3.1.22 \(\int (b \coth ^2(c+d x))^{4/3} \, dx\) [22]

3.1.22.1 Optimal result

Integrand size = 14, antiderivative size = 297 \[ \int \left (b \coth ^2(c+d x)\right )^{4/3} \, dx=\frac {\sqrt {3} b \arctan \left (\frac {1-2 \sqrt [3]{\coth (c+d x)}}{\sqrt {3}}\right ) \sqrt [3]{b \coth ^2(c+d x)}}{2 d \coth ^{\frac {2}{3}}(c+d x)}-\frac {\sqrt {3} b \arctan \left (\frac {1+2 \sqrt [3]{\coth (c+d x)}}{\sqrt {3}}\right ) \sqrt [3]{b \coth ^2(c+d x)}}{2 d \coth ^{\frac {2}{3}}(c+d x)}+\frac {b \text {arctanh}\left (\sqrt [3]{\coth (c+d x)}\right ) \sqrt [3]{b \coth ^2(c+d x)}}{d \coth ^{\frac {2}{3}}(c+d x)}-\frac {3 b \coth (c+d x) \sqrt [3]{b \coth ^2(c+d x)}}{5 d}-\frac {b \sqrt [3]{b \coth ^2(c+d x)} \log \left (1-\sqrt [3]{\coth (c+d x)}+\coth ^{\frac {2}{3}}(c+d x)\right )}{4 d \coth ^{\frac {2}{3}}(c+d x)}+\frac {b \sqrt [3]{b \coth ^2(c+d x)} \log \left (1+\sqrt [3]{\coth (c+d x)}+\coth ^{\frac {2}{3}}(c+d x)\right )}{4 d \coth ^{\frac {2}{3}}(c+d x)} \]

output

b*arctanh(coth(d*x+c)^(1/3))*(b*coth(d*x+c)^2)^(1/3)/d/coth(d*x+c)^(2/3)-3 
/5*b*coth(d*x+c)*(b*coth(d*x+c)^2)^(1/3)/d-1/4*b*(b*coth(d*x+c)^2)^(1/3)*l 
n(1-coth(d*x+c)^(1/3)+coth(d*x+c)^(2/3))/d/coth(d*x+c)^(2/3)+1/4*b*(b*coth 
(d*x+c)^2)^(1/3)*ln(1+coth(d*x+c)^(1/3)+coth(d*x+c)^(2/3))/d/coth(d*x+c)^( 
2/3)+1/2*b*arctan(1/3*(1-2*coth(d*x+c)^(1/3))*3^(1/2))*(b*coth(d*x+c)^2)^( 
1/3)*3^(1/2)/d/coth(d*x+c)^(2/3)-1/2*b*arctan(1/3*(1+2*coth(d*x+c)^(1/3))* 
3^(1/2))*(b*coth(d*x+c)^2)^(1/3)*3^(1/2)/d/coth(d*x+c)^(2/3)
 

3.1.22.2 Mathematica [A] (verified)

Time = 0.36 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.56 \[ \int \left (b \coth ^2(c+d x)\right )^{4/3} \, dx=-\frac {\left (b \coth ^2(c+d x)\right )^{4/3} \left (-20 \text {arctanh}\left (\sqrt [3]{\coth (c+d x)}\right )+12 \coth ^{\frac {5}{3}}(c+d x)-5 \left (2 \sqrt {3} \arctan \left (\frac {1-2 \sqrt [3]{\coth (c+d x)}}{\sqrt {3}}\right )-2 \sqrt {3} \arctan \left (\frac {1+2 \sqrt [3]{\coth (c+d x)}}{\sqrt {3}}\right )-\log \left (1-\sqrt [3]{\coth (c+d x)}+\coth ^{\frac {2}{3}}(c+d x)\right )+\log \left (1+\sqrt [3]{\coth (c+d x)}+\coth ^{\frac {2}{3}}(c+d x)\right )\right )\right )}{20 d \coth ^{\frac {8}{3}}(c+d x)} \]

input

Integrate[(b*Coth[c + d*x]^2)^(4/3),x]
 

output

-1/20*((b*Coth[c + d*x]^2)^(4/3)*(-20*ArcTanh[Coth[c + d*x]^(1/3)] + 12*Co 
th[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)])))/(d*Coth[c + d*x]^(8/3))
 

3.1.22.3 Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.63, 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.

\(\displaystyle \int \left (b \coth ^2(c+d x)\right )^{4/3} \, dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4141

\(\displaystyle \frac {b \sqrt [3]{b \coth ^2(c+d x)} \int \coth ^{\frac {8}{3}}(c+d x)dx}{\coth ^{\frac {2}{3}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {b \sqrt [3]{b \coth ^2(c+d x)} \int \left (-i \tan \left (i c+i d x+\frac {\pi }{2}\right )\right )^{8/3}dx}{\coth ^{\frac {2}{3}}(c+d x)}\)

\(\Big \downarrow \) 3954

\(\displaystyle \frac {b \sqrt [3]{b \coth ^2(c+d x)} \left (\int \coth ^{\frac {2}{3}}(c+d x)dx-\frac {3 \coth ^{\frac {5}{3}}(c+d x)}{5 d}\right )}{\coth ^{\frac {2}{3}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {b \sqrt [3]{b \coth ^2(c+d x)} \left (-\frac {3 \coth ^{\frac {5}{3}}(c+d x)}{5 d}+\int \left (-i \tan \left (i c+i d x+\frac {\pi }{2}\right )\right )^{2/3}dx\right )}{\coth ^{\frac {2}{3}}(c+d x)}\)

\(\Big \downarrow \) 3957

\(\displaystyle \frac {b \sqrt [3]{b \coth ^2(c+d x)} \left (-\frac {\int -\frac {\coth ^{\frac {2}{3}}(c+d x)}{1-\coth ^2(c+d x)}d\coth (c+d x)}{d}-\frac {3 \coth ^{\frac {5}{3}}(c+d x)}{5 d}\right )}{\coth ^{\frac {2}{3}}(c+d x)}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {b \sqrt [3]{b \coth ^2(c+d x)} \left (\frac {\int \frac {\coth ^{\frac {2}{3}}(c+d x)}{1-\coth ^2(c+d x)}d\coth (c+d x)}{d}-\frac {3 \coth ^{\frac {5}{3}}(c+d x)}{5 d}\right )}{\coth ^{\frac {2}{3}}(c+d x)}\)

\(\Big \downarrow \) 266

\(\displaystyle \frac {b \sqrt [3]{b \coth ^2(c+d x)} \left (\frac {3 \int \frac {\coth ^{\frac {4}{3}}(c+d x)}{1-\coth ^2(c+d x)}d\sqrt [3]{\coth (c+d x)}}{d}-\frac {3 \coth ^{\frac {5}{3}}(c+d x)}{5 d}\right )}{\coth ^{\frac {2}{3}}(c+d x)}\)

\(\Big \downarrow \) 825

\(\displaystyle \frac {b \sqrt [3]{b \coth ^2(c+d x)} \left (\frac {3 \left (\frac {1}{3} \int \frac {1}{1-\coth ^{\frac {2}{3}}(c+d x)}d\sqrt [3]{\coth (c+d x)}+\frac {1}{3} \int -\frac {\sqrt [3]{\coth (c+d x)}+1}{2 \left (\coth ^{\frac {2}{3}}(c+d x)-\sqrt [3]{\coth (c+d x)}+1\right )}d\sqrt [3]{\coth (c+d x)}+\frac {1}{3} \int -\frac {1-\sqrt [3]{\coth (c+d x)}}{2 \left (\coth ^{\frac {2}{3}}(c+d x)+\sqrt [3]{\coth (c+d x)}+1\right )}d\sqrt [3]{\coth (c+d x)}\right )}{d}-\frac {3 \coth ^{\frac {5}{3}}(c+d x)}{5 d}\right )}{\coth ^{\frac {2}{3}}(c+d x)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {b \sqrt [3]{b \coth ^2(c+d x)} \left (\frac {3 \left (\frac {1}{3} \int \frac {1}{1-\coth ^{\frac {2}{3}}(c+d x)}d\sqrt [3]{\coth (c+d x)}-\frac {1}{6} \int \frac {\sqrt [3]{\coth (c+d x)}+1}{\coth ^{\frac {2}{3}}(c+d x)-\sqrt [3]{\coth (c+d x)}+1}d\sqrt [3]{\coth (c+d x)}-\frac {1}{6} \int \frac {1-\sqrt [3]{\coth (c+d x)}}{\coth ^{\frac {2}{3}}(c+d x)+\sqrt [3]{\coth (c+d x)}+1}d\sqrt [3]{\coth (c+d x)}\right )}{d}-\frac {3 \coth ^{\frac {5}{3}}(c+d x)}{5 d}\right )}{\coth ^{\frac {2}{3}}(c+d x)}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {b \sqrt [3]{b \coth ^2(c+d x)} \left (\frac {3 \left (-\frac {1}{6} \int \frac {\sqrt [3]{\coth (c+d x)}+1}{\coth ^{\frac {2}{3}}(c+d x)-\sqrt [3]{\coth (c+d x)}+1}d\sqrt [3]{\coth (c+d x)}-\frac {1}{6} \int \frac {1-\sqrt [3]{\coth (c+d x)}}{\coth ^{\frac {2}{3}}(c+d x)+\sqrt [3]{\coth (c+d x)}+1}d\sqrt [3]{\coth (c+d x)}+\frac {1}{3} \text {arctanh}\left (\sqrt [3]{\coth (c+d x)}\right )\right )}{d}-\frac {3 \coth ^{\frac {5}{3}}(c+d x)}{5 d}\right )}{\coth ^{\frac {2}{3}}(c+d x)}\)

\(\Big \downarrow \) 1142

\(\displaystyle \frac {b \sqrt [3]{b \coth ^2(c+d x)} \left (\frac {3 \left (\frac {1}{6} \left (-\frac {3}{2} \int \frac {1}{\coth ^{\frac {2}{3}}(c+d x)-\sqrt [3]{\coth (c+d x)}+1}d\sqrt [3]{\coth (c+d x)}-\frac {1}{2} \int -\frac {1-2 \sqrt [3]{\coth (c+d x)}}{\coth ^{\frac {2}{3}}(c+d x)-\sqrt [3]{\coth (c+d x)}+1}d\sqrt [3]{\coth (c+d x)}\right )+\frac {1}{6} \left (\frac {1}{2} \int \frac {2 \sqrt [3]{\coth (c+d x)}+1}{\coth ^{\frac {2}{3}}(c+d x)+\sqrt [3]{\coth (c+d x)}+1}d\sqrt [3]{\coth (c+d x)}-\frac {3}{2} \int \frac {1}{\coth ^{\frac {2}{3}}(c+d x)+\sqrt [3]{\coth (c+d x)}+1}d\sqrt [3]{\coth (c+d x)}\right )+\frac {1}{3} \text {arctanh}\left (\sqrt [3]{\coth (c+d x)}\right )\right )}{d}-\frac {3 \coth ^{\frac {5}{3}}(c+d x)}{5 d}\right )}{\coth ^{\frac {2}{3}}(c+d x)}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {b \sqrt [3]{b \coth ^2(c+d x)} \left (\frac {3 \left (\frac {1}{6} \left (\frac {1}{2} \int \frac {1-2 \sqrt [3]{\coth (c+d x)}}{\coth ^{\frac {2}{3}}(c+d x)-\sqrt [3]{\coth (c+d x)}+1}d\sqrt [3]{\coth (c+d x)}-\frac {3}{2} \int \frac {1}{\coth ^{\frac {2}{3}}(c+d x)-\sqrt [3]{\coth (c+d x)}+1}d\sqrt [3]{\coth (c+d x)}\right )+\frac {1}{6} \left (\frac {1}{2} \int \frac {2 \sqrt [3]{\coth (c+d x)}+1}{\coth ^{\frac {2}{3}}(c+d x)+\sqrt [3]{\coth (c+d x)}+1}d\sqrt [3]{\coth (c+d x)}-\frac {3}{2} \int \frac {1}{\coth ^{\frac {2}{3}}(c+d x)+\sqrt [3]{\coth (c+d x)}+1}d\sqrt [3]{\coth (c+d x)}\right )+\frac {1}{3} \text {arctanh}\left (\sqrt [3]{\coth (c+d x)}\right )\right )}{d}-\frac {3 \coth ^{\frac {5}{3}}(c+d x)}{5 d}\right )}{\coth ^{\frac {2}{3}}(c+d x)}\)

\(\Big \downarrow \) 1083

\(\displaystyle \frac {b \sqrt [3]{b \coth ^2(c+d x)} \left (\frac {3 \left (\frac {1}{6} \left (3 \int \frac {1}{-\coth ^{\frac {2}{3}}(c+d x)-3}d\left (2 \sqrt [3]{\coth (c+d x)}-1\right )+\frac {1}{2} \int \frac {1-2 \sqrt [3]{\coth (c+d x)}}{\coth ^{\frac {2}{3}}(c+d x)-\sqrt [3]{\coth (c+d x)}+1}d\sqrt [3]{\coth (c+d x)}\right )+\frac {1}{6} \left (3 \int \frac {1}{-\coth ^{\frac {2}{3}}(c+d x)-3}d\left (2 \sqrt [3]{\coth (c+d x)}+1\right )+\frac {1}{2} \int \frac {2 \sqrt [3]{\coth (c+d x)}+1}{\coth ^{\frac {2}{3}}(c+d x)+\sqrt [3]{\coth (c+d x)}+1}d\sqrt [3]{\coth (c+d x)}\right )+\frac {1}{3} \text {arctanh}\left (\sqrt [3]{\coth (c+d x)}\right )\right )}{d}-\frac {3 \coth ^{\frac {5}{3}}(c+d x)}{5 d}\right )}{\coth ^{\frac {2}{3}}(c+d x)}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {b \sqrt [3]{b \coth ^2(c+d x)} \left (\frac {3 \left (\frac {1}{6} \left (\frac {1}{2} \int \frac {1-2 \sqrt [3]{\coth (c+d x)}}{\coth ^{\frac {2}{3}}(c+d x)-\sqrt [3]{\coth (c+d x)}+1}d\sqrt [3]{\coth (c+d x)}-\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{\coth (c+d x)}-1}{\sqrt {3}}\right )\right )+\frac {1}{6} \left (\frac {1}{2} \int \frac {2 \sqrt [3]{\coth (c+d x)}+1}{\coth ^{\frac {2}{3}}(c+d x)+\sqrt [3]{\coth (c+d x)}+1}d\sqrt [3]{\coth (c+d x)}-\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{\coth (c+d x)}+1}{\sqrt {3}}\right )\right )+\frac {1}{3} \text {arctanh}\left (\sqrt [3]{\coth (c+d x)}\right )\right )}{d}-\frac {3 \coth ^{\frac {5}{3}}(c+d x)}{5 d}\right )}{\coth ^{\frac {2}{3}}(c+d x)}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {b \sqrt [3]{b \coth ^2(c+d x)} \left (\frac {3 \left (\frac {1}{6} \left (-\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{\coth (c+d x)}-1}{\sqrt {3}}\right )-\frac {1}{2} \log \left (\coth ^{\frac {2}{3}}(c+d x)-\sqrt [3]{\coth (c+d x)}+1\right )\right )+\frac {1}{6} \left (\frac {1}{2} \log \left (\coth ^{\frac {2}{3}}(c+d x)+\sqrt [3]{\coth (c+d x)}+1\right )-\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{\coth (c+d x)}+1}{\sqrt {3}}\right )\right )+\frac {1}{3} \text {arctanh}\left (\sqrt [3]{\coth (c+d x)}\right )\right )}{d}-\frac {3 \coth ^{\frac {5}{3}}(c+d x)}{5 d}\right )}{\coth ^{\frac {2}{3}}(c+d x)}\)

input

Int[(b*Coth[c + d*x]^2)^(4/3),x]
 

output

(b*(b*Coth[c + d*x]^2)^(1/3)*((-3*Coth[c + d*x]^(5/3))/(5*d) + (3*(ArcTanh 
[Coth[c + d*x]^(1/3)]/3 + (-(Sqrt[3]*ArcTan[(-1 + 2*Coth[c + d*x]^(1/3))/S 
qrt[3]]) - Log[1 - Coth[c + d*x]^(1/3) + Coth[c + d*x]^(2/3)]/2)/6 + (-(Sq 
rt[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]^(2/3)
 

3.1.22.3.1 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 825

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]
 

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]])
 

3.1.22.4 Maple [F]

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

input

int((coth(d*x+c)^2*b)^(4/3),x)
 

output

int((coth(d*x+c)^2*b)^(4/3),x)
 

3.1.22.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1994 vs. \(2 (245) = 490\).

Time = 0.29 (sec) , antiderivative size = 1994, normalized size of antiderivative = 6.71 \[ \int \left (b \coth ^2(c+d x)\right )^{4/3} \, dx=\text {Too large to display} \]

input

integrate((b*coth(d*x+c)^2)^(4/3),x, algorithm="fricas")
 

output

-1/20*(10*(sqrt(3)*b*cosh(d*x + c)^2 + 2*sqrt(3)*b*cosh(d*x + c)*sinh(d*x 
+ c) + sqrt(3)*b*sinh(d*x + c)^2 - sqrt(3)*b)*(-b)^(1/3)*arctan(1/3*(sqrt( 
3)*b*cosh(d*x + c)^2 + 2*sqrt(3)*b*cosh(d*x + c)*sinh(d*x + c) + sqrt(3)*b 
*sinh(d*x + c)^2 + 2*(sqrt(3)*cosh(d*x + c)^2 + 2*sqrt(3)*cosh(d*x + c)*si 
nh(d*x + c) + sqrt(3)*sinh(d*x + c)^2 - sqrt(3))*(-b)^(2/3)*((b*cosh(d*x + 
 c)^2 + b*sinh(d*x + c)^2 + b)/(cosh(d*x + c)^2 + sinh(d*x + c)^2 - 1))^(1 
/3) + sqrt(3)*b)/(b*cosh(d*x + c)^2 + 2*b*cosh(d*x + c)*sinh(d*x + c) + b* 
sinh(d*x + c)^2 + b)) - 10*(sqrt(3)*b*cosh(d*x + c)^2 + 2*sqrt(3)*b*cosh(d 
*x + c)*sinh(d*x + c) + sqrt(3)*b*sinh(d*x + c)^2 - sqrt(3)*b)*b^(1/3)*arc 
tan(-1/3*(sqrt(3)*b*cosh(d*x + c)^2 + 2*sqrt(3)*b*cosh(d*x + c)*sinh(d*x + 
 c) + sqrt(3)*b*sinh(d*x + c)^2 - 2*(sqrt(3)*cosh(d*x + c)^2 + 2*sqrt(3)*c 
osh(d*x + c)*sinh(d*x + c) + sqrt(3)*sinh(d*x + c)^2 - sqrt(3))*b^(2/3)*(( 
b*cosh(d*x + c)^2 + b*sinh(d*x + c)^2 + b)/(cosh(d*x + c)^2 + sinh(d*x + c 
)^2 - 1))^(1/3) + sqrt(3)*b)/(b*cosh(d*x + c)^2 + 2*b*cosh(d*x + c)*sinh(d 
*x + c) + b*sinh(d*x + c)^2 + b)) + 5*(b*cosh(d*x + c)^2 + 2*b*cosh(d*x + 
c)*sinh(d*x + c) + b*sinh(d*x + c)^2 - b)*(-b)^(1/3)*log(((cosh(d*x + c)^4 
 + 4*cosh(d*x + c)*sinh(d*x + c)^3 + sinh(d*x + c)^4 + 2*(3*cosh(d*x + c)^ 
2 + 1)*sinh(d*x + c)^2 + 2*cosh(d*x + c)^2 + 4*(cosh(d*x + c)^3 + cosh(d*x 
 + c))*sinh(d*x + c) + 1)*(-b)^(2/3) - (cosh(d*x + c)^4 + 4*cosh(d*x + c)^ 
3*sinh(d*x + c) + 6*cosh(d*x + c)^2*sinh(d*x + c)^2 + 4*cosh(d*x + c)*s...
 

3.1.22.6 Sympy [F]

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

input

integrate((b*coth(d*x+c)**2)**(4/3),x)
 

output

Integral((b*coth(c + d*x)**2)**(4/3), x)
 

3.1.22.7 Maxima [F]

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

input

integrate((b*coth(d*x+c)^2)^(4/3),x, algorithm="maxima")
 

output

integrate((b*coth(d*x + c)^2)^(4/3), x)
 

3.1.22.8 Giac [F]

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

input

integrate((b*coth(d*x+c)^2)^(4/3),x, algorithm="giac")
 

output

integrate((b*coth(d*x + c)^2)^(4/3), x)
 

3.1.22.9 Mupad [F(-1)]

Timed out. \[ \int \left (b \coth ^2(c+d x)\right )^{4/3} \, dx=\int {\left (b\,{\mathrm {coth}\left (c+d\,x\right )}^2\right )}^{4/3} \,d x \]

input

int((b*coth(c + d*x)^2)^(4/3),x)
 

output

int((b*coth(c + d*x)^2)^(4/3), x)