\(\int x^2 (d+c d x)^2 (a+b \text {arctanh}(c x))^2 \, dx\) [77]

Optimal result

Integrand size = 22, antiderivative size = 312 \[ \int x^2 (d+c d x)^2 (a+b \text {arctanh}(c x))^2 \, dx=\frac {a b d^2 x}{c^2}+\frac {19 b^2 d^2 x}{30 c^2}+\frac {b^2 d^2 x^2}{6 c}+\frac {1}{30} b^2 d^2 x^3-\frac {19 b^2 d^2 \text {arctanh}(c x)}{30 c^3}+\frac {b^2 d^2 x \text {arctanh}(c x)}{c^2}+\frac {8 b d^2 x^2 (a+b \text {arctanh}(c x))}{15 c}+\frac {1}{3} b d^2 x^3 (a+b \text {arctanh}(c x))+\frac {1}{10} b c d^2 x^4 (a+b \text {arctanh}(c x))+\frac {d^2 (a+b \text {arctanh}(c x))^2}{30 c^3}+\frac {1}{3} d^2 x^3 (a+b \text {arctanh}(c x))^2+\frac {1}{2} c d^2 x^4 (a+b \text {arctanh}(c x))^2+\frac {1}{5} c^2 d^2 x^5 (a+b \text {arctanh}(c x))^2-\frac {16 b d^2 (a+b \text {arctanh}(c x)) \log \left (\frac {2}{1-c x}\right )}{15 c^3}+\frac {2 b^2 d^2 \log \left (1-c^2 x^2\right )}{3 c^3}-\frac {8 b^2 d^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{15 c^3} \] Output:

a*b*d^2*x/c^2+19/30*b^2*d^2*x/c^2+1/6*b^2*d^2*x^2/c+1/30*b^2*d^2*x^3-19/30 
*b^2*d^2*arctanh(c*x)/c^3+b^2*d^2*x*arctanh(c*x)/c^2+8/15*b*d^2*x^2*(a+b*a 
rctanh(c*x))/c+1/3*b*d^2*x^3*(a+b*arctanh(c*x))+1/10*b*c*d^2*x^4*(a+b*arct 
anh(c*x))+1/30*d^2*(a+b*arctanh(c*x))^2/c^3+1/3*d^2*x^3*(a+b*arctanh(c*x)) 
^2+1/2*c*d^2*x^4*(a+b*arctanh(c*x))^2+1/5*c^2*d^2*x^5*(a+b*arctanh(c*x))^2 
-16/15*b*d^2*(a+b*arctanh(c*x))*ln(2/(-c*x+1))/c^3+2/3*b^2*d^2*ln(-c^2*x^2 
+1)/c^3-8/15*b^2*d^2*polylog(2,1-2/(-c*x+1))/c^3
 

Mathematica [A] (verified)

Time = 0.64 (sec) , antiderivative size = 297, normalized size of antiderivative = 0.95 \[ \int x^2 (d+c d x)^2 (a+b \text {arctanh}(c x))^2 \, dx=\frac {d^2 \left (-9 a b-5 b^2+30 a b c x+19 b^2 c x+16 a b c^2 x^2+5 b^2 c^2 x^2+10 a^2 c^3 x^3+10 a b c^3 x^3+b^2 c^3 x^3+15 a^2 c^4 x^4+3 a b c^4 x^4+6 a^2 c^5 x^5+b^2 \left (-31+10 c^3 x^3+15 c^4 x^4+6 c^5 x^5\right ) \text {arctanh}(c x)^2+b \text {arctanh}(c x) \left (2 a c^3 x^3 \left (10+15 c x+6 c^2 x^2\right )+b \left (-19+30 c x+16 c^2 x^2+10 c^3 x^3+3 c^4 x^4\right )-32 b \log \left (1+e^{-2 \text {arctanh}(c x)}\right )\right )+15 a b \log (1-c x)-15 a b \log (1+c x)+20 b^2 \log \left (1-c^2 x^2\right )+16 a b \log \left (-1+c^2 x^2\right )+16 b^2 \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )\right )}{30 c^3} \] Input:

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

Output:

(d^2*(-9*a*b - 5*b^2 + 30*a*b*c*x + 19*b^2*c*x + 16*a*b*c^2*x^2 + 5*b^2*c^ 
2*x^2 + 10*a^2*c^3*x^3 + 10*a*b*c^3*x^3 + b^2*c^3*x^3 + 15*a^2*c^4*x^4 + 3 
*a*b*c^4*x^4 + 6*a^2*c^5*x^5 + b^2*(-31 + 10*c^3*x^3 + 15*c^4*x^4 + 6*c^5* 
x^5)*ArcTanh[c*x]^2 + b*ArcTanh[c*x]*(2*a*c^3*x^3*(10 + 15*c*x + 6*c^2*x^2 
) + b*(-19 + 30*c*x + 16*c^2*x^2 + 10*c^3*x^3 + 3*c^4*x^4) - 32*b*Log[1 + 
E^(-2*ArcTanh[c*x])]) + 15*a*b*Log[1 - c*x] - 15*a*b*Log[1 + c*x] + 20*b^2 
*Log[1 - c^2*x^2] + 16*a*b*Log[-1 + c^2*x^2] + 16*b^2*PolyLog[2, -E^(-2*Ar 
cTanh[c*x])]))/(30*c^3)
 

Rubi [A] (verified)

Time = 1.39 (sec) , antiderivative size = 312, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {6502, 2009}

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.

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

\(\Big \downarrow \) 6502

\(\displaystyle \int \left (c^2 d^2 x^4 (a+b \text {arctanh}(c x))^2+2 c d^2 x^3 (a+b \text {arctanh}(c x))^2+d^2 x^2 (a+b \text {arctanh}(c x))^2\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {d^2 (a+b \text {arctanh}(c x))^2}{30 c^3}-\frac {16 b d^2 \log \left (\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))}{15 c^3}+\frac {1}{5} c^2 d^2 x^5 (a+b \text {arctanh}(c x))^2+\frac {1}{2} c d^2 x^4 (a+b \text {arctanh}(c x))^2+\frac {1}{10} b c d^2 x^4 (a+b \text {arctanh}(c x))+\frac {1}{3} d^2 x^3 (a+b \text {arctanh}(c x))^2+\frac {1}{3} b d^2 x^3 (a+b \text {arctanh}(c x))+\frac {8 b d^2 x^2 (a+b \text {arctanh}(c x))}{15 c}+\frac {a b d^2 x}{c^2}-\frac {19 b^2 d^2 \text {arctanh}(c x)}{30 c^3}+\frac {b^2 d^2 x \text {arctanh}(c x)}{c^2}-\frac {8 b^2 d^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{15 c^3}+\frac {19 b^2 d^2 x}{30 c^2}+\frac {2 b^2 d^2 \log \left (1-c^2 x^2\right )}{3 c^3}+\frac {b^2 d^2 x^2}{6 c}+\frac {1}{30} b^2 d^2 x^3\)

Input:

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

Output:

(a*b*d^2*x)/c^2 + (19*b^2*d^2*x)/(30*c^2) + (b^2*d^2*x^2)/(6*c) + (b^2*d^2 
*x^3)/30 - (19*b^2*d^2*ArcTanh[c*x])/(30*c^3) + (b^2*d^2*x*ArcTanh[c*x])/c 
^2 + (8*b*d^2*x^2*(a + b*ArcTanh[c*x]))/(15*c) + (b*d^2*x^3*(a + b*ArcTanh 
[c*x]))/3 + (b*c*d^2*x^4*(a + b*ArcTanh[c*x]))/10 + (d^2*(a + b*ArcTanh[c* 
x])^2)/(30*c^3) + (d^2*x^3*(a + b*ArcTanh[c*x])^2)/3 + (c*d^2*x^4*(a + b*A 
rcTanh[c*x])^2)/2 + (c^2*d^2*x^5*(a + b*ArcTanh[c*x])^2)/5 - (16*b*d^2*(a 
+ b*ArcTanh[c*x])*Log[2/(1 - c*x)])/(15*c^3) + (2*b^2*d^2*Log[1 - c^2*x^2] 
)/(3*c^3) - (8*b^2*d^2*PolyLog[2, 1 - 2/(1 - c*x)])/(15*c^3)
 

Defintions of rubi rules used

rule 2009

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

rule 6502

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e 
_.)*(x_))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*ArcTanh[c*x])^p, ( 
f*x)^m*(d + e*x)^q, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[p, 0] 
 && IntegerQ[q] && (GtQ[q, 0] || NeQ[a, 0] || IntegerQ[m])
 

Maple [A] (verified)

Time = 0.67 (sec) , antiderivative size = 345, normalized size of antiderivative = 1.11

method	result	size
parts	\(d^{2} a^{2} \left (\frac {1}{5} c^{2} x^{5}+\frac {1}{2} c \,x^{4}+\frac {1}{3} x^{3}\right )+\frac {d^{2} b^{2} \left (\frac {\operatorname {arctanh}\left (c x \right )^{2} c^{5} x^{5}}{5}+\frac {\operatorname {arctanh}\left (c x \right )^{2} c^{4} x^{4}}{2}+\frac {\operatorname {arctanh}\left (c x \right )^{2} c^{3} x^{3}}{3}+\frac {\operatorname {arctanh}\left (c x \right ) c^{4} x^{4}}{10}+\frac {\operatorname {arctanh}\left (c x \right ) c^{3} x^{3}}{3}+\frac {8 \,\operatorname {arctanh}\left (c x \right ) c^{2} x^{2}}{15}+\operatorname {arctanh}\left (c x \right ) c x +\frac {31 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )}{30}+\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )}{30}-\frac {\ln \left (c x +1\right )^{2}}{120}+\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{60}-\frac {8 \operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{15}+\frac {x^{3} c^{3}}{30}+\frac {c^{2} x^{2}}{6}+\frac {19 c x}{30}+\frac {59 \ln \left (c x -1\right )}{60}+\frac {7 \ln \left (c x +1\right )}{20}+\frac {31 \ln \left (c x -1\right )^{2}}{120}-\frac {31 \ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{60}\right )}{c^{3}}+\frac {2 d^{2} a b \left (\frac {\operatorname {arctanh}\left (c x \right ) c^{5} x^{5}}{5}+\frac {\operatorname {arctanh}\left (c x \right ) c^{4} x^{4}}{2}+\frac {\operatorname {arctanh}\left (c x \right ) c^{3} x^{3}}{3}+\frac {c^{4} x^{4}}{20}+\frac {x^{3} c^{3}}{6}+\frac {4 c^{2} x^{2}}{15}+\frac {c x}{2}+\frac {31 \ln \left (c x -1\right )}{60}+\frac {\ln \left (c x +1\right )}{60}\right )}{c^{3}}\)	\(345\)
derivativedivides	\(\frac {d^{2} a^{2} \left (\frac {1}{5} c^{5} x^{5}+\frac {1}{2} c^{4} x^{4}+\frac {1}{3} x^{3} c^{3}\right )+d^{2} b^{2} \left (\frac {\operatorname {arctanh}\left (c x \right )^{2} c^{5} x^{5}}{5}+\frac {\operatorname {arctanh}\left (c x \right )^{2} c^{4} x^{4}}{2}+\frac {\operatorname {arctanh}\left (c x \right )^{2} c^{3} x^{3}}{3}+\frac {\operatorname {arctanh}\left (c x \right ) c^{4} x^{4}}{10}+\frac {\operatorname {arctanh}\left (c x \right ) c^{3} x^{3}}{3}+\frac {8 \,\operatorname {arctanh}\left (c x \right ) c^{2} x^{2}}{15}+\operatorname {arctanh}\left (c x \right ) c x +\frac {31 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )}{30}+\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )}{30}-\frac {\ln \left (c x +1\right )^{2}}{120}+\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{60}-\frac {8 \operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{15}+\frac {x^{3} c^{3}}{30}+\frac {c^{2} x^{2}}{6}+\frac {19 c x}{30}+\frac {59 \ln \left (c x -1\right )}{60}+\frac {7 \ln \left (c x +1\right )}{20}+\frac {31 \ln \left (c x -1\right )^{2}}{120}-\frac {31 \ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{60}\right )+2 d^{2} a b \left (\frac {\operatorname {arctanh}\left (c x \right ) c^{5} x^{5}}{5}+\frac {\operatorname {arctanh}\left (c x \right ) c^{4} x^{4}}{2}+\frac {\operatorname {arctanh}\left (c x \right ) c^{3} x^{3}}{3}+\frac {c^{4} x^{4}}{20}+\frac {x^{3} c^{3}}{6}+\frac {4 c^{2} x^{2}}{15}+\frac {c x}{2}+\frac {31 \ln \left (c x -1\right )}{60}+\frac {\ln \left (c x +1\right )}{60}\right )}{c^{3}}\)	\(348\)
default	\(\frac {d^{2} a^{2} \left (\frac {1}{5} c^{5} x^{5}+\frac {1}{2} c^{4} x^{4}+\frac {1}{3} x^{3} c^{3}\right )+d^{2} b^{2} \left (\frac {\operatorname {arctanh}\left (c x \right )^{2} c^{5} x^{5}}{5}+\frac {\operatorname {arctanh}\left (c x \right )^{2} c^{4} x^{4}}{2}+\frac {\operatorname {arctanh}\left (c x \right )^{2} c^{3} x^{3}}{3}+\frac {\operatorname {arctanh}\left (c x \right ) c^{4} x^{4}}{10}+\frac {\operatorname {arctanh}\left (c x \right ) c^{3} x^{3}}{3}+\frac {8 \,\operatorname {arctanh}\left (c x \right ) c^{2} x^{2}}{15}+\operatorname {arctanh}\left (c x \right ) c x +\frac {31 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )}{30}+\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )}{30}-\frac {\ln \left (c x +1\right )^{2}}{120}+\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{60}-\frac {8 \operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{15}+\frac {x^{3} c^{3}}{30}+\frac {c^{2} x^{2}}{6}+\frac {19 c x}{30}+\frac {59 \ln \left (c x -1\right )}{60}+\frac {7 \ln \left (c x +1\right )}{20}+\frac {31 \ln \left (c x -1\right )^{2}}{120}-\frac {31 \ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{60}\right )+2 d^{2} a b \left (\frac {\operatorname {arctanh}\left (c x \right ) c^{5} x^{5}}{5}+\frac {\operatorname {arctanh}\left (c x \right ) c^{4} x^{4}}{2}+\frac {\operatorname {arctanh}\left (c x \right ) c^{3} x^{3}}{3}+\frac {c^{4} x^{4}}{20}+\frac {x^{3} c^{3}}{6}+\frac {4 c^{2} x^{2}}{15}+\frac {c x}{2}+\frac {31 \ln \left (c x -1\right )}{60}+\frac {\ln \left (c x +1\right )}{60}\right )}{c^{3}}\)	\(348\)
risch	\(-\frac {d^{2} c^{2} a b \ln \left (-c x +1\right ) x^{5}}{5}-\frac {d^{2} c a b \ln \left (-c x +1\right ) x^{4}}{2}-\frac {d^{2} b^{2} \ln \left (-c x +1\right ) x^{3}}{6}-\frac {31 d^{2} b^{2} \ln \left (-c x +1\right )^{2}}{120 c^{3}}+\frac {d^{2} b^{2} \ln \left (-c x +1\right )^{2} x^{3}}{12}+\frac {d^{2} c^{2} x^{5} a^{2}}{5}+\frac {d^{2} c \,x^{4} a^{2}}{2}+\frac {d^{2} b \,x^{3} a}{3}+\frac {7 d^{2} b^{2} \ln \left (-c x -1\right )}{20 c^{3}}+\frac {8 d^{2} b^{2} \operatorname {dilog}\left (-\frac {c x}{2}+\frac {1}{2}\right )}{15 c^{3}}-\frac {5 d^{2} b^{2}}{6 c^{3}}+\frac {59 d^{2} b^{2} \ln \left (-c x +1\right )}{60 c^{3}}-\frac {31 d^{2} a^{2}}{30 c^{3}}+\frac {d^{2} x^{3} a^{2}}{3}+\frac {d^{2} b \ln \left (-c x -1\right ) a}{30 c^{3}}+\left (-\frac {d^{2} b^{2} x^{3} \left (6 c^{2} x^{2}+15 c x +10\right ) \ln \left (-c x +1\right )}{60}+\frac {d^{2} b \left (12 a \,c^{5} x^{5}+30 a \,c^{4} x^{4}+3 b \,c^{4} x^{4}+20 a \,c^{3} x^{3}+10 b \,c^{3} x^{3}+16 b \,c^{2} x^{2}+30 b c x +31 b \ln \left (-c x +1\right )\right )}{60 c^{3}}\right ) \ln \left (c x +1\right )-\frac {59 d^{2} b a}{30 c^{3}}-\frac {8 d^{2} b^{2} \ln \left (\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (-c x +1\right )}{15 c^{3}}+\frac {8 d^{2} b^{2} \ln \left (\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{15 c^{3}}+\frac {19 b^{2} d^{2} x}{30 c^{2}}+\frac {b^{2} d^{2} x^{2}}{6 c}+\frac {b^{2} d^{2} x^{3}}{30}+\frac {d^{2} b c a \,x^{4}}{10}+\frac {8 d^{2} b a \,x^{2}}{15 c}-\frac {d^{2} a b \ln \left (-c x +1\right ) x^{3}}{3}+\frac {d^{2} c^{2} b^{2} \ln \left (-c x +1\right )^{2} x^{5}}{20}+\frac {d^{2} c \,b^{2} \ln \left (-c x +1\right )^{2} x^{4}}{8}+\frac {31 d^{2} a b \ln \left (-c x +1\right )}{30 c^{3}}-\frac {d^{2} b^{2} \ln \left (-c x +1\right ) x}{2 c^{2}}-\frac {d^{2} b^{2} c \ln \left (-c x +1\right ) x^{4}}{20}-\frac {4 d^{2} b^{2} \ln \left (-c x +1\right ) x^{2}}{15 c}+\frac {d^{2} b^{2} \left (6 c^{5} x^{5}+15 c^{4} x^{4}+10 x^{3} c^{3}+1\right ) \ln \left (c x +1\right )^{2}}{120 c^{3}}+\frac {a b \,d^{2} x}{c^{2}}\)	\(670\)

Input:

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

Output:

d^2*a^2*(1/5*c^2*x^5+1/2*c*x^4+1/3*x^3)+d^2*b^2/c^3*(1/5*arctanh(c*x)^2*c^ 
5*x^5+1/2*arctanh(c*x)^2*c^4*x^4+1/3*arctanh(c*x)^2*c^3*x^3+1/10*arctanh(c 
*x)*c^4*x^4+1/3*arctanh(c*x)*c^3*x^3+8/15*arctanh(c*x)*c^2*x^2+arctanh(c*x 
)*c*x+31/30*arctanh(c*x)*ln(c*x-1)+1/30*arctanh(c*x)*ln(c*x+1)-1/120*ln(c* 
x+1)^2+1/60*(ln(c*x+1)-ln(1/2*c*x+1/2))*ln(-1/2*c*x+1/2)-8/15*dilog(1/2*c* 
x+1/2)+1/30*x^3*c^3+1/6*c^2*x^2+19/30*c*x+59/60*ln(c*x-1)+7/20*ln(c*x+1)+3 
1/120*ln(c*x-1)^2-31/60*ln(c*x-1)*ln(1/2*c*x+1/2))+2*d^2*a*b/c^3*(1/5*arct 
anh(c*x)*c^5*x^5+1/2*arctanh(c*x)*c^4*x^4+1/3*arctanh(c*x)*c^3*x^3+1/20*c^ 
4*x^4+1/6*x^3*c^3+4/15*c^2*x^2+1/2*c*x+31/60*ln(c*x-1)+1/60*ln(c*x+1))
 

Fricas [F]

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

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

Output:

integral(a^2*c^2*d^2*x^4 + 2*a^2*c*d^2*x^3 + a^2*d^2*x^2 + (b^2*c^2*d^2*x^ 
4 + 2*b^2*c*d^2*x^3 + b^2*d^2*x^2)*arctanh(c*x)^2 + 2*(a*b*c^2*d^2*x^4 + 2 
*a*b*c*d^2*x^3 + a*b*d^2*x^2)*arctanh(c*x), x)
 

Sympy [F]

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

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

Output:

d**2*(Integral(a**2*x**2, x) + Integral(2*a**2*c*x**3, x) + Integral(a**2* 
c**2*x**4, x) + Integral(b**2*x**2*atanh(c*x)**2, x) + Integral(2*a*b*x**2 
*atanh(c*x), x) + Integral(2*b**2*c*x**3*atanh(c*x)**2, x) + Integral(b**2 
*c**2*x**4*atanh(c*x)**2, x) + Integral(4*a*b*c*x**3*atanh(c*x), x) + Inte 
gral(2*a*b*c**2*x**4*atanh(c*x), x))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 604 vs. \(2 (281) = 562\).

Time = 0.29 (sec) , antiderivative size = 604, normalized size of antiderivative = 1.94 \[ \int x^2 (d+c d x)^2 (a+b \text {arctanh}(c x))^2 \, dx=\frac {1}{5} \, a^{2} c^{2} d^{2} x^{5} + \frac {1}{2} \, a^{2} c d^{2} x^{4} + \frac {1}{10} \, {\left (4 \, x^{5} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {c^{2} x^{4} + 2 \, x^{2}}{c^{4}} + \frac {2 \, \log \left (c^{2} x^{2} - 1\right )}{c^{6}}\right )}\right )} a b c^{2} d^{2} + \frac {1}{3} \, a^{2} d^{2} x^{3} + \frac {1}{6} \, {\left (6 \, x^{4} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {2 \, {\left (c^{2} x^{3} + 3 \, x\right )}}{c^{4}} - \frac {3 \, \log \left (c x + 1\right )}{c^{5}} + \frac {3 \, \log \left (c x - 1\right )}{c^{5}}\right )}\right )} a b c d^{2} + \frac {1}{3} \, {\left (2 \, x^{3} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {x^{2}}{c^{2}} + \frac {\log \left (c^{2} x^{2} - 1\right )}{c^{4}}\right )}\right )} a b d^{2} + \frac {8 \, {\left (\log \left (c x + 1\right ) \log \left (-\frac {1}{2} \, c x + \frac {1}{2}\right ) + {\rm Li}_2\left (\frac {1}{2} \, c x + \frac {1}{2}\right )\right )} b^{2} d^{2}}{15 \, c^{3}} + \frac {7 \, b^{2} d^{2} \log \left (c x + 1\right )}{20 \, c^{3}} + \frac {59 \, b^{2} d^{2} \log \left (c x - 1\right )}{60 \, c^{3}} + \frac {4 \, b^{2} c^{3} d^{2} x^{3} + 20 \, b^{2} c^{2} d^{2} x^{2} + 76 \, b^{2} c d^{2} x + {\left (6 \, b^{2} c^{5} d^{2} x^{5} + 15 \, b^{2} c^{4} d^{2} x^{4} + 10 \, b^{2} c^{3} d^{2} x^{3} + b^{2} d^{2}\right )} \log \left (c x + 1\right )^{2} + {\left (6 \, b^{2} c^{5} d^{2} x^{5} + 15 \, b^{2} c^{4} d^{2} x^{4} + 10 \, b^{2} c^{3} d^{2} x^{3} - 31 \, b^{2} d^{2}\right )} \log \left (-c x + 1\right )^{2} + 2 \, {\left (3 \, b^{2} c^{4} d^{2} x^{4} + 10 \, b^{2} c^{3} d^{2} x^{3} + 16 \, b^{2} c^{2} d^{2} x^{2} + 30 \, b^{2} c d^{2} x\right )} \log \left (c x + 1\right ) - 2 \, {\left (3 \, b^{2} c^{4} d^{2} x^{4} + 10 \, b^{2} c^{3} d^{2} x^{3} + 16 \, b^{2} c^{2} d^{2} x^{2} + 30 \, b^{2} c d^{2} x + {\left (6 \, b^{2} c^{5} d^{2} x^{5} + 15 \, b^{2} c^{4} d^{2} x^{4} + 10 \, b^{2} c^{3} d^{2} x^{3} + b^{2} d^{2}\right )} \log \left (c x + 1\right )\right )} \log \left (-c x + 1\right )}{120 \, c^{3}} \] Input:

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

Output:

1/5*a^2*c^2*d^2*x^5 + 1/2*a^2*c*d^2*x^4 + 1/10*(4*x^5*arctanh(c*x) + c*((c 
^2*x^4 + 2*x^2)/c^4 + 2*log(c^2*x^2 - 1)/c^6))*a*b*c^2*d^2 + 1/3*a^2*d^2*x 
^3 + 1/6*(6*x^4*arctanh(c*x) + c*(2*(c^2*x^3 + 3*x)/c^4 - 3*log(c*x + 1)/c 
^5 + 3*log(c*x - 1)/c^5))*a*b*c*d^2 + 1/3*(2*x^3*arctanh(c*x) + c*(x^2/c^2 
 + log(c^2*x^2 - 1)/c^4))*a*b*d^2 + 8/15*(log(c*x + 1)*log(-1/2*c*x + 1/2) 
 + dilog(1/2*c*x + 1/2))*b^2*d^2/c^3 + 7/20*b^2*d^2*log(c*x + 1)/c^3 + 59/ 
60*b^2*d^2*log(c*x - 1)/c^3 + 1/120*(4*b^2*c^3*d^2*x^3 + 20*b^2*c^2*d^2*x^ 
2 + 76*b^2*c*d^2*x + (6*b^2*c^5*d^2*x^5 + 15*b^2*c^4*d^2*x^4 + 10*b^2*c^3* 
d^2*x^3 + b^2*d^2)*log(c*x + 1)^2 + (6*b^2*c^5*d^2*x^5 + 15*b^2*c^4*d^2*x^ 
4 + 10*b^2*c^3*d^2*x^3 - 31*b^2*d^2)*log(-c*x + 1)^2 + 2*(3*b^2*c^4*d^2*x^ 
4 + 10*b^2*c^3*d^2*x^3 + 16*b^2*c^2*d^2*x^2 + 30*b^2*c*d^2*x)*log(c*x + 1) 
 - 2*(3*b^2*c^4*d^2*x^4 + 10*b^2*c^3*d^2*x^3 + 16*b^2*c^2*d^2*x^2 + 30*b^2 
*c*d^2*x + (6*b^2*c^5*d^2*x^5 + 15*b^2*c^4*d^2*x^4 + 10*b^2*c^3*d^2*x^3 + 
b^2*d^2)*log(c*x + 1))*log(-c*x + 1))/c^3
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int x^2 (d+c d x)^2 (a+b \text {arctanh}(c x))^2 \, dx=\frac {d^{2} \left (32 \,\mathrm {log}\left (c^{2} x -c \right ) a b +6 a^{2} c^{5} x^{5}+19 b^{2} c x +2 \mathit {atanh} \left (c x \right ) a b +5 b^{2} c^{2} x^{2}+10 \mathit {atanh} \left (c x \right ) b^{2} c^{3} x^{3}+30 \mathit {atanh} \left (c x \right ) b^{2} c x +10 a b \,c^{3} x^{3}+30 a b c x +b^{2} c^{3} x^{3}+6 \mathit {atanh} \left (c x \right )^{2} b^{2} c^{5} x^{5}-16 \mathit {atanh} \left (c x \right )^{2} b^{2} c x +3 \mathit {atanh} \left (c x \right ) b^{2} c^{4} x^{4}+16 \mathit {atanh} \left (c x \right ) b^{2} c^{2} x^{2}+3 a b \,c^{4} x^{4}+16 a b \,c^{2} x^{2}+15 \mathit {atanh} \left (c x \right )^{2} b^{2} c^{4} x^{4}+10 \mathit {atanh} \left (c x \right )^{2} b^{2} c^{3} x^{3}+10 a^{2} c^{3} x^{3}+30 \mathit {atanh} \left (c x \right ) a b \,c^{4} x^{4}+20 \mathit {atanh} \left (c x \right ) a b \,c^{3} x^{3}-15 \mathit {atanh} \left (c x \right )^{2} b^{2}+21 \mathit {atanh} \left (c x \right ) b^{2}+40 \,\mathrm {log}\left (c^{2} x -c \right ) b^{2}+16 \left (\int \mathit {atanh} \left (c x \right )^{2}d x \right ) b^{2} c +15 a^{2} c^{4} x^{4}+12 \mathit {atanh} \left (c x \right ) a b \,c^{5} x^{5}\right )}{30 c^{3}} \] Input:

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

Output:

(d**2*(6*atanh(c*x)**2*b**2*c**5*x**5 + 15*atanh(c*x)**2*b**2*c**4*x**4 + 
10*atanh(c*x)**2*b**2*c**3*x**3 - 16*atanh(c*x)**2*b**2*c*x - 15*atanh(c*x 
)**2*b**2 + 12*atanh(c*x)*a*b*c**5*x**5 + 30*atanh(c*x)*a*b*c**4*x**4 + 20 
*atanh(c*x)*a*b*c**3*x**3 + 2*atanh(c*x)*a*b + 3*atanh(c*x)*b**2*c**4*x**4 
 + 10*atanh(c*x)*b**2*c**3*x**3 + 16*atanh(c*x)*b**2*c**2*x**2 + 30*atanh( 
c*x)*b**2*c*x + 21*atanh(c*x)*b**2 + 16*int(atanh(c*x)**2,x)*b**2*c + 32*l 
og(c**2*x - c)*a*b + 40*log(c**2*x - c)*b**2 + 6*a**2*c**5*x**5 + 15*a**2* 
c**4*x**4 + 10*a**2*c**3*x**3 + 3*a*b*c**4*x**4 + 10*a*b*c**3*x**3 + 16*a* 
b*c**2*x**2 + 30*a*b*c*x + b**2*c**3*x**3 + 5*b**2*c**2*x**2 + 19*b**2*c*x 
))/(30*c**3)