\(\int (a+b \coth ^{-1}(c+d x))^3 \, dx\) [35]

Optimal result

Integrand size = 12, antiderivative size = 132 \[ \int \left (a+b \coth ^{-1}(c+d x)\right )^3 \, dx=\frac {\left (a+b \coth ^{-1}(c+d x)\right )^3}{d}+\frac {(c+d x) \left (a+b \coth ^{-1}(c+d x)\right )^3}{d}-\frac {3 b \left (a+b \coth ^{-1}(c+d x)\right )^2 \log \left (\frac {2}{1-c-d x}\right )}{d}-\frac {3 b^2 \left (a+b \coth ^{-1}(c+d x)\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{1-c-d x}\right )}{d}+\frac {3 b^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1-c-d x}\right )}{2 d} \] Output:

(a+b*arccoth(d*x+c))^3/d+(d*x+c)*(a+b*arccoth(d*x+c))^3/d-3*b*(a+b*arccoth 
(d*x+c))^2*ln(2/(-d*x-c+1))/d-3*b^2*(a+b*arccoth(d*x+c))*polylog(2,1-2/(-d 
*x-c+1))/d+3/2*b^3*polylog(3,1-2/(-d*x-c+1))/d
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.23 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.58 \[ \int \left (a+b \coth ^{-1}(c+d x)\right )^3 \, dx=\frac {2 a^3 (c+d x)+6 a^2 b (c+d x) \coth ^{-1}(c+d x)+3 a^2 b \log \left (1-(c+d x)^2\right )+6 a b^2 \left (\coth ^{-1}(c+d x) \left ((-1+c+d x) \coth ^{-1}(c+d x)-2 \log \left (1-e^{-2 \coth ^{-1}(c+d x)}\right )\right )+\operatorname {PolyLog}\left (2,e^{-2 \coth ^{-1}(c+d x)}\right )\right )+2 b^3 \left (-\frac {i \pi ^3}{8}+\coth ^{-1}(c+d x)^3+(c+d x) \coth ^{-1}(c+d x)^3-3 \coth ^{-1}(c+d x)^2 \log \left (1-e^{2 \coth ^{-1}(c+d x)}\right )-3 \coth ^{-1}(c+d x) \operatorname {PolyLog}\left (2,e^{2 \coth ^{-1}(c+d x)}\right )+\frac {3}{2} \operatorname {PolyLog}\left (3,e^{2 \coth ^{-1}(c+d x)}\right )\right )}{2 d} \] Input:

Integrate[(a + b*ArcCoth[c + d*x])^3,x]
 

Output:

(2*a^3*(c + d*x) + 6*a^2*b*(c + d*x)*ArcCoth[c + d*x] + 3*a^2*b*Log[1 - (c 
 + d*x)^2] + 6*a*b^2*(ArcCoth[c + d*x]*((-1 + c + d*x)*ArcCoth[c + d*x] - 
2*Log[1 - E^(-2*ArcCoth[c + d*x])]) + PolyLog[2, E^(-2*ArcCoth[c + d*x])]) 
 + 2*b^3*((-1/8*I)*Pi^3 + ArcCoth[c + d*x]^3 + (c + d*x)*ArcCoth[c + d*x]^ 
3 - 3*ArcCoth[c + d*x]^2*Log[1 - E^(2*ArcCoth[c + d*x])] - 3*ArcCoth[c + d 
*x]*PolyLog[2, E^(2*ArcCoth[c + d*x])] + (3*PolyLog[3, E^(2*ArcCoth[c + d* 
x])])/2))/(2*d)
 

Rubi [A] (verified)

Time = 0.75 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.98, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6654, 6437, 6547, 6471, 6621, 7164}

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.

\(\displaystyle \int \left (a+b \coth ^{-1}(c+d x)\right )^3 \, dx\)

\(\Big \downarrow \) 6654

\(\displaystyle \frac {\int \left (a+b \coth ^{-1}(c+d x)\right )^3d(c+d x)}{d}\)

\(\Big \downarrow \) 6437

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

\(\Big \downarrow \) 6547

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

\(\Big \downarrow \) 6471

\(\displaystyle \frac {(c+d x) \left (a+b \coth ^{-1}(c+d x)\right )^3-3 b \left (-2 b \int \frac {\left (a+b \coth ^{-1}(c+d x)\right ) \log \left (\frac {2}{-c-d x+1}\right )}{1-(c+d x)^2}d(c+d x)-\frac {\left (a+b \coth ^{-1}(c+d x)\right )^3}{3 b}+\log \left (\frac {2}{-c-d x+1}\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2\right )}{d}\)

\(\Big \downarrow \) 6621

\(\displaystyle \frac {(c+d x) \left (a+b \coth ^{-1}(c+d x)\right )^3-3 b \left (-2 b \left (\frac {1}{2} b \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{-c-d x+1}\right )}{1-(c+d x)^2}d(c+d x)-\frac {1}{2} \operatorname {PolyLog}\left (2,1-\frac {2}{-c-d x+1}\right ) \left (a+b \coth ^{-1}(c+d x)\right )\right )-\frac {\left (a+b \coth ^{-1}(c+d x)\right )^3}{3 b}+\log \left (\frac {2}{-c-d x+1}\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2\right )}{d}\)

\(\Big \downarrow \) 7164

\(\displaystyle \frac {(c+d x) \left (a+b \coth ^{-1}(c+d x)\right )^3-3 b \left (-2 b \left (\frac {1}{4} b \operatorname {PolyLog}\left (3,1-\frac {2}{-c-d x+1}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,1-\frac {2}{-c-d x+1}\right ) \left (a+b \coth ^{-1}(c+d x)\right )\right )-\frac {\left (a+b \coth ^{-1}(c+d x)\right )^3}{3 b}+\log \left (\frac {2}{-c-d x+1}\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2\right )}{d}\)

Input:

Int[(a + b*ArcCoth[c + d*x])^3,x]
 

Output:

((c + d*x)*(a + b*ArcCoth[c + d*x])^3 - 3*b*(-1/3*(a + b*ArcCoth[c + d*x]) 
^3/b + (a + b*ArcCoth[c + d*x])^2*Log[2/(1 - c - d*x)] - 2*b*(-1/2*((a + b 
*ArcCoth[c + d*x])*PolyLog[2, 1 - 2/(1 - c - d*x)]) + (b*PolyLog[3, 1 - 2/ 
(1 - c - d*x)])/4)))/d
 

Defintions of rubi rules used

rule 6437

Int[((a_.) + ArcCoth[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a 
 + b*ArcCoth[c*x^n])^p, x] - Simp[b*c*n*p   Int[x^n*((a + b*ArcCoth[c*x^n]) 
^(p - 1)/(1 - c^2*x^(2*n))), x], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[p, 0] 
 && (EqQ[n, 1] || EqQ[p, 1])
 

rule 6471

Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol 
] :> Simp[(-(a + b*ArcCoth[c*x])^p)*(Log[2/(1 + e*(x/d))]/e), x] + Simp[b*c 
*(p/e)   Int[(a + b*ArcCoth[c*x])^(p - 1)*(Log[2/(1 + e*(x/d))]/(1 - c^2*x^ 
2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 - e^2 
, 0]
 

rule 6547

Int[(((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), 
 x_Symbol] :> Simp[(a + b*ArcCoth[c*x])^(p + 1)/(b*e*(p + 1)), x] + Simp[1/ 
(c*d)   Int[(a + b*ArcCoth[c*x])^p/(1 - c*x), x], x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]
 

rule 6621

Int[(Log[u_]*((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^ 
2), x_Symbol] :> Simp[(-(a + b*ArcCoth[c*x])^p)*(PolyLog[2, 1 - u]/(2*c*d)) 
, x] + Simp[b*(p/2)   Int[(a + b*ArcCoth[c*x])^(p - 1)*(PolyLog[2, 1 - u]/( 
d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d 
 + e, 0] && EqQ[(1 - u)^2 - (1 - 2/(1 - c*x))^2, 0]
 

rule 6654

Int[((a_.) + ArcCoth[(c_) + (d_.)*(x_)]*(b_.))^(p_.), x_Symbol] :> Simp[1/d 
   Subst[Int[(a + b*ArcCoth[x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d 
}, x] && IGtQ[p, 0]
 

rule 7164

Int[(u_)*PolyLog[n_, v_], x_Symbol] :> With[{w = DerivativeDivides[v, u*v, 
x]}, Simp[w*PolyLog[n + 1, v], x] /;  !FalseQ[w]] /; FreeQ[n, x]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(371\) vs. \(2(130)=260\).

Time = 1.01 (sec) , antiderivative size = 372, normalized size of antiderivative = 2.82

method	result	size
derivativedivides	\(\frac {\left (d x +c \right ) a^{3}+b^{3} \left (\operatorname {arccoth}\left (d x +c \right )^{3} \left (d x +c -1\right )+2 \operatorname {arccoth}\left (d x +c \right )^{3}-3 \operatorname {arccoth}\left (d x +c \right )^{2} \ln \left (1+\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )-6 \,\operatorname {arccoth}\left (d x +c \right ) \operatorname {polylog}\left (2, -\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )+6 \operatorname {polylog}\left (3, -\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )-3 \operatorname {arccoth}\left (d x +c \right )^{2} \ln \left (1-\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )-6 \,\operatorname {arccoth}\left (d x +c \right ) \operatorname {polylog}\left (2, \frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )+6 \operatorname {polylog}\left (3, \frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )\right )+3 a \,b^{2} \left (\operatorname {arccoth}\left (d x +c \right )^{2} \left (d x +c -1\right )+2 \operatorname {arccoth}\left (d x +c \right )^{2}-2 \,\operatorname {arccoth}\left (d x +c \right ) \ln \left (1-\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )-2 \operatorname {polylog}\left (2, \frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )-2 \,\operatorname {arccoth}\left (d x +c \right ) \ln \left (1+\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )-2 \operatorname {polylog}\left (2, -\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )\right )+3 a^{2} b \left (\left (d x +c \right ) \operatorname {arccoth}\left (d x +c \right )+\frac {\ln \left (\left (d x +c \right )^{2}-1\right )}{2}\right )}{d}\)	\(372\)
default	\(\frac {\left (d x +c \right ) a^{3}+b^{3} \left (\operatorname {arccoth}\left (d x +c \right )^{3} \left (d x +c -1\right )+2 \operatorname {arccoth}\left (d x +c \right )^{3}-3 \operatorname {arccoth}\left (d x +c \right )^{2} \ln \left (1+\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )-6 \,\operatorname {arccoth}\left (d x +c \right ) \operatorname {polylog}\left (2, -\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )+6 \operatorname {polylog}\left (3, -\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )-3 \operatorname {arccoth}\left (d x +c \right )^{2} \ln \left (1-\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )-6 \,\operatorname {arccoth}\left (d x +c \right ) \operatorname {polylog}\left (2, \frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )+6 \operatorname {polylog}\left (3, \frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )\right )+3 a \,b^{2} \left (\operatorname {arccoth}\left (d x +c \right )^{2} \left (d x +c -1\right )+2 \operatorname {arccoth}\left (d x +c \right )^{2}-2 \,\operatorname {arccoth}\left (d x +c \right ) \ln \left (1-\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )-2 \operatorname {polylog}\left (2, \frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )-2 \,\operatorname {arccoth}\left (d x +c \right ) \ln \left (1+\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )-2 \operatorname {polylog}\left (2, -\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )\right )+3 a^{2} b \left (\left (d x +c \right ) \operatorname {arccoth}\left (d x +c \right )+\frac {\ln \left (\left (d x +c \right )^{2}-1\right )}{2}\right )}{d}\)	\(372\)
parts	\(a^{3} x +\frac {b^{3} \left (\operatorname {arccoth}\left (d x +c \right )^{3} \left (d x +c -1\right )+2 \operatorname {arccoth}\left (d x +c \right )^{3}-3 \operatorname {arccoth}\left (d x +c \right )^{2} \ln \left (1+\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )-6 \,\operatorname {arccoth}\left (d x +c \right ) \operatorname {polylog}\left (2, -\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )+6 \operatorname {polylog}\left (3, -\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )-3 \operatorname {arccoth}\left (d x +c \right )^{2} \ln \left (1-\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )-6 \,\operatorname {arccoth}\left (d x +c \right ) \operatorname {polylog}\left (2, \frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )+6 \operatorname {polylog}\left (3, \frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )\right )}{d}+\frac {3 a \,b^{2} \left (\operatorname {arccoth}\left (d x +c \right )^{2} \left (d x +c -1\right )+2 \operatorname {arccoth}\left (d x +c \right )^{2}-2 \,\operatorname {arccoth}\left (d x +c \right ) \ln \left (1-\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )-2 \operatorname {polylog}\left (2, \frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )-2 \,\operatorname {arccoth}\left (d x +c \right ) \ln \left (1+\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )-2 \operatorname {polylog}\left (2, -\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )\right )}{d}+\frac {3 a^{2} b \left (\left (d x +c \right ) \operatorname {arccoth}\left (d x +c \right )+\frac {\ln \left (\left (d x +c \right )^{2}-1\right )}{2}\right )}{d}\)	\(373\)

Input:

int((a+b*arccoth(d*x+c))^3,x,method=_RETURNVERBOSE)
 

Output:

1/d*((d*x+c)*a^3+b^3*(arccoth(d*x+c)^3*(d*x+c-1)+2*arccoth(d*x+c)^3-3*arcc 
oth(d*x+c)^2*ln(1+1/((d*x+c-1)/(d*x+c+1))^(1/2))-6*arccoth(d*x+c)*polylog( 
2,-1/((d*x+c-1)/(d*x+c+1))^(1/2))+6*polylog(3,-1/((d*x+c-1)/(d*x+c+1))^(1/ 
2))-3*arccoth(d*x+c)^2*ln(1-1/((d*x+c-1)/(d*x+c+1))^(1/2))-6*arccoth(d*x+c 
)*polylog(2,1/((d*x+c-1)/(d*x+c+1))^(1/2))+6*polylog(3,1/((d*x+c-1)/(d*x+c 
+1))^(1/2)))+3*a*b^2*(arccoth(d*x+c)^2*(d*x+c-1)+2*arccoth(d*x+c)^2-2*arcc 
oth(d*x+c)*ln(1-1/((d*x+c-1)/(d*x+c+1))^(1/2))-2*polylog(2,1/((d*x+c-1)/(d 
*x+c+1))^(1/2))-2*arccoth(d*x+c)*ln(1+1/((d*x+c-1)/(d*x+c+1))^(1/2))-2*pol 
ylog(2,-1/((d*x+c-1)/(d*x+c+1))^(1/2)))+3*a^2*b*((d*x+c)*arccoth(d*x+c)+1/ 
2*ln((d*x+c)^2-1)))
 

Fricas [F]

\[ \int \left (a+b \coth ^{-1}(c+d x)\right )^3 \, dx=\int { {\left (b \operatorname {arcoth}\left (d x + c\right ) + a\right )}^{3} \,d x } \] Input:

integrate((a+b*arccoth(d*x+c))^3,x, algorithm="fricas")
 

Output:

integral(b^3*arccoth(d*x + c)^3 + 3*a*b^2*arccoth(d*x + c)^2 + 3*a^2*b*arc 
coth(d*x + c) + a^3, x)
 

Sympy [F]

\[ \int \left (a+b \coth ^{-1}(c+d x)\right )^3 \, dx=\int \left (a + b \operatorname {acoth}{\left (c + d x \right )}\right )^{3}\, dx \] Input:

integrate((a+b*acoth(d*x+c))**3,x)
 

Output:

Integral((a + b*acoth(c + d*x))**3, x)
 

Maxima [F]

\[ \int \left (a+b \coth ^{-1}(c+d x)\right )^3 \, dx=\int { {\left (b \operatorname {arcoth}\left (d x + c\right ) + a\right )}^{3} \,d x } \] Input:

integrate((a+b*arccoth(d*x+c))^3,x, algorithm="maxima")
 

Output:

a^3*x + 3/2*(2*(d*x + c)*arccoth(d*x + c) + log(-(d*x + c)^2 + 1))*a^2*b/d 
 + 1/8*((b^3*d*x + b^3*(c + 1))*log(d*x + c + 1)^3 + 3*(2*a*b^2*d*x - (b^3 
*d*x + b^3*(c - 1))*log(d*x + c - 1))*log(d*x + c + 1)^2)/d + integrate(-1 
/8*((b^3*d*x + b^3*(c + 1))*log(d*x + c - 1)^3 - 6*(a*b^2*d*x + a*b^2*(c + 
 1))*log(d*x + c - 1)^2 + 3*(4*a*b^2*d*x - (b^3*d*x + b^3*(c + 1))*log(d*x 
 + c - 1)^2 + 2*(2*a*b^2*(c + 1) - b^3*(c - 1) + (2*a*b^2*d - b^3*d)*x)*lo 
g(d*x + c - 1))*log(d*x + c + 1))/(d*x + c + 1), x)
 

Giac [F]

\[ \int \left (a+b \coth ^{-1}(c+d x)\right )^3 \, dx=\int { {\left (b \operatorname {arcoth}\left (d x + c\right ) + a\right )}^{3} \,d x } \] Input:

integrate((a+b*arccoth(d*x+c))^3,x, algorithm="giac")
 

Output:

integrate((b*arccoth(d*x + c) + a)^3, x)
 

Mupad [F(-1)]

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

int((a + b*acoth(c + d*x))^3,x)
 

Output:

int((a + b*acoth(c + d*x))^3, x)
                                                                                    
                                                                                    
 

Reduce [F]

\[ \int \left (a+b \coth ^{-1}(c+d x)\right )^3 \, dx=\frac {\mathit {acoth} \left (d x +c \right )^{3} b^{3} d x +3 \mathit {acoth} \left (d x +c \right )^{2} a \,b^{2} d x +3 \mathit {acoth} \left (d x +c \right ) a^{2} b c +3 \mathit {acoth} \left (d x +c \right ) a^{2} b d x +3 \mathit {acoth} \left (d x +c \right ) a^{2} b -6 \left (\int \frac {\mathit {acoth} \left (d x +c \right ) x}{d^{2} x^{2}+2 c d x +c^{2}-1}d x \right ) a \,b^{2} d^{2}-3 \left (\int \frac {\mathit {acoth} \left (d x +c \right )^{2} x}{d^{2} x^{2}+2 c d x +c^{2}-1}d x \right ) b^{3} d^{2}-3 \,\mathrm {log}\left (d x +c -1\right ) a^{2} b +a^{3} d x}{d} \] Input:

int((a+b*acoth(d*x+c))^3,x)
 

Output:

(acoth(c + d*x)**3*b**3*d*x + 3*acoth(c + d*x)**2*a*b**2*d*x + 3*acoth(c + 
 d*x)*a**2*b*c + 3*acoth(c + d*x)*a**2*b*d*x + 3*acoth(c + d*x)*a**2*b - 6 
*int((acoth(c + d*x)*x)/(c**2 + 2*c*d*x + d**2*x**2 - 1),x)*a*b**2*d**2 - 
3*int((acoth(c + d*x)**2*x)/(c**2 + 2*c*d*x + d**2*x**2 - 1),x)*b**3*d**2 
- 3*log(c + d*x - 1)*a**2*b + a**3*d*x)/d