Integrand size = 12, antiderivative size = 218 \[ \int \frac {1}{(b \coth (c+d x))^{2/3}} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b \coth (c+d x)}}{\sqrt [3]{b}}}{\sqrt {3}}\right )}{2 b^{2/3} d}+\frac {\sqrt {3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{b \coth (c+d x)}}{\sqrt [3]{b}}}{\sqrt {3}}\right )}{2 b^{2/3} d}+\frac {\text {arctanh}\left (\frac {\sqrt [3]{b \coth (c+d x)}}{\sqrt [3]{b}}\right )}{b^{2/3} d}-\frac {\log \left (b^{2/3}-\sqrt [3]{b} \sqrt [3]{b \coth (c+d x)}+(b \coth (c+d x))^{2/3}\right )}{4 b^{2/3} d}+\frac {\log \left (b^{2/3}+\sqrt [3]{b} \sqrt [3]{b \coth (c+d x)}+(b \coth (c+d x))^{2/3}\right )}{4 b^{2/3} d} \]
arctanh((b*coth(d*x+c))^(1/3)/b^(1/3))/b^(2/3)/d-1/4*ln(b^(2/3)-b^(1/3)*(b *coth(d*x+c))^(1/3)+(b*coth(d*x+c))^(2/3))/b^(2/3)/d+1/4*ln(b^(2/3)+b^(1/3 )*(b*coth(d*x+c))^(1/3)+(b*coth(d*x+c))^(2/3))/b^(2/3)/d-1/2*arctan(1/3*(1 -2*(b*coth(d*x+c))^(1/3)/b^(1/3))*3^(1/2))*3^(1/2)/b^(2/3)/d+1/2*arctan(1/ 3*(1+2*(b*coth(d*x+c))^(1/3)/b^(1/3))*3^(1/2))*3^(1/2)/b^(2/3)/d
Time = 0.25 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.81 \[ \int \frac {1}{(b \coth (c+d x))^{2/3}} \, dx=-\frac {\sqrt [3]{b \coth (c+d x)} \left (\log \left (1-\sqrt [6]{\coth ^2(c+d x)}\right )-\log \left (1+\sqrt [6]{\coth ^2(c+d x)}\right )+\sqrt [3]{-1} \left (-\sqrt [3]{-1} \log \left (1-\sqrt [3]{-1} \sqrt [6]{\coth ^2(c+d x)}\right )+\sqrt [3]{-1} \log \left (1+\sqrt [3]{-1} \sqrt [6]{\coth ^2(c+d x)}\right )-\log \left (1-(-1)^{2/3} \sqrt [6]{\coth ^2(c+d x)}\right )+\log \left (1+(-1)^{2/3} \sqrt [6]{\coth ^2(c+d x)}\right )\right )\right )}{2 b d \sqrt [6]{\coth ^2(c+d x)}} \]
-1/2*((b*Coth[c + d*x])^(1/3)*(Log[1 - (Coth[c + d*x]^2)^(1/6)] - Log[1 + (Coth[c + d*x]^2)^(1/6)] + (-1)^(1/3)*(-((-1)^(1/3)*Log[1 - (-1)^(1/3)*(Co th[c + d*x]^2)^(1/6)]) + (-1)^(1/3)*Log[1 + (-1)^(1/3)*(Coth[c + d*x]^2)^( 1/6)] - Log[1 - (-1)^(2/3)*(Coth[c + d*x]^2)^(1/6)] + Log[1 + (-1)^(2/3)*( Coth[c + d*x]^2)^(1/6)])))/(b*d*(Coth[c + d*x]^2)^(1/6))
Time = 0.38 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.81, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {3042, 3957, 25, 266, 754, 27, 219, 1142, 25, 1082, 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.
| \(\displaystyle \int \frac {1}{(b \coth (c+d x))^{2/3}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\left (-i b \tan \left (i c+i d x+\frac {\pi }{2}\right )\right )^{2/3}}dx\) |
| \(\Big \downarrow \) 3957 |
| \(\displaystyle -\frac {b \int -\frac {1}{(b \coth (c+d x))^{2/3} \left (b^2-b^2 \coth ^2(c+d x)\right )}d(b \coth (c+d x))}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \int \frac {1}{(b \coth (c+d x))^{2/3} \left (b^2-b^2 \coth ^2(c+d x)\right )}d(b \coth (c+d x))}{d}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {3 b \int \frac {1}{b^2-b^6 \coth ^6(c+d x)}d\sqrt [3]{b \coth (c+d x)}}{d}\) |
| \(\Big \downarrow \) 754 |
| \(\displaystyle \frac {3 b \left (\frac {\int \frac {1}{b^{2/3}-b^2 \coth ^2(c+d x)}d\sqrt [3]{b \coth (c+d x)}}{3 b^{4/3}}+\frac {\int \frac {2 \sqrt [3]{b}-\sqrt [3]{b \coth (c+d x)}}{2 \left (b^2 \coth ^2(c+d x)-b^{4/3} \coth (c+d x)+b^{2/3}\right )}d\sqrt [3]{b \coth (c+d x)}}{3 b^{5/3}}+\frac {\int \frac {2 \sqrt [3]{b}+\sqrt [3]{b \coth (c+d x)}}{2 \left (b^2 \coth ^2(c+d x)+b^{4/3} \coth (c+d x)+b^{2/3}\right )}d\sqrt [3]{b \coth (c+d x)}}{3 b^{5/3}}\right )}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3 b \left (\frac {\int \frac {1}{b^{2/3}-b^2 \coth ^2(c+d x)}d\sqrt [3]{b \coth (c+d x)}}{3 b^{4/3}}+\frac {\int \frac {2 \sqrt [3]{b}-\sqrt [3]{b \coth (c+d x)}}{b^2 \coth ^2(c+d x)-b^{4/3} \coth (c+d x)+b^{2/3}}d\sqrt [3]{b \coth (c+d x)}}{6 b^{5/3}}+\frac {\int \frac {2 \sqrt [3]{b}+\sqrt [3]{b \coth (c+d x)}}{b^2 \coth ^2(c+d x)+b^{4/3} \coth (c+d x)+b^{2/3}}d\sqrt [3]{b \coth (c+d x)}}{6 b^{5/3}}\right )}{d}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {3 b \left (\frac {\int \frac {2 \sqrt [3]{b}-\sqrt [3]{b \coth (c+d x)}}{b^2 \coth ^2(c+d x)-b^{4/3} \coth (c+d x)+b^{2/3}}d\sqrt [3]{b \coth (c+d x)}}{6 b^{5/3}}+\frac {\int \frac {2 \sqrt [3]{b}+\sqrt [3]{b \coth (c+d x)}}{b^2 \coth ^2(c+d x)+b^{4/3} \coth (c+d x)+b^{2/3}}d\sqrt [3]{b \coth (c+d x)}}{6 b^{5/3}}+\frac {\text {arctanh}\left (b^{2/3} \coth (c+d x)\right )}{3 b^{5/3}}\right )}{d}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {3 b \left (\frac {\frac {3}{2} \sqrt [3]{b} \int \frac {1}{b^2 \coth ^2(c+d x)-b^{4/3} \coth (c+d x)+b^{2/3}}d\sqrt [3]{b \coth (c+d x)}-\frac {1}{2} \int -\frac {\sqrt [3]{b}-2 \sqrt [3]{b \coth (c+d x)}}{b^2 \coth ^2(c+d x)-b^{4/3} \coth (c+d x)+b^{2/3}}d\sqrt [3]{b \coth (c+d x)}}{6 b^{5/3}}+\frac {\frac {3}{2} \sqrt [3]{b} \int \frac {1}{b^2 \coth ^2(c+d x)+b^{4/3} \coth (c+d x)+b^{2/3}}d\sqrt [3]{b \coth (c+d x)}+\frac {1}{2} \int \frac {\sqrt [3]{b}+2 \sqrt [3]{b \coth (c+d x)}}{b^2 \coth ^2(c+d x)+b^{4/3} \coth (c+d x)+b^{2/3}}d\sqrt [3]{b \coth (c+d x)}}{6 b^{5/3}}+\frac {\text {arctanh}\left (b^{2/3} \coth (c+d x)\right )}{3 b^{5/3}}\right )}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {3 b \left (\frac {\frac {3}{2} \sqrt [3]{b} \int \frac {1}{b^2 \coth ^2(c+d x)-b^{4/3} \coth (c+d x)+b^{2/3}}d\sqrt [3]{b \coth (c+d x)}+\frac {1}{2} \int \frac {\sqrt [3]{b}-2 \sqrt [3]{b \coth (c+d x)}}{b^2 \coth ^2(c+d x)-b^{4/3} \coth (c+d x)+b^{2/3}}d\sqrt [3]{b \coth (c+d x)}}{6 b^{5/3}}+\frac {\frac {3}{2} \sqrt [3]{b} \int \frac {1}{b^2 \coth ^2(c+d x)+b^{4/3} \coth (c+d x)+b^{2/3}}d\sqrt [3]{b \coth (c+d x)}+\frac {1}{2} \int \frac {\sqrt [3]{b}+2 \sqrt [3]{b \coth (c+d x)}}{b^2 \coth ^2(c+d x)+b^{4/3} \coth (c+d x)+b^{2/3}}d\sqrt [3]{b \coth (c+d x)}}{6 b^{5/3}}+\frac {\text {arctanh}\left (b^{2/3} \coth (c+d x)\right )}{3 b^{5/3}}\right )}{d}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {3 b \left (\frac {3 \int \frac {1}{-b^2 \coth ^2(c+d x)-3}d\left (1-2 b^{2/3} \coth (c+d x)\right )+\frac {1}{2} \int \frac {\sqrt [3]{b}-2 \sqrt [3]{b \coth (c+d x)}}{b^2 \coth ^2(c+d x)-b^{4/3} \coth (c+d x)+b^{2/3}}d\sqrt [3]{b \coth (c+d x)}}{6 b^{5/3}}+\frac {\frac {1}{2} \int \frac {\sqrt [3]{b}+2 \sqrt [3]{b \coth (c+d x)}}{b^2 \coth ^2(c+d x)+b^{4/3} \coth (c+d x)+b^{2/3}}d\sqrt [3]{b \coth (c+d x)}-3 \int \frac {1}{-b^2 \coth ^2(c+d x)-3}d\left (2 b^{2/3} \coth (c+d x)+1\right )}{6 b^{5/3}}+\frac {\text {arctanh}\left (b^{2/3} \coth (c+d x)\right )}{3 b^{5/3}}\right )}{d}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {3 b \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{b}-2 \sqrt [3]{b \coth (c+d x)}}{b^2 \coth ^2(c+d x)-b^{4/3} \coth (c+d x)+b^{2/3}}d\sqrt [3]{b \coth (c+d x)}-\sqrt {3} \arctan \left (\frac {1-2 b^{2/3} \coth (c+d x)}{\sqrt {3}}\right )}{6 b^{5/3}}+\frac {\frac {1}{2} \int \frac {\sqrt [3]{b}+2 \sqrt [3]{b \coth (c+d x)}}{b^2 \coth ^2(c+d x)+b^{4/3} \coth (c+d x)+b^{2/3}}d\sqrt [3]{b \coth (c+d x)}+\sqrt {3} \arctan \left (\frac {2 b^{2/3} \coth (c+d x)+1}{\sqrt {3}}\right )}{6 b^{5/3}}+\frac {\text {arctanh}\left (b^{2/3} \coth (c+d x)\right )}{3 b^{5/3}}\right )}{d}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {3 b \left (\frac {-\sqrt {3} \arctan \left (\frac {1-2 b^{2/3} \coth (c+d x)}{\sqrt {3}}\right )-\frac {1}{2} \log \left (-b^{4/3} \coth (c+d x)+b^{2/3}+b^2 \coth ^2(c+d x)\right )}{6 b^{5/3}}+\frac {\sqrt {3} \arctan \left (\frac {2 b^{2/3} \coth (c+d x)+1}{\sqrt {3}}\right )+\frac {1}{2} \log \left (b^{4/3} \coth (c+d x)+b^{2/3}+b^2 \coth ^2(c+d x)\right )}{6 b^{5/3}}+\frac {\text {arctanh}\left (b^{2/3} \coth (c+d x)\right )}{3 b^{5/3}}\right )}{d}\) |
(3*b*(ArcTanh[b^(2/3)*Coth[c + d*x]]/(3*b^(5/3)) + (-(Sqrt[3]*ArcTan[(1 - 2*b^(2/3)*Coth[c + d*x])/Sqrt[3]]) - Log[b^(2/3) - b^(4/3)*Coth[c + d*x] + b^2*Coth[c + d*x]^2]/2)/(6*b^(5/3)) + (Sqrt[3]*ArcTan[(1 + 2*b^(2/3)*Coth [c + d*x])/Sqrt[3]] + Log[b^(2/3) + b^(4/3)*Coth[c + d*x] + b^2*Coth[c + d *x]^2]/2)/(6*b^(5/3))))/d
3.1.13.3.1 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[((a_) + (b_.)*(x_)^(n_))^(-1), x_Symbol] :> Module[{r = Numerator[Rt[-a /b, n]], s = Denominator[Rt[-a/b, n]], k, u}, Simp[u = Int[(r - s*Cos[(2*k* Pi)/n]*x)/(r^2 - 2*r*s*Cos[(2*k*Pi)/n]*x + s^2*x^2), x] + Int[(r + s*Cos[(2 *k*Pi)/n]*x)/(r^2 + 2*r*s*Cos[(2*k*Pi)/n]*x + s^2*x^2), x]; 2*(r^2/(a*n)) Int[1/(r^2 - s^2*x^2), x] + 2*(r/(a*n)) Sum[u, {k, 1, (n - 2)/4}], x]] / ; FreeQ[{a, b}, x] && IGtQ[(n - 2)/4, 0] && NegQ[a/b]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{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/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]
Time = 0.10 (sec) , antiderivative size = 193, normalized size of antiderivative = 0.89
| method | result | size |
| derivativedivides | \(-\frac {\ln \left (\left (b \coth \left (d x +c \right )\right )^{\frac {1}{3}}-b^{\frac {1}{3}}\right )}{2 d \,b^{\frac {2}{3}}}+\frac {\ln \left (b^{\frac {2}{3}}+b^{\frac {1}{3}} \left (b \coth \left (d x +c \right )\right )^{\frac {1}{3}}+\left (b \coth \left (d x +c \right )\right )^{\frac {2}{3}}\right )}{4 b^{\frac {2}{3}} d}+\frac {\arctan \left (\frac {\left (1+\frac {2 \left (b \coth \left (d x +c \right )\right )^{\frac {1}{3}}}{b^{\frac {1}{3}}}\right ) \sqrt {3}}{3}\right ) \sqrt {3}}{2 b^{\frac {2}{3}} d}+\frac {\ln \left (\left (b \coth \left (d x +c \right )\right )^{\frac {1}{3}}+b^{\frac {1}{3}}\right )}{2 d \,b^{\frac {2}{3}}}-\frac {\ln \left (b^{\frac {2}{3}}-b^{\frac {1}{3}} \left (b \coth \left (d x +c \right )\right )^{\frac {1}{3}}+\left (b \coth \left (d x +c \right )\right )^{\frac {2}{3}}\right )}{4 b^{\frac {2}{3}} d}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b \coth \left (d x +c \right )\right )^{\frac {1}{3}}}{b^{\frac {1}{3}}}-1\right )}{3}\right )}{2 d \,b^{\frac {2}{3}}}\) | \(193\) |
| default | \(-\frac {\ln \left (\left (b \coth \left (d x +c \right )\right )^{\frac {1}{3}}-b^{\frac {1}{3}}\right )}{2 d \,b^{\frac {2}{3}}}+\frac {\ln \left (b^{\frac {2}{3}}+b^{\frac {1}{3}} \left (b \coth \left (d x +c \right )\right )^{\frac {1}{3}}+\left (b \coth \left (d x +c \right )\right )^{\frac {2}{3}}\right )}{4 b^{\frac {2}{3}} d}+\frac {\arctan \left (\frac {\left (1+\frac {2 \left (b \coth \left (d x +c \right )\right )^{\frac {1}{3}}}{b^{\frac {1}{3}}}\right ) \sqrt {3}}{3}\right ) \sqrt {3}}{2 b^{\frac {2}{3}} d}+\frac {\ln \left (\left (b \coth \left (d x +c \right )\right )^{\frac {1}{3}}+b^{\frac {1}{3}}\right )}{2 d \,b^{\frac {2}{3}}}-\frac {\ln \left (b^{\frac {2}{3}}-b^{\frac {1}{3}} \left (b \coth \left (d x +c \right )\right )^{\frac {1}{3}}+\left (b \coth \left (d x +c \right )\right )^{\frac {2}{3}}\right )}{4 b^{\frac {2}{3}} d}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b \coth \left (d x +c \right )\right )^{\frac {1}{3}}}{b^{\frac {1}{3}}}-1\right )}{3}\right )}{2 d \,b^{\frac {2}{3}}}\) | \(193\) |
-1/2/d/b^(2/3)*ln((b*coth(d*x+c))^(1/3)-b^(1/3))+1/4*ln(b^(2/3)+b^(1/3)*(b *coth(d*x+c))^(1/3)+(b*coth(d*x+c))^(2/3))/b^(2/3)/d+1/2*arctan(1/3*(1+2*( b*coth(d*x+c))^(1/3)/b^(1/3))*3^(1/2))*3^(1/2)/b^(2/3)/d+1/2/d/b^(2/3)*ln( (b*coth(d*x+c))^(1/3)+b^(1/3))-1/4*ln(b^(2/3)-b^(1/3)*(b*coth(d*x+c))^(1/3 )+(b*coth(d*x+c))^(2/3))/b^(2/3)/d+1/2/d/b^(2/3)*3^(1/2)*arctan(1/3*3^(1/2 )*(2*(b*coth(d*x+c))^(1/3)/b^(1/3)-1))
Leaf count of result is larger than twice the leaf count of optimal. 356 vs. \(2 (166) = 332\).
Time = 0.26 (sec) , antiderivative size = 356, normalized size of antiderivative = 1.63 \[ \int \frac {1}{(b \coth (c+d x))^{2/3}} \, dx=\frac {2 \, \sqrt {3} b \sqrt {-\left (-b^{2}\right )^{\frac {1}{3}}} \arctan \left (-\frac {\sqrt {3} \left (-b^{2}\right )^{\frac {1}{3}} b \sqrt {-\left (-b^{2}\right )^{\frac {1}{3}}} - 2 \, \sqrt {3} \left (-b^{2}\right )^{\frac {2}{3}} \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {1}{3}} \sqrt {-\left (-b^{2}\right )^{\frac {1}{3}}}}{3 \, b^{2}}\right ) + 2 \, \sqrt {3} {\left (b^{2}\right )}^{\frac {1}{6}} b \arctan \left (-\frac {\sqrt {3} {\left (b^{2}\right )}^{\frac {1}{6}} {\left ({\left (b^{2}\right )}^{\frac {1}{3}} b - 2 \, {\left (b^{2}\right )}^{\frac {2}{3}} \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {1}{3}}\right )}}{3 \, b^{2}}\right ) + \left (-b^{2}\right )^{\frac {2}{3}} \log \left (b \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {2}{3}} - \left (-b^{2}\right )^{\frac {1}{3}} b + \left (-b^{2}\right )^{\frac {2}{3}} \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {1}{3}}\right ) - {\left (b^{2}\right )}^{\frac {2}{3}} \log \left (b \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {2}{3}} + {\left (b^{2}\right )}^{\frac {1}{3}} b - {\left (b^{2}\right )}^{\frac {2}{3}} \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {1}{3}}\right ) - 2 \, \left (-b^{2}\right )^{\frac {2}{3}} \log \left (b \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {1}{3}} - \left (-b^{2}\right )^{\frac {2}{3}}\right ) + 2 \, {\left (b^{2}\right )}^{\frac {2}{3}} \log \left (b \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {1}{3}} + {\left (b^{2}\right )}^{\frac {2}{3}}\right )}{4 \, b^{2} d} \]
1/4*(2*sqrt(3)*b*sqrt(-(-b^2)^(1/3))*arctan(-1/3*(sqrt(3)*(-b^2)^(1/3)*b*s qrt(-(-b^2)^(1/3)) - 2*sqrt(3)*(-b^2)^(2/3)*(b*cosh(d*x + c)/sinh(d*x + c) )^(1/3)*sqrt(-(-b^2)^(1/3)))/b^2) + 2*sqrt(3)*(b^2)^(1/6)*b*arctan(-1/3*sq rt(3)*(b^2)^(1/6)*((b^2)^(1/3)*b - 2*(b^2)^(2/3)*(b*cosh(d*x + c)/sinh(d*x + c))^(1/3))/b^2) + (-b^2)^(2/3)*log(b*(b*cosh(d*x + c)/sinh(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)) - (b^2)^(2/3)*log(b*(b*cosh(d*x + c)/sinh(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)) - 2*(-b^2)^(2/3)*log (b*(b*cosh(d*x + c)/sinh(d*x + c))^(1/3) - (-b^2)^(2/3)) + 2*(b^2)^(2/3)*l og(b*(b*cosh(d*x + c)/sinh(d*x + c))^(1/3) + (b^2)^(2/3)))/(b^2*d)
\[ \int \frac {1}{(b \coth (c+d x))^{2/3}} \, dx=\int \frac {1}{\left (b \coth {\left (c + d x \right )}\right )^{\frac {2}{3}}}\, dx \]
\[ \int \frac {1}{(b \coth (c+d x))^{2/3}} \, dx=\int { \frac {1}{\left (b \coth \left (d x + c\right )\right )^{\frac {2}{3}}} \,d x } \]
Exception generated. \[ \int \frac {1}{(b \coth (c+d x))^{2/3}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Minimal poly. in rootof must be fra ction free Error: Bad Argument ValueMinimal poly. in rootof must be fracti on free E
Time = 2.07 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.90 \[ \int \frac {1}{(b \coth (c+d x))^{2/3}} \, dx=\frac {\mathrm {atanh}\left (\frac {{\left (b\,\mathrm {coth}\left (c+d\,x\right )\right )}^{1/3}}{b^{1/3}}\right )}{b^{2/3}\,d}-\frac {\mathrm {atan}\left (\frac {b^{10/3}\,{\left (b\,\mathrm {coth}\left (c+d\,x\right )\right )}^{1/3}\,243{}\mathrm {i}}{-243\,b^{11/3}+\sqrt {3}\,b^{11/3}\,243{}\mathrm {i}}-\frac {243\,\sqrt {3}\,b^{10/3}\,{\left (b\,\mathrm {coth}\left (c+d\,x\right )\right )}^{1/3}}{-243\,b^{11/3}+\sqrt {3}\,b^{11/3}\,243{}\mathrm {i}}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2\,b^{2/3}\,d}-\frac {\mathrm {atan}\left (\frac {b^{10/3}\,{\left (b\,\mathrm {coth}\left (c+d\,x\right )\right )}^{1/3}\,243{}\mathrm {i}}{243\,b^{11/3}+\sqrt {3}\,b^{11/3}\,243{}\mathrm {i}}+\frac {243\,\sqrt {3}\,b^{10/3}\,{\left (b\,\mathrm {coth}\left (c+d\,x\right )\right )}^{1/3}}{243\,b^{11/3}+\sqrt {3}\,b^{11/3}\,243{}\mathrm {i}}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2\,b^{2/3}\,d} \]
atanh((b*coth(c + d*x))^(1/3)/b^(1/3))/(b^(2/3)*d) - (atan((b^(10/3)*(b*co th(c + d*x))^(1/3)*243i)/(3^(1/2)*b^(11/3)*243i - 243*b^(11/3)) - (243*3^( 1/2)*b^(10/3)*(b*coth(c + d*x))^(1/3))/(3^(1/2)*b^(11/3)*243i - 243*b^(11/ 3)))*(3^(1/2)*1i + 1)*1i)/(2*b^(2/3)*d) - (atan((b^(10/3)*(b*coth(c + d*x) )^(1/3)*243i)/(3^(1/2)*b^(11/3)*243i + 243*b^(11/3)) + (243*3^(1/2)*b^(10/ 3)*(b*coth(c + d*x))^(1/3))/(3^(1/2)*b^(11/3)*243i + 243*b^(11/3)))*(3^(1/ 2)*1i - 1)*1i)/(2*b^(2/3)*d)