\(\int \frac {(a+b \text {arctanh}(c x))^3}{x^4} \, dx\) [33]

Optimal result

Integrand size = 14, antiderivative size = 200 \[ \int \frac {(a+b \text {arctanh}(c x))^3}{x^4} \, dx=-\frac {b^2 c^2 (a+b \text {arctanh}(c x))}{x}+\frac {1}{2} b c^3 (a+b \text {arctanh}(c x))^2-\frac {b c (a+b \text {arctanh}(c x))^2}{2 x^2}+\frac {1}{3} c^3 (a+b \text {arctanh}(c x))^3-\frac {(a+b \text {arctanh}(c x))^3}{3 x^3}+b^3 c^3 \log (x)-\frac {1}{2} b^3 c^3 \log \left (1-c^2 x^2\right )+b c^3 (a+b \text {arctanh}(c x))^2 \log \left (2-\frac {2}{1+c x}\right )-b^2 c^3 (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+c x}\right )-\frac {1}{2} b^3 c^3 \operatorname {PolyLog}\left (3,-1+\frac {2}{1+c x}\right ) \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.64 (sec) , antiderivative size = 323, normalized size of antiderivative = 1.62 \[ \int \frac {(a+b \text {arctanh}(c x))^3}{x^4} \, dx=-\frac {8 a^3+12 a^2 b c x+24 a^2 b \text {arctanh}(c x)-24 a^2 b c^3 x^3 \log (x)+12 a^2 b c^3 x^3 \log \left (1-c^2 x^2\right )+24 a b^2 \left (c^2 x^2+\left (1-c^3 x^3\right ) \text {arctanh}(c x)^2-c x \text {arctanh}(c x) \left (-1+c^2 x^2+2 c^2 x^2 \log \left (1-e^{-2 \text {arctanh}(c x)}\right )\right )+c^3 x^3 \operatorname {PolyLog}\left (2,e^{-2 \text {arctanh}(c x)}\right )\right )+b^3 \left (-i c^3 \pi ^3 x^3+24 c^2 x^2 \text {arctanh}(c x)+12 c x \text {arctanh}(c x)^2-12 c^3 x^3 \text {arctanh}(c x)^2+8 \text {arctanh}(c x)^3+8 c^3 x^3 \text {arctanh}(c x)^3-24 c^3 x^3 \text {arctanh}(c x)^2 \log \left (1-e^{2 \text {arctanh}(c x)}\right )-24 c^3 x^3 \log \left (\frac {c x}{\sqrt {1-c^2 x^2}}\right )-24 c^3 x^3 \text {arctanh}(c x) \operatorname {PolyLog}\left (2,e^{2 \text {arctanh}(c x)}\right )+12 c^3 x^3 \operatorname {PolyLog}\left (3,e^{2 \text {arctanh}(c x)}\right )\right )}{24 x^3} \] Input:

Integrate[(a + b*ArcTanh[c*x])^3/x^4,x]
 

Output:

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

Rubi [A] (verified)

Time = 1.93 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.98, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {6452, 6544, 6452, 6544, 6452, 243, 47, 14, 16, 6510, 6550, 6494, 6618, 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.

\(\displaystyle \int \frac {(a+b \text {arctanh}(c x))^3}{x^4} \, dx\)

\(\Big \downarrow \) 6452

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

\(\Big \downarrow \) 6544

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

\(\Big \downarrow \) 6452

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

\(\Big \downarrow \) 6544

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

\(\Big \downarrow \) 6452

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 47

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

\(\Big \downarrow \) 14

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

\(\Big \downarrow \) 16

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

\(\Big \downarrow \) 6510

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

\(\Big \downarrow \) 6550

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

\(\Big \downarrow \) 6494

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

\(\Big \downarrow \) 6618

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

\(\Big \downarrow \) 7164

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

Input:

Int[(a + b*ArcTanh[c*x])^3/x^4,x]
 

Output:

-1/3*(a + b*ArcTanh[c*x])^3/x^3 + b*c*(-1/2*(a + b*ArcTanh[c*x])^2/x^2 + b 
*c*(-((a + b*ArcTanh[c*x])/x) + (c*(a + b*ArcTanh[c*x])^2)/(2*b) + (b*c*(L 
og[x^2] - Log[1 - c^2*x^2]))/2) + c^2*((a + b*ArcTanh[c*x])^3/(3*b) + (a + 
 b*ArcTanh[c*x])^2*Log[2 - 2/(1 + c*x)] - 2*b*c*(((a + b*ArcTanh[c*x])*Pol 
yLog[2, -1 + 2/(1 + c*x)])/(2*c) + (b*PolyLog[3, -1 + 2/(1 + c*x)])/(4*c)) 
))
 

Defintions of rubi rules used

rule 14

Int[(a_.)/(x_), x_Symbol] :> Simp[a*Log[x], x] /; FreeQ[a, x]
 

rule 16

Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + 
b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
 

rule 47

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Simp[b/(b*c 
 - a*d)   Int[1/(a + b*x), x], x] - Simp[d/(b*c - a*d)   Int[1/(c + d*x), x 
], x] /; FreeQ[{a, b, c, d}, x]
 

rule 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

rule 6452

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

rule 6494

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_))), x 
_Symbol] :> Simp[(a + b*ArcTanh[c*x])^p*(Log[2 - 2/(1 + e*(x/d))]/d), x] - 
Simp[b*c*(p/d)   Int[(a + b*ArcTanh[c*x])^(p - 1)*(Log[2 - 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 6510

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

rule 6544

Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + ( 
e_.)*(x_)^2), x_Symbol] :> Simp[1/d   Int[(f*x)^m*(a + b*ArcTanh[c*x])^p, x 
], x] - Simp[e/(d*f^2)   Int[(f*x)^(m + 2)*((a + b*ArcTanh[c*x])^p/(d + e*x 
^2)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && LtQ[m, -1]
 

rule 6550

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

rule 6618

Int[(Log[u_]*((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^ 
2), x_Symbol] :> Simp[(a + b*ArcTanh[c*x])^p*(PolyLog[2, 1 - u]/(2*c*d)), x 
] - Simp[b*(p/2)   Int[(a + b*ArcTanh[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 = 12.88 (sec) , antiderivative size = 1581, normalized size of antiderivative = 7.90

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1581\)
default	\(\text {Expression too large to display}\)	\(1581\)
parts	\(\text {Expression too large to display}\)	\(1583\)

Input:

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

Output:

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

Fricas [F]

\[ \int \frac {(a+b \text {arctanh}(c x))^3}{x^4} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{3}}{x^{4}} \,d x } \] Input:

integrate((a+b*arctanh(c*x))^3/x^4,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^4, x)
 

Sympy [F]

\[ \int \frac {(a+b \text {arctanh}(c x))^3}{x^4} \, dx=\int \frac {\left (a + b \operatorname {atanh}{\left (c x \right )}\right )^{3}}{x^{4}}\, dx \] Input:

integrate((a+b*atanh(c*x))**3/x**4,x)
 

Output:

Integral((a + b*atanh(c*x))**3/x**4, x)
 

Maxima [F]

\[ \int \frac {(a+b \text {arctanh}(c x))^3}{x^4} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{3}}{x^{4}} \,d x } \] Input:

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

Output:

-1/2*((c^2*log(c^2*x^2 - 1) - c^2*log(x^2) + 1/x^2)*c + 2*arctanh(c*x)/x^3 
)*a^2*b - 1/3*a^3/x^3 - 1/24*((b^3*c^3*x^3 - b^3)*log(-c*x + 1)^3 + 3*(b^3 
*c*x + 2*a*b^2 + (b^3*c^3*x^3 + b^3)*log(c*x + 1))*log(-c*x + 1)^2)/x^3 - 
integrate(-1/8*((b^3*c*x - b^3)*log(c*x + 1)^3 + 6*(a*b^2*c*x - a*b^2)*log 
(c*x + 1)^2 + (2*b^3*c^2*x^2 + 4*a*b^2*c*x - 3*(b^3*c*x - b^3)*log(c*x + 1 
)^2 + 2*(b^3*c^4*x^4 + 6*a*b^2 - (6*a*b^2*c - b^3*c)*x)*log(c*x + 1))*log( 
-c*x + 1))/(c*x^5 - x^4), x)
 

Giac [F]

\[ \int \frac {(a+b \text {arctanh}(c x))^3}{x^4} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{3}}{x^{4}} \,d x } \] Input:

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

Output:

integrate((b*arctanh(c*x) + a)^3/x^4, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \text {arctanh}(c x))^3}{x^4} \, dx=\int \frac {{\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^3}{x^4} \,d x \] Input:

int((a + b*atanh(c*x))^3/x^4,x)
 

Output:

int((a + b*atanh(c*x))^3/x^4, x)
 

Reduce [F]

\[ \int \frac {(a+b \text {arctanh}(c x))^3}{x^4} \, dx=\frac {-2 \mathit {atanh} \left (c x \right )^{3} b^{3}-6 \mathit {atanh} \left (c x \right )^{2} a \,b^{2}+3 \mathit {atanh} \left (c x \right )^{2} b^{3} c^{3} x^{3}-3 \mathit {atanh} \left (c x \right )^{2} b^{3} c x -6 \mathit {atanh} \left (c x \right ) a^{2} b \,c^{3} x^{3}-6 \mathit {atanh} \left (c x \right ) a^{2} b +6 \mathit {atanh} \left (c x \right ) a \,b^{2} c^{3} x^{3}-6 \mathit {atanh} \left (c x \right ) a \,b^{2} c x -6 \mathit {atanh} \left (c x \right ) b^{3} c^{3} x^{3}-6 \mathit {atanh} \left (c x \right ) b^{3} c^{2} x^{2}-12 \left (\int \frac {\mathit {atanh} \left (c x \right )}{c^{2} x^{3}-x}d x \right ) a \,b^{2} c^{3} x^{3}-6 \left (\int \frac {\mathit {atanh} \left (c x \right )^{2}}{c^{2} x^{3}-x}d x \right ) b^{3} c^{3} x^{3}-6 \,\mathrm {log}\left (c^{2} x -c \right ) a^{2} b \,c^{3} x^{3}-6 \,\mathrm {log}\left (c^{2} x -c \right ) b^{3} c^{3} x^{3}+6 \,\mathrm {log}\left (x \right ) a^{2} b \,c^{3} x^{3}+6 \,\mathrm {log}\left (x \right ) b^{3} c^{3} x^{3}-2 a^{3}-3 a^{2} b c x -6 a \,b^{2} c^{2} x^{2}}{6 x^{3}} \] Input:

int((a+b*atanh(c*x))^3/x^4,x)
                                                                                    
                                                                                    
 

Output:

( - 2*atanh(c*x)**3*b**3 - 6*atanh(c*x)**2*a*b**2 + 3*atanh(c*x)**2*b**3*c 
**3*x**3 - 3*atanh(c*x)**2*b**3*c*x - 6*atanh(c*x)*a**2*b*c**3*x**3 - 6*at 
anh(c*x)*a**2*b + 6*atanh(c*x)*a*b**2*c**3*x**3 - 6*atanh(c*x)*a*b**2*c*x 
- 6*atanh(c*x)*b**3*c**3*x**3 - 6*atanh(c*x)*b**3*c**2*x**2 - 12*int(atanh 
(c*x)/(c**2*x**3 - x),x)*a*b**2*c**3*x**3 - 6*int(atanh(c*x)**2/(c**2*x**3 
 - x),x)*b**3*c**3*x**3 - 6*log(c**2*x - c)*a**2*b*c**3*x**3 - 6*log(c**2* 
x - c)*b**3*c**3*x**3 + 6*log(x)*a**2*b*c**3*x**3 + 6*log(x)*b**3*c**3*x** 
3 - 2*a**3 - 3*a**2*b*c*x - 6*a*b**2*c**2*x**2)/(6*x**3)