\(\int \sqrt [3]{b \coth ^4(c+d x)} \, dx\) [46]

Optimal result

Integrand size = 14, antiderivative size = 236 \[ \int \sqrt [3]{b \coth ^4(c+d x)} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {1-2 \sqrt [3]{\coth (c+d x)}}{\sqrt {3}}\right ) \sqrt [3]{b \coth ^4(c+d x)}}{2 d \coth ^{\frac {4}{3}}(c+d x)}+\frac {\sqrt {3} \arctan \left (\frac {1+2 \sqrt [3]{\coth (c+d x)}}{\sqrt {3}}\right ) \sqrt [3]{b \coth ^4(c+d x)}}{2 d \coth ^{\frac {4}{3}}(c+d x)}+\frac {\text {arctanh}\left (\sqrt [3]{\coth (c+d x)}\right ) \sqrt [3]{b \coth ^4(c+d x)}}{d \coth ^{\frac {4}{3}}(c+d x)}+\frac {\text {arctanh}\left (\frac {\sqrt [3]{\coth (c+d x)}}{1+\coth ^{\frac {2}{3}}(c+d x)}\right ) \sqrt [3]{b \coth ^4(c+d x)}}{2 d \coth ^{\frac {4}{3}}(c+d x)}-\frac {3 \sqrt [3]{b \coth ^4(c+d x)} \tanh (c+d x)}{d} \] Output:

-1/2*3^(1/2)*arctan(1/3*(1-2*coth(d*x+c)^(1/3))*3^(1/2))*(b*coth(d*x+c)^4) 
^(1/3)/d/coth(d*x+c)^(4/3)+1/2*3^(1/2)*arctan(1/3*(1+2*coth(d*x+c)^(1/3))* 
3^(1/2))*(b*coth(d*x+c)^4)^(1/3)/d/coth(d*x+c)^(4/3)+arctanh(coth(d*x+c)^( 
1/3))*(b*coth(d*x+c)^4)^(1/3)/d/coth(d*x+c)^(4/3)+1/2*arctanh(coth(d*x+c)^ 
(1/3)/(1+coth(d*x+c)^(2/3)))*(b*coth(d*x+c)^4)^(1/3)/d/coth(d*x+c)^(4/3)-3 
*(b*coth(d*x+c)^4)^(1/3)*tanh(d*x+c)/d
 

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 200, normalized size of antiderivative = 0.85 \[ \int \sqrt [3]{b \coth ^4(c+d x)} \, dx=-\frac {\sqrt [3]{b \coth ^4(c+d x)} \left (6 \sqrt [6]{\coth ^2(c+d x)}+\log \left (1-\sqrt [6]{\coth ^2(c+d x)}\right )-\log \left (1+\sqrt [6]{\coth ^2(c+d x)}\right )-(-1)^{2/3} \log \left (1-\sqrt [3]{-1} \sqrt [6]{\coth ^2(c+d x)}\right )+(-1)^{2/3} \log \left (1+\sqrt [3]{-1} \sqrt [6]{\coth ^2(c+d x)}\right )-\sqrt [3]{-1} \log \left (1-(-1)^{2/3} \sqrt [6]{\coth ^2(c+d x)}\right )+\sqrt [3]{-1} \log \left (1+(-1)^{2/3} \sqrt [6]{\coth ^2(c+d x)}\right )\right ) \tanh (c+d x)}{2 d \sqrt [6]{\coth ^2(c+d x)}} \] Input:

Integrate[(b*Coth[c + d*x]^4)^(1/3),x]
 

Output:

-1/2*((b*Coth[c + d*x]^4)^(1/3)*(6*(Coth[c + d*x]^2)^(1/6) + Log[1 - (Coth 
[c + d*x]^2)^(1/6)] - Log[1 + (Coth[c + d*x]^2)^(1/6)] - (-1)^(2/3)*Log[1 
- (-1)^(1/3)*(Coth[c + d*x]^2)^(1/6)] + (-1)^(2/3)*Log[1 + (-1)^(1/3)*(Cot 
h[c + d*x]^2)^(1/6)] - (-1)^(1/3)*Log[1 - (-1)^(2/3)*(Coth[c + d*x]^2)^(1/ 
6)] + (-1)^(1/3)*Log[1 + (-1)^(2/3)*(Coth[c + d*x]^2)^(1/6)])*Tanh[c + d*x 
])/(d*(Coth[c + d*x]^2)^(1/6))
 

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.77, 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, 754, 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.

\(\displaystyle \int \sqrt [3]{b \coth ^4(c+d x)} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \sqrt [3]{b \tan \left (i c+i d x+\frac {\pi }{2}\right )^4}dx\)

\(\Big \downarrow \) 4141

\(\displaystyle \frac {\sqrt [3]{b \coth ^4(c+d x)} \int \coth ^{\frac {4}{3}}(c+d x)dx}{\coth ^{\frac {4}{3}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\sqrt [3]{b \coth ^4(c+d x)} \int \left (-i \tan \left (i c+i d x+\frac {\pi }{2}\right )\right )^{4/3}dx}{\coth ^{\frac {4}{3}}(c+d x)}\)

\(\Big \downarrow \) 3954

\(\displaystyle \frac {\sqrt [3]{b \coth ^4(c+d x)} \left (\int \frac {1}{\coth ^{\frac {2}{3}}(c+d x)}dx-\frac {3 \sqrt [3]{\coth (c+d x)}}{d}\right )}{\coth ^{\frac {4}{3}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\sqrt [3]{b \coth ^4(c+d x)} \left (-\frac {3 \sqrt [3]{\coth (c+d x)}}{d}+\int \frac {1}{\left (-i \tan \left (i c+i d x+\frac {\pi }{2}\right )\right )^{2/3}}dx\right )}{\coth ^{\frac {4}{3}}(c+d x)}\)

\(\Big \downarrow \) 3957

\(\displaystyle \frac {\sqrt [3]{b \coth ^4(c+d x)} \left (-\frac {\int -\frac {1}{\coth ^{\frac {2}{3}}(c+d x) \left (1-\coth ^2(c+d x)\right )}d\coth (c+d x)}{d}-\frac {3 \sqrt [3]{\coth (c+d x)}}{d}\right )}{\coth ^{\frac {4}{3}}(c+d x)}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\sqrt [3]{b \coth ^4(c+d x)} \left (\frac {\int \frac {1}{\coth ^{\frac {2}{3}}(c+d x) \left (1-\coth ^2(c+d x)\right )}d\coth (c+d x)}{d}-\frac {3 \sqrt [3]{\coth (c+d x)}}{d}\right )}{\coth ^{\frac {4}{3}}(c+d x)}\)

\(\Big \downarrow \) 266

\(\displaystyle \frac {\sqrt [3]{b \coth ^4(c+d x)} \left (\frac {3 \int \frac {1}{1-\coth ^2(c+d x)}d\sqrt [3]{\coth (c+d x)}}{d}-\frac {3 \sqrt [3]{\coth (c+d x)}}{d}\right )}{\coth ^{\frac {4}{3}}(c+d x)}\)

\(\Big \downarrow \) 754

\(\displaystyle \frac {\sqrt [3]{b \coth ^4(c+d x)} \left (\frac {3 \left (\frac {1}{3} \int \frac {1}{1-\coth ^{\frac {2}{3}}(c+d x)}d\sqrt [3]{\coth (c+d x)}+\frac {1}{3} \int \frac {2-\sqrt [3]{\coth (c+d x)}}{2 \left (\coth ^{\frac {2}{3}}(c+d x)-\sqrt [3]{\coth (c+d x)}+1\right )}d\sqrt [3]{\coth (c+d x)}+\frac {1}{3} \int \frac {\sqrt [3]{\coth (c+d x)}+2}{2 \left (\coth ^{\frac {2}{3}}(c+d x)+\sqrt [3]{\coth (c+d x)}+1\right )}d\sqrt [3]{\coth (c+d x)}\right )}{d}-\frac {3 \sqrt [3]{\coth (c+d x)}}{d}\right )}{\coth ^{\frac {4}{3}}(c+d x)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt [3]{b \coth ^4(c+d x)} \left (\frac {3 \left (\frac {1}{3} \int \frac {1}{1-\coth ^{\frac {2}{3}}(c+d x)}d\sqrt [3]{\coth (c+d x)}+\frac {1}{6} \int \frac {2-\sqrt [3]{\coth (c+d x)}}{\coth ^{\frac {2}{3}}(c+d x)-\sqrt [3]{\coth (c+d x)}+1}d\sqrt [3]{\coth (c+d x)}+\frac {1}{6} \int \frac {\sqrt [3]{\coth (c+d x)}+2}{\coth ^{\frac {2}{3}}(c+d x)+\sqrt [3]{\coth (c+d x)}+1}d\sqrt [3]{\coth (c+d x)}\right )}{d}-\frac {3 \sqrt [3]{\coth (c+d x)}}{d}\right )}{\coth ^{\frac {4}{3}}(c+d x)}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\sqrt [3]{b \coth ^4(c+d x)} \left (\frac {3 \left (\frac {1}{6} \int \frac {2-\sqrt [3]{\coth (c+d x)}}{\coth ^{\frac {2}{3}}(c+d x)-\sqrt [3]{\coth (c+d x)}+1}d\sqrt [3]{\coth (c+d x)}+\frac {1}{6} \int \frac {\sqrt [3]{\coth (c+d x)}+2}{\coth ^{\frac {2}{3}}(c+d x)+\sqrt [3]{\coth (c+d x)}+1}d\sqrt [3]{\coth (c+d x)}+\frac {1}{3} \text {arctanh}\left (\sqrt [3]{\coth (c+d x)}\right )\right )}{d}-\frac {3 \sqrt [3]{\coth (c+d x)}}{d}\right )}{\coth ^{\frac {4}{3}}(c+d x)}\)

\(\Big \downarrow \) 1142

\(\displaystyle \frac {\sqrt [3]{b \coth ^4(c+d x)} \left (\frac {3 \left (\frac {1}{6} \left (\frac {3}{2} \int \frac {1}{\coth ^{\frac {2}{3}}(c+d x)-\sqrt [3]{\coth (c+d x)}+1}d\sqrt [3]{\coth (c+d x)}-\frac {1}{2} \int -\frac {1-2 \sqrt [3]{\coth (c+d x)}}{\coth ^{\frac {2}{3}}(c+d x)-\sqrt [3]{\coth (c+d x)}+1}d\sqrt [3]{\coth (c+d x)}\right )+\frac {1}{6} \left (\frac {3}{2} \int \frac {1}{\coth ^{\frac {2}{3}}(c+d x)+\sqrt [3]{\coth (c+d x)}+1}d\sqrt [3]{\coth (c+d x)}+\frac {1}{2} \int \frac {2 \sqrt [3]{\coth (c+d x)}+1}{\coth ^{\frac {2}{3}}(c+d x)+\sqrt [3]{\coth (c+d x)}+1}d\sqrt [3]{\coth (c+d x)}\right )+\frac {1}{3} \text {arctanh}\left (\sqrt [3]{\coth (c+d x)}\right )\right )}{d}-\frac {3 \sqrt [3]{\coth (c+d x)}}{d}\right )}{\coth ^{\frac {4}{3}}(c+d x)}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\sqrt [3]{b \coth ^4(c+d x)} \left (\frac {3 \left (\frac {1}{6} \left (\frac {3}{2} \int \frac {1}{\coth ^{\frac {2}{3}}(c+d x)-\sqrt [3]{\coth (c+d x)}+1}d\sqrt [3]{\coth (c+d x)}+\frac {1}{2} \int \frac {1-2 \sqrt [3]{\coth (c+d x)}}{\coth ^{\frac {2}{3}}(c+d x)-\sqrt [3]{\coth (c+d x)}+1}d\sqrt [3]{\coth (c+d x)}\right )+\frac {1}{6} \left (\frac {3}{2} \int \frac {1}{\coth ^{\frac {2}{3}}(c+d x)+\sqrt [3]{\coth (c+d x)}+1}d\sqrt [3]{\coth (c+d x)}+\frac {1}{2} \int \frac {2 \sqrt [3]{\coth (c+d x)}+1}{\coth ^{\frac {2}{3}}(c+d x)+\sqrt [3]{\coth (c+d x)}+1}d\sqrt [3]{\coth (c+d x)}\right )+\frac {1}{3} \text {arctanh}\left (\sqrt [3]{\coth (c+d x)}\right )\right )}{d}-\frac {3 \sqrt [3]{\coth (c+d x)}}{d}\right )}{\coth ^{\frac {4}{3}}(c+d x)}\)

\(\Big \downarrow \) 1083

\(\displaystyle \frac {\sqrt [3]{b \coth ^4(c+d x)} \left (\frac {3 \left (\frac {1}{6} \left (\frac {1}{2} \int \frac {1-2 \sqrt [3]{\coth (c+d x)}}{\coth ^{\frac {2}{3}}(c+d x)-\sqrt [3]{\coth (c+d x)}+1}d\sqrt [3]{\coth (c+d x)}-3 \int \frac {1}{-\coth ^{\frac {2}{3}}(c+d x)-3}d\left (2 \sqrt [3]{\coth (c+d x)}-1\right )\right )+\frac {1}{6} \left (\frac {1}{2} \int \frac {2 \sqrt [3]{\coth (c+d x)}+1}{\coth ^{\frac {2}{3}}(c+d x)+\sqrt [3]{\coth (c+d x)}+1}d\sqrt [3]{\coth (c+d x)}-3 \int \frac {1}{-\coth ^{\frac {2}{3}}(c+d x)-3}d\left (2 \sqrt [3]{\coth (c+d x)}+1\right )\right )+\frac {1}{3} \text {arctanh}\left (\sqrt [3]{\coth (c+d x)}\right )\right )}{d}-\frac {3 \sqrt [3]{\coth (c+d x)}}{d}\right )}{\coth ^{\frac {4}{3}}(c+d x)}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {\sqrt [3]{b \coth ^4(c+d x)} \left (\frac {3 \left (\frac {1}{6} \left (\frac {1}{2} \int \frac {1-2 \sqrt [3]{\coth (c+d x)}}{\coth ^{\frac {2}{3}}(c+d x)-\sqrt [3]{\coth (c+d x)}+1}d\sqrt [3]{\coth (c+d x)}+\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{\coth (c+d x)}-1}{\sqrt {3}}\right )\right )+\frac {1}{6} \left (\frac {1}{2} \int \frac {2 \sqrt [3]{\coth (c+d x)}+1}{\coth ^{\frac {2}{3}}(c+d x)+\sqrt [3]{\coth (c+d x)}+1}d\sqrt [3]{\coth (c+d x)}+\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{\coth (c+d x)}+1}{\sqrt {3}}\right )\right )+\frac {1}{3} \text {arctanh}\left (\sqrt [3]{\coth (c+d x)}\right )\right )}{d}-\frac {3 \sqrt [3]{\coth (c+d x)}}{d}\right )}{\coth ^{\frac {4}{3}}(c+d x)}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {\sqrt [3]{b \coth ^4(c+d x)} \left (\frac {3 \left (\frac {1}{6} \left (\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{\coth (c+d x)}-1}{\sqrt {3}}\right )-\frac {1}{2} \log \left (\coth ^{\frac {2}{3}}(c+d x)-\sqrt [3]{\coth (c+d x)}+1\right )\right )+\frac {1}{6} \left (\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{\coth (c+d x)}+1}{\sqrt {3}}\right )+\frac {1}{2} \log \left (\coth ^{\frac {2}{3}}(c+d x)+\sqrt [3]{\coth (c+d x)}+1\right )\right )+\frac {1}{3} \text {arctanh}\left (\sqrt [3]{\coth (c+d x)}\right )\right )}{d}-\frac {3 \sqrt [3]{\coth (c+d x)}}{d}\right )}{\coth ^{\frac {4}{3}}(c+d x)}\)

Input:

Int[(b*Coth[c + d*x]^4)^(1/3),x]
 

Output:

((b*Coth[c + d*x]^4)^(1/3)*((-3*Coth[c + d*x]^(1/3))/d + (3*(ArcTanh[Coth[ 
c + d*x]^(1/3)]/3 + (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 + (Sqrt[3]*ArcTa 
n[(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))/Coth[c + d*x]^(4/3)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 217

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 219

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])
 

rule 266

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]
 

rule 754

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]
 

rule 1083

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]
 

rule 1103

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]
 

rule 1142

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]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3954

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]
 

rule 3957

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]
 

rule 4141

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]])
 

Maple [F]

\[\int \left (b \coth \left (d x +c \right )^{4}\right )^{\frac {1}{3}}d x\]

Input:

int((b*coth(d*x+c)^4)^(1/3),x)
 

Output:

int((b*coth(d*x+c)^4)^(1/3),x)
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.22 \[ \int \sqrt [3]{b \coth ^4(c+d x)} \, dx=-\frac {2 \, \sqrt {3} \left (-b\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} b + 2 \, \sqrt {3} \left (-b\right )^{\frac {2}{3}} \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {1}{3}}}{3 \, b}\right ) - 2 \, \sqrt {3} b^{\frac {1}{3}} \arctan \left (-\frac {\sqrt {3} b - 2 \, \sqrt {3} b^{\frac {2}{3}} \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {1}{3}}}{3 \, b}\right ) + \left (-b\right )^{\frac {1}{3}} \log \left (\left (-b\right )^{\frac {2}{3}} - \left (-b\right )^{\frac {1}{3}} \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {1}{3}} + \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {2}{3}}\right ) + b^{\frac {1}{3}} \log \left (b^{\frac {2}{3}} - b^{\frac {1}{3}} \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {1}{3}} + \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {2}{3}}\right ) - 2 \, \left (-b\right )^{\frac {1}{3}} \log \left (\left (-b\right )^{\frac {1}{3}} + \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {1}{3}}\right ) - 2 \, b^{\frac {1}{3}} \log \left (b^{\frac {1}{3}} + \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {1}{3}}\right ) + 12 \, \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {1}{3}}}{4 \, d} \] Input:

integrate((b*coth(d*x+c)^4)^(1/3),x, algorithm="fricas")
 

Output:

-1/4*(2*sqrt(3)*(-b)^(1/3)*arctan(1/3*(sqrt(3)*b + 2*sqrt(3)*(-b)^(2/3)*(b 
*cosh(d*x + c)/sinh(d*x + c))^(1/3))/b) - 2*sqrt(3)*b^(1/3)*arctan(-1/3*(s 
qrt(3)*b - 2*sqrt(3)*b^(2/3)*(b*cosh(d*x + c)/sinh(d*x + c))^(1/3))/b) + ( 
-b)^(1/3)*log((-b)^(2/3) - (-b)^(1/3)*(b*cosh(d*x + c)/sinh(d*x + c))^(1/3 
) + (b*cosh(d*x + c)/sinh(d*x + c))^(2/3)) + b^(1/3)*log(b^(2/3) - b^(1/3) 
*(b*cosh(d*x + c)/sinh(d*x + c))^(1/3) + (b*cosh(d*x + c)/sinh(d*x + c))^( 
2/3)) - 2*(-b)^(1/3)*log((-b)^(1/3) + (b*cosh(d*x + c)/sinh(d*x + c))^(1/3 
)) - 2*b^(1/3)*log(b^(1/3) + (b*cosh(d*x + c)/sinh(d*x + c))^(1/3)) + 12*( 
b*cosh(d*x + c)/sinh(d*x + c))^(1/3))/d
 

Sympy [F]

\[ \int \sqrt [3]{b \coth ^4(c+d x)} \, dx=\int \sqrt [3]{b \coth ^{4}{\left (c + d x \right )}}\, dx \] Input:

integrate((b*coth(d*x+c)**4)**(1/3),x)
 

Output:

Integral((b*coth(c + d*x)**4)**(1/3), x)
 

Maxima [F]

\[ \int \sqrt [3]{b \coth ^4(c+d x)} \, dx=\int { \left (b \coth \left (d x + c\right )^{4}\right )^{\frac {1}{3}} \,d x } \] Input:

integrate((b*coth(d*x+c)^4)^(1/3),x, algorithm="maxima")
 

Output:

integrate((b*coth(d*x + c)^4)^(1/3), x)
 

Giac [F]

\[ \int \sqrt [3]{b \coth ^4(c+d x)} \, dx=\int { \left (b \coth \left (d x + c\right )^{4}\right )^{\frac {1}{3}} \,d x } \] Input:

integrate((b*coth(d*x+c)^4)^(1/3),x, algorithm="giac")
 

Output:

integrate((b*coth(d*x + c)^4)^(1/3), x)
 

Mupad [F(-1)]

Timed out. \[ \int \sqrt [3]{b \coth ^4(c+d x)} \, dx=\int {\left (b\,{\mathrm {coth}\left (c+d\,x\right )}^4\right )}^{1/3} \,d x \] Input:

int((b*coth(c + d*x)^4)^(1/3),x)
 

Output:

int((b*coth(c + d*x)^4)^(1/3), x)
 

Reduce [F]

\[ \int \sqrt [3]{b \coth ^4(c+d x)} \, dx=\frac {b^{\frac {1}{3}} \left (-3 \coth \left (d x +c \right )^{\frac {1}{3}}+\left (\int \frac {1}{\coth \left (d x +c \right )^{\frac {2}{3}}}d x \right ) d \right )}{d} \] Input:

int((b*coth(d*x+c)^4)^(1/3),x)
 

Output:

(b**(1/3)*( - 3*coth(c + d*x)**(1/3) + int(coth(c + d*x)**(1/3)/coth(c + d 
*x),x)*d))/d