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

3.1.33.1 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))
 

3.1.33.2 Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.74 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.52 \[ \int \frac {(a+b \text {arctanh}(c x))^3}{x^4} \, dx=-\frac {a^3}{3 x^3}-\frac {a^2 b c}{2 x^2}-\frac {a^2 b \text {arctanh}(c x)}{x^3}+a^2 b c^3 \log (x)-\frac {1}{2} a^2 b c^3 \log \left (1-c^2 x^2\right )+\frac {a b^2 \left (-c^2 x^2+\left (-1+c^3 x^3\right ) \text {arctanh}(c x)^2+c x \text {arctanh}(c x) \left (-1+c^2 x^2+2 c^2 x^2 \log \left (1-e^{-2 \text {arctanh}(c x)}\right )\right )-c^3 x^3 \operatorname {PolyLog}\left (2,e^{-2 \text {arctanh}(c x)}\right )\right )}{x^3}+b^3 c^3 \left (\frac {i \pi ^3}{24}-\frac {\text {arctanh}(c x)}{c x}+\frac {1}{2} \text {arctanh}(c x)^2-\frac {\text {arctanh}(c x)^2}{2 c^2 x^2}-\frac {1}{3} \text {arctanh}(c x)^3-\frac {\text {arctanh}(c x)^3}{3 c^3 x^3}+\text {arctanh}(c x)^2 \log \left (1-e^{2 \text {arctanh}(c x)}\right )+\log (c x)-\frac {1}{2} \log \left (1-c^2 x^2\right )+\text {arctanh}(c x) \operatorname {PolyLog}\left (2,e^{2 \text {arctanh}(c x)}\right )-\frac {1}{2} \operatorname {PolyLog}\left (3,e^{2 \text {arctanh}(c x)}\right )\right ) \]

input

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

output

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

3.1.33.3 Rubi [A] (verified)

Time = 1.85 (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)) 
))
 

3.1.33.3.1 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]
 

3.1.33.4 Maple [C] (warning: unable to verify)

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

Time = 58.16 (sec) , antiderivative size = 1542, normalized size of antiderivative = 7.71

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1542\)
default	\(\text {Expression too large to display}\)	\(1542\)
parts	\(\text {Expression too large to display}\)	\(1544\)

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*(-1/3/c^3/x^3*arctanh(c*x)^3-1/2*arctanh(c*x)^2* 
ln(c*x+1)-1/2*arctanh(c*x)^2*ln(c*x-1)-1/2/c^2/x^2*arctanh(c*x)^2+ln(c*x)* 
arctanh(c*x)^2+arctanh(c*x)^2*ln((c*x+1)/(-c^2*x^2+1)^(1/2))-arctanh(c*x)^ 
2*ln((c*x+1)^2/(-c^2*x^2+1)-1)+1/12*arctanh(c*x)*(-6*I*csgn(I/(1-(c*x+1)^2 
/(c^2*x^2-1)))^2*arctanh(c*x)*Pi*c*x+6*I*csgn(I*(-(c*x+1)^2/(c^2*x^2-1)-1) 
/(1-(c*x+1)^2/(c^2*x^2-1)))*csgn(I*(-(c*x+1)^2/(c^2*x^2-1)-1))*csgn(I/(1-( 
c*x+1)^2/(c^2*x^2-1)))*arctanh(c*x)*Pi*c*x-6*I*csgn(I*(-(c*x+1)^2/(c^2*x^2 
-1)-1)/(1-(c*x+1)^2/(c^2*x^2-1)))^2*csgn(I*(-(c*x+1)^2/(c^2*x^2-1)-1))*arc 
tanh(c*x)*Pi*c*x-6*I*csgn(I*(-(c*x+1)^2/(c^2*x^2-1)-1)/(1-(c*x+1)^2/(c^2*x 
^2-1)))^2*csgn(I/(1-(c*x+1)^2/(c^2*x^2-1)))*arctanh(c*x)*Pi*c*x+3*I*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)*Pi*c*x+6* 
I*arctanh(c*x)*Pi*c*x+3*I*csgn(I*(c*x+1)^2/(c^2*x^2-1))^3*arctanh(c*x)*Pi* 
c*x+3*I*csgn(I*(c*x+1)/(-c^2*x^2+1)^(1/2))^2*csgn(I*(c*x+1)^2/(c^2*x^2-1)) 
*arctanh(c*x)*Pi*c*x+3*I*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)*Pi*c*x+6*I*csgn(I/ 
(1-(c*x+1)^2/(c^2*x^2-1)))^3*arctanh(c*x)*Pi*c*x-3*I*csgn(I*(c*x+1)^2/(c^2 
*x^2-1)/(1-(c*x+1)^2/(c^2*x^2-1)))^2*csgn(I*(c*x+1)^2/(c^2*x^2-1))*arctanh 
(c*x)*Pi*c*x-3*I*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)))*csgn(I*(c*x+1)^2/(c^2*x^2-1))*arctanh(c*x 
)*Pi*c*x+6*I*csgn(I*(-(c*x+1)^2/(c^2*x^2-1)-1)/(1-(c*x+1)^2/(c^2*x^2-1)...
 

3.1.33.5 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)
 

3.1.33.6 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)
 

3.1.33.7 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)
 

3.1.33.8 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)
 

3.1.33.9 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)