\(\int \frac {1}{x^3 \coth ^{-1}(\tanh (a+b x))^3} \, dx\) [183]

Optimal result

Integrand size = 13, antiderivative size = 170 \[ \int \frac {1}{x^3 \coth ^{-1}(\tanh (a+b x))^3} \, dx=-\frac {3 b^2}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3 \coth ^{-1}(\tanh (a+b x))^2}+\frac {2 b}{x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \coth ^{-1}(\tanh (a+b x))^2}+\frac {1}{2 x^2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))^2}+\frac {6 b^2}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^4 \coth ^{-1}(\tanh (a+b x))}-\frac {6 b^2 \log (x)}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^5}+\frac {6 b^2 \log \left (\coth ^{-1}(\tanh (a+b x))\right )}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^5} \]

[Out]

-3*b^2/(b*x-arccoth(tanh(b*x+a)))^3/arccoth(tanh(b*x+a))^2+2*b/x/(b*x-arccoth(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(ta
nh(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

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {2202, 2194, 2191, 2188, 29} \[ \int \frac {1}{x^3 \coth ^{-1}(\tanh (a+b x))^3} \, dx=\frac {6 b^2}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^4 \coth ^{-1}(\tanh (a+b x))}-\frac {3 b^2}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3 \coth ^{-1}(\tanh (a+b x))^2}-\frac {6 b^2 \log (x)}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^5}+\frac {6 b^2 \log \left (\coth ^{-1}(\tanh (a+b x))\right )}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^5}+\frac {1}{2 x^2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))^2}+\frac {2 b}{x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \coth ^{-1}(\tanh (a+b x))^2} \]

[In]

Int[1/(x^3*ArcCoth[Tanh[a + b*x]]^3),x]

[Out]

(-3*b^2)/((b*x - ArcCoth[Tanh[a + b*x]])^3*ArcCoth[Tanh[a + b*x]]^2) + (2*b)/(x*(b*x - ArcCoth[Tanh[a + b*x]])
^2*ArcCoth[Tanh[a + b*x]]^2) + 1/(2*x^2*(b*x - ArcCoth[Tanh[a + b*x]])*ArcCoth[Tanh[a + b*x]]^2) + (6*b^2)/((b
*x - ArcCoth[Tanh[a + b*x]])^4*ArcCoth[Tanh[a + b*x]]) - (6*b^2*Log[x])/(b*x - ArcCoth[Tanh[a + b*x]])^5 + (6*
b^2*Log[ArcCoth[Tanh[a + b*x]]])/(b*x - ArcCoth[Tanh[a + b*x]])^5

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 2188

Int[(u_)^(m_.), x_Symbol] :> With[{c = Simplify[D[u, x]]}, Dist[1/c, Subst[Int[x^m, x], x, u], x]] /; FreeQ[m,
x] && PiecewiseLinearQ[u, x]

Rule 2191

Int[1/((u_)*(v_)), x_Symbol] :> With[{a = Simplify[D[u, x]], b = Simplify[D[v, x]]}, Dist[b/(b*u - a*v), Int[1
/v, x], x] - Dist[a/(b*u - a*v), Int[1/u, x], x] /; NeQ[b*u - a*v, 0]] /; PiecewiseLinearQ[u, v, x]

Rule 2194

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] - Dist[a*((n + 1)/((n + 1)*(b*u - a*v))), Int[v^(n + 1)/u, x], x] /; NeQ[b*u - a*v, 0]] /; Pi
ecewiseLinearQ[u, v, x] && LtQ[n, -1]

Rule 2202

Int[(u_)^(m_)*(v_)^(n_), x_Symbol] :> With[{a = Simplify[D[u, x]], b = Simplify[D[v, x]]}, Simp[(-u^(m + 1))*(
v^(n + 1)/((m + 1)*(b*u - a*v))), x] + Dist[b*((m + n + 2)/((m + 1)*(b*u - a*v))), Int[u^(m + 1)*v^n, x], x] /
; NeQ[b*u - a*v, 0]] /; PiecewiseLinearQ[u, v, x] && NeQ[m + n + 2, 0] && LtQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = \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) \int \frac {1}{x^2 \coth ^{-1}(\tanh (a+b x))^3} \, dx}{b x-\coth ^{-1}(\tanh (a+b x))} \\ & = \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 {\left (6 b^2\right ) \int \frac {1}{x \coth ^{-1}(\tanh (a+b x))^3} \, dx}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \left (-b x+\coth ^{-1}(\tanh (a+b x))\right )} \\ & = -\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 {\left (6 b^2\right ) \int \frac {1}{x \coth ^{-1}(\tanh (a+b x))^2} \, dx}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \left (-b x+\coth ^{-1}(\tanh (a+b x))\right )} \\ & = -\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 {\left (6 b^2\right ) \int \frac {1}{x \coth ^{-1}(\tanh (a+b x))} \, dx}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \left (-b x+\coth ^{-1}(\tanh (a+b x))\right )^2} \\ & = -\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 {\left (6 b^2\right ) \int \frac {1}{x} \, dx}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3 \left (-b x+\coth ^{-1}(\tanh (a+b x))\right )^2}+\frac {\left (6 b^3\right ) \int \frac {1}{\coth ^{-1}(\tanh (a+b x))} \, dx}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3 \left (-b x+\coth ^{-1}(\tanh (a+b x))\right )^2} \\ & = -\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 {\left (6 b^2\right ) \text {Subst}\left (\int \frac {1}{x} \, dx,x,\coth ^{-1}(\tanh (a+b x))\right )}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3 \left (-b x+\coth ^{-1}(\tanh (a+b x))\right )^2} \\ & = -\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} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.63 \[ \int \frac {1}{x^3 \coth ^{-1}(\tanh (a+b x))^3} \, dx=\frac {-b^4 x^4+8 b^3 x^3 \coth ^{-1}(\tanh (a+b x))-8 b x \coth ^{-1}(\tanh (a+b x))^3+\coth ^{-1}(\tanh (a+b x))^4-12 b^2 x^2 \coth ^{-1}(\tanh (a+b x))^2 \left (\log (x)-\log \left (\coth ^{-1}(\tanh (a+b x))\right )\right )}{2 x^2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^5 \coth ^{-1}(\tanh (a+b x))^2} \]

[In]

Integrate[1/(x^3*ArcCoth[Tanh[a + b*x]]^3),x]

[Out]

(-(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 - 1
2*b^2*x^2*ArcCoth[Tanh[a + b*x]]^2*(Log[x] - Log[ArcCoth[Tanh[a + b*x]]]))/(2*x^2*(b*x - ArcCoth[Tanh[a + b*x]
])^5*ArcCoth[Tanh[a + b*x]]^2)

Maple [F(-1)]

Timed out.

\[\int \frac {1}{x^{3} \operatorname {arccoth}\left (\tanh \left (b x +a \right )\right )^{3}}d x\]

[In]

int(1/x^3/arccoth(tanh(b*x+a))^3,x)

[Out]

int(1/x^3/arccoth(tanh(b*x+a))^3,x)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.26 (sec) , antiderivative size = 461, normalized size of antiderivative = 2.71 \[ \int \frac {1}{x^3 \coth ^{-1}(\tanh (a+b x))^3} \, dx=\frac {4 \, {\left (192 \, a b^{3} x^{3} + 288 \, a^{2} b^{2} x^{2} + 64 \, a^{3} b x - \pi ^{4} - 8 i \, \pi ^{3} {\left (b x - a\right )} - 16 \, a^{4} - 24 \, \pi ^{2} {\left (3 \, b^{2} x^{2} + 2 \, a b x - a^{2}\right )} + 32 i \, \pi {\left (3 \, b^{3} x^{3} + 9 \, a b^{2} x^{2} + 3 \, a^{2} b x - a^{3}\right )} - 48 \, {\left (4 \, b^{4} x^{4} + 8 \, a b^{3} x^{3} - \pi ^{2} b^{2} x^{2} + 4 \, a^{2} b^{2} x^{2} + 4 i \, \pi {\left (b^{3} x^{3} + a b^{2} x^{2}\right )}\right )} \log \left (i \, \pi + 2 \, b x + 2 \, a\right ) + 48 \, {\left (4 \, b^{4} x^{4} + 8 \, a b^{3} x^{3} - \pi ^{2} b^{2} x^{2} + 4 \, a^{2} b^{2} x^{2} + 4 i \, \pi {\left (b^{3} x^{3} + a b^{2} x^{2}\right )}\right )} \log \left (x\right )\right )}}{128 \, a^{5} b^{2} x^{4} + 256 \, a^{6} b x^{3} - i \, \pi ^{7} x^{2} + 128 \, a^{7} x^{2} - 2 \, \pi ^{6} {\left (2 \, b x^{3} + 7 \, a x^{2}\right )} + 4 i \, \pi ^{5} {\left (b^{2} x^{4} + 12 \, a b x^{3} + 21 \, a^{2} x^{2}\right )} + 40 \, \pi ^{4} {\left (a b^{2} x^{4} + 6 \, a^{2} b x^{3} + 7 \, a^{3} x^{2}\right )} - 80 i \, \pi ^{3} {\left (2 \, a^{2} b^{2} x^{4} + 8 \, a^{3} b x^{3} + 7 \, a^{4} x^{2}\right )} - 32 \, \pi ^{2} {\left (10 \, a^{3} b^{2} x^{4} + 30 \, a^{4} b x^{3} + 21 \, a^{5} x^{2}\right )} + 64 i \, \pi {\left (5 \, a^{4} b^{2} x^{4} + 12 \, a^{5} b x^{3} + 7 \, a^{6} x^{2}\right )}} \]

[In]

integrate(1/x^3/arccoth(tanh(b*x+a))^3,x, algorithm="fricas")

[Out]

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
+ 128*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))

Sympy [F]

\[ \int \frac {1}{x^3 \coth ^{-1}(\tanh (a+b x))^3} \, dx=\int \frac {1}{x^{3} \operatorname {acoth}^{3}{\left (\tanh {\left (a + b x \right )} \right )}}\, dx \]

[In]

integrate(1/x**3/acoth(tanh(b*x+a))**3,x)

[Out]

Integral(1/(x**3*acoth(tanh(a + b*x))**3), x)

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.78 (sec) , antiderivative size = 331, normalized size of antiderivative = 1.95 \[ \int \frac {1}{x^3 \coth ^{-1}(\tanh (a+b x))^3} \, dx=\frac {192 \, b^{2} \log \left (-i \, \pi + 2 \, b x + 2 \, a\right )}{i \, \pi ^{5} - 10 \, \pi ^{4} a - 40 i \, \pi ^{3} a^{2} + 80 \, \pi ^{2} a^{3} + 80 i \, \pi a^{4} - 32 \, a^{5}} - \frac {192 \, b^{2} \log \left (x\right )}{i \, \pi ^{5} - 10 \, \pi ^{4} a - 40 i \, \pi ^{3} a^{2} + 80 \, \pi ^{2} a^{3} + 80 i \, \pi a^{4} - 32 \, a^{5}} + \frac {4 \, {\left (96 \, b^{3} x^{3} - i \, \pi ^{3} + 6 \, \pi ^{2} a + 12 i \, \pi a^{2} - 8 \, a^{3} - 72 \, {\left (i \, \pi b^{2} - 2 \, a b^{2}\right )} x^{2} - 8 \, {\left (\pi ^{2} b + 4 i \, \pi a b - 4 \, a^{2} b\right )} x\right )}}{4 \, {\left (\pi ^{4} b^{2} + 8 i \, \pi ^{3} a b^{2} - 24 \, \pi ^{2} a^{2} b^{2} - 32 i \, \pi a^{3} b^{2} + 16 \, a^{4} b^{2}\right )} x^{4} - 4 \, {\left (i \, \pi ^{5} b - 10 \, \pi ^{4} a b - 40 i \, \pi ^{3} a^{2} b + 80 \, \pi ^{2} a^{3} b + 80 i \, \pi a^{4} b - 32 \, a^{5} b\right )} x^{3} - {\left (\pi ^{6} + 12 i \, \pi ^{5} a - 60 \, \pi ^{4} a^{2} - 160 i \, \pi ^{3} a^{3} + 240 \, \pi ^{2} a^{4} + 192 i \, \pi a^{5} - 64 \, a^{6}\right )} x^{2}} \]

[In]

integrate(1/x^3/arccoth(tanh(b*x+a))^3,x, algorithm="maxima")

[Out]

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) - 1
92*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*(p
i^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 - 4
0*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)

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.31 (sec) , antiderivative size = 343, normalized size of antiderivative = 2.02 \[ \int \frac {1}{x^3 \coth ^{-1}(\tanh (a+b x))^3} \, dx=\frac {192 i \, b^{2} \log \left (i \, \pi + 2 \, b x + 2 \, a\right )}{\pi ^{5} - 10 i \, \pi ^{4} a - 40 \, \pi ^{3} a^{2} + 80 i \, \pi ^{2} a^{3} + 80 \, \pi a^{4} - 32 i \, a^{5}} - \frac {192 i \, b^{2} \log \left (x\right )}{\pi ^{5} - 10 i \, \pi ^{4} a - 40 \, \pi ^{3} a^{2} + 80 i \, \pi ^{2} a^{3} + 80 \, \pi a^{4} - 32 i \, a^{5}} - \frac {4 \, {\left (i \, \pi - 12 \, b x + 2 \, a\right )}}{\pi ^{4} x^{2} - 8 i \, \pi ^{3} a x^{2} - 24 \, \pi ^{2} a^{2} x^{2} + 32 i \, \pi a^{3} x^{2} + 16 \, a^{4} x^{2}} + \frac {16 \, {\left (12 \, b^{3} x + 7 i \, \pi b^{2} + 14 \, a b^{2}\right )}}{4 \, \pi ^{4} b^{2} x^{2} - 32 i \, \pi ^{3} a b^{2} x^{2} - 96 \, \pi ^{2} a^{2} b^{2} x^{2} + 128 i \, \pi a^{3} b^{2} x^{2} + 64 \, a^{4} b^{2} x^{2} + 4 i \, \pi ^{5} b x + 40 \, \pi ^{4} a b x - 160 i \, \pi ^{3} a^{2} b x - 320 \, \pi ^{2} a^{3} b x + 320 i \, \pi a^{4} b x + 128 \, a^{5} b x - \pi ^{6} + 12 i \, \pi ^{5} a + 60 \, \pi ^{4} a^{2} - 160 i \, \pi ^{3} a^{3} - 240 \, \pi ^{2} a^{4} + 192 i \, \pi a^{5} + 64 \, a^{6}} \]

[In]

integrate(1/x^3/arccoth(tanh(b*x+a))^3,x, algorithm="giac")

[Out]

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 + 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)

Mupad [B] (verification not implemented)

Time = 10.02 (sec) , antiderivative size = 1251, normalized size of antiderivative = 7.36 \[ \int \frac {1}{x^3 \coth ^{-1}(\tanh (a+b x))^3} \, dx=\text {Too large to display} \]

[In]

int(1/(x^3*acoth(tanh(a + b*x))^3),x)

[Out]

(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 -
(288*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)) + log(-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 - 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)))/(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))/(exp(2*a)*exp(2*b*x) - 1)) + log(-2/(exp(2*a)*exp(2*b*x) - 1)) +
2*b*x) + 4*a^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)^3 - ((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)^5 + 40*a^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)^3 - 80*a^3*(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 - 32*a^5 - 10*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 + 80*a^4*(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))/(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)^5))/(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)^5