\(\int x^2 (a+b \text {arctanh}(c x))^2 \, dx\) [16]

Optimal result

Integrand size = 14, antiderivative size = 130 \[ \int x^2 (a+b \text {arctanh}(c x))^2 \, dx=\frac {b^2 x}{3 c^2}-\frac {b^2 \text {arctanh}(c x)}{3 c^3}+\frac {b x^2 (a+b \text {arctanh}(c x))}{3 c}+\frac {(a+b \text {arctanh}(c x))^2}{3 c^3}+\frac {1}{3} x^3 (a+b \text {arctanh}(c x))^2-\frac {2 b (a+b \text {arctanh}(c x)) \log \left (\frac {2}{1-c x}\right )}{3 c^3}-\frac {b^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{3 c^3} \] Output:

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

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.94 \[ \int x^2 (a+b \text {arctanh}(c x))^2 \, dx=\frac {b^2 c x+a b c^2 x^2+a^2 c^3 x^3+b^2 \left (-1+c^3 x^3\right ) \text {arctanh}(c x)^2+b \text {arctanh}(c x) \left (-b+b c^2 x^2+2 a c^3 x^3-2 b \log \left (1+e^{-2 \text {arctanh}(c x)}\right )\right )+a b \log \left (-1+c^2 x^2\right )+b^2 \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )}{3 c^3} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.39 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.09, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {6452, 6542, 6452, 262, 219, 6546, 6470, 2849, 2752}

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.

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

\(\Big \downarrow \) 6452

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

\(\Big \downarrow \) 6542

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

\(\Big \downarrow \) 6452

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

\(\Big \downarrow \) 262

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 6546

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

\(\Big \downarrow \) 6470

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

\(\Big \downarrow \) 2849

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

\(\Big \downarrow \) 2752

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 262

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

rule 2752

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLo 
g[2, 1 - c*x], x] /; FreeQ[{c, d, e}, x] && EqQ[e + c*d, 0]
 

rule 2849

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Simp 
[-e/g   Subst[Int[Log[2*d*x]/(1 - 2*d*x), x], x, 1/(d + e*x)], x] /; FreeQ[ 
{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]
 

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 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]
 

Maple [A] (verified)

Time = 0.71 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.55

method	result	size
parts	\(\frac {a^{2} x^{3}}{3}+\frac {b^{2} \left (\frac {c^{3} x^{3} \operatorname {arctanh}\left (c x \right )^{2}}{3}+\frac {c^{2} x^{2} \operatorname {arctanh}\left (c x \right )}{3}+\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )}{3}+\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )}{3}+\frac {c x}{3}+\frac {\ln \left (c x -1\right )}{6}-\frac {\ln \left (c x +1\right )}{6}+\frac {\ln \left (c x -1\right )^{2}}{12}-\frac {\operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{3}-\frac {\ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{6}-\frac {\ln \left (c x +1\right )^{2}}{12}+\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{6}\right )}{c^{3}}+\frac {2 a b \left (\frac {c^{3} x^{3} \operatorname {arctanh}\left (c x \right )}{3}+\frac {c^{2} x^{2}}{6}+\frac {\ln \left (c x -1\right )}{6}+\frac {\ln \left (c x +1\right )}{6}\right )}{c^{3}}\)	\(201\)
derivativedivides	\(\frac {\frac {a^{2} c^{3} x^{3}}{3}+b^{2} \left (\frac {c^{3} x^{3} \operatorname {arctanh}\left (c x \right )^{2}}{3}+\frac {c^{2} x^{2} \operatorname {arctanh}\left (c x \right )}{3}+\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )}{3}+\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )}{3}+\frac {c x}{3}+\frac {\ln \left (c x -1\right )}{6}-\frac {\ln \left (c x +1\right )}{6}+\frac {\ln \left (c x -1\right )^{2}}{12}-\frac {\operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{3}-\frac {\ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{6}-\frac {\ln \left (c x +1\right )^{2}}{12}+\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{6}\right )+2 a b \left (\frac {c^{3} x^{3} \operatorname {arctanh}\left (c x \right )}{3}+\frac {c^{2} x^{2}}{6}+\frac {\ln \left (c x -1\right )}{6}+\frac {\ln \left (c x +1\right )}{6}\right )}{c^{3}}\)	\(202\)
default	\(\frac {\frac {a^{2} c^{3} x^{3}}{3}+b^{2} \left (\frac {c^{3} x^{3} \operatorname {arctanh}\left (c x \right )^{2}}{3}+\frac {c^{2} x^{2} \operatorname {arctanh}\left (c x \right )}{3}+\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )}{3}+\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )}{3}+\frac {c x}{3}+\frac {\ln \left (c x -1\right )}{6}-\frac {\ln \left (c x +1\right )}{6}+\frac {\ln \left (c x -1\right )^{2}}{12}-\frac {\operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{3}-\frac {\ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{6}-\frac {\ln \left (c x +1\right )^{2}}{12}+\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{6}\right )+2 a b \left (\frac {c^{3} x^{3} \operatorname {arctanh}\left (c x \right )}{3}+\frac {c^{2} x^{2}}{6}+\frac {\ln \left (c x -1\right )}{6}+\frac {\ln \left (c x +1\right )}{6}\right )}{c^{3}}\)	\(202\)
risch	\(-\frac {b^{2} \ln \left (-c x +1\right )^{2}}{12 c^{3}}+\frac {b^{2} \ln \left (c x +1\right ) x^{2}}{6 c}+\frac {11 b^{2} \ln \left (-c x +1\right )}{18 c^{3}}+\frac {b^{2} \ln \left (-c x +1\right )^{2} x^{3}}{12}+\frac {a^{2} x^{3}}{3}+\frac {b a \ln \left (c x +1\right )}{3 c^{3}}-\frac {a b \ln \left (-c x +1\right ) x^{3}}{3}-\frac {b^{2} \ln \left (-c x +1\right ) x^{2}}{6 c}+\frac {a b \ln \left (-c x +1\right )}{3 c^{3}}-\frac {b^{2} \ln \left (-c x +1\right ) \ln \left (c x +1\right ) x^{3}}{6}-\frac {b^{2} \ln \left (-c x +1\right ) \ln \left (c x +1\right )}{6 c^{3}}+\frac {b^{2} \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (c x +1\right )}{3 c^{3}}-\frac {b^{2} \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{3 c^{3}}+\frac {b a \ln \left (c x +1\right ) x^{3}}{3}+\frac {b^{2} \ln \left (c x +1\right )^{2} x^{3}}{12}+\frac {b^{2} \ln \left (c x +1\right )^{2}}{12 c^{3}}-\frac {b^{2} \ln \left (c x +1\right )}{6 c^{3}}-\frac {a^{2}}{3 c^{3}}-\frac {17 b^{2}}{54 c^{3}}-\frac {b^{2} \operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{3 c^{3}}-\frac {4 b^{2} \ln \left (c x -1\right )}{9 c^{3}}-\frac {11 a b}{9 c^{3}}+\frac {a b \,x^{2}}{3 c}+\frac {b^{2} x}{3 c^{2}}\)	\(350\)

Input:

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

Output:

1/3*a^2*x^3+b^2/c^3*(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(1/2*c*x+1/2))*ln(-1/2*c*x+1/ 
2))+2*a*b/c^3*(1/3*c^3*x^3*arctanh(c*x)+1/6*c^2*x^2+1/6*ln(c*x-1)+1/6*ln(c 
*x+1))
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

1/3*a^2*x^3 + 1/3*(2*x^3*arctanh(c*x) + c*(x^2/c^2 + log(c^2*x^2 - 1)/c^4) 
)*a*b - 1/216*(2*c^4*(2*(c^2*x^3 + 3*x)/c^6 - 3*log(c*x + 1)/c^7 + 3*log(c 
*x - 1)/c^7) - 3*c^3*(x^2/c^4 + log(c^2*x^2 - 1)/c^6) - 648*c^3*integrate( 
1/9*x^3*log(c*x + 1)/(c^4*x^2 - c^2), x) + 9*c^2*(2*x/c^4 - log(c*x + 1)/c 
^5 + log(c*x - 1)/c^5) - 324*c*integrate(1/9*x*log(c*x + 1)/(c^4*x^2 - c^2 
), x) - 6*(3*c^3*x^3*log(c*x + 1)^2 + (2*c^3*x^3 - 3*c^2*x^2 + 6*c*x - 6*( 
c^3*x^3 + 1)*log(c*x + 1))*log(-c*x + 1))/c^3 - (2*(c*x - 1)^3*(9*log(-c*x 
 + 1)^2 - 6*log(-c*x + 1) + 2) + 27*(c*x - 1)^2*(2*log(-c*x + 1)^2 - 2*log 
(-c*x + 1) + 1) + 54*(c*x - 1)*(log(-c*x + 1)^2 - 2*log(-c*x + 1) + 2))/c^ 
3 + 18*log(9*c^4*x^2 - 9*c^2)/c^3 - 324*integrate(1/9*log(c*x + 1)/(c^4*x^ 
2 - c^2), x))*b^2
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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