Integrand size = 14, antiderivative size = 289 \[ \int \frac {1}{\left (b \coth ^2(c+d x)\right )^{2/3}} \, dx=-\frac {3 \coth (c+d x)}{d \left (b \coth ^2(c+d x)\right )^{2/3}}+\frac {\sqrt {3} \arctan \left (\frac {1-2 \sqrt [3]{\coth (c+d x)}}{\sqrt {3}}\right ) \coth ^{\frac {4}{3}}(c+d x)}{2 d \left (b \coth ^2(c+d x)\right )^{2/3}}-\frac {\sqrt {3} \arctan \left (\frac {1+2 \sqrt [3]{\coth (c+d x)}}{\sqrt {3}}\right ) \coth ^{\frac {4}{3}}(c+d x)}{2 d \left (b \coth ^2(c+d x)\right )^{2/3}}+\frac {\text {arctanh}\left (\sqrt [3]{\coth (c+d x)}\right ) \coth ^{\frac {4}{3}}(c+d x)}{d \left (b \coth ^2(c+d x)\right )^{2/3}}-\frac {\coth ^{\frac {4}{3}}(c+d x) \log \left (1-\sqrt [3]{\coth (c+d x)}+\coth ^{\frac {2}{3}}(c+d x)\right )}{4 d \left (b \coth ^2(c+d x)\right )^{2/3}}+\frac {\coth ^{\frac {4}{3}}(c+d x) \log \left (1+\sqrt [3]{\coth (c+d x)}+\coth ^{\frac {2}{3}}(c+d x)\right )}{4 d \left (b \coth ^2(c+d x)\right )^{2/3}} \]
-3*coth(d*x+c)/d/(b*coth(d*x+c)^2)^(2/3)+arctanh(coth(d*x+c)^(1/3))*coth(d *x+c)^(4/3)/d/(b*coth(d*x+c)^2)^(2/3)-1/4*coth(d*x+c)^(4/3)*ln(1-coth(d*x+ c)^(1/3)+coth(d*x+c)^(2/3))/d/(b*coth(d*x+c)^2)^(2/3)+1/4*coth(d*x+c)^(4/3 )*ln(1+coth(d*x+c)^(1/3)+coth(d*x+c)^(2/3))/d/(b*coth(d*x+c)^2)^(2/3)+1/2* arctan(1/3*(1-2*coth(d*x+c)^(1/3))*3^(1/2))*coth(d*x+c)^(4/3)*3^(1/2)/d/(b *coth(d*x+c)^2)^(2/3)-1/2*arctan(1/3*(1+2*coth(d*x+c)^(1/3))*3^(1/2))*coth (d*x+c)^(4/3)*3^(1/2)/d/(b*coth(d*x+c)^2)^(2/3)
Time = 0.28 (sec) , antiderivative size = 248, normalized size of antiderivative = 0.86 \[ \int \frac {1}{\left (b \coth ^2(c+d x)\right )^{2/3}} \, dx=-\frac {\coth (c+d x) \left (6+\sqrt [6]{\coth ^2(c+d x)} \log \left (1-\sqrt [6]{\coth ^2(c+d x)}\right )-\sqrt [6]{\coth ^2(c+d x)} \log \left (1+\sqrt [6]{\coth ^2(c+d x)}\right )+\sqrt [3]{-1} \sqrt [6]{\coth ^2(c+d x)} \log \left (1-\sqrt [3]{-1} \sqrt [6]{\coth ^2(c+d x)}\right )-\sqrt [3]{-1} \sqrt [6]{\coth ^2(c+d x)} \log \left (1+\sqrt [3]{-1} \sqrt [6]{\coth ^2(c+d x)}\right )+(-1)^{2/3} \sqrt [6]{\coth ^2(c+d x)} \log \left (1-(-1)^{2/3} \sqrt [6]{\coth ^2(c+d x)}\right )-(-1)^{2/3} \sqrt [6]{\coth ^2(c+d x)} \log \left (1+(-1)^{2/3} \sqrt [6]{\coth ^2(c+d x)}\right )\right )}{2 d \left (b \coth ^2(c+d x)\right )^{2/3}} \]
-1/2*(Coth[c + d*x]*(6 + (Coth[c + d*x]^2)^(1/6)*Log[1 - (Coth[c + d*x]^2) ^(1/6)] - (Coth[c + d*x]^2)^(1/6)*Log[1 + (Coth[c + d*x]^2)^(1/6)] + (-1)^ (1/3)*(Coth[c + d*x]^2)^(1/6)*Log[1 - (-1)^(1/3)*(Coth[c + d*x]^2)^(1/6)] - (-1)^(1/3)*(Coth[c + d*x]^2)^(1/6)*Log[1 + (-1)^(1/3)*(Coth[c + d*x]^2)^ (1/6)] + (-1)^(2/3)*(Coth[c + d*x]^2)^(1/6)*Log[1 - (-1)^(2/3)*(Coth[c + d *x]^2)^(1/6)] - (-1)^(2/3)*(Coth[c + d*x]^2)^(1/6)*Log[1 + (-1)^(2/3)*(Cot h[c + d*x]^2)^(1/6)]))/(d*(b*Coth[c + d*x]^2)^(2/3))
Time = 0.50 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.64, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.143, Rules used = {3042, 4141, 3042, 3955, 3042, 3957, 25, 266, 825, 27, 219, 1142, 25, 1083, 217, 1103}
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.
| \(\displaystyle \int \frac {1}{\left (b \coth ^2(c+d x)\right )^{2/3}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\left (-b \tan \left (i c+i d x+\frac {\pi }{2}\right )^2\right )^{2/3}}dx\) |
| \(\Big \downarrow \) 4141 |
| \(\displaystyle \frac {\coth ^{\frac {4}{3}}(c+d x) \int \frac {1}{\coth ^{\frac {4}{3}}(c+d x)}dx}{\left (b \coth ^2(c+d x)\right )^{2/3}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\coth ^{\frac {4}{3}}(c+d x) \int \frac {1}{\left (-i \tan \left (i c+i d x+\frac {\pi }{2}\right )\right )^{4/3}}dx}{\left (b \coth ^2(c+d x)\right )^{2/3}}\) |
| \(\Big \downarrow \) 3955 |
| \(\displaystyle \frac {\coth ^{\frac {4}{3}}(c+d x) \left (\int \coth ^{\frac {2}{3}}(c+d x)dx-\frac {3}{d \sqrt [3]{\coth (c+d x)}}\right )}{\left (b \coth ^2(c+d x)\right )^{2/3}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\coth ^{\frac {4}{3}}(c+d x) \left (-\frac {3}{d \sqrt [3]{\coth (c+d x)}}+\int \left (-i \tan \left (i c+i d x+\frac {\pi }{2}\right )\right )^{2/3}dx\right )}{\left (b \coth ^2(c+d x)\right )^{2/3}}\) |
| \(\Big \downarrow \) 3957 |
| \(\displaystyle \frac {\coth ^{\frac {4}{3}}(c+d x) \left (-\frac {\int -\frac {\coth ^{\frac {2}{3}}(c+d x)}{1-\coth ^2(c+d x)}d\coth (c+d x)}{d}-\frac {3}{d \sqrt [3]{\coth (c+d x)}}\right )}{\left (b \coth ^2(c+d x)\right )^{2/3}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\coth ^{\frac {4}{3}}(c+d x) \left (\frac {\int \frac {\coth ^{\frac {2}{3}}(c+d x)}{1-\coth ^2(c+d x)}d\coth (c+d x)}{d}-\frac {3}{d \sqrt [3]{\coth (c+d x)}}\right )}{\left (b \coth ^2(c+d x)\right )^{2/3}}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {\coth ^{\frac {4}{3}}(c+d x) \left (\frac {3 \int \frac {\coth ^{\frac {4}{3}}(c+d x)}{1-\coth ^2(c+d x)}d\sqrt [3]{\coth (c+d x)}}{d}-\frac {3}{d \sqrt [3]{\coth (c+d x)}}\right )}{\left (b \coth ^2(c+d x)\right )^{2/3}}\) |
| \(\Big \downarrow \) 825 |
| \(\displaystyle \frac {\coth ^{\frac {4}{3}}(c+d x) \left (\frac {3 \left (\frac {1}{3} \int \frac {1}{1-\coth ^{\frac {2}{3}}(c+d x)}d\sqrt [3]{\coth (c+d x)}+\frac {1}{3} \int -\frac {\sqrt [3]{\coth (c+d x)}+1}{2 \left (\coth ^{\frac {2}{3}}(c+d x)-\sqrt [3]{\coth (c+d x)}+1\right )}d\sqrt [3]{\coth (c+d x)}+\frac {1}{3} \int -\frac {1-\sqrt [3]{\coth (c+d x)}}{2 \left (\coth ^{\frac {2}{3}}(c+d x)+\sqrt [3]{\coth (c+d x)}+1\right )}d\sqrt [3]{\coth (c+d x)}\right )}{d}-\frac {3}{d \sqrt [3]{\coth (c+d x)}}\right )}{\left (b \coth ^2(c+d x)\right )^{2/3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\coth ^{\frac {4}{3}}(c+d x) \left (\frac {3 \left (\frac {1}{3} \int \frac {1}{1-\coth ^{\frac {2}{3}}(c+d x)}d\sqrt [3]{\coth (c+d x)}-\frac {1}{6} \int \frac {\sqrt [3]{\coth (c+d x)}+1}{\coth ^{\frac {2}{3}}(c+d x)-\sqrt [3]{\coth (c+d x)}+1}d\sqrt [3]{\coth (c+d x)}-\frac {1}{6} \int \frac {1-\sqrt [3]{\coth (c+d x)}}{\coth ^{\frac {2}{3}}(c+d x)+\sqrt [3]{\coth (c+d x)}+1}d\sqrt [3]{\coth (c+d x)}\right )}{d}-\frac {3}{d \sqrt [3]{\coth (c+d x)}}\right )}{\left (b \coth ^2(c+d x)\right )^{2/3}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\coth ^{\frac {4}{3}}(c+d x) \left (\frac {3 \left (-\frac {1}{6} \int \frac {\sqrt [3]{\coth (c+d x)}+1}{\coth ^{\frac {2}{3}}(c+d x)-\sqrt [3]{\coth (c+d x)}+1}d\sqrt [3]{\coth (c+d x)}-\frac {1}{6} \int \frac {1-\sqrt [3]{\coth (c+d x)}}{\coth ^{\frac {2}{3}}(c+d x)+\sqrt [3]{\coth (c+d x)}+1}d\sqrt [3]{\coth (c+d x)}+\frac {1}{3} \text {arctanh}\left (\sqrt [3]{\coth (c+d x)}\right )\right )}{d}-\frac {3}{d \sqrt [3]{\coth (c+d x)}}\right )}{\left (b \coth ^2(c+d x)\right )^{2/3}}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {\coth ^{\frac {4}{3}}(c+d x) \left (\frac {3 \left (\frac {1}{6} \left (-\frac {3}{2} \int \frac {1}{\coth ^{\frac {2}{3}}(c+d x)-\sqrt [3]{\coth (c+d x)}+1}d\sqrt [3]{\coth (c+d x)}-\frac {1}{2} \int -\frac {1-2 \sqrt [3]{\coth (c+d x)}}{\coth ^{\frac {2}{3}}(c+d x)-\sqrt [3]{\coth (c+d x)}+1}d\sqrt [3]{\coth (c+d x)}\right )+\frac {1}{6} \left (\frac {1}{2} \int \frac {2 \sqrt [3]{\coth (c+d x)}+1}{\coth ^{\frac {2}{3}}(c+d x)+\sqrt [3]{\coth (c+d x)}+1}d\sqrt [3]{\coth (c+d x)}-\frac {3}{2} \int \frac {1}{\coth ^{\frac {2}{3}}(c+d x)+\sqrt [3]{\coth (c+d x)}+1}d\sqrt [3]{\coth (c+d x)}\right )+\frac {1}{3} \text {arctanh}\left (\sqrt [3]{\coth (c+d x)}\right )\right )}{d}-\frac {3}{d \sqrt [3]{\coth (c+d x)}}\right )}{\left (b \coth ^2(c+d x)\right )^{2/3}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\coth ^{\frac {4}{3}}(c+d x) \left (\frac {3 \left (\frac {1}{6} \left (\frac {1}{2} \int \frac {1-2 \sqrt [3]{\coth (c+d x)}}{\coth ^{\frac {2}{3}}(c+d x)-\sqrt [3]{\coth (c+d x)}+1}d\sqrt [3]{\coth (c+d x)}-\frac {3}{2} \int \frac {1}{\coth ^{\frac {2}{3}}(c+d x)-\sqrt [3]{\coth (c+d x)}+1}d\sqrt [3]{\coth (c+d x)}\right )+\frac {1}{6} \left (\frac {1}{2} \int \frac {2 \sqrt [3]{\coth (c+d x)}+1}{\coth ^{\frac {2}{3}}(c+d x)+\sqrt [3]{\coth (c+d x)}+1}d\sqrt [3]{\coth (c+d x)}-\frac {3}{2} \int \frac {1}{\coth ^{\frac {2}{3}}(c+d x)+\sqrt [3]{\coth (c+d x)}+1}d\sqrt [3]{\coth (c+d x)}\right )+\frac {1}{3} \text {arctanh}\left (\sqrt [3]{\coth (c+d x)}\right )\right )}{d}-\frac {3}{d \sqrt [3]{\coth (c+d x)}}\right )}{\left (b \coth ^2(c+d x)\right )^{2/3}}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {\coth ^{\frac {4}{3}}(c+d x) \left (\frac {3 \left (\frac {1}{6} \left (3 \int \frac {1}{-\coth ^{\frac {2}{3}}(c+d x)-3}d\left (2 \sqrt [3]{\coth (c+d x)}-1\right )+\frac {1}{2} \int \frac {1-2 \sqrt [3]{\coth (c+d x)}}{\coth ^{\frac {2}{3}}(c+d x)-\sqrt [3]{\coth (c+d x)}+1}d\sqrt [3]{\coth (c+d x)}\right )+\frac {1}{6} \left (3 \int \frac {1}{-\coth ^{\frac {2}{3}}(c+d x)-3}d\left (2 \sqrt [3]{\coth (c+d x)}+1\right )+\frac {1}{2} \int \frac {2 \sqrt [3]{\coth (c+d x)}+1}{\coth ^{\frac {2}{3}}(c+d x)+\sqrt [3]{\coth (c+d x)}+1}d\sqrt [3]{\coth (c+d x)}\right )+\frac {1}{3} \text {arctanh}\left (\sqrt [3]{\coth (c+d x)}\right )\right )}{d}-\frac {3}{d \sqrt [3]{\coth (c+d x)}}\right )}{\left (b \coth ^2(c+d x)\right )^{2/3}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\coth ^{\frac {4}{3}}(c+d x) \left (\frac {3 \left (\frac {1}{6} \left (\frac {1}{2} \int \frac {1-2 \sqrt [3]{\coth (c+d x)}}{\coth ^{\frac {2}{3}}(c+d x)-\sqrt [3]{\coth (c+d x)}+1}d\sqrt [3]{\coth (c+d x)}-\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{\coth (c+d x)}-1}{\sqrt {3}}\right )\right )+\frac {1}{6} \left (\frac {1}{2} \int \frac {2 \sqrt [3]{\coth (c+d x)}+1}{\coth ^{\frac {2}{3}}(c+d x)+\sqrt [3]{\coth (c+d x)}+1}d\sqrt [3]{\coth (c+d x)}-\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{\coth (c+d x)}+1}{\sqrt {3}}\right )\right )+\frac {1}{3} \text {arctanh}\left (\sqrt [3]{\coth (c+d x)}\right )\right )}{d}-\frac {3}{d \sqrt [3]{\coth (c+d x)}}\right )}{\left (b \coth ^2(c+d x)\right )^{2/3}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\coth ^{\frac {4}{3}}(c+d x) \left (\frac {3 \left (\frac {1}{6} \left (-\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{\coth (c+d x)}-1}{\sqrt {3}}\right )-\frac {1}{2} \log \left (\coth ^{\frac {2}{3}}(c+d x)-\sqrt [3]{\coth (c+d x)}+1\right )\right )+\frac {1}{6} \left (\frac {1}{2} \log \left (\coth ^{\frac {2}{3}}(c+d x)+\sqrt [3]{\coth (c+d x)}+1\right )-\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{\coth (c+d x)}+1}{\sqrt {3}}\right )\right )+\frac {1}{3} \text {arctanh}\left (\sqrt [3]{\coth (c+d x)}\right )\right )}{d}-\frac {3}{d \sqrt [3]{\coth (c+d x)}}\right )}{\left (b \coth ^2(c+d x)\right )^{2/3}}\) |
(Coth[c + d*x]^(4/3)*(-3/(d*Coth[c + d*x]^(1/3)) + (3*(ArcTanh[Coth[c + d* x]^(1/3)]/3 + (-(Sqrt[3]*ArcTan[(-1 + 2*Coth[c + d*x]^(1/3))/Sqrt[3]]) - L og[1 - Coth[c + d*x]^(1/3) + Coth[c + d*x]^(2/3)]/2)/6 + (-(Sqrt[3]*ArcTan [(1 + 2*Coth[c + d*x]^(1/3))/Sqrt[3]]) + Log[1 + Coth[c + d*x]^(1/3) + Cot h[c + d*x]^(2/3)]/2)/6))/d))/(b*Coth[c + d*x]^2)^(2/3)
3.1.26.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De nominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) ^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I ntBinomialQ[a, b, c, 2, m, p, x]
Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Module[{r = Numerator [Rt[-a/b, n]], s = Denominator[Rt[-a/b, n]], k, u}, Simp[u = Int[(r*Cos[2*k *m*(Pi/n)] - s*Cos[2*k*(m + 1)*(Pi/n)]*x)/(r^2 - 2*r*s*Cos[2*k*(Pi/n)]*x + s^2*x^2), x] + Int[(r*Cos[2*k*m*(Pi/n)] + s*Cos[2*k*(m + 1)*(Pi/n)]*x)/(r^2 + 2*r*s*Cos[2*k*(Pi/n)]*x + s^2*x^2), x]; 2*(r^(m + 2)/(a*n*s^m)) Int[1/ (r^2 - s^2*x^2), x] + 2*(r^(m + 1)/(a*n*s^m)) Sum[u, {k, 1, (n - 2)/4}], x]] /; FreeQ[{a, b}, x] && IGtQ[(n - 2)/4, 0] && IGtQ[m, 0] && LtQ[m, n - 1 ] && NegQ[a/b]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*Tan[c + d*x] )^(n + 1)/(b*d*(n + 1)), x] - Simp[1/b^2 Int[(b*Tan[c + d*x])^(n + 2), x] , x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1]
Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b/d Subst[Int [x^n/(b^2 + x^2), x], x, b*Tan[c + d*x]], x] /; FreeQ[{b, c, d, n}, x] && !IntegerQ[n]
Int[(u_.)*((b_.)*tan[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[(b*ff^n)^IntPart[p]*((b*Tan[e + f*x]^ n)^FracPart[p]/(Tan[e + f*x]/ff)^(n*FracPart[p])) Int[ActivateTrig[u]*(Ta n[e + f*x]/ff)^(n*p), x], x]] /; FreeQ[{b, e, f, n, p}, x] && !IntegerQ[p] && IntegerQ[n] && (EqQ[u, 1] || MatchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) / ; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig]])
\[\int \frac {1}{\left (\coth \left (d x +c \right )^{2} b \right )^{\frac {2}{3}}}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 2066 vs. \(2 (239) = 478\).
Time = 0.31 (sec) , antiderivative size = 2066, normalized size of antiderivative = 7.15 \[ \int \frac {1}{\left (b \coth ^2(c+d x)\right )^{2/3}} \, dx=\text {Too large to display} \]
1/4*(2*sqrt(3)*(b*cosh(d*x + c)^2 + 2*b*cosh(d*x + c)*sinh(d*x + c) + b*si nh(d*x + c)^2 + b)*sqrt(-(-b^2)^(1/3))*arctan(1/3*(2*sqrt(3)*(-b^2)^(2/3)* (cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2 - 1)*sq rt(-(-b^2)^(1/3))*((b*cosh(d*x + c)^2 + b*sinh(d*x + c)^2 + b)/(cosh(d*x + c)^2 + sinh(d*x + c)^2 - 1))^(1/3) - sqrt(3)*(b*cosh(d*x + c)^2 + 2*b*cos h(d*x + c)*sinh(d*x + c) + b*sinh(d*x + c)^2 + b)*(-b^2)^(1/3)*sqrt(-(-b^2 )^(1/3)))/(b^2*cosh(d*x + c)^2 + 2*b^2*cosh(d*x + c)*sinh(d*x + c) + b^2*s inh(d*x + c)^2 + b^2)) + 2*sqrt(3)*(b*cosh(d*x + c)^2 + 2*b*cosh(d*x + c)* sinh(d*x + c) + b*sinh(d*x + c)^2 + b)*(b^2)^(1/6)*arctan(1/3*sqrt(3)*(b^2 )^(1/6)*(2*(b^2)^(2/3)*(cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2 - 1)*((b*cosh(d*x + c)^2 + b*sinh(d*x + c)^2 + b)/(cosh(d* x + c)^2 + sinh(d*x + c)^2 - 1))^(1/3) - (b*cosh(d*x + c)^2 + 2*b*cosh(d*x + c)*sinh(d*x + c) + b*sinh(d*x + c)^2 + b)*(b^2)^(1/3))/(b^2*cosh(d*x + c)^2 + 2*b^2*cosh(d*x + c)*sinh(d*x + c) + b^2*sinh(d*x + c)^2 + b^2)) + ( -b^2)^(2/3)*(cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2 + 1)*log(((cosh(d*x + c)^4 + 4*cosh(d*x + c)^3*sinh(d*x + c) + 6*cosh (d*x + c)^2*sinh(d*x + c)^2 + 4*cosh(d*x + c)*sinh(d*x + c)^3 + sinh(d*x + c)^4 - 1)*(-b^2)^(2/3)*((b*cosh(d*x + c)^2 + b*sinh(d*x + c)^2 + b)/(cosh (d*x + c)^2 + sinh(d*x + c)^2 - 1))^(1/3) + (b*cosh(d*x + c)^4 + 4*b*cosh( d*x + c)*sinh(d*x + c)^3 + b*sinh(d*x + c)^4 - 2*b*cosh(d*x + c)^2 + 2*...
\[ \int \frac {1}{\left (b \coth ^2(c+d x)\right )^{2/3}} \, dx=\int \frac {1}{\left (b \coth ^{2}{\left (c + d x \right )}\right )^{\frac {2}{3}}}\, dx \]
\[ \int \frac {1}{\left (b \coth ^2(c+d x)\right )^{2/3}} \, dx=\int { \frac {1}{\left (b \coth \left (d x + c\right )^{2}\right )^{\frac {2}{3}}} \,d x } \]
\[ \int \frac {1}{\left (b \coth ^2(c+d x)\right )^{2/3}} \, dx=\int { \frac {1}{\left (b \coth \left (d x + c\right )^{2}\right )^{\frac {2}{3}}} \,d x } \]
Timed out. \[ \int \frac {1}{\left (b \coth ^2(c+d x)\right )^{2/3}} \, dx=\int \frac {1}{{\left (b\,{\mathrm {coth}\left (c+d\,x\right )}^2\right )}^{2/3}} \,d x \]