\(\int (a+b \text {arctanh}(\frac {c}{x}))^3 \, dx\) [153]

Optimal result

Integrand size = 12, antiderivative size = 108 \[ \int \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^3 \, dx=c \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^3+x \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^3-3 b c \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^2 \log \left (\frac {2 c}{c-x}\right )-3 b^2 c \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right ) \operatorname {PolyLog}\left (2,1-\frac {2 c}{c-x}\right )+\frac {3}{2} b^3 c \operatorname {PolyLog}\left (3,1-\frac {2 c}{c-x}\right ) \] Output:

c*(a+b*arccoth(x/c))^3+x*(a+b*arccoth(x/c))^3-3*b*c*(a+b*arccoth(x/c))^2*l 
n(2*c/(c-x))-3*b^2*c*(a+b*arccoth(x/c))*polylog(2,1-2*c/(c-x))+3/2*b^3*c*p 
olylog(3,1-2*c/(c-x))
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.20 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.83 \[ \int \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^3 \, dx=a^3 x+3 a^2 b x \text {arctanh}\left (\frac {c}{x}\right )+\frac {3}{2} a^2 b c \log \left (-c^2+x^2\right )-3 a b^2 \left (\text {arctanh}\left (\frac {c}{x}\right ) \left ((c-x) \text {arctanh}\left (\frac {c}{x}\right )+2 c \log \left (1-e^{-2 \text {arctanh}\left (\frac {c}{x}\right )}\right )\right )-c \operatorname {PolyLog}\left (2,e^{-2 \text {arctanh}\left (\frac {c}{x}\right )}\right )\right )+\frac {1}{8} b^3 \left (-i c \pi ^3+8 c \text {arctanh}\left (\frac {c}{x}\right )^3+8 x \text {arctanh}\left (\frac {c}{x}\right )^3-24 c \text {arctanh}\left (\frac {c}{x}\right )^2 \log \left (1-e^{2 \text {arctanh}\left (\frac {c}{x}\right )}\right )-24 c \text {arctanh}\left (\frac {c}{x}\right ) \operatorname {PolyLog}\left (2,e^{2 \text {arctanh}\left (\frac {c}{x}\right )}\right )+12 c \operatorname {PolyLog}\left (3,e^{2 \text {arctanh}\left (\frac {c}{x}\right )}\right )\right ) \] Input:

Integrate[(a + b*ArcTanh[c/x])^3,x]
 

Output:

a^3*x + 3*a^2*b*x*ArcTanh[c/x] + (3*a^2*b*c*Log[-c^2 + x^2])/2 - 3*a*b^2*( 
ArcTanh[c/x]*((c - x)*ArcTanh[c/x] + 2*c*Log[1 - E^(-2*ArcTanh[c/x])]) - c 
*PolyLog[2, E^(-2*ArcTanh[c/x])]) + (b^3*((-I)*c*Pi^3 + 8*c*ArcTanh[c/x]^3 
 + 8*x*ArcTanh[c/x]^3 - 24*c*ArcTanh[c/x]^2*Log[1 - E^(2*ArcTanh[c/x])] - 
24*c*ArcTanh[c/x]*PolyLog[2, E^(2*ArcTanh[c/x])] + 12*c*PolyLog[3, E^(2*Ar 
cTanh[c/x])]))/8
 

Rubi [A] (verified)

Time = 1.07 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.12, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {6440, 6437, 27, 6547, 27, 6471, 27, 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.

\(\displaystyle \int \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^3 \, dx\)

\(\Big \downarrow \) 6440

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

\(\Big \downarrow \) 6437

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6547

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6471

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6621

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

\(\Big \downarrow \) 7164

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

Input:

Int[(a + b*ArcTanh[c/x])^3,x]
 

Output:

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

Defintions of rubi rules used

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 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 6440

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

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 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 [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 27.39 (sec) , antiderivative size = 1475, normalized size of antiderivative = 13.66

method	result	size
parts	\(\text {Expression too large to display}\)	\(1475\)
derivativedivides	\(\text {Expression too large to display}\)	\(1478\)
default	\(\text {Expression too large to display}\)	\(1478\)

Input:

int((a+b*arctanh(c/x))^3,x,method=_RETURNVERBOSE)
 

Output:

x*a^3-b^3*c*(-1/c*x*arctanh(c/x)^3+3*ln(c/x)*arctanh(c/x)^2-3/2*arctanh(c/ 
x)^2*ln(c/x-1)-3/2*arctanh(c/x)^2*ln(1+c/x)+3*arctanh(c/x)^2*ln((1+c/x)/(1 
-c^2/x^2)^(1/2))-arctanh(c/x)^3+3/4*(I*Pi*csgn(I*(1+c/x)^2/(c^2/x^2-1))^3+ 
2*I*Pi-2*I*Pi*csgn(I/(1-(1+c/x)^2/(c^2/x^2-1)))*csgn(I*(-(1+c/x)^2/(c^2/x^ 
2-1)-1)/(1-(1+c/x)^2/(c^2/x^2-1)))^2+2*I*Pi*csgn(I*(-(1+c/x)^2/(c^2/x^2-1) 
-1)/(1-(1+c/x)^2/(c^2/x^2-1)))^3+2*I*Pi*csgn(I/(1-(1+c/x)^2/(c^2/x^2-1)))^ 
3+I*Pi*csgn(I/(1-(1+c/x)^2/(c^2/x^2-1)))*csgn(I*(1+c/x)^2/(c^2/x^2-1)/(1-( 
1+c/x)^2/(c^2/x^2-1)))^2-2*I*Pi*csgn(I*(-(1+c/x)^2/(c^2/x^2-1)-1))*csgn(I* 
(-(1+c/x)^2/(c^2/x^2-1)-1)/(1-(1+c/x)^2/(c^2/x^2-1)))^2-2*I*Pi*csgn(I/(1-( 
1+c/x)^2/(c^2/x^2-1)))^2-I*Pi*csgn(I*(1+c/x)^2/(c^2/x^2-1))*csgn(I*(1+c/x) 
^2/(c^2/x^2-1)/(1-(1+c/x)^2/(c^2/x^2-1)))^2+I*Pi*csgn(I*(1+c/x)^2/(c^2/x^2 
-1)/(1-(1+c/x)^2/(c^2/x^2-1)))^3+2*I*Pi*csgn(I*(-(1+c/x)^2/(c^2/x^2-1)-1)) 
*csgn(I/(1-(1+c/x)^2/(c^2/x^2-1)))*csgn(I*(-(1+c/x)^2/(c^2/x^2-1)-1)/(1-(1 
+c/x)^2/(c^2/x^2-1)))+2*I*Pi*csgn(I*(1+c/x)/(1-c^2/x^2)^(1/2))*csgn(I*(1+c 
/x)^2/(c^2/x^2-1))^2+I*Pi*csgn(I*(1+c/x)/(1-c^2/x^2)^(1/2))^2*csgn(I*(1+c/ 
x)^2/(c^2/x^2-1))-I*Pi*csgn(I/(1-(1+c/x)^2/(c^2/x^2-1)))*csgn(I*(1+c/x)^2/ 
(c^2/x^2-1))*csgn(I*(1+c/x)^2/(c^2/x^2-1)/(1-(1+c/x)^2/(c^2/x^2-1)))+4*ln( 
2))*arctanh(c/x)^2-3*arctanh(c/x)^2*ln((1+c/x)^2/(1-c^2/x^2)-1)+3*arctanh( 
c/x)^2*ln(1+(1+c/x)/(1-c^2/x^2)^(1/2))+6*arctanh(c/x)*polylog(2,-(1+c/x)/( 
1-c^2/x^2)^(1/2))-6*polylog(3,-(1+c/x)/(1-c^2/x^2)^(1/2))+3*arctanh(c/x...
 

Fricas [F]

\[ \int \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^3 \, dx=\int { {\left (b \operatorname {artanh}\left (\frac {c}{x}\right ) + a\right )}^{3} \,d x } \] Input:

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

Output:

integral(b^3*arctanh(c/x)^3 + 3*a*b^2*arctanh(c/x)^2 + 3*a^2*b*arctanh(c/x 
) + a^3, x)
 

Sympy [F]

\[ \int \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^3 \, dx=\int \left (a + b \operatorname {atanh}{\left (\frac {c}{x} \right )}\right )^{3}\, dx \] Input:

integrate((a+b*atanh(c/x))**3,x)
 

Output:

Integral((a + b*atanh(c/x))**3, x)
 

Maxima [F]

\[ \int \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^3 \, dx=\int { {\left (b \operatorname {artanh}\left (\frac {c}{x}\right ) + a\right )}^{3} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^3 \, dx=\int { {\left (b \operatorname {artanh}\left (\frac {c}{x}\right ) + a\right )}^{3} \,d x } \] Input:

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

Output:

integrate((b*arctanh(c/x) + a)^3, x)
 

Mupad [F(-1)]

Timed out. \[ \int \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^3 \, dx=\int {\left (a+b\,\mathrm {atanh}\left (\frac {c}{x}\right )\right )}^3 \,d x \] Input:

int((a + b*atanh(c/x))^3,x)
 

Output:

int((a + b*atanh(c/x))^3, x)
 

Reduce [F]

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

int((a+b*atanh(c/x))^3,x)
 

Output:

 - 3*atanh(c/x)*a**2*b*c + 3*atanh(c/x)*a**2*b*x + int(atanh(c/x)**3,x)*b* 
*3 + 3*int(atanh(c/x)**2,x)*a*b**2 + 3*log( - c - x)*a**2*b*c + a**3*x