\(\int x^2 (a+b \text {arctanh}(\frac {c}{x}))^2 \, dx\) [144]

Optimal result

Integrand size = 16, antiderivative size = 142 \[ \int x^2 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2 \, dx=\frac {1}{3} b^2 c^2 x-\frac {1}{3} b^2 c^3 \coth ^{-1}\left (\frac {x}{c}\right )+\frac {1}{3} b c x^2 \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )-\frac {1}{3} c^3 \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^2+\frac {1}{3} x^3 \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^2-\frac {2}{3} b c^3 \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right ) \log \left (2-\frac {2}{1+\frac {c}{x}}\right )+\frac {1}{3} b^2 c^3 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+\frac {c}{x}}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.02 \[ \int x^2 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2 \, dx=\frac {1}{3} \left (b^2 c^2 x+a b c x^2+a^2 x^3+b^2 \left (-c^3+x^3\right ) \text {arctanh}\left (\frac {c}{x}\right )^2+b \text {arctanh}\left (\frac {c}{x}\right ) \left (-b c^3+b c x^2+2 a x^3-2 b c^3 \log \left (1-e^{-2 \text {arctanh}\left (\frac {c}{x}\right )}\right )\right )+a b c^3 \log \left (1-\frac {c^2}{x^2}\right )-2 a b c^3 \log \left (\frac {c}{x}\right )+b^2 c^3 \operatorname {PolyLog}\left (2,e^{-2 \text {arctanh}\left (\frac {c}{x}\right )}\right )\right ) \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.88 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.92, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.562, Rules used = {6454, 6452, 6544, 6452, 264, 219, 6550, 6494, 2897}

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.

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

\(\Big \downarrow \) 6454

\(\displaystyle -\int x^4 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2d\frac {1}{x}\)

\(\Big \downarrow \) 6452

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

\(\Big \downarrow \) 6544

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

\(\Big \downarrow \) 6452

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

\(\Big \downarrow \) 264

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 6550

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

\(\Big \downarrow \) 6494

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

\(\Big \downarrow \) 2897

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

Input:

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

Output:

(x^3*(a + b*ArcTanh[c/x])^2)/3 - (2*b*c*(-1/2*(x^2*(a + b*ArcTanh[c/x])) + 
 (b*c*(-x + c*ArcTanh[c/x]))/2 + c^2*((a + b*ArcTanh[c/x])^2/(2*b) + (a + 
b*ArcTanh[c/x])*Log[2 - 2/(1 + c/x)] - (b*PolyLog[2, -1 + 2/(1 + c/x)])/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 264

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^( 
m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c 
^2*(m + 1)))   Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p 
}, x] && LtQ[m, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
 

rule 2897

Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[Pq^m*((1 - u)/ 
D[u, x])]}, Simp[C*PolyLog[2, 1 - u], x] /; FreeQ[C, x]] /; IntegerQ[m] && 
PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponents[u, 
 x][[2]], Expon[Pq, x]]
 

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 6454

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

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 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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(297\) vs. \(2(128)=256\).

Time = 1.48 (sec) , antiderivative size = 298, normalized size of antiderivative = 2.10

method	result	size
parts	\(\frac {a^{2} x^{3}}{3}-b^{2} c^{3} \left (-\frac {x^{3} \operatorname {arctanh}\left (\frac {c}{x}\right )^{2}}{3 c^{3}}-\frac {\operatorname {arctanh}\left (\frac {c}{x}\right ) \ln \left (\frac {c}{x}-1\right )}{3}-\frac {x^{2} \operatorname {arctanh}\left (\frac {c}{x}\right )}{3 c^{2}}+\frac {2 \ln \left (\frac {c}{x}\right ) \operatorname {arctanh}\left (\frac {c}{x}\right )}{3}-\frac {\operatorname {arctanh}\left (\frac {c}{x}\right ) \ln \left (1+\frac {c}{x}\right )}{3}-\frac {\ln \left (\frac {c}{x}-1\right )}{6}-\frac {x}{3 c}+\frac {\ln \left (1+\frac {c}{x}\right )}{6}+\frac {\operatorname {dilog}\left (\frac {c}{2 x}+\frac {1}{2}\right )}{3}+\frac {\ln \left (\frac {c}{x}-1\right ) \ln \left (\frac {c}{2 x}+\frac {1}{2}\right )}{6}-\frac {\ln \left (\frac {c}{x}-1\right )^{2}}{12}+\frac {\ln \left (1+\frac {c}{x}\right )^{2}}{12}-\frac {\left (\ln \left (1+\frac {c}{x}\right )-\ln \left (\frac {c}{2 x}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c}{2 x}+\frac {1}{2}\right )}{6}-\frac {\operatorname {dilog}\left (\frac {c}{x}\right )}{3}-\frac {\operatorname {dilog}\left (1+\frac {c}{x}\right )}{3}-\frac {\ln \left (\frac {c}{x}\right ) \ln \left (1+\frac {c}{x}\right )}{3}\right )-2 a b \,c^{3} \left (-\frac {x^{3} \operatorname {arctanh}\left (\frac {c}{x}\right )}{3 c^{3}}-\frac {\ln \left (\frac {c}{x}-1\right )}{6}-\frac {\ln \left (1+\frac {c}{x}\right )}{6}-\frac {x^{2}}{6 c^{2}}+\frac {\ln \left (\frac {c}{x}\right )}{3}\right )\)	\(298\)
derivativedivides	\(-c^{3} \left (-\frac {a^{2} x^{3}}{3 c^{3}}+b^{2} \left (-\frac {x^{3} \operatorname {arctanh}\left (\frac {c}{x}\right )^{2}}{3 c^{3}}-\frac {\operatorname {arctanh}\left (\frac {c}{x}\right ) \ln \left (\frac {c}{x}-1\right )}{3}-\frac {x^{2} \operatorname {arctanh}\left (\frac {c}{x}\right )}{3 c^{2}}+\frac {2 \ln \left (\frac {c}{x}\right ) \operatorname {arctanh}\left (\frac {c}{x}\right )}{3}-\frac {\operatorname {arctanh}\left (\frac {c}{x}\right ) \ln \left (1+\frac {c}{x}\right )}{3}-\frac {\ln \left (\frac {c}{x}-1\right )}{6}-\frac {x}{3 c}+\frac {\ln \left (1+\frac {c}{x}\right )}{6}+\frac {\operatorname {dilog}\left (\frac {c}{2 x}+\frac {1}{2}\right )}{3}+\frac {\ln \left (\frac {c}{x}-1\right ) \ln \left (\frac {c}{2 x}+\frac {1}{2}\right )}{6}-\frac {\ln \left (\frac {c}{x}-1\right )^{2}}{12}+\frac {\ln \left (1+\frac {c}{x}\right )^{2}}{12}-\frac {\left (\ln \left (1+\frac {c}{x}\right )-\ln \left (\frac {c}{2 x}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c}{2 x}+\frac {1}{2}\right )}{6}-\frac {\operatorname {dilog}\left (\frac {c}{x}\right )}{3}-\frac {\operatorname {dilog}\left (1+\frac {c}{x}\right )}{3}-\frac {\ln \left (\frac {c}{x}\right ) \ln \left (1+\frac {c}{x}\right )}{3}\right )+2 a b \left (-\frac {x^{3} \operatorname {arctanh}\left (\frac {c}{x}\right )}{3 c^{3}}-\frac {\ln \left (\frac {c}{x}-1\right )}{6}-\frac {\ln \left (1+\frac {c}{x}\right )}{6}-\frac {x^{2}}{6 c^{2}}+\frac {\ln \left (\frac {c}{x}\right )}{3}\right )\right )\)	\(299\)
default	\(-c^{3} \left (-\frac {a^{2} x^{3}}{3 c^{3}}+b^{2} \left (-\frac {x^{3} \operatorname {arctanh}\left (\frac {c}{x}\right )^{2}}{3 c^{3}}-\frac {\operatorname {arctanh}\left (\frac {c}{x}\right ) \ln \left (\frac {c}{x}-1\right )}{3}-\frac {x^{2} \operatorname {arctanh}\left (\frac {c}{x}\right )}{3 c^{2}}+\frac {2 \ln \left (\frac {c}{x}\right ) \operatorname {arctanh}\left (\frac {c}{x}\right )}{3}-\frac {\operatorname {arctanh}\left (\frac {c}{x}\right ) \ln \left (1+\frac {c}{x}\right )}{3}-\frac {\ln \left (\frac {c}{x}-1\right )}{6}-\frac {x}{3 c}+\frac {\ln \left (1+\frac {c}{x}\right )}{6}+\frac {\operatorname {dilog}\left (\frac {c}{2 x}+\frac {1}{2}\right )}{3}+\frac {\ln \left (\frac {c}{x}-1\right ) \ln \left (\frac {c}{2 x}+\frac {1}{2}\right )}{6}-\frac {\ln \left (\frac {c}{x}-1\right )^{2}}{12}+\frac {\ln \left (1+\frac {c}{x}\right )^{2}}{12}-\frac {\left (\ln \left (1+\frac {c}{x}\right )-\ln \left (\frac {c}{2 x}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c}{2 x}+\frac {1}{2}\right )}{6}-\frac {\operatorname {dilog}\left (\frac {c}{x}\right )}{3}-\frac {\operatorname {dilog}\left (1+\frac {c}{x}\right )}{3}-\frac {\ln \left (\frac {c}{x}\right ) \ln \left (1+\frac {c}{x}\right )}{3}\right )+2 a b \left (-\frac {x^{3} \operatorname {arctanh}\left (\frac {c}{x}\right )}{3 c^{3}}-\frac {\ln \left (\frac {c}{x}-1\right )}{6}-\frac {\ln \left (1+\frac {c}{x}\right )}{6}-\frac {x^{2}}{6 c^{2}}+\frac {\ln \left (\frac {c}{x}\right )}{3}\right )\right )\)	\(299\)
risch	\(\text {Expression too large to display}\)	\(6706\)

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*arctanh(c/x)*ln(c/x-1 
)-1/3/c^2*x^2*arctanh(c/x)+2/3*ln(c/x)*arctanh(c/x)-1/3*arctanh(c/x)*ln(1+ 
c/x)-1/6*ln(c/x-1)-1/3*x/c+1/6*ln(1+c/x)+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+1/12*ln(1+c/x)^2-1/6*(ln(1+c/x)-ln(1/ 
2*c/x+1/2))*ln(-1/2*c/x+1/2)-1/3*dilog(c/x)-1/3*dilog(1+c/x)-1/3*ln(c/x)*l 
n(1+c/x))-2*a*b*c^3*(-1/3/c^3*x^3*arctanh(c/x)-1/6*ln(c/x-1)-1/6*ln(1+c/x) 
-1/6/c^2*x^2+1/3*ln(c/x))
 

Fricas [F]

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

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

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

Output:

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

Maxima [F]

\[ \int x^2 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2 \, dx=\int { {\left (b \operatorname {artanh}\left (\frac {c}{x}\right ) + a\right )}^{2} x^{2} \,d x } \] 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^2*log(-c^2 + x^2) + x^2)*c)*a*b 
 + 1/12*(6*c^4*integrate(-1/3*log(c + x)/(c^2 - x^2), x) + x^3*log(c + x)^ 
2 + 6*c^3*integrate(-1/3*x*log(c + x)/(c^2 - x^2), x) - (c*log(c + x) - c* 
log(-c + x) - 2*x)*c^2 - (c^3 - x^3)*log(-c + x)^2 + (c^2*log(-c^2 + x^2) 
+ x^2)*c + 12*c*integrate(-1/3*x^3*log(c + x)/(c^2 - x^2), x) - 2*(c*x^2 + 
 (c^3 + x^3)*log(c + x))*log(-c + x))*b^2
 

Giac [F]

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

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

Output:

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

Reduce [F]

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

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

Output:

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