\(\int (b \coth (c+d x))^{2/3} \, dx\) [10]

Optimal result

Integrand size = 12, antiderivative size = 218 \[ \int (b \coth (c+d x))^{2/3} \, dx=\frac {\sqrt {3} b^{2/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b \coth (c+d x)}}{\sqrt [3]{b}}}{\sqrt {3}}\right )}{2 d}-\frac {\sqrt {3} b^{2/3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{b \coth (c+d x)}}{\sqrt [3]{b}}}{\sqrt {3}}\right )}{2 d}+\frac {b^{2/3} \text {arctanh}\left (\frac {\sqrt [3]{b \coth (c+d x)}}{\sqrt [3]{b}}\right )}{d}-\frac {b^{2/3} \log \left (b^{2/3}-\sqrt [3]{b} \sqrt [3]{b \coth (c+d x)}+(b \coth (c+d x))^{2/3}\right )}{4 d}+\frac {b^{2/3} \log \left (b^{2/3}+\sqrt [3]{b} \sqrt [3]{b \coth (c+d x)}+(b \coth (c+d x))^{2/3}\right )}{4 d} \]

[Out]

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)*arc
tan(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)*arctan(1/3*(1+2*(b*coth(d*x+c))^(1/
3)/b^(1/3))*3^(1/2))*3^(1/2)/d

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {3557, 335, 302, 648, 632, 210, 642, 212} \[ \int (b \coth (c+d x))^{2/3} \, dx=\frac {\sqrt {3} b^{2/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b \coth (c+d x)}}{\sqrt [3]{b}}}{\sqrt {3}}\right )}{2 d}-\frac {\sqrt {3} b^{2/3} \arctan \left (\frac {\frac {2 \sqrt [3]{b \coth (c+d x)}}{\sqrt [3]{b}}+1}{\sqrt {3}}\right )}{2 d}+\frac {b^{2/3} \text {arctanh}\left (\frac {\sqrt [3]{b \coth (c+d x)}}{\sqrt [3]{b}}\right )}{d}-\frac {b^{2/3} \log \left (b^{2/3}-\sqrt [3]{b} \sqrt [3]{b \coth (c+d x)}+(b \coth (c+d x))^{2/3}\right )}{4 d}+\frac {b^{2/3} \log \left (b^{2/3}+\sqrt [3]{b} \sqrt [3]{b \coth (c+d x)}+(b \coth (c+d x))^{2/3}\right )}{4 d} \]

[In]

Int[(b*Coth[c + d*x])^(2/3),x]

[Out]

(Sqrt[3]*b^(2/3)*ArcTan[(1 - (2*(b*Coth[c + d*x])^(1/3))/b^(1/3))/Sqrt[3]])/(2*d) - (Sqrt[3]*b^(2/3)*ArcTan[(1
+ (2*(b*Coth[c + d*x])^(1/3))/b^(1/3))/Sqrt[3]])/(2*d) + (b^(2/3)*ArcTanh[(b*Coth[c + d*x])^(1/3)/b^(1/3)])/d
- (b^(2/3)*Log[b^(2/3) - b^(1/3)*(b*Coth[c + d*x])^(1/3) + (b*Coth[c + d*x])^(2/3)])/(4*d) + (b^(2/3)*Log[b^(
2/3) + b^(1/3)*(b*Coth[c + d*x])^(1/3) + (b*Coth[c + d*x])^(2/3)])/(4*d)

Rule 210

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])

Rule 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 302

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] + Dist[2*(r^(m + 1)/(a*n*s^m)), Sum[u, {k, 1, (
n - 2)/4}], x], x]] /; FreeQ[{a, b}, x] && IGtQ[(n - 2)/4, 0] && IGtQ[m, 0] && LtQ[m, n - 1] && NegQ[a/b]

Rule 335

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[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]

Rule 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 3557

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[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]

Rubi steps \begin{align*} \text {integral}& = -\frac {b \text {Subst}\left (\int \frac {x^{2/3}}{-b^2+x^2} \, dx,x,b \coth (c+d x)\right )}{d} \\ & = -\frac {(3 b) \text {Subst}\left (\int \frac {x^4}{-b^2+x^6} \, dx,x,\sqrt [3]{b \coth (c+d x)}\right )}{d} \\ & = \frac {b^{2/3} \text {Subst}\left (\int \frac {-\frac {\sqrt [3]{b}}{2}-\frac {x}{2}}{b^{2/3}-\sqrt [3]{b} x+x^2} \, dx,x,\sqrt [3]{b \coth (c+d x)}\right )}{d}+\frac {b^{2/3} \text {Subst}\left (\int \frac {-\frac {\sqrt [3]{b}}{2}+\frac {x}{2}}{b^{2/3}+\sqrt [3]{b} x+x^2} \, dx,x,\sqrt [3]{b \coth (c+d x)}\right )}{d}+\frac {b \text {Subst}\left (\int \frac {1}{b^{2/3}-x^2} \, dx,x,\sqrt [3]{b \coth (c+d x)}\right )}{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} \text {Subst}\left (\int \frac {-\sqrt [3]{b}+2 x}{b^{2/3}-\sqrt [3]{b} x+x^2} \, dx,x,\sqrt [3]{b \coth (c+d x)}\right )}{4 d}+\frac {b^{2/3} \text {Subst}\left (\int \frac {\sqrt [3]{b}+2 x}{b^{2/3}+\sqrt [3]{b} x+x^2} \, dx,x,\sqrt [3]{b \coth (c+d x)}\right )}{4 d}-\frac {(3 b) \text {Subst}\left (\int \frac {1}{b^{2/3}-\sqrt [3]{b} x+x^2} \, dx,x,\sqrt [3]{b \coth (c+d x)}\right )}{4 d}-\frac {(3 b) \text {Subst}\left (\int \frac {1}{b^{2/3}+\sqrt [3]{b} x+x^2} \, dx,x,\sqrt [3]{b \coth (c+d x)}\right )}{4 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}-\frac {\left (3 b^{2/3}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b \coth (c+d x)}}{\sqrt [3]{b}}\right )}{2 d}+\frac {\left (3 b^{2/3}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{b \coth (c+d x)}}{\sqrt [3]{b}}\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 {\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} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.68 \[ \int (b \coth (c+d x))^{2/3} \, dx=\frac {(b \coth (c+d x))^{2/3} \left (2 \sqrt {3} \arctan \left (\frac {1-2 \sqrt [3]{\coth (c+d x)}}{\sqrt {3}}\right )-2 \sqrt {3} \arctan \left (\frac {1+2 \sqrt [3]{\coth (c+d x)}}{\sqrt {3}}\right )+4 \text {arctanh}\left (\sqrt [3]{\coth (c+d x)}\right )-\log \left (1-\sqrt [3]{\coth (c+d x)}+\coth ^{\frac {2}{3}}(c+d x)\right )+\log \left (1+\sqrt [3]{\coth (c+d x)}+\coth ^{\frac {2}{3}}(c+d x)\right )\right )}{4 d \coth ^{\frac {2}{3}}(c+d x)} \]

[In]

Integrate[(b*Coth[c + d*x])^(2/3),x]

[Out]

((b*Coth[c + d*x])^(2/3)*(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]] + 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))

Maple [A] (verified)

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	\(-\frac {3 b \left (\frac {\ln \left (\left (b \coth \left (d x +c \right )\right )^{\frac {1}{3}}-b^{\frac {1}{3}}\right )}{6 b^{\frac {1}{3}}}-\frac {\ln \left (b^{\frac {2}{3}}+b^{\frac {1}{3}} \left (b \coth \left (d x +c \right )\right )^{\frac {1}{3}}+\left (b \coth \left (d x +c \right )\right )^{\frac {2}{3}}\right )}{12 b^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (1+\frac {2 \left (b \coth \left (d x +c \right )\right )^{\frac {1}{3}}}{b^{\frac {1}{3}}}\right ) \sqrt {3}}{3}\right )}{6 b^{\frac {1}{3}}}-\frac {\ln \left (\left (b \coth \left (d x +c \right )\right )^{\frac {1}{3}}+b^{\frac {1}{3}}\right )}{6 b^{\frac {1}{3}}}+\frac {\ln \left (b^{\frac {2}{3}}-b^{\frac {1}{3}} \left (b \coth \left (d x +c \right )\right )^{\frac {1}{3}}+\left (b \coth \left (d x +c \right )\right )^{\frac {2}{3}}\right )}{12 b^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b \coth \left (d x +c \right )\right )^{\frac {1}{3}}}{b^{\frac {1}{3}}}-1\right )}{3}\right )}{6 b^{\frac {1}{3}}}\right )}{d}\)	\(181\)

[In]

int((b*coth(d*x+c))^(2/3),x,method=_RETURNVERBOSE)

[Out]

-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)*l
n((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)))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.42 \[ \int (b \coth (c+d x))^{2/3} \, dx=-\frac {2 \, \sqrt {3} \left (-b^{2}\right )^{\frac {1}{3}} \arctan \left (-\frac {\sqrt {3} b - 2 \, \sqrt {3} \left (-b^{2}\right )^{\frac {1}{3}} \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {1}{3}}}{3 \, b}\right ) + 2 \, \sqrt {3} {\left (b^{2}\right )}^{\frac {1}{3}} \arctan \left (-\frac {\sqrt {3} b - 2 \, \sqrt {3} {\left (b^{2}\right )}^{\frac {1}{3}} \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {1}{3}}}{3 \, b}\right ) + \left (-b^{2}\right )^{\frac {1}{3}} \log \left (b \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {2}{3}} - \left (-b^{2}\right )^{\frac {1}{3}} b + \left (-b^{2}\right )^{\frac {2}{3}} \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {1}{3}}\right ) + {\left (b^{2}\right )}^{\frac {1}{3}} \log \left (b \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {2}{3}} + {\left (b^{2}\right )}^{\frac {1}{3}} b - {\left (b^{2}\right )}^{\frac {2}{3}} \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {1}{3}}\right ) - 2 \, \left (-b^{2}\right )^{\frac {1}{3}} \log \left (b \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {1}{3}} - \left (-b^{2}\right )^{\frac {2}{3}}\right ) - 2 \, {\left (b^{2}\right )}^{\frac {1}{3}} \log \left (b \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {1}{3}} + {\left (b^{2}\right )}^{\frac {2}{3}}\right )}{4 \, d} \]

[In]

integrate((b*coth(d*x+c))^(2/3),x, algorithm="fricas")

[Out]

-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)*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) + (-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*cosh(d*x + c)/sinh(d*x + c))^(1/3)
- (-b^2)^(2/3)) - 2*(b^2)^(1/3)*log(b*(b*cosh(d*x + c)/sinh(d*x + c))^(1/3) + (b^2)^(2/3)))/d

Sympy [F]

\[ \int (b \coth (c+d x))^{2/3} \, dx=\int \left (b \coth {\left (c + d x \right )}\right )^{\frac {2}{3}}\, dx \]

[In]

integrate((b*coth(d*x+c))**(2/3),x)

[Out]

Integral((b*coth(c + d*x))**(2/3), x)

Maxima [F]

\[ \int (b \coth (c+d x))^{2/3} \, dx=\int { \left (b \coth \left (d x + c\right )\right )^{\frac {2}{3}} \,d x } \]

[In]

integrate((b*coth(d*x+c))^(2/3),x, algorithm="maxima")

[Out]

integrate((b*coth(d*x + c))^(2/3), x)

Giac [F(-2)]

Exception generated. \[ \int (b \coth (c+d x))^{2/3} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((b*coth(d*x+c))^(2/3),x, algorithm="giac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Minimal poly. in rootof must be fraction free Error: Bad Argument ValueMinimal poly. in rootof must be frac
tion free E

Mupad [B] (verification not implemented)

Time = 2.21 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.07 \[ \int (b \coth (c+d x))^{2/3} \, dx=-\frac {b^{2/3}\,\mathrm {atan}\left (\frac {{\left (b\,\mathrm {coth}\left (c+d\,x\right )\right )}^{1/3}\,1{}\mathrm {i}}{b^{1/3}}\right )\,1{}\mathrm {i}}{d}-\frac {b^{2/3}\,\ln \left (\frac {972\,b^9}{d^3}-\frac {972\,b^{26/3}\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (b\,\mathrm {coth}\left (c+d\,x\right )\right )}^{1/3}}{d^3}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{2\,d}-\frac {b^{2/3}\,\ln \left (\frac {972\,b^9}{d^3}-\frac {972\,b^{26/3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (b\,\mathrm {coth}\left (c+d\,x\right )\right )}^{1/3}}{d^3}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{2\,d}+\frac {b^{2/3}\,\ln \left (\frac {972\,b^9}{d^3}+\frac {1944\,b^{26/3}\,\left (-\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )\,{\left (b\,\mathrm {coth}\left (c+d\,x\right )\right )}^{1/3}}{d^3}\right )\,\left (-\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )}{d}+\frac {b^{2/3}\,\ln \left (\frac {972\,b^9}{d^3}+\frac {1944\,b^{26/3}\,\left (\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )\,{\left (b\,\mathrm {coth}\left (c+d\,x\right )\right )}^{1/3}}{d^3}\right )\,\left (\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )}{d} \]

[In]

int((b*coth(c + d*x))^(2/3),x)

[Out]

(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