3.1.9 \(\int (b \coth (c+d x))^{4/3} \, dx\) [9]

3.1.9.1 Optimal result

Integrand size = 12, antiderivative size = 236 \[ \int (b \coth (c+d x))^{4/3} \, dx=-\frac {\sqrt {3} b^{4/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b \coth (c+d x)}}{\sqrt [3]{b}}}{\sqrt {3}}\right )}{2 d}+\frac {\sqrt {3} b^{4/3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{b \coth (c+d x)}}{\sqrt [3]{b}}}{\sqrt {3}}\right )}{2 d}+\frac {b^{4/3} \text {arctanh}\left (\frac {\sqrt [3]{b \coth (c+d x)}}{\sqrt [3]{b}}\right )}{d}-\frac {3 b \sqrt [3]{b \coth (c+d x)}}{d}-\frac {b^{4/3} \log \left (b^{2/3}-\sqrt [3]{b} \sqrt [3]{b \coth (c+d x)}+(b \coth (c+d x))^{2/3}\right )}{4 d}+\frac {b^{4/3} \log \left (b^{2/3}+\sqrt [3]{b} \sqrt [3]{b \coth (c+d x)}+(b \coth (c+d x))^{2/3}\right )}{4 d} \]

output

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

3.1.9.2 Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 193, normalized size of antiderivative = 0.82 \[ \int (b \coth (c+d x))^{4/3} \, dx=-\frac {b \sqrt [3]{b \coth (c+d x)} \left (6 \sqrt [6]{\coth ^2(c+d x)}+\log \left (1-\sqrt [6]{\coth ^2(c+d x)}\right )-\log \left (1+\sqrt [6]{\coth ^2(c+d x)}\right )-(-1)^{2/3} \log \left (1-\sqrt [3]{-1} \sqrt [6]{\coth ^2(c+d x)}\right )+(-1)^{2/3} \log \left (1+\sqrt [3]{-1} \sqrt [6]{\coth ^2(c+d x)}\right )-\sqrt [3]{-1} \log \left (1-(-1)^{2/3} \sqrt [6]{\coth ^2(c+d x)}\right )+\sqrt [3]{-1} \log \left (1+(-1)^{2/3} \sqrt [6]{\coth ^2(c+d x)}\right )\right )}{2 d \sqrt [6]{\coth ^2(c+d x)}} \]

input

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

output

-1/2*(b*(b*Coth[c + d*x])^(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)]))/(d*(Coth[c 
+ d*x]^2)^(1/6))
 

3.1.9.3 Rubi [A] (warning: unable to verify)

Time = 0.46 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.84, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.167, Rules used = {3042, 3954, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3954

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3957

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 266

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

\(\Big \downarrow \) 754

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

\(\Big \downarrow \) 27

\(\displaystyle \frac {3 b^3 \left (\frac {\int \frac {1}{b^{2/3}-b^2 \coth ^2(c+d x)}d\sqrt [3]{b \coth (c+d x)}}{3 b^{4/3}}+\frac {\int \frac {2 \sqrt [3]{b}-\sqrt [3]{b \coth (c+d x)}}{b^2 \coth ^2(c+d x)-b^{4/3} \coth (c+d x)+b^{2/3}}d\sqrt [3]{b \coth (c+d x)}}{6 b^{5/3}}+\frac {\int \frac {2 \sqrt [3]{b}+\sqrt [3]{b \coth (c+d x)}}{b^2 \coth ^2(c+d x)+b^{4/3} \coth (c+d x)+b^{2/3}}d\sqrt [3]{b \coth (c+d x)}}{6 b^{5/3}}\right )}{d}-\frac {3 b \sqrt [3]{b \coth (c+d x)}}{d}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {3 b^3 \left (\frac {\int \frac {2 \sqrt [3]{b}-\sqrt [3]{b \coth (c+d x)}}{b^2 \coth ^2(c+d x)-b^{4/3} \coth (c+d x)+b^{2/3}}d\sqrt [3]{b \coth (c+d x)}}{6 b^{5/3}}+\frac {\int \frac {2 \sqrt [3]{b}+\sqrt [3]{b \coth (c+d x)}}{b^2 \coth ^2(c+d x)+b^{4/3} \coth (c+d x)+b^{2/3}}d\sqrt [3]{b \coth (c+d x)}}{6 b^{5/3}}+\frac {\text {arctanh}\left (b^{2/3} \coth (c+d x)\right )}{3 b^{5/3}}\right )}{d}-\frac {3 b \sqrt [3]{b \coth (c+d x)}}{d}\)

\(\Big \downarrow \) 1142

\(\displaystyle \frac {3 b^3 \left (\frac {\frac {3}{2} \sqrt [3]{b} \int \frac {1}{b^2 \coth ^2(c+d x)-b^{4/3} \coth (c+d x)+b^{2/3}}d\sqrt [3]{b \coth (c+d x)}-\frac {1}{2} \int -\frac {\sqrt [3]{b}-2 \sqrt [3]{b \coth (c+d x)}}{b^2 \coth ^2(c+d x)-b^{4/3} \coth (c+d x)+b^{2/3}}d\sqrt [3]{b \coth (c+d x)}}{6 b^{5/3}}+\frac {\frac {3}{2} \sqrt [3]{b} \int \frac {1}{b^2 \coth ^2(c+d x)+b^{4/3} \coth (c+d x)+b^{2/3}}d\sqrt [3]{b \coth (c+d x)}+\frac {1}{2} \int \frac {\sqrt [3]{b}+2 \sqrt [3]{b \coth (c+d x)}}{b^2 \coth ^2(c+d x)+b^{4/3} \coth (c+d x)+b^{2/3}}d\sqrt [3]{b \coth (c+d x)}}{6 b^{5/3}}+\frac {\text {arctanh}\left (b^{2/3} \coth (c+d x)\right )}{3 b^{5/3}}\right )}{d}-\frac {3 b \sqrt [3]{b \coth (c+d x)}}{d}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {3 b^3 \left (\frac {\frac {3}{2} \sqrt [3]{b} \int \frac {1}{b^2 \coth ^2(c+d x)-b^{4/3} \coth (c+d x)+b^{2/3}}d\sqrt [3]{b \coth (c+d x)}+\frac {1}{2} \int \frac {\sqrt [3]{b}-2 \sqrt [3]{b \coth (c+d x)}}{b^2 \coth ^2(c+d x)-b^{4/3} \coth (c+d x)+b^{2/3}}d\sqrt [3]{b \coth (c+d x)}}{6 b^{5/3}}+\frac {\frac {3}{2} \sqrt [3]{b} \int \frac {1}{b^2 \coth ^2(c+d x)+b^{4/3} \coth (c+d x)+b^{2/3}}d\sqrt [3]{b \coth (c+d x)}+\frac {1}{2} \int \frac {\sqrt [3]{b}+2 \sqrt [3]{b \coth (c+d x)}}{b^2 \coth ^2(c+d x)+b^{4/3} \coth (c+d x)+b^{2/3}}d\sqrt [3]{b \coth (c+d x)}}{6 b^{5/3}}+\frac {\text {arctanh}\left (b^{2/3} \coth (c+d x)\right )}{3 b^{5/3}}\right )}{d}-\frac {3 b \sqrt [3]{b \coth (c+d x)}}{d}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {3 b^3 \left (\frac {3 \int \frac {1}{-b^2 \coth ^2(c+d x)-3}d\left (1-2 b^{2/3} \coth (c+d x)\right )+\frac {1}{2} \int \frac {\sqrt [3]{b}-2 \sqrt [3]{b \coth (c+d x)}}{b^2 \coth ^2(c+d x)-b^{4/3} \coth (c+d x)+b^{2/3}}d\sqrt [3]{b \coth (c+d x)}}{6 b^{5/3}}+\frac {\frac {1}{2} \int \frac {\sqrt [3]{b}+2 \sqrt [3]{b \coth (c+d x)}}{b^2 \coth ^2(c+d x)+b^{4/3} \coth (c+d x)+b^{2/3}}d\sqrt [3]{b \coth (c+d x)}-3 \int \frac {1}{-b^2 \coth ^2(c+d x)-3}d\left (2 b^{2/3} \coth (c+d x)+1\right )}{6 b^{5/3}}+\frac {\text {arctanh}\left (b^{2/3} \coth (c+d x)\right )}{3 b^{5/3}}\right )}{d}-\frac {3 b \sqrt [3]{b \coth (c+d x)}}{d}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {3 b^3 \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{b}-2 \sqrt [3]{b \coth (c+d x)}}{b^2 \coth ^2(c+d x)-b^{4/3} \coth (c+d x)+b^{2/3}}d\sqrt [3]{b \coth (c+d x)}-\sqrt {3} \arctan \left (\frac {1-2 b^{2/3} \coth (c+d x)}{\sqrt {3}}\right )}{6 b^{5/3}}+\frac {\frac {1}{2} \int \frac {\sqrt [3]{b}+2 \sqrt [3]{b \coth (c+d x)}}{b^2 \coth ^2(c+d x)+b^{4/3} \coth (c+d x)+b^{2/3}}d\sqrt [3]{b \coth (c+d x)}+\sqrt {3} \arctan \left (\frac {2 b^{2/3} \coth (c+d x)+1}{\sqrt {3}}\right )}{6 b^{5/3}}+\frac {\text {arctanh}\left (b^{2/3} \coth (c+d x)\right )}{3 b^{5/3}}\right )}{d}-\frac {3 b \sqrt [3]{b \coth (c+d x)}}{d}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {3 b^3 \left (\frac {-\sqrt {3} \arctan \left (\frac {1-2 b^{2/3} \coth (c+d x)}{\sqrt {3}}\right )-\frac {1}{2} \log \left (-b^{4/3} \coth (c+d x)+b^{2/3}+b^2 \coth ^2(c+d x)\right )}{6 b^{5/3}}+\frac {\sqrt {3} \arctan \left (\frac {2 b^{2/3} \coth (c+d x)+1}{\sqrt {3}}\right )+\frac {1}{2} \log \left (b^{4/3} \coth (c+d x)+b^{2/3}+b^2 \coth ^2(c+d x)\right )}{6 b^{5/3}}+\frac {\text {arctanh}\left (b^{2/3} \coth (c+d x)\right )}{3 b^{5/3}}\right )}{d}-\frac {3 b \sqrt [3]{b \coth (c+d x)}}{d}\)

input

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

output

(-3*b*(b*Coth[c + d*x])^(1/3))/d + (3*b^3*(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
 

3.1.9.3.1 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 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 1082

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]
 

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]
 

3.1.9.4 Maple [A] (verified)

Time = 0.07 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.89

method	result	size
derivativedivides	\(-\frac {3 b \left (b \coth \left (d x +c \right )\right )^{\frac {1}{3}}}{d}+\frac {b^{\frac {4}{3}} \ln \left (\left (b \coth \left (d x +c \right )\right )^{\frac {1}{3}}+b^{\frac {1}{3}}\right )}{2 d}-\frac {b^{\frac {4}{3}} \ln \left (b^{\frac {2}{3}}-b^{\frac {1}{3}} \left (b \coth \left (d x +c \right )\right )^{\frac {1}{3}}+\left (b \coth \left (d x +c \right )\right )^{\frac {2}{3}}\right )}{4 d}+\frac {b^{\frac {4}{3}} \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b \coth \left (d x +c \right )\right )^{\frac {1}{3}}}{b^{\frac {1}{3}}}-1\right )}{3}\right )}{2 d}-\frac {b^{\frac {4}{3}} \ln \left (\left (b \coth \left (d x +c \right )\right )^{\frac {1}{3}}-b^{\frac {1}{3}}\right )}{2 d}+\frac {b^{\frac {4}{3}} \ln \left (b^{\frac {2}{3}}+b^{\frac {1}{3}} \left (b \coth \left (d x +c \right )\right )^{\frac {1}{3}}+\left (b \coth \left (d x +c \right )\right )^{\frac {2}{3}}\right )}{4 d}+\frac {b^{\frac {4}{3}} \arctan \left (\frac {\left (1+\frac {2 \left (b \coth \left (d x +c \right )\right )^{\frac {1}{3}}}{b^{\frac {1}{3}}}\right ) \sqrt {3}}{3}\right ) \sqrt {3}}{2 d}\)	\(209\)
default	\(-\frac {3 b \left (b \coth \left (d x +c \right )\right )^{\frac {1}{3}}}{d}+\frac {b^{\frac {4}{3}} \ln \left (\left (b \coth \left (d x +c \right )\right )^{\frac {1}{3}}+b^{\frac {1}{3}}\right )}{2 d}-\frac {b^{\frac {4}{3}} \ln \left (b^{\frac {2}{3}}-b^{\frac {1}{3}} \left (b \coth \left (d x +c \right )\right )^{\frac {1}{3}}+\left (b \coth \left (d x +c \right )\right )^{\frac {2}{3}}\right )}{4 d}+\frac {b^{\frac {4}{3}} \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b \coth \left (d x +c \right )\right )^{\frac {1}{3}}}{b^{\frac {1}{3}}}-1\right )}{3}\right )}{2 d}-\frac {b^{\frac {4}{3}} \ln \left (\left (b \coth \left (d x +c \right )\right )^{\frac {1}{3}}-b^{\frac {1}{3}}\right )}{2 d}+\frac {b^{\frac {4}{3}} \ln \left (b^{\frac {2}{3}}+b^{\frac {1}{3}} \left (b \coth \left (d x +c \right )\right )^{\frac {1}{3}}+\left (b \coth \left (d x +c \right )\right )^{\frac {2}{3}}\right )}{4 d}+\frac {b^{\frac {4}{3}} \arctan \left (\frac {\left (1+\frac {2 \left (b \coth \left (d x +c \right )\right )^{\frac {1}{3}}}{b^{\frac {1}{3}}}\right ) \sqrt {3}}{3}\right ) \sqrt {3}}{2 d}\)	\(209\)

input

int((b*coth(d*x+c))^(4/3),x,method=_RETURNVERBOSE)
 

output

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

3.1.9.5 Fricas [A] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.24 \[ \int (b \coth (c+d x))^{4/3} \, dx=-\frac {2 \, \sqrt {3} \left (-b\right )^{\frac {1}{3}} b \arctan \left (\frac {\sqrt {3} b + 2 \, \sqrt {3} \left (-b\right )^{\frac {2}{3}} \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {1}{3}}}{3 \, b}\right ) - 2 \, \sqrt {3} b^{\frac {4}{3}} \arctan \left (-\frac {\sqrt {3} b - 2 \, \sqrt {3} b^{\frac {2}{3}} \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {1}{3}}}{3 \, b}\right ) + \left (-b\right )^{\frac {1}{3}} b \log \left (\left (-b\right )^{\frac {2}{3}} - \left (-b\right )^{\frac {1}{3}} \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {1}{3}} + \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {2}{3}}\right ) + b^{\frac {4}{3}} \log \left (b^{\frac {2}{3}} - b^{\frac {1}{3}} \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {1}{3}} + \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {2}{3}}\right ) - 2 \, \left (-b\right )^{\frac {1}{3}} b \log \left (\left (-b\right )^{\frac {1}{3}} + \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {1}{3}}\right ) - 2 \, b^{\frac {4}{3}} \log \left (b^{\frac {1}{3}} + \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {1}{3}}\right ) + 12 \, b \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {1}{3}}}{4 \, d} \]

input

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

output

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

3.1.9.6 Sympy [F]

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

input

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

output

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

3.1.9.7 Maxima [F]

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

input

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

output

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

3.1.9.8 Giac [F(-2)]

Exception generated. \[ \int (b \coth (c+d x))^{4/3} \, dx=\text {Exception raised: TypeError} \]

input

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

output

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
 

3.1.9.9 Mupad [B] (verification not implemented)

Time = 2.43 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.06 \[ \int (b \coth (c+d x))^{4/3} \, dx=-\frac {3\,b\,{\left (b\,\mathrm {coth}\left (c+d\,x\right )\right )}^{1/3}}{d}-\frac {b^{4/3}\,\mathrm {atan}\left (\frac {{\left (b\,\mathrm {coth}\left (c+d\,x\right )\right )}^{1/3}\,1{}\mathrm {i}}{b^{1/3}}\right )\,1{}\mathrm {i}}{d}-\frac {b^{4/3}\,\ln \left (\frac {486\,b^{37/3}\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{d^4}-\frac {486\,b^{12}\,{\left (b\,\mathrm {coth}\left (c+d\,x\right )\right )}^{1/3}}{d^4}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{2\,d}-\frac {b^{4/3}\,\ln \left (\frac {486\,b^{37/3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{d^4}-\frac {486\,b^{12}\,{\left (b\,\mathrm {coth}\left (c+d\,x\right )\right )}^{1/3}}{d^4}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{2\,d}+\frac {b^{4/3}\,\ln \left (\frac {972\,b^{37/3}\,\left (-\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )}{d^4}+\frac {486\,b^{12}\,{\left (b\,\mathrm {coth}\left (c+d\,x\right )\right )}^{1/3}}{d^4}\right )\,\left (-\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )}{d}+\frac {b^{4/3}\,\ln \left (\frac {972\,b^{37/3}\,\left (\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )}{d^4}+\frac {486\,b^{12}\,{\left (b\,\mathrm {coth}\left (c+d\,x\right )\right )}^{1/3}}{d^4}\right )\,\left (\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )}{d} \]

input

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

output

(b^(4/3)*log((972*b^(37/3)*((3^(1/2)*1i)/4 - 1/4))/d^4 + (486*b^12*(b*coth 
(c + d*x))^(1/3))/d^4)*((3^(1/2)*1i)/4 - 1/4))/d - (b^(4/3)*atan(((b*coth( 
c + d*x))^(1/3)*1i)/b^(1/3))*1i)/d - (b^(4/3)*log((486*b^(37/3)*((3^(1/2)* 
1i)/2 - 1/2))/d^4 - (486*b^12*(b*coth(c + d*x))^(1/3))/d^4)*((3^(1/2)*1i)/ 
2 - 1/2))/(2*d) - (b^(4/3)*log((486*b^(37/3)*((3^(1/2)*1i)/2 + 1/2))/d^4 - 
 (486*b^12*(b*coth(c + d*x))^(1/3))/d^4)*((3^(1/2)*1i)/2 + 1/2))/(2*d) - ( 
3*b*(b*coth(c + d*x))^(1/3))/d + (b^(4/3)*log((972*b^(37/3)*((3^(1/2)*1i)/ 
4 + 1/4))/d^4 + (486*b^12*(b*coth(c + d*x))^(1/3))/d^4)*((3^(1/2)*1i)/4 + 
1/4))/d