Integrand size = 14, antiderivative size = 291 \[ \int \left (b \coth ^4(c+d x)\right )^{2/3} \, dx=\frac {\sqrt {3} \arctan \left (\frac {1-2 \sqrt [3]{\coth (c+d x)}}{\sqrt {3}}\right ) \left (b \coth ^4(c+d x)\right )^{2/3}}{2 d \coth ^{\frac {8}{3}}(c+d x)}-\frac {\sqrt {3} \arctan \left (\frac {1+2 \sqrt [3]{\coth (c+d x)}}{\sqrt {3}}\right ) \left (b \coth ^4(c+d x)\right )^{2/3}}{2 d \coth ^{\frac {8}{3}}(c+d x)}+\frac {\text {arctanh}\left (\sqrt [3]{\coth (c+d x)}\right ) \left (b \coth ^4(c+d x)\right )^{2/3}}{d \coth ^{\frac {8}{3}}(c+d x)}-\frac {\left (b \coth ^4(c+d x)\right )^{2/3} \log \left (1-\sqrt [3]{\coth (c+d x)}+\coth ^{\frac {2}{3}}(c+d x)\right )}{4 d \coth ^{\frac {8}{3}}(c+d x)}+\frac {\left (b \coth ^4(c+d x)\right )^{2/3} \log \left (1+\sqrt [3]{\coth (c+d x)}+\coth ^{\frac {2}{3}}(c+d x)\right )}{4 d \coth ^{\frac {8}{3}}(c+d x)}-\frac {3 \left (b \coth ^4(c+d x)\right )^{2/3} \tanh (c+d x)}{5 d} \]
arctanh(coth(d*x+c)^(1/3))*(b*coth(d*x+c)^4)^(2/3)/d/coth(d*x+c)^(8/3)-1/4 *(b*coth(d*x+c)^4)^(2/3)*ln(1-coth(d*x+c)^(1/3)+coth(d*x+c)^(2/3))/d/coth( d*x+c)^(8/3)+1/4*(b*coth(d*x+c)^4)^(2/3)*ln(1+coth(d*x+c)^(1/3)+coth(d*x+c )^(2/3))/d/coth(d*x+c)^(8/3)+1/2*arctan(1/3*(1-2*coth(d*x+c)^(1/3))*3^(1/2 ))*(b*coth(d*x+c)^4)^(2/3)*3^(1/2)/d/coth(d*x+c)^(8/3)-1/2*arctan(1/3*(1+2 *coth(d*x+c)^(1/3))*3^(1/2))*(b*coth(d*x+c)^4)^(2/3)*3^(1/2)/d/coth(d*x+c) ^(8/3)-3/5*(b*coth(d*x+c)^4)^(2/3)*tanh(d*x+c)/d
Time = 0.40 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.57 \[ \int \left (b \coth ^4(c+d x)\right )^{2/3} \, dx=\frac {\left (b \coth ^4(c+d x)\right )^{2/3} \left (20 \text {arctanh}\left (\sqrt [3]{\coth (c+d x)}\right )-12 \coth ^{\frac {5}{3}}(c+d x)+5 \left (2 \sqrt {3} \arctan \left (\frac {1-2 \sqrt [3]{\coth (c+d x)}}{\sqrt {3}}\right )-2 \sqrt {3} \arctan \left (\frac {1+2 \sqrt [3]{\coth (c+d x)}}{\sqrt {3}}\right )-\log \left (1-\sqrt [3]{\coth (c+d x)}+\coth ^{\frac {2}{3}}(c+d x)\right )+\log \left (1+\sqrt [3]{\coth (c+d x)}+\coth ^{\frac {2}{3}}(c+d x)\right )\right )\right )}{20 d \coth ^{\frac {8}{3}}(c+d x)} \]
((b*Coth[c + d*x]^4)^(2/3)*(20*ArcTanh[Coth[c + d*x]^(1/3)] - 12*Coth[c + d*x]^(5/3) + 5*(2*Sqrt[3]*ArcTan[(1 - 2*Coth[c + d*x]^(1/3))/Sqrt[3]] - 2* Sqrt[3]*ArcTan[(1 + 2*Coth[c + d*x]^(1/3))/Sqrt[3]] - Log[1 - Coth[c + d*x ]^(1/3) + Coth[c + d*x]^(2/3)] + Log[1 + Coth[c + d*x]^(1/3) + Coth[c + d* x]^(2/3)])))/(20*d*Coth[c + d*x]^(8/3))
Time = 0.51 (sec) , antiderivative size = 186, 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, 3954, 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.
\(\displaystyle \int \left (b \coth ^4(c+d x)\right )^{2/3} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \left (b \tan \left (i c+i d x+\frac {\pi }{2}\right )^4\right )^{2/3}dx\) |
\(\Big \downarrow \) 4141 |
\(\displaystyle \frac {\left (b \coth ^4(c+d x)\right )^{2/3} \int \coth ^{\frac {8}{3}}(c+d x)dx}{\coth ^{\frac {8}{3}}(c+d x)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\left (b \coth ^4(c+d x)\right )^{2/3} \int \left (-i \tan \left (i c+i d x+\frac {\pi }{2}\right )\right )^{8/3}dx}{\coth ^{\frac {8}{3}}(c+d x)}\) |
\(\Big \downarrow \) 3954 |
\(\displaystyle \frac {\left (b \coth ^4(c+d x)\right )^{2/3} \left (\int \coth ^{\frac {2}{3}}(c+d x)dx-\frac {3 \coth ^{\frac {5}{3}}(c+d x)}{5 d}\right )}{\coth ^{\frac {8}{3}}(c+d x)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\left (b \coth ^4(c+d x)\right )^{2/3} \left (-\frac {3 \coth ^{\frac {5}{3}}(c+d x)}{5 d}+\int \left (-i \tan \left (i c+i d x+\frac {\pi }{2}\right )\right )^{2/3}dx\right )}{\coth ^{\frac {8}{3}}(c+d x)}\) |
\(\Big \downarrow \) 3957 |
\(\displaystyle \frac {\left (b \coth ^4(c+d x)\right )^{2/3} \left (-\frac {\int -\frac {\coth ^{\frac {2}{3}}(c+d x)}{1-\coth ^2(c+d x)}d\coth (c+d x)}{d}-\frac {3 \coth ^{\frac {5}{3}}(c+d x)}{5 d}\right )}{\coth ^{\frac {8}{3}}(c+d x)}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\left (b \coth ^4(c+d x)\right )^{2/3} \left (\frac {\int \frac {\coth ^{\frac {2}{3}}(c+d x)}{1-\coth ^2(c+d x)}d\coth (c+d x)}{d}-\frac {3 \coth ^{\frac {5}{3}}(c+d x)}{5 d}\right )}{\coth ^{\frac {8}{3}}(c+d x)}\) |
\(\Big \downarrow \) 266 |
\(\displaystyle \frac {\left (b \coth ^4(c+d x)\right )^{2/3} \left (\frac {3 \int \frac {\coth ^{\frac {4}{3}}(c+d x)}{1-\coth ^2(c+d x)}d\sqrt [3]{\coth (c+d x)}}{d}-\frac {3 \coth ^{\frac {5}{3}}(c+d x)}{5 d}\right )}{\coth ^{\frac {8}{3}}(c+d x)}\) |
\(\Big \downarrow \) 825 |
\(\displaystyle \frac {\left (b \coth ^4(c+d x)\right )^{2/3} \left (\frac {3 \left (\frac {1}{3} \int \frac {1}{1-\coth ^{\frac {2}{3}}(c+d x)}d\sqrt [3]{\coth (c+d x)}+\frac {1}{3} \int -\frac {\sqrt [3]{\coth (c+d x)}+1}{2 \left (\coth ^{\frac {2}{3}}(c+d x)-\sqrt [3]{\coth (c+d x)}+1\right )}d\sqrt [3]{\coth (c+d x)}+\frac {1}{3} \int -\frac {1-\sqrt [3]{\coth (c+d x)}}{2 \left (\coth ^{\frac {2}{3}}(c+d x)+\sqrt [3]{\coth (c+d x)}+1\right )}d\sqrt [3]{\coth (c+d x)}\right )}{d}-\frac {3 \coth ^{\frac {5}{3}}(c+d x)}{5 d}\right )}{\coth ^{\frac {8}{3}}(c+d x)}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\left (b \coth ^4(c+d x)\right )^{2/3} \left (\frac {3 \left (\frac {1}{3} \int \frac {1}{1-\coth ^{\frac {2}{3}}(c+d x)}d\sqrt [3]{\coth (c+d x)}-\frac {1}{6} \int \frac {\sqrt [3]{\coth (c+d x)}+1}{\coth ^{\frac {2}{3}}(c+d x)-\sqrt [3]{\coth (c+d x)}+1}d\sqrt [3]{\coth (c+d x)}-\frac {1}{6} \int \frac {1-\sqrt [3]{\coth (c+d x)}}{\coth ^{\frac {2}{3}}(c+d x)+\sqrt [3]{\coth (c+d x)}+1}d\sqrt [3]{\coth (c+d x)}\right )}{d}-\frac {3 \coth ^{\frac {5}{3}}(c+d x)}{5 d}\right )}{\coth ^{\frac {8}{3}}(c+d x)}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {\left (b \coth ^4(c+d x)\right )^{2/3} \left (\frac {3 \left (-\frac {1}{6} \int \frac {\sqrt [3]{\coth (c+d x)}+1}{\coth ^{\frac {2}{3}}(c+d x)-\sqrt [3]{\coth (c+d x)}+1}d\sqrt [3]{\coth (c+d x)}-\frac {1}{6} \int \frac {1-\sqrt [3]{\coth (c+d x)}}{\coth ^{\frac {2}{3}}(c+d x)+\sqrt [3]{\coth (c+d x)}+1}d\sqrt [3]{\coth (c+d x)}+\frac {1}{3} \text {arctanh}\left (\sqrt [3]{\coth (c+d x)}\right )\right )}{d}-\frac {3 \coth ^{\frac {5}{3}}(c+d x)}{5 d}\right )}{\coth ^{\frac {8}{3}}(c+d x)}\) |
\(\Big \downarrow \) 1142 |
\(\displaystyle \frac {\left (b \coth ^4(c+d x)\right )^{2/3} \left (\frac {3 \left (\frac {1}{6} \left (-\frac {3}{2} \int \frac {1}{\coth ^{\frac {2}{3}}(c+d x)-\sqrt [3]{\coth (c+d x)}+1}d\sqrt [3]{\coth (c+d x)}-\frac {1}{2} \int -\frac {1-2 \sqrt [3]{\coth (c+d x)}}{\coth ^{\frac {2}{3}}(c+d x)-\sqrt [3]{\coth (c+d x)}+1}d\sqrt [3]{\coth (c+d x)}\right )+\frac {1}{6} \left (\frac {1}{2} \int \frac {2 \sqrt [3]{\coth (c+d x)}+1}{\coth ^{\frac {2}{3}}(c+d x)+\sqrt [3]{\coth (c+d x)}+1}d\sqrt [3]{\coth (c+d x)}-\frac {3}{2} \int \frac {1}{\coth ^{\frac {2}{3}}(c+d x)+\sqrt [3]{\coth (c+d x)}+1}d\sqrt [3]{\coth (c+d x)}\right )+\frac {1}{3} \text {arctanh}\left (\sqrt [3]{\coth (c+d x)}\right )\right )}{d}-\frac {3 \coth ^{\frac {5}{3}}(c+d x)}{5 d}\right )}{\coth ^{\frac {8}{3}}(c+d x)}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\left (b \coth ^4(c+d x)\right )^{2/3} \left (\frac {3 \left (\frac {1}{6} \left (\frac {1}{2} \int \frac {1-2 \sqrt [3]{\coth (c+d x)}}{\coth ^{\frac {2}{3}}(c+d x)-\sqrt [3]{\coth (c+d x)}+1}d\sqrt [3]{\coth (c+d x)}-\frac {3}{2} \int \frac {1}{\coth ^{\frac {2}{3}}(c+d x)-\sqrt [3]{\coth (c+d x)}+1}d\sqrt [3]{\coth (c+d x)}\right )+\frac {1}{6} \left (\frac {1}{2} \int \frac {2 \sqrt [3]{\coth (c+d x)}+1}{\coth ^{\frac {2}{3}}(c+d x)+\sqrt [3]{\coth (c+d x)}+1}d\sqrt [3]{\coth (c+d x)}-\frac {3}{2} \int \frac {1}{\coth ^{\frac {2}{3}}(c+d x)+\sqrt [3]{\coth (c+d x)}+1}d\sqrt [3]{\coth (c+d x)}\right )+\frac {1}{3} \text {arctanh}\left (\sqrt [3]{\coth (c+d x)}\right )\right )}{d}-\frac {3 \coth ^{\frac {5}{3}}(c+d x)}{5 d}\right )}{\coth ^{\frac {8}{3}}(c+d x)}\) |
\(\Big \downarrow \) 1083 |
\(\displaystyle \frac {\left (b \coth ^4(c+d x)\right )^{2/3} \left (\frac {3 \left (\frac {1}{6} \left (3 \int \frac {1}{-\coth ^{\frac {2}{3}}(c+d x)-3}d\left (2 \sqrt [3]{\coth (c+d x)}-1\right )+\frac {1}{2} \int \frac {1-2 \sqrt [3]{\coth (c+d x)}}{\coth ^{\frac {2}{3}}(c+d x)-\sqrt [3]{\coth (c+d x)}+1}d\sqrt [3]{\coth (c+d x)}\right )+\frac {1}{6} \left (3 \int \frac {1}{-\coth ^{\frac {2}{3}}(c+d x)-3}d\left (2 \sqrt [3]{\coth (c+d x)}+1\right )+\frac {1}{2} \int \frac {2 \sqrt [3]{\coth (c+d x)}+1}{\coth ^{\frac {2}{3}}(c+d x)+\sqrt [3]{\coth (c+d x)}+1}d\sqrt [3]{\coth (c+d x)}\right )+\frac {1}{3} \text {arctanh}\left (\sqrt [3]{\coth (c+d x)}\right )\right )}{d}-\frac {3 \coth ^{\frac {5}{3}}(c+d x)}{5 d}\right )}{\coth ^{\frac {8}{3}}(c+d x)}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {\left (b \coth ^4(c+d x)\right )^{2/3} \left (\frac {3 \left (\frac {1}{6} \left (\frac {1}{2} \int \frac {1-2 \sqrt [3]{\coth (c+d x)}}{\coth ^{\frac {2}{3}}(c+d x)-\sqrt [3]{\coth (c+d x)}+1}d\sqrt [3]{\coth (c+d x)}-\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{\coth (c+d x)}-1}{\sqrt {3}}\right )\right )+\frac {1}{6} \left (\frac {1}{2} \int \frac {2 \sqrt [3]{\coth (c+d x)}+1}{\coth ^{\frac {2}{3}}(c+d x)+\sqrt [3]{\coth (c+d x)}+1}d\sqrt [3]{\coth (c+d x)}-\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{\coth (c+d x)}+1}{\sqrt {3}}\right )\right )+\frac {1}{3} \text {arctanh}\left (\sqrt [3]{\coth (c+d x)}\right )\right )}{d}-\frac {3 \coth ^{\frac {5}{3}}(c+d x)}{5 d}\right )}{\coth ^{\frac {8}{3}}(c+d x)}\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle \frac {\left (b \coth ^4(c+d x)\right )^{2/3} \left (\frac {3 \left (\frac {1}{6} \left (-\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{\coth (c+d x)}-1}{\sqrt {3}}\right )-\frac {1}{2} \log \left (\coth ^{\frac {2}{3}}(c+d x)-\sqrt [3]{\coth (c+d x)}+1\right )\right )+\frac {1}{6} \left (\frac {1}{2} \log \left (\coth ^{\frac {2}{3}}(c+d x)+\sqrt [3]{\coth (c+d x)}+1\right )-\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{\coth (c+d x)}+1}{\sqrt {3}}\right )\right )+\frac {1}{3} \text {arctanh}\left (\sqrt [3]{\coth (c+d x)}\right )\right )}{d}-\frac {3 \coth ^{\frac {5}{3}}(c+d x)}{5 d}\right )}{\coth ^{\frac {8}{3}}(c+d x)}\) |
((b*Coth[c + d*x]^4)^(2/3)*((-3*Coth[c + d*x]^(5/3))/(5*d) + (3*(ArcTanh[C oth[c + d*x]^(1/3)]/3 + (-(Sqrt[3]*ArcTan[(-1 + 2*Coth[c + d*x]^(1/3))/Sqr t[3]]) - Log[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) + Coth[c + d*x]^(2/3)]/2)/6))/d))/Coth[c + d*x]^(8/3)
3.1.45.3.1 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*((b*Tan[c + d *x])^(n - 1)/(d*(n - 1))), x] - Simp[b^2 Int[(b*Tan[c + d*x])^(n - 2), x] , x] /; FreeQ[{b, c, d}, x] && GtQ[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]])
\[\int \left (b \coth \left (d x +c \right )^{4}\right )^{\frac {2}{3}}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 618 vs. \(2 (239) = 478\).
Time = 0.28 (sec) , antiderivative size = 618, normalized size of antiderivative = 2.12 \[ \int \left (b \coth ^4(c+d x)\right )^{2/3} \, dx=-\frac {10 \, {\left (\sqrt {3} \cosh \left (d x + c\right )^{2} + 2 \, \sqrt {3} \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sqrt {3} \sinh \left (d x + c\right )^{2} - \sqrt {3}\right )} \left (-b^{2}\right )^{\frac {1}{3}} \arctan \left (-\frac {\sqrt {3} b - 2 \, \sqrt {3} \left (-b^{2}\right )^{\frac {1}{3}} \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {1}{3}}}{3 \, b}\right ) + 10 \, {\left (\sqrt {3} \cosh \left (d x + c\right )^{2} + 2 \, \sqrt {3} \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sqrt {3} \sinh \left (d x + c\right )^{2} - \sqrt {3}\right )} {\left (b^{2}\right )}^{\frac {1}{3}} \arctan \left (-\frac {\sqrt {3} b - 2 \, \sqrt {3} {\left (b^{2}\right )}^{\frac {1}{3}} \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {1}{3}}}{3 \, b}\right ) + 5 \, \left (-b^{2}\right )^{\frac {1}{3}} {\left (\cosh \left (d x + c\right )^{2} + 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2} - 1\right )} \log \left (b \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {2}{3}} - \left (-b^{2}\right )^{\frac {1}{3}} b + \left (-b^{2}\right )^{\frac {2}{3}} \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {1}{3}}\right ) + 5 \, {\left (b^{2}\right )}^{\frac {1}{3}} {\left (\cosh \left (d x + c\right )^{2} + 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2} - 1\right )} \log \left (b \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {2}{3}} + {\left (b^{2}\right )}^{\frac {1}{3}} b - {\left (b^{2}\right )}^{\frac {2}{3}} \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {1}{3}}\right ) - 10 \, \left (-b^{2}\right )^{\frac {1}{3}} {\left (\cosh \left (d x + c\right )^{2} + 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2} - 1\right )} \log \left (b \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {1}{3}} - \left (-b^{2}\right )^{\frac {2}{3}}\right ) - 10 \, {\left (b^{2}\right )}^{\frac {1}{3}} {\left (\cosh \left (d x + c\right )^{2} + 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2} - 1\right )} \log \left (b \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {1}{3}} + {\left (b^{2}\right )}^{\frac {2}{3}}\right ) + 12 \, {\left (\cosh \left (d x + c\right )^{2} + 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2} + 1\right )} \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {2}{3}}}{20 \, {\left (d \cosh \left (d x + c\right )^{2} + 2 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + d \sinh \left (d x + c\right )^{2} - d\right )}} \]
-1/20*(10*(sqrt(3)*cosh(d*x + c)^2 + 2*sqrt(3)*cosh(d*x + c)*sinh(d*x + c) + sqrt(3)*sinh(d*x + c)^2 - sqrt(3))*(-b^2)^(1/3)*arctan(-1/3*(sqrt(3)*b - 2*sqrt(3)*(-b^2)^(1/3)*(b*cosh(d*x + c)/sinh(d*x + c))^(1/3))/b) + 10*(s qrt(3)*cosh(d*x + c)^2 + 2*sqrt(3)*cosh(d*x + c)*sinh(d*x + c) + sqrt(3)*s inh(d*x + c)^2 - sqrt(3))*(b^2)^(1/3)*arctan(-1/3*(sqrt(3)*b - 2*sqrt(3)*( b^2)^(1/3)*(b*cosh(d*x + c)/sinh(d*x + c))^(1/3))/b) + 5*(-b^2)^(1/3)*(cos h(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2 - 1)*log(b* (b*cosh(d*x + c)/sinh(d*x + c))^(2/3) - (-b^2)^(1/3)*b + (-b^2)^(2/3)*(b*c osh(d*x + c)/sinh(d*x + c))^(1/3)) + 5*(b^2)^(1/3)*(cosh(d*x + c)^2 + 2*co sh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2 - 1)*log(b*(b*cosh(d*x + c)/si nh(d*x + c))^(2/3) + (b^2)^(1/3)*b - (b^2)^(2/3)*(b*cosh(d*x + c)/sinh(d*x + c))^(1/3)) - 10*(-b^2)^(1/3)*(cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d* x + c) + sinh(d*x + c)^2 - 1)*log(b*(b*cosh(d*x + c)/sinh(d*x + c))^(1/3) - (-b^2)^(2/3)) - 10*(b^2)^(1/3)*(cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d *x + c) + sinh(d*x + c)^2 - 1)*log(b*(b*cosh(d*x + c)/sinh(d*x + c))^(1/3) + (b^2)^(2/3)) + 12*(cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + si nh(d*x + c)^2 + 1)*(b*cosh(d*x + c)/sinh(d*x + c))^(2/3))/(d*cosh(d*x + c) ^2 + 2*d*cosh(d*x + c)*sinh(d*x + c) + d*sinh(d*x + c)^2 - d)
\[ \int \left (b \coth ^4(c+d x)\right )^{2/3} \, dx=\int \left (b \coth ^{4}{\left (c + d x \right )}\right )^{\frac {2}{3}}\, dx \]
\[ \int \left (b \coth ^4(c+d x)\right )^{2/3} \, dx=\int { \left (b \coth \left (d x + c\right )^{4}\right )^{\frac {2}{3}} \,d x } \]
\[ \int \left (b \coth ^4(c+d x)\right )^{2/3} \, dx=\int { \left (b \coth \left (d x + c\right )^{4}\right )^{\frac {2}{3}} \,d x } \]
Timed out. \[ \int \left (b \coth ^4(c+d x)\right )^{2/3} \, dx=\int {\left (b\,{\mathrm {coth}\left (c+d\,x\right )}^4\right )}^{2/3} \,d x \]