\(\int \frac {x^2 (a+b \text {arctanh}(c x))^2}{d+c d x} \, dx\) [96]

Optimal result

Integrand size = 22, antiderivative size = 247 \[ \int \frac {x^2 (a+b \text {arctanh}(c x))^2}{d+c d x} \, dx=\frac {a b x}{c^2 d}+\frac {b^2 x \text {arctanh}(c x)}{c^2 d}-\frac {3 (a+b \text {arctanh}(c x))^2}{2 c^3 d}-\frac {x (a+b \text {arctanh}(c x))^2}{c^2 d}+\frac {x^2 (a+b \text {arctanh}(c x))^2}{2 c d}+\frac {2 b (a+b \text {arctanh}(c x)) \log \left (\frac {2}{1-c x}\right )}{c^3 d}-\frac {(a+b \text {arctanh}(c x))^2 \log \left (\frac {2}{1+c x}\right )}{c^3 d}+\frac {b^2 \log \left (1-c^2 x^2\right )}{2 c^3 d}+\frac {b^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{c^3 d}+\frac {b (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+c x}\right )}{c^3 d}+\frac {b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1+c x}\right )}{2 c^3 d} \] Output:

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

Mathematica [A] (verified)

Time = 0.47 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.05 \[ \int \frac {x^2 (a+b \text {arctanh}(c x))^2}{d+c d x} \, dx=\frac {-2 a^2 c x+2 a b c x+a^2 c^2 x^2-2 a b \text {arctanh}(c x)-4 a b c x \text {arctanh}(c x)+2 b^2 c x \text {arctanh}(c x)+2 a b c^2 x^2 \text {arctanh}(c x)+b^2 \text {arctanh}(c x)^2-2 b^2 c x \text {arctanh}(c x)^2+b^2 c^2 x^2 \text {arctanh}(c x)^2-4 a b \text {arctanh}(c x) \log \left (1+e^{-2 \text {arctanh}(c x)}\right )+4 b^2 \text {arctanh}(c x) \log \left (1+e^{-2 \text {arctanh}(c x)}\right )-2 b^2 \text {arctanh}(c x)^2 \log \left (1+e^{-2 \text {arctanh}(c x)}\right )+2 a^2 \log (1+c x)-2 a b \log \left (1-c^2 x^2\right )+b^2 \log \left (1-c^2 x^2\right )+2 b (a-b+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )+b^2 \operatorname {PolyLog}\left (3,-e^{-2 \text {arctanh}(c x)}\right )}{2 c^3 d} \] Input:

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

Output:

(-2*a^2*c*x + 2*a*b*c*x + a^2*c^2*x^2 - 2*a*b*ArcTanh[c*x] - 4*a*b*c*x*Arc 
Tanh[c*x] + 2*b^2*c*x*ArcTanh[c*x] + 2*a*b*c^2*x^2*ArcTanh[c*x] + b^2*ArcT 
anh[c*x]^2 - 2*b^2*c*x*ArcTanh[c*x]^2 + b^2*c^2*x^2*ArcTanh[c*x]^2 - 4*a*b 
*ArcTanh[c*x]*Log[1 + E^(-2*ArcTanh[c*x])] + 4*b^2*ArcTanh[c*x]*Log[1 + E^ 
(-2*ArcTanh[c*x])] - 2*b^2*ArcTanh[c*x]^2*Log[1 + E^(-2*ArcTanh[c*x])] + 2 
*a^2*Log[1 + c*x] - 2*a*b*Log[1 - c^2*x^2] + b^2*Log[1 - c^2*x^2] + 2*b*(a 
 - b + b*ArcTanh[c*x])*PolyLog[2, -E^(-2*ArcTanh[c*x])] + b^2*PolyLog[3, - 
E^(-2*ArcTanh[c*x])])/(2*c^3*d)
 

Rubi [A] (verified)

Time = 3.59 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.10, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.682, Rules used = {6492, 27, 6452, 6492, 6436, 6470, 6542, 2009, 6510, 6546, 6470, 2849, 2752, 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.

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

\(\Big \downarrow \) 6492

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6452

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

\(\Big \downarrow \) 6492

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

\(\Big \downarrow \) 6436

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

\(\Big \downarrow \) 6470

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

\(\Big \downarrow \) 6542

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 6510

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

\(\Big \downarrow \) 6546

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

\(\Big \downarrow \) 6470

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

\(\Big \downarrow \) 2849

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

\(\Big \downarrow \) 2752

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

\(\Big \downarrow \) 6618

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

\(\Big \downarrow \) 7164

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2009

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

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 6436

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

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 6492

Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.))/((d_) + 
(e_.)*(x_)), x_Symbol] :> Simp[f/e   Int[(f*x)^(m - 1)*(a + b*ArcTanh[c*x]) 
^p, x], x] - Simp[d*(f/e)   Int[(f*x)^(m - 1)*((a + b*ArcTanh[c*x])^p/(d + 
e*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 
- e^2, 0] && GtQ[m, 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 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 = 3.58 (sec) , antiderivative size = 905, normalized size of antiderivative = 3.66

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(905\)
default	\(\text {Expression too large to display}\)	\(905\)
parts	\(\text {Expression too large to display}\)	\(913\)

Input:

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

Output:

1/c^3*(a^2/d*(1/2*c^2*x^2-c*x+ln(c*x+1))+b^2/d*(1/2*arctanh(c*x)^2*c^2*x^2 
-arctanh(c*x)^2*c*x+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))-2*arctanh(c*x)^2 
*ln((c*x+1)/(-c^2*x^2+1)^(1/2))+2/3*arctanh(c*x)^3-ln(2)*arctanh(c*x)^2+2* 
dilog(1+I*(c*x+1)/(-c^2*x^2+1)^(1/2))+2*dilog(1-I*(c*x+1)/(-c^2*x^2+1)^(1/ 
2))-3/2*arctanh(c*x)^2-1/2*I*Pi*csgn(I*(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))^2*csgn(I*(c*x+1)^2/(c^2*x 
^2-1))*arctanh(c*x)^2+1/2*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)))*csgn(I/(1-(c*x+1)^2/(c^2*x^2-1) 
))*arctanh(c*x)^2+(c*x+1)*arctanh(c*x)-ln(1+(c*x+1)^2/(-c^2*x^2+1))+2*arct 
anh(c*x)*ln(1+I*(c*x+1)/(-c^2*x^2+1)^(1/2))+2*arctanh(c*x)*ln(1-I*(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-(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)^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*(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*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)+2/d*a*b*(1/2*arctanh(c*x)* 
c^2*x^2-arctanh(c*x)*c*x+arctanh(c*x)*ln(c*x+1)-1/4*ln(c*x+1)^2+1/2*(ln(c* 
x+1)-ln(1/2*c*x+1/2))*ln(-1/2*c*x+1/2)-1/2*dilog(1/2*c*x+1/2)+1/2*c*x+1/2- 
3/4*ln(c*x+1)-1/4*ln(c*x-1)))
 

Fricas [F]

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

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

Sympy [F]

\[ \int \frac {x^2 (a+b \text {arctanh}(c x))^2}{d+c d x} \, dx=\frac {\int \frac {a^{2} x^{2}}{c x + 1}\, dx + \int \frac {b^{2} x^{2} \operatorname {atanh}^{2}{\left (c x \right )}}{c x + 1}\, dx + \int \frac {2 a b x^{2} \operatorname {atanh}{\left (c x \right )}}{c x + 1}\, dx}{d} \] Input:

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

Output:

(Integral(a**2*x**2/(c*x + 1), x) + Integral(b**2*x**2*atanh(c*x)**2/(c*x 
+ 1), x) + Integral(2*a*b*x**2*atanh(c*x)/(c*x + 1), x))/d
 

Maxima [F]

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

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

Output:

1/2*a^2*((c*x^2 - 2*x)/(c^2*d) + 2*log(c*x + 1)/(c^3*d)) + 1/8*(b^2*c^2*x^ 
2 - 2*b^2*c*x + 2*b^2*log(c*x + 1))*log(-c*x + 1)^2/(c^3*d) - integrate(-1 
/4*((b^2*c^3*x^3 - b^2*c^2*x^2)*log(c*x + 1)^2 + 4*(a*b*c^3*x^3 - a*b*c^2* 
x^2)*log(c*x + 1) + (2*b^2*c*x - (4*a*b*c^3 + b^2*c^3)*x^3 + (4*a*b*c^2 + 
b^2*c^2)*x^2 - 2*(b^2*c^3*x^3 - b^2*c^2*x^2 + b^2*c*x + b^2)*log(c*x + 1)) 
*log(-c*x + 1))/(c^4*d*x^2 - c^2*d), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {x^2 (a+b \text {arctanh}(c x))^2}{d+c d x} \, dx=\frac {4 \left (\int \frac {\mathit {atanh} \left (c x \right ) x^{2}}{c x +1}d x \right ) a b \,c^{3}+2 \left (\int \frac {\mathit {atanh} \left (c x \right )^{2} x^{2}}{c x +1}d x \right ) b^{2} c^{3}+2 \,\mathrm {log}\left (c x +1\right ) a^{2}+a^{2} c^{2} x^{2}-2 a^{2} c x}{2 c^{3} d} \] Input:

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

Output:

(4*int((atanh(c*x)*x**2)/(c*x + 1),x)*a*b*c**3 + 2*int((atanh(c*x)**2*x**2 
)/(c*x + 1),x)*b**2*c**3 + 2*log(c*x + 1)*a**2 + a**2*c**2*x**2 - 2*a**2*c 
*x)/(2*c**3*d)