*b^2*x^2 + 64*a^3*b*x - pi^4 - 8*I*pi^3*(b*x - a ) - 16*a^4 - 24*pi^2*(3*b^2*x^2 + 2*a*b*x - a^2) + 32*I*pi*(3*b^3*x^3 + 9* a*b^2*x^2 + 3*a^2*b*x - a^3) - 48*(4*b^4*x^4 + 8*a*b^3*x^3 - pi^2*b^2*x^2 + 4*a^2*b^2*x^2 + 4*I*pi*(b^3*x^3 + a*b^2*x^2))*log(I*pi + 2*b*x + 2*a) + 48*(4*b^4*x^4 + 8*a*b^3*x^3 - pi^2*b^2*x^2 + 4*a^2*b^2*x^2 + 4*I*pi*(b^3*x ^3 + a*b^2*x^2))*log(x))/(128*a^5*b^2*x^4 + 256*a^6*b*x^3 - I*pi^7*x^2 + 1 28*a^7*x^2 - 2*pi^6*(2*b*x^3 + 7*a*x^2) + 4*I*pi^5*(b^2*x^4 + 12*a*b*x^3 + 21*a^2*x^2) + 40*pi^4*(a*b^2*x^4 + 6*a^2*b*x^3 + 7*a^3*x^2) - 80*I*pi^3*( 2*a^2*b^2*x^4 + 8*a^3*b*x^3 + 7*a^4*x^2) - 32*pi^2*(10*a^3*b^2*x^4 + 30*a^ 4*b*x^3 + 21*a^5*x^2) + 64*I*pi*(5*a^4*b^2*x^4 + 12*a^5*b*x^3 + 7*a^6*x^2) )
\[ \int \frac {1}{x^3 \coth ^{-1}(\tanh (a+b x))^3} \, dx=\int \frac {1}{x^{3} \operatorname {acoth}^{3}{\left (\tanh {\left (a + b x \right )} \right )}}\, dx \]
Result contains complex when optimal does not.
Time = 0.78 (sec) , antiderivative size = 331, normalized size of antiderivative = 1.95 \[ \int \frac {1}{x^3 \coth ^{-1}(\tanh (a+b x))^3} \, dx=\frac {192 \, b^{2} \log \left (-i \, \pi + 2 \, b x + 2 \, a\right )}{i \, \pi ^{5} - 10 \, \pi ^{4} a - 40 i \, \pi ^{3} a^{2} + 80 \, \pi ^{2} a^{3} + 80 i \, \pi a^{4} - 32 \, a^{5}} - \frac {192 \, b^{2} \log \left (x\right )}{i \, \pi ^{5} - 10 \, \pi ^{4} a - 40 i \, \pi ^{3} a^{2} + 80 \, \pi ^{2} a^{3} + 80 i \, \pi a^{4} - 32 \, a^{5}} + \frac {4 \, {\left (96 \, b^{3} x^{3} - i \, \pi ^{3} + 6 \, \pi ^{2} a + 12 i \, \pi a^{2} - 8 \, a^{3} - 72 \, {\left (i \, \pi b^{2} - 2 \, a b^{2}\right )} x^{2} - 8 \, {\left (\pi ^{2} b + 4 i \, \pi a b - 4 \, a^{2} b\right )} x\right )}}{4 \, {\left (\pi ^{4} b^{2} + 8 i \, \pi ^{3} a b^{2} - 24 \, \pi ^{2} a^{2} b^{2} - 32 i \, \pi a^{3} b^{2} + 16 \, a^{4} b^{2}\right )} x^{4} - 4 \, {\left (i \, \pi ^{5} b - 10 \, \pi ^{4} a b - 40 i \, \pi ^{3} a^{2} b + 80 \, \pi ^{2} a^{3} b + 80 i \, \pi a^{4} b - 32 \, a^{5} b\right )} x^{3} - {\left (\pi ^{6} + 12 i \, \pi ^{5} a - 60 \, \pi ^{4} a^{2} - 160 i \, \pi ^{3} a^{3} + 240 \, \pi ^{2} a^{4} + 192 i \, \pi a^{5} - 64 \, a^{6}\right )} x^{2}} \]
192*b^2*log(-I*pi + 2*b*x + 2*a)/(I*pi^5 - 10*pi^4*a - 40*I*pi^3*a^2 + 80* pi^2*a^3 + 80*I*pi*a^4 - 32*a^5) - 192*b^2*log(x)/(I*pi^5 - 10*pi^4*a - 40 *I*pi^3*a^2 + 80*pi^2*a^3 + 80*I*pi*a^4 - 32*a^5) + 4*(96*b^3*x^3 - I*pi^3 + 6*pi^2*a + 12*I*pi*a^2 - 8*a^3 - 72*(I*pi*b^2 - 2*a*b^2)*x^2 - 8*(pi^2* b + 4*I*pi*a*b - 4*a^2*b)*x)/(4*(pi^4*b^2 + 8*I*pi^3*a*b^2 - 24*pi^2*a^2*b ^2 - 32*I*pi*a^3*b^2 + 16*a^4*b^2)*x^4 - 4*(I*pi^5*b - 10*pi^4*a*b - 40*I* pi^3*a^2*b + 80*pi^2*a^3*b + 80*I*pi*a^4*b - 32*a^5*b)*x^3 - (pi^6 + 12*I* pi^5*a - 60*pi^4*a^2 - 160*I*pi^3*a^3 + 240*pi^2*a^4 + 192*I*pi*a^5 - 64*a ^6)*x^2)
Result contains complex when optimal does not.
Time = 0.31 (sec) , antiderivative size = 343, normalized size of antiderivative = 2.02 \[ \int \frac {1}{x^3 \coth ^{-1}(\tanh (a+b x))^3} \, dx=\frac {192 i \, b^{2} \log \left (i \, \pi + 2 \, b x + 2 \, a\right )}{\pi ^{5} - 10 i \, \pi ^{4} a - 40 \, \pi ^{3} a^{2} + 80 i \, \pi ^{2} a^{3} + 80 \, \pi a^{4} - 32 i \, a^{5}} - \frac {192 i \, b^{2} \log \left (x\right )}{\pi ^{5} - 10 i \, \pi ^{4} a - 40 \, \pi ^{3} a^{2} + 80 i \, \pi ^{2} a^{3} + 80 \, \pi a^{4} - 32 i \, a^{5}} - \frac {4 \, {\left (i \, \pi - 12 \, b x + 2 \, a\right )}}{\pi ^{4} x^{2} - 8 i \, \pi ^{3} a x^{2} - 24 \, \pi ^{2} a^{2} x^{2} + 32 i \, \pi a^{3} x^{2} + 16 \, a^{4} x^{2}} + \frac {16 \, {\left (12 \, b^{3} x + 7 i \, \pi b^{2} + 14 \, a b^{2}\right )}}{4 \, \pi ^{4} b^{2} x^{2} - 32 i \, \pi ^{3} a b^{2} x^{2} - 96 \, \pi ^{2} a^{2} b^{2} x^{2} + 128 i \, \pi a^{3} b^{2} x^{2} + 64 \, a^{4} b^{2} x^{2} + 4 i \, \pi ^{5} b x + 40 \, \pi ^{4} a b x - 160 i \, \pi ^{3} a^{2} b x - 320 \, \pi ^{2} a^{3} b x + 320 i \, \pi a^{4} b x + 128 \, a^{5} b x - \pi ^{6} + 12 i \, \pi ^{5} a + 60 \, \pi ^{4} a^{2} - 160 i \, \pi ^{3} a^{3} - 240 \, \pi ^{2} a^{4} + 192 i \, \pi a^{5} + 64 \, a^{6}} \]
192*I*b^2*log(I*pi + 2*b*x + 2*a)/(pi^5 - 10*I*pi^4*a - 40*pi^3*a^2 + 80*I *pi^2*a^3 + 80*pi*a^4 - 32*I*a^5) - 192*I*b^2*log(x)/(pi^5 - 10*I*pi^4*a - 40*pi^3*a^2 + 80*I*pi^2*a^3 + 80*pi*a^4 - 32*I*a^5) - 4*(I*pi - 12*b*x + 2*a)/(pi^4*x^2 - 8*I*pi^3*a*x^2 - 24*pi^2*a^2*x^2 + 32*I*pi*a^3*x^2 + 16*a ^4*x^2) + 16*(12*b^3*x + 7*I*pi*b^2 + 14*a*b^2)/(4*pi^4*b^2*x^2 - 32*I*pi^ 3*a*b^2*x^2 - 96*pi^2*a^2*b^2*x^2 + 128*I*pi*a^3*b^2*x^2 + 64*a^4*b^2*x^2 + 4*I*pi^5*b*x + 40*pi^4*a*b*x - 160*I*pi^3*a^2*b*x - 320*pi^2*a^3*b*x + 3 20*I*pi*a^4*b*x + 128*a^5*b*x - pi^6 + 12*I*pi^5*a + 60*pi^4*a^2 - 160*I*p i^3*a^3 - 240*pi^2*a^4 + 192*I*pi*a^5 + 64*a^6)
Time = 10.02 (sec) , antiderivative size = 1251, normalized size of antiderivative = 7.36 \[ \int \frac {1}{x^3 \coth ^{-1}(\tanh (a+b x))^3} \, dx=\text {Too large to display} \]
(4/(log(-2/(exp(2*a)*exp(2*b*x) - 1)) - log((2*exp(2*a)*exp(2*b*x))/(exp(2 *a)*exp(2*b*x) - 1)) + 2*b*x) + (32*b*x)/(log(-2/(exp(2*a)*exp(2*b*x) - 1) ) - log((2*exp(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) - 1)) + 2*b*x)^2 - (2 88*b^2*x^2)/((log(-2/(exp(2*a)*exp(2*b*x) - 1)) - log((2*exp(2*a)*exp(2*b* x))/(exp(2*a)*exp(2*b*x) - 1)) + 2*b*x)*((2*a - log((2*exp(2*a)*exp(2*b*x) )/(exp(2*a)*exp(2*b*x) - 1)) + log(-2/(exp(2*a)*exp(2*b*x) - 1)) + 2*b*x)^ 2 - 4*a*(2*a - log((2*exp(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) - 1)) + lo g(-2/(exp(2*a)*exp(2*b*x) - 1)) + 2*b*x) + 4*a^2)) + (384*b^3*x^3)/((log(- 2/(exp(2*a)*exp(2*b*x) - 1)) - log((2*exp(2*a)*exp(2*b*x))/(exp(2*a)*exp(2 *b*x) - 1)) + 2*b*x)^2*((2*a - log((2*exp(2*a)*exp(2*b*x))/(exp(2*a)*exp(2 *b*x) - 1)) + log(-2/(exp(2*a)*exp(2*b*x) - 1)) + 2*b*x)^2 - 4*a*(2*a - lo g((2*exp(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) - 1)) + log(-2/(exp(2*a)*ex p(2*b*x) - 1)) + 2*b*x) + 4*a^2)))/(x^3*(8*a*b - 4*b*(2*a - log((2*exp(2*a )*exp(2*b*x))/(exp(2*a)*exp(2*b*x) - 1)) + log(-2/(exp(2*a)*exp(2*b*x) - 1 )) + 2*b*x)) + x^2*((2*a - log((2*exp(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x ) - 1)) + log(-2/(exp(2*a)*exp(2*b*x) - 1)) + 2*b*x)^2 - 4*a*(2*a - log((2 *exp(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) - 1)) + log(-2/(exp(2*a)*exp(2* b*x) - 1)) + 2*b*x) + 4*a^2) + 4*b^2*x^4) - (384*b^2*atanh((4*b*x*((2*a - log((2*exp(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) - 1)) + log(-2/(exp(2*a)* exp(2*b*x) - 1)) + 2*b*x)^2 - 4*a*(2*a - log((2*exp(2*a)*exp(2*b*x))/(e...