Integrand size = 12, antiderivative size = 218 \[ \int (b \coth (c+d x))^{2/3} \, dx=\frac {\sqrt {3} b^{2/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b \coth (c+d x)}}{\sqrt [3]{b}}}{\sqrt {3}}\right )}{2 d}-\frac {\sqrt {3} b^{2/3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{b \coth (c+d x)}}{\sqrt [3]{b}}}{\sqrt {3}}\right )}{2 d}+\frac {b^{2/3} \text {arctanh}\left (\frac {\sqrt [3]{b \coth (c+d x)}}{\sqrt [3]{b}}\right )}{d}-\frac {b^{2/3} \log \left (b^{2/3}-\sqrt [3]{b} \sqrt [3]{b \coth (c+d x)}+(b \coth (c+d x))^{2/3}\right )}{4 d}+\frac {b^{2/3} \log \left (b^{2/3}+\sqrt [3]{b} \sqrt [3]{b \coth (c+d x)}+(b \coth (c+d x))^{2/3}\right )}{4 d} \]
b^(2/3)*arctanh((b*coth(d*x+c))^(1/3)/b^(1/3))/d-1/4*b^(2/3)*ln(b^(2/3)-b^ (1/3)*(b*coth(d*x+c))^(1/3)+(b*coth(d*x+c))^(2/3))/d+1/4*b^(2/3)*ln(b^(2/3 )+b^(1/3)*(b*coth(d*x+c))^(1/3)+(b*coth(d*x+c))^(2/3))/d+1/2*b^(2/3)*arcta n(1/3*(1-2*(b*coth(d*x+c))^(1/3)/b^(1/3))*3^(1/2))*3^(1/2)/d-1/2*b^(2/3)*a rctan(1/3*(1+2*(b*coth(d*x+c))^(1/3)/b^(1/3))*3^(1/2))*3^(1/2)/d
Time = 0.20 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.68 \[ \int (b \coth (c+d x))^{2/3} \, dx=\frac {(b \coth (c+d x))^{2/3} \left (2 \sqrt {3} \arctan \left (\frac {1-2 \sqrt [3]{\coth (c+d x)}}{\sqrt {3}}\right )-2 \sqrt {3} \arctan \left (\frac {1+2 \sqrt [3]{\coth (c+d x)}}{\sqrt {3}}\right )+4 \text {arctanh}\left (\sqrt [3]{\coth (c+d x)}\right )-\log \left (1-\sqrt [3]{\coth (c+d x)}+\coth ^{\frac {2}{3}}(c+d x)\right )+\log \left (1+\sqrt [3]{\coth (c+d x)}+\coth ^{\frac {2}{3}}(c+d x)\right )\right )}{4 d \coth ^{\frac {2}{3}}(c+d x)} \]
((b*Coth[c + d*x])^(2/3)*(2*Sqrt[3]*ArcTan[(1 - 2*Coth[c + d*x]^(1/3))/Sqr t[3]] - 2*Sqrt[3]*ArcTan[(1 + 2*Coth[c + d*x]^(1/3))/Sqrt[3]] + 4*ArcTanh[ Coth[c + d*x]^(1/3)] - 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)]))/(4*d*Coth[c + d*x]^ (2/3))
Time = 0.39 (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, 825, 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.
| \(\displaystyle \int (b \coth (c+d x))^{2/3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (-i b \tan \left (i c+i d x+\frac {\pi }{2}\right )\right )^{2/3}dx\) |
| \(\Big \downarrow \) 3957 |
| \(\displaystyle -\frac {b \int -\frac {(b \coth (c+d x))^{2/3}}{b^2-b^2 \coth ^2(c+d x)}d(b \coth (c+d x))}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \int \frac {(b \coth (c+d x))^{2/3}}{b^2-b^2 \coth ^2(c+d x)}d(b \coth (c+d x))}{d}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {3 b \int \frac {b^4 \coth ^4(c+d x)}{b^2-b^6 \coth ^6(c+d x)}d\sqrt [3]{b \coth (c+d x)}}{d}\) |
| \(\Big \downarrow \) 825 |
| \(\displaystyle \frac {3 b \left (\frac {1}{3} \int \frac {1}{b^{2/3}-b^2 \coth ^2(c+d x)}d\sqrt [3]{b \coth (c+d x)}+\frac {\int -\frac {\sqrt [3]{b}-\sqrt [3]{b \coth (c+d x)}}{2 \left (b^2 \coth ^2(c+d x)+b^{4/3} \coth (c+d x)+b^{2/3}\right )}d\sqrt [3]{b \coth (c+d x)}}{3 \sqrt [3]{b}}+\frac {\int -\frac {\sqrt [3]{b}+\sqrt [3]{b \coth (c+d x)}}{2 \left (b^2 \coth ^2(c+d x)-b^{4/3} \coth (c+d x)+b^{2/3}\right )}d\sqrt [3]{b \coth (c+d x)}}{3 \sqrt [3]{b}}\right )}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3 b \left (\frac {1}{3} \int \frac {1}{b^{2/3}-b^2 \coth ^2(c+d x)}d\sqrt [3]{b \coth (c+d x)}-\frac {\int \frac {\sqrt [3]{b}-\sqrt [3]{b \coth (c+d x)}}{b^2 \coth ^2(c+d x)+b^{4/3} \coth (c+d x)+b^{2/3}}d\sqrt [3]{b \coth (c+d x)}}{6 \sqrt [3]{b}}-\frac {\int \frac {\sqrt [3]{b}+\sqrt [3]{b \coth (c+d x)}}{b^2 \coth ^2(c+d x)-b^{4/3} \coth (c+d x)+b^{2/3}}d\sqrt [3]{b \coth (c+d x)}}{6 \sqrt [3]{b}}\right )}{d}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {3 b \left (-\frac {\int \frac {\sqrt [3]{b}-\sqrt [3]{b \coth (c+d x)}}{b^2 \coth ^2(c+d x)+b^{4/3} \coth (c+d x)+b^{2/3}}d\sqrt [3]{b \coth (c+d x)}}{6 \sqrt [3]{b}}-\frac {\int \frac {\sqrt [3]{b}+\sqrt [3]{b \coth (c+d x)}}{b^2 \coth ^2(c+d x)-b^{4/3} \coth (c+d x)+b^{2/3}}d\sqrt [3]{b \coth (c+d x)}}{6 \sqrt [3]{b}}+\frac {\text {arctanh}\left (b^{2/3} \coth (c+d x)\right )}{3 \sqrt [3]{b}}\right )}{d}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {3 b \left (-\frac {\frac {3}{2} \sqrt [3]{b} \int \frac {1}{b^2 \coth ^2(c+d x)-b^{4/3} \coth (c+d x)+b^{2/3}}d\sqrt [3]{b \coth (c+d x)}+\frac {1}{2} \int -\frac {\sqrt [3]{b}-2 \sqrt [3]{b \coth (c+d x)}}{b^2 \coth ^2(c+d x)-b^{4/3} \coth (c+d x)+b^{2/3}}d\sqrt [3]{b \coth (c+d x)}}{6 \sqrt [3]{b}}-\frac {\frac {3}{2} \sqrt [3]{b} \int \frac {1}{b^2 \coth ^2(c+d x)+b^{4/3} \coth (c+d x)+b^{2/3}}d\sqrt [3]{b \coth (c+d x)}-\frac {1}{2} \int \frac {\sqrt [3]{b}+2 \sqrt [3]{b \coth (c+d x)}}{b^2 \coth ^2(c+d x)+b^{4/3} \coth (c+d x)+b^{2/3}}d\sqrt [3]{b \coth (c+d x)}}{6 \sqrt [3]{b}}+\frac {\text {arctanh}\left (b^{2/3} \coth (c+d x)\right )}{3 \sqrt [3]{b}}\right )}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {3 b \left (-\frac {\frac {3}{2} \sqrt [3]{b} \int \frac {1}{b^2 \coth ^2(c+d x)-b^{4/3} \coth (c+d x)+b^{2/3}}d\sqrt [3]{b \coth (c+d x)}-\frac {1}{2} \int \frac {\sqrt [3]{b}-2 \sqrt [3]{b \coth (c+d x)}}{b^2 \coth ^2(c+d x)-b^{4/3} \coth (c+d x)+b^{2/3}}d\sqrt [3]{b \coth (c+d x)}}{6 \sqrt [3]{b}}-\frac {\frac {3}{2} \sqrt [3]{b} \int \frac {1}{b^2 \coth ^2(c+d x)+b^{4/3} \coth (c+d x)+b^{2/3}}d\sqrt [3]{b \coth (c+d x)}-\frac {1}{2} \int \frac {\sqrt [3]{b}+2 \sqrt [3]{b \coth (c+d x)}}{b^2 \coth ^2(c+d x)+b^{4/3} \coth (c+d x)+b^{2/3}}d\sqrt [3]{b \coth (c+d x)}}{6 \sqrt [3]{b}}+\frac {\text {arctanh}\left (b^{2/3} \coth (c+d x)\right )}{3 \sqrt [3]{b}}\right )}{d}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {3 b \left (-\frac {3 \int \frac {1}{-b^2 \coth ^2(c+d x)-3}d\left (1-2 b^{2/3} \coth (c+d x)\right )-\frac {1}{2} \int \frac {\sqrt [3]{b}-2 \sqrt [3]{b \coth (c+d x)}}{b^2 \coth ^2(c+d x)-b^{4/3} \coth (c+d x)+b^{2/3}}d\sqrt [3]{b \coth (c+d x)}}{6 \sqrt [3]{b}}-\frac {-3 \int \frac {1}{-b^2 \coth ^2(c+d x)-3}d\left (2 b^{2/3} \coth (c+d x)+1\right )-\frac {1}{2} \int \frac {\sqrt [3]{b}+2 \sqrt [3]{b \coth (c+d x)}}{b^2 \coth ^2(c+d x)+b^{4/3} \coth (c+d x)+b^{2/3}}d\sqrt [3]{b \coth (c+d x)}}{6 \sqrt [3]{b}}+\frac {\text {arctanh}\left (b^{2/3} \coth (c+d x)\right )}{3 \sqrt [3]{b}}\right )}{d}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {3 b \left (-\frac {-\frac {1}{2} \int \frac {\sqrt [3]{b}-2 \sqrt [3]{b \coth (c+d x)}}{b^2 \coth ^2(c+d x)-b^{4/3} \coth (c+d x)+b^{2/3}}d\sqrt [3]{b \coth (c+d x)}-\sqrt {3} \arctan \left (\frac {1-2 b^{2/3} \coth (c+d x)}{\sqrt {3}}\right )}{6 \sqrt [3]{b}}-\frac {\sqrt {3} \arctan \left (\frac {2 b^{2/3} \coth (c+d x)+1}{\sqrt {3}}\right )-\frac {1}{2} \int \frac {\sqrt [3]{b}+2 \sqrt [3]{b \coth (c+d x)}}{b^2 \coth ^2(c+d x)+b^{4/3} \coth (c+d x)+b^{2/3}}d\sqrt [3]{b \coth (c+d x)}}{6 \sqrt [3]{b}}+\frac {\text {arctanh}\left (b^{2/3} \coth (c+d x)\right )}{3 \sqrt [3]{b}}\right )}{d}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {3 b \left (-\frac {\frac {1}{2} \log \left (-b^{4/3} \coth (c+d x)+b^{2/3}+b^2 \coth ^2(c+d x)\right )-\sqrt {3} \arctan \left (\frac {1-2 b^{2/3} \coth (c+d x)}{\sqrt {3}}\right )}{6 \sqrt [3]{b}}-\frac {\sqrt {3} \arctan \left (\frac {2 b^{2/3} \coth (c+d x)+1}{\sqrt {3}}\right )-\frac {1}{2} \log \left (b^{4/3} \coth (c+d x)+b^{2/3}+b^2 \coth ^2(c+d x)\right )}{6 \sqrt [3]{b}}+\frac {\text {arctanh}\left (b^{2/3} \coth (c+d x)\right )}{3 \sqrt [3]{b}}\right )}{d}\) |
(3*b*(ArcTanh[b^(2/3)*Coth[c + d*x]]/(3*b^(1/3)) - (-(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^(1/3)) - (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^(1/3))))/d
3.1.10.3.1 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] :> 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 = 0.11 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.83
| method | result | size |
| derivativedivides | \(-\frac {3 b \left (\frac {\ln \left (\left (b \coth \left (d x +c \right )\right )^{\frac {1}{3}}-b^{\frac {1}{3}}\right )}{6 b^{\frac {1}{3}}}-\frac {\ln \left (b^{\frac {2}{3}}+b^{\frac {1}{3}} \left (b \coth \left (d x +c \right )\right )^{\frac {1}{3}}+\left (b \coth \left (d x +c \right )\right )^{\frac {2}{3}}\right )}{12 b^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (1+\frac {2 \left (b \coth \left (d x +c \right )\right )^{\frac {1}{3}}}{b^{\frac {1}{3}}}\right ) \sqrt {3}}{3}\right )}{6 b^{\frac {1}{3}}}-\frac {\ln \left (\left (b \coth \left (d x +c \right )\right )^{\frac {1}{3}}+b^{\frac {1}{3}}\right )}{6 b^{\frac {1}{3}}}+\frac {\ln \left (b^{\frac {2}{3}}-b^{\frac {1}{3}} \left (b \coth \left (d x +c \right )\right )^{\frac {1}{3}}+\left (b \coth \left (d x +c \right )\right )^{\frac {2}{3}}\right )}{12 b^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b \coth \left (d x +c \right )\right )^{\frac {1}{3}}}{b^{\frac {1}{3}}}-1\right )}{3}\right )}{6 b^{\frac {1}{3}}}\right )}{d}\) | \(181\) |
| default | \(-\frac {3 b \left (\frac {\ln \left (\left (b \coth \left (d x +c \right )\right )^{\frac {1}{3}}-b^{\frac {1}{3}}\right )}{6 b^{\frac {1}{3}}}-\frac {\ln \left (b^{\frac {2}{3}}+b^{\frac {1}{3}} \left (b \coth \left (d x +c \right )\right )^{\frac {1}{3}}+\left (b \coth \left (d x +c \right )\right )^{\frac {2}{3}}\right )}{12 b^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (1+\frac {2 \left (b \coth \left (d x +c \right )\right )^{\frac {1}{3}}}{b^{\frac {1}{3}}}\right ) \sqrt {3}}{3}\right )}{6 b^{\frac {1}{3}}}-\frac {\ln \left (\left (b \coth \left (d x +c \right )\right )^{\frac {1}{3}}+b^{\frac {1}{3}}\right )}{6 b^{\frac {1}{3}}}+\frac {\ln \left (b^{\frac {2}{3}}-b^{\frac {1}{3}} \left (b \coth \left (d x +c \right )\right )^{\frac {1}{3}}+\left (b \coth \left (d x +c \right )\right )^{\frac {2}{3}}\right )}{12 b^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b \coth \left (d x +c \right )\right )^{\frac {1}{3}}}{b^{\frac {1}{3}}}-1\right )}{3}\right )}{6 b^{\frac {1}{3}}}\right )}{d}\) | \(181\) |
-3/d*b*(1/6/b^(1/3)*ln((b*coth(d*x+c))^(1/3)-b^(1/3))-1/12/b^(1/3)*ln(b^(2 /3)+b^(1/3)*(b*coth(d*x+c))^(1/3)+(b*coth(d*x+c))^(2/3))+1/6*3^(1/2)/b^(1/ 3)*arctan(1/3*(1+2*(b*coth(d*x+c))^(1/3)/b^(1/3))*3^(1/2))-1/6/b^(1/3)*ln( (b*coth(d*x+c))^(1/3)+b^(1/3))+1/12/b^(1/3)*ln(b^(2/3)-b^(1/3)*(b*coth(d*x +c))^(1/3)+(b*coth(d*x+c))^(2/3))+1/6*3^(1/2)/b^(1/3)*arctan(1/3*3^(1/2)*( 2*(b*coth(d*x+c))^(1/3)/b^(1/3)-1)))
Time = 0.27 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.42 \[ \int (b \coth (c+d x))^{2/3} \, dx=-\frac {2 \, \sqrt {3} \left (-b^{2}\right )^{\frac {1}{3}} \arctan \left (-\frac {\sqrt {3} b - 2 \, \sqrt {3} \left (-b^{2}\right )^{\frac {1}{3}} \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {1}{3}}}{3 \, b}\right ) + 2 \, \sqrt {3} {\left (b^{2}\right )}^{\frac {1}{3}} \arctan \left (-\frac {\sqrt {3} b - 2 \, \sqrt {3} {\left (b^{2}\right )}^{\frac {1}{3}} \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {1}{3}}}{3 \, b}\right ) + \left (-b^{2}\right )^{\frac {1}{3}} \log \left (b \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {2}{3}} - \left (-b^{2}\right )^{\frac {1}{3}} b + \left (-b^{2}\right )^{\frac {2}{3}} \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {1}{3}}\right ) + {\left (b^{2}\right )}^{\frac {1}{3}} \log \left (b \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {2}{3}} + {\left (b^{2}\right )}^{\frac {1}{3}} b - {\left (b^{2}\right )}^{\frac {2}{3}} \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {1}{3}}\right ) - 2 \, \left (-b^{2}\right )^{\frac {1}{3}} \log \left (b \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {1}{3}} - \left (-b^{2}\right )^{\frac {2}{3}}\right ) - 2 \, {\left (b^{2}\right )}^{\frac {1}{3}} \log \left (b \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {1}{3}} + {\left (b^{2}\right )}^{\frac {2}{3}}\right )}{4 \, d} \]
-1/4*(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) + 2*sqrt(3)*(b^2)^(1/3)*arcta n(-1/3*(sqrt(3)*b - 2*sqrt(3)*(b^2)^(1/3)*(b*cosh(d*x + c)/sinh(d*x + c))^ (1/3))/b) + (-b^2)^(1/3)*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)^ (1/3)*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)^(1/3)*log(b*(b*cos h(d*x + c)/sinh(d*x + c))^(1/3) - (-b^2)^(2/3)) - 2*(b^2)^(1/3)*log(b*(b*c osh(d*x + c)/sinh(d*x + c))^(1/3) + (b^2)^(2/3)))/d
\[ \int (b \coth (c+d x))^{2/3} \, dx=\int \left (b \coth {\left (c + d x \right )}\right )^{\frac {2}{3}}\, dx \]
\[ \int (b \coth (c+d x))^{2/3} \, dx=\int { \left (b \coth \left (d x + c\right )\right )^{\frac {2}{3}} \,d x } \]
Exception generated. \[ \int (b \coth (c+d x))^{2/3} \, dx=\text {Exception raised: TypeError} \]
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 = 2.21 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.07 \[ \int (b \coth (c+d x))^{2/3} \, dx=-\frac {b^{2/3}\,\mathrm {atan}\left (\frac {{\left (b\,\mathrm {coth}\left (c+d\,x\right )\right )}^{1/3}\,1{}\mathrm {i}}{b^{1/3}}\right )\,1{}\mathrm {i}}{d}-\frac {b^{2/3}\,\ln \left (\frac {972\,b^9}{d^3}-\frac {972\,b^{26/3}\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (b\,\mathrm {coth}\left (c+d\,x\right )\right )}^{1/3}}{d^3}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{2\,d}-\frac {b^{2/3}\,\ln \left (\frac {972\,b^9}{d^3}-\frac {972\,b^{26/3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (b\,\mathrm {coth}\left (c+d\,x\right )\right )}^{1/3}}{d^3}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{2\,d}+\frac {b^{2/3}\,\ln \left (\frac {972\,b^9}{d^3}+\frac {1944\,b^{26/3}\,\left (-\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )\,{\left (b\,\mathrm {coth}\left (c+d\,x\right )\right )}^{1/3}}{d^3}\right )\,\left (-\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )}{d}+\frac {b^{2/3}\,\ln \left (\frac {972\,b^9}{d^3}+\frac {1944\,b^{26/3}\,\left (\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )\,{\left (b\,\mathrm {coth}\left (c+d\,x\right )\right )}^{1/3}}{d^3}\right )\,\left (\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )}{d} \]
(b^(2/3)*log((972*b^9)/d^3 + (1944*b^(26/3)*((3^(1/2)*1i)/4 - 1/4)*(b*coth (c + d*x))^(1/3))/d^3)*((3^(1/2)*1i)/4 - 1/4))/d - (b^(2/3)*log((972*b^9)/ d^3 - (972*b^(26/3)*((3^(1/2)*1i)/2 - 1/2)*(b*coth(c + d*x))^(1/3))/d^3)*( (3^(1/2)*1i)/2 - 1/2))/(2*d) - (b^(2/3)*log((972*b^9)/d^3 - (972*b^(26/3)* ((3^(1/2)*1i)/2 + 1/2)*(b*coth(c + d*x))^(1/3))/d^3)*((3^(1/2)*1i)/2 + 1/2 ))/(2*d) - (b^(2/3)*atan(((b*coth(c + d*x))^(1/3)*1i)/b^(1/3))*1i)/d + (b^ (2/3)*log((972*b^9)/d^3 + (1944*b^(26/3)*((3^(1/2)*1i)/4 + 1/4)*(b*coth(c + d*x))^(1/3))/d^3)*((3^(1/2)*1i)/4 + 1/4))/d