\(\int x^2 (a+b \text {arctanh}(c x))^3 \, dx\) [27]

Optimal result

Integrand size = 14, antiderivative size = 197 \[ \int x^2 (a+b \text {arctanh}(c x))^3 \, dx=\frac {a b^2 x}{c^2}+\frac {b^3 x \text {arctanh}(c x)}{c^2}-\frac {b (a+b \text {arctanh}(c x))^2}{2 c^3}+\frac {b x^2 (a+b \text {arctanh}(c x))^2}{2 c}+\frac {(a+b \text {arctanh}(c x))^3}{3 c^3}+\frac {1}{3} x^3 (a+b \text {arctanh}(c x))^3-\frac {b (a+b \text {arctanh}(c x))^2 \log \left (\frac {2}{1-c x}\right )}{c^3}+\frac {b^3 \log \left (1-c^2 x^2\right )}{2 c^3}-\frac {b^2 (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{c^3}+\frac {b^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1-c x}\right )}{2 c^3} \] Output:

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

Mathematica [A] (verified)

Time = 0.37 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.27 \[ \int x^2 (a+b \text {arctanh}(c x))^3 \, dx=\frac {3 a^2 b c^2 x^2+2 a^3 c^3 x^3+6 a^2 b c^3 x^3 \text {arctanh}(c x)+3 a^2 b \log \left (1-c^2 x^2\right )+6 a b^2 \left (c x+\left (-1+c^3 x^3\right ) \text {arctanh}(c x)^2+\text {arctanh}(c x) \left (-1+c^2 x^2-2 \log \left (1+e^{-2 \text {arctanh}(c x)}\right )\right )+\operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )\right )+b^3 \left (6 c x \text {arctanh}(c x)-3 \text {arctanh}(c x)^2+3 c^2 x^2 \text {arctanh}(c x)^2-2 \text {arctanh}(c x)^3+2 c^3 x^3 \text {arctanh}(c x)^3-6 \text {arctanh}(c x)^2 \log \left (1+e^{-2 \text {arctanh}(c x)}\right )+3 \log \left (1-c^2 x^2\right )+6 \text {arctanh}(c x) \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )+3 \operatorname {PolyLog}\left (3,-e^{-2 \text {arctanh}(c x)}\right )\right )}{6 c^3} \] Input:

Integrate[x^2*(a + b*ArcTanh[c*x])^3,x]
 

Output:

(3*a^2*b*c^2*x^2 + 2*a^3*c^3*x^3 + 6*a^2*b*c^3*x^3*ArcTanh[c*x] + 3*a^2*b* 
Log[1 - c^2*x^2] + 6*a*b^2*(c*x + (-1 + c^3*x^3)*ArcTanh[c*x]^2 + ArcTanh[ 
c*x]*(-1 + c^2*x^2 - 2*Log[1 + E^(-2*ArcTanh[c*x])]) + PolyLog[2, -E^(-2*A 
rcTanh[c*x])]) + b^3*(6*c*x*ArcTanh[c*x] - 3*ArcTanh[c*x]^2 + 3*c^2*x^2*Ar 
cTanh[c*x]^2 - 2*ArcTanh[c*x]^3 + 2*c^3*x^3*ArcTanh[c*x]^3 - 6*ArcTanh[c*x 
]^2*Log[1 + E^(-2*ArcTanh[c*x])] + 3*Log[1 - c^2*x^2] + 6*ArcTanh[c*x]*Pol 
yLog[2, -E^(-2*ArcTanh[c*x])] + 3*PolyLog[3, -E^(-2*ArcTanh[c*x])]))/(6*c^ 
3)
 

Rubi [A] (verified)

Time = 1.96 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.10, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {6452, 6542, 6452, 6542, 2009, 6510, 6546, 6470, 6620, 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.

\(\displaystyle \int x^2 (a+b \text {arctanh}(c x))^3 \, dx\)

\(\Big \downarrow \) 6452

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

\(\Big \downarrow \) 6542

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

\(\Big \downarrow \) 6452

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

\(\Big \downarrow \) 6542

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 6510

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

\(\Big \downarrow \) 6546

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

\(\Big \downarrow \) 6470

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

\(\Big \downarrow \) 6620

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

\(\Big \downarrow \) 7164

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

Input:

Int[x^2*(a + b*ArcTanh[c*x])^3,x]
 

Output:

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

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

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 6470

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol 
] :> Simp[(-(a + b*ArcTanh[c*x])^p)*(Log[2/(1 + e*(x/d))]/e), x] + Simp[b*c 
*(p/e)   Int[(a + b*ArcTanh[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 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 6542

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

rule 6546

Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), 
 x_Symbol] :> Simp[(a + b*ArcTanh[c*x])^(p + 1)/(b*e*(p + 1)), x] + Simp[1/ 
(c*d)   Int[(a + b*ArcTanh[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 6620

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 = 10.26 (sec) , antiderivative size = 973, normalized size of antiderivative = 4.94

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(973\)
default	\(\text {Expression too large to display}\)	\(973\)
parts	\(\text {Expression too large to display}\)	\(975\)

Input:

int(x^2*(a+b*arctanh(c*x))^3,x,method=_RETURNVERBOSE)
 

Output:

1/c^3*(1/3*a^3*c^3*x^3+b^3*(1/3*c^3*x^3*arctanh(c*x)^3+1/2*c^2*x^2*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)*polylog(2,-(c*x+1)^2/(-c^2*x^2+1))+1/2*polylog(3,-(c*x+1)^2/(-c^2*x^2 
+1))-arctanh(c*x)^2*ln((c*x+1)/(-c^2*x^2+1)^(1/2))+1/3*arctanh(c*x)^3+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)^2/(-c^2*x^2+1))-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)^2-1/2*I*Pi*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^2-1)))^2*arcta 
nh(c*x)^2+(c*x+1)*arctanh(c*x)-1/4*I*Pi*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)^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*I*Pi*csgn(I*(c 
*x+1)/(-c^2*x^2+1)^(1/2))*csgn(I*(c*x+1)^2/(c^2*x^2-1))^2*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 
/(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-ln(2)*arctanh(c*x)^2-1/2*I*Pi*csgn(I/(1-(c*x+1) 
^2/(c^2*x^2-1)))^3*arctanh(c*x)^2)+3*a*b^2*(1/3*c^3*x^3*arctanh(c*x)^2+1/3 
*c^2*x^2*arctanh(c*x)+1/3*arctanh(c*x)*ln(c*x-1)+1/3*arctanh(c*x)*ln(c*x+1 
)+1/3*c*x+1/6*ln(c*x-1)-1/6*ln(c*x+1)+1/12*ln(c*x-1)^2-1/3*dilog(1/2*c*x+1 
/2)-1/6*ln(c*x-1)*ln(1/2*c*x+1/2)-1/12*ln(c*x+1)^2+1/6*(ln(c*x+1)-ln(1/...
 

Fricas [F]

\[ \int x^2 (a+b \text {arctanh}(c x))^3 \, dx=\int { {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{3} x^{2} \,d x } \] Input:

integrate(x^2*(a+b*arctanh(c*x))^3,x, algorithm="fricas")
 

Output:

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

Sympy [F]

\[ \int x^2 (a+b \text {arctanh}(c x))^3 \, dx=\int x^{2} \left (a + b \operatorname {atanh}{\left (c x \right )}\right )^{3}\, dx \] Input:

integrate(x**2*(a+b*atanh(c*x))**3,x)
 

Output:

Integral(x**2*(a + b*atanh(c*x))**3, x)
 

Maxima [F]

\[ \int x^2 (a+b \text {arctanh}(c x))^3 \, dx=\int { {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{3} x^{2} \,d x } \] Input:

integrate(x^2*(a+b*arctanh(c*x))^3,x, algorithm="maxima")
 

Output:

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

Giac [F]

\[ \int x^2 (a+b \text {arctanh}(c x))^3 \, dx=\int { {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{3} x^{2} \,d x } \] Input:

integrate(x^2*(a+b*arctanh(c*x))^3,x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

int(x^2*(a + b*atanh(c*x))^3,x)
 

Output:

int(x^2*(a + b*atanh(c*x))^3, x)
 

Reduce [F]

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

int(x^2*(a+b*atanh(c*x))^3,x)
 

Output:

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