\(\int x^2 (d+c d x)^3 (a+b \text {arctanh}(c x))^2 \, dx\) [85]

Optimal result

Integrand size = 22, antiderivative size = 377 \[ \int x^2 (d+c d x)^3 (a+b \text {arctanh}(c x))^2 \, dx=\frac {11 a b d^3 x}{6 c^2}+\frac {37 b^2 d^3 x}{30 c^2}+\frac {61 b^2 d^3 x^2}{180 c}+\frac {1}{10} b^2 d^3 x^3+\frac {1}{60} b^2 c d^3 x^4-\frac {37 b^2 d^3 \text {arctanh}(c x)}{30 c^3}+\frac {11 b^2 d^3 x \text {arctanh}(c x)}{6 c^2}+\frac {14 b d^3 x^2 (a+b \text {arctanh}(c x))}{15 c}+\frac {11}{18} b d^3 x^3 (a+b \text {arctanh}(c x))+\frac {3}{10} b c d^3 x^4 (a+b \text {arctanh}(c x))+\frac {1}{15} b c^2 d^3 x^5 (a+b \text {arctanh}(c x))+\frac {d^3 (a+b \text {arctanh}(c x))^2}{60 c^3}+\frac {1}{3} d^3 x^3 (a+b \text {arctanh}(c x))^2+\frac {3}{4} c d^3 x^4 (a+b \text {arctanh}(c x))^2+\frac {3}{5} c^2 d^3 x^5 (a+b \text {arctanh}(c x))^2+\frac {1}{6} c^3 d^3 x^6 (a+b \text {arctanh}(c x))^2-\frac {28 b d^3 (a+b \text {arctanh}(c x)) \log \left (\frac {2}{1-c x}\right )}{15 c^3}+\frac {113 b^2 d^3 \log \left (1-c^2 x^2\right )}{90 c^3}-\frac {14 b^2 d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{15 c^3} \] Output:

11/6*a*b*d^3*x/c^2+37/30*b^2*d^3*x/c^2+61/180*b^2*d^3*x^2/c+1/10*b^2*d^3*x 
^3+1/60*b^2*c*d^3*x^4-37/30*b^2*d^3*arctanh(c*x)/c^3+11/6*b^2*d^3*x*arctan 
h(c*x)/c^2+14/15*b*d^3*x^2*(a+b*arctanh(c*x))/c+11/18*b*d^3*x^3*(a+b*arcta 
nh(c*x))+3/10*b*c*d^3*x^4*(a+b*arctanh(c*x))+1/15*b*c^2*d^3*x^5*(a+b*arcta 
nh(c*x))+1/60*d^3*(a+b*arctanh(c*x))^2/c^3+1/3*d^3*x^3*(a+b*arctanh(c*x))^ 
2+3/4*c*d^3*x^4*(a+b*arctanh(c*x))^2+3/5*c^2*d^3*x^5*(a+b*arctanh(c*x))^2+ 
1/6*c^3*d^3*x^6*(a+b*arctanh(c*x))^2-28/15*b*d^3*(a+b*arctanh(c*x))*ln(2/( 
-c*x+1))/c^3+113/90*b^2*d^3*ln(-c^2*x^2+1)/c^3-14/15*b^2*d^3*polylog(2,1-2 
/(-c*x+1))/c^3
 

Mathematica [A] (verified)

Time = 0.80 (sec) , antiderivative size = 356, normalized size of antiderivative = 0.94 \[ \int x^2 (d+c d x)^3 (a+b \text {arctanh}(c x))^2 \, dx=\frac {d^3 \left (-162 a b-64 b^2+330 a b c x+222 b^2 c x+168 a b c^2 x^2+61 b^2 c^2 x^2+60 a^2 c^3 x^3+110 a b c^3 x^3+18 b^2 c^3 x^3+135 a^2 c^4 x^4+54 a b c^4 x^4+3 b^2 c^4 x^4+108 a^2 c^5 x^5+12 a b c^5 x^5+30 a^2 c^6 x^6+3 b^2 \left (-111+20 c^3 x^3+45 c^4 x^4+36 c^5 x^5+10 c^6 x^6\right ) \text {arctanh}(c x)^2+2 b \text {arctanh}(c x) \left (3 a c^3 x^3 \left (20+45 c x+36 c^2 x^2+10 c^3 x^3\right )+b \left (-111+165 c x+84 c^2 x^2+55 c^3 x^3+27 c^4 x^4+6 c^5 x^5\right )-168 b \log \left (1+e^{-2 \text {arctanh}(c x)}\right )\right )+165 a b \log (1-c x)-165 a b \log (1+c x)+226 b^2 \log \left (1-c^2 x^2\right )+168 a b \log \left (-1+c^2 x^2\right )+168 b^2 \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )\right )}{180 c^3} \] Input:

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

Output:

(d^3*(-162*a*b - 64*b^2 + 330*a*b*c*x + 222*b^2*c*x + 168*a*b*c^2*x^2 + 61 
*b^2*c^2*x^2 + 60*a^2*c^3*x^3 + 110*a*b*c^3*x^3 + 18*b^2*c^3*x^3 + 135*a^2 
*c^4*x^4 + 54*a*b*c^4*x^4 + 3*b^2*c^4*x^4 + 108*a^2*c^5*x^5 + 12*a*b*c^5*x 
^5 + 30*a^2*c^6*x^6 + 3*b^2*(-111 + 20*c^3*x^3 + 45*c^4*x^4 + 36*c^5*x^5 + 
 10*c^6*x^6)*ArcTanh[c*x]^2 + 2*b*ArcTanh[c*x]*(3*a*c^3*x^3*(20 + 45*c*x + 
 36*c^2*x^2 + 10*c^3*x^3) + b*(-111 + 165*c*x + 84*c^2*x^2 + 55*c^3*x^3 + 
27*c^4*x^4 + 6*c^5*x^5) - 168*b*Log[1 + E^(-2*ArcTanh[c*x])]) + 165*a*b*Lo 
g[1 - c*x] - 165*a*b*Log[1 + c*x] + 226*b^2*Log[1 - c^2*x^2] + 168*a*b*Log 
[-1 + c^2*x^2] + 168*b^2*PolyLog[2, -E^(-2*ArcTanh[c*x])]))/(180*c^3)
 

Rubi [A] (verified)

Time = 2.09 (sec) , antiderivative size = 377, 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.

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

\(\Big \downarrow \) 6502

\(\displaystyle \int \left (c^3 d^3 x^5 (a+b \text {arctanh}(c x))^2+3 c^2 d^3 x^4 (a+b \text {arctanh}(c x))^2+3 c d^3 x^3 (a+b \text {arctanh}(c x))^2+d^3 x^2 (a+b \text {arctanh}(c x))^2\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{6} c^3 d^3 x^6 (a+b \text {arctanh}(c x))^2+\frac {d^3 (a+b \text {arctanh}(c x))^2}{60 c^3}-\frac {28 b d^3 \log \left (\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))}{15 c^3}+\frac {3}{5} c^2 d^3 x^5 (a+b \text {arctanh}(c x))^2+\frac {1}{15} b c^2 d^3 x^5 (a+b \text {arctanh}(c x))+\frac {3}{4} c d^3 x^4 (a+b \text {arctanh}(c x))^2+\frac {3}{10} b c d^3 x^4 (a+b \text {arctanh}(c x))+\frac {1}{3} d^3 x^3 (a+b \text {arctanh}(c x))^2+\frac {11}{18} b d^3 x^3 (a+b \text {arctanh}(c x))+\frac {14 b d^3 x^2 (a+b \text {arctanh}(c x))}{15 c}+\frac {11 a b d^3 x}{6 c^2}-\frac {37 b^2 d^3 \text {arctanh}(c x)}{30 c^3}+\frac {11 b^2 d^3 x \text {arctanh}(c x)}{6 c^2}-\frac {14 b^2 d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{15 c^3}+\frac {37 b^2 d^3 x}{30 c^2}+\frac {113 b^2 d^3 \log \left (1-c^2 x^2\right )}{90 c^3}+\frac {1}{60} b^2 c d^3 x^4+\frac {61 b^2 d^3 x^2}{180 c}+\frac {1}{10} b^2 d^3 x^3\)

Input:

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

Output:

(11*a*b*d^3*x)/(6*c^2) + (37*b^2*d^3*x)/(30*c^2) + (61*b^2*d^3*x^2)/(180*c 
) + (b^2*d^3*x^3)/10 + (b^2*c*d^3*x^4)/60 - (37*b^2*d^3*ArcTanh[c*x])/(30* 
c^3) + (11*b^2*d^3*x*ArcTanh[c*x])/(6*c^2) + (14*b*d^3*x^2*(a + b*ArcTanh[ 
c*x]))/(15*c) + (11*b*d^3*x^3*(a + b*ArcTanh[c*x]))/18 + (3*b*c*d^3*x^4*(a 
 + b*ArcTanh[c*x]))/10 + (b*c^2*d^3*x^5*(a + b*ArcTanh[c*x]))/15 + (d^3*(a 
 + b*ArcTanh[c*x])^2)/(60*c^3) + (d^3*x^3*(a + b*ArcTanh[c*x])^2)/3 + (3*c 
*d^3*x^4*(a + b*ArcTanh[c*x])^2)/4 + (3*c^2*d^3*x^5*(a + b*ArcTanh[c*x])^2 
)/5 + (c^3*d^3*x^6*(a + b*ArcTanh[c*x])^2)/6 - (28*b*d^3*(a + b*ArcTanh[c* 
x])*Log[2/(1 - c*x)])/(15*c^3) + (113*b^2*d^3*Log[1 - c^2*x^2])/(90*c^3) - 
 (14*b^2*d^3*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 = 0.92 (sec) , antiderivative size = 408, normalized size of antiderivative = 1.08

method	result	size
parts	\(d^{3} a^{2} \left (\frac {1}{6} c^{3} x^{6}+\frac {3}{5} c^{2} x^{5}+\frac {3}{4} c \,x^{4}+\frac {1}{3} x^{3}\right )+\frac {d^{3} b^{2} \left (\frac {\operatorname {arctanh}\left (c x \right )^{2} c^{6} x^{6}}{6}+\frac {3 \operatorname {arctanh}\left (c x \right )^{2} c^{5} x^{5}}{5}+\frac {3 \operatorname {arctanh}\left (c x \right )^{2} c^{4} x^{4}}{4}+\frac {\operatorname {arctanh}\left (c x \right )^{2} c^{3} x^{3}}{3}+\frac {\operatorname {arctanh}\left (c x \right ) c^{5} x^{5}}{15}+\frac {3 \,\operatorname {arctanh}\left (c x \right ) c^{4} x^{4}}{10}+\frac {11 \,\operatorname {arctanh}\left (c x \right ) c^{3} x^{3}}{18}+\frac {14 \,\operatorname {arctanh}\left (c x \right ) c^{2} x^{2}}{15}+\frac {11 \,\operatorname {arctanh}\left (c x \right ) c x}{6}+\frac {37 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )}{20}+\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )}{60}+\frac {37 \ln \left (c x -1\right )^{2}}{80}-\frac {14 \operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{15}-\frac {37 \ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{40}-\frac {\ln \left (c x +1\right )^{2}}{240}+\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 )}{120}+\frac {c^{4} x^{4}}{60}+\frac {x^{3} c^{3}}{10}+\frac {61 c^{2} x^{2}}{180}+\frac {37 c x}{30}+\frac {337 \ln \left (c x -1\right )}{180}+\frac {23 \ln \left (c x +1\right )}{36}\right )}{c^{3}}+\frac {2 d^{3} a b \left (\frac {\operatorname {arctanh}\left (c x \right ) c^{6} x^{6}}{6}+\frac {3 \,\operatorname {arctanh}\left (c x \right ) c^{5} x^{5}}{5}+\frac {3 \,\operatorname {arctanh}\left (c x \right ) c^{4} x^{4}}{4}+\frac {\operatorname {arctanh}\left (c x \right ) c^{3} x^{3}}{3}+\frac {c^{5} x^{5}}{30}+\frac {3 c^{4} x^{4}}{20}+\frac {11 x^{3} c^{3}}{36}+\frac {7 c^{2} x^{2}}{15}+\frac {11 c x}{12}+\frac {37 \ln \left (c x -1\right )}{40}+\frac {\ln \left (c x +1\right )}{120}\right )}{c^{3}}\)	\(408\)
derivativedivides	\(\frac {d^{3} a^{2} \left (\frac {1}{6} c^{6} x^{6}+\frac {3}{5} c^{5} x^{5}+\frac {3}{4} c^{4} x^{4}+\frac {1}{3} x^{3} c^{3}\right )+d^{3} b^{2} \left (\frac {\operatorname {arctanh}\left (c x \right )^{2} c^{6} x^{6}}{6}+\frac {3 \operatorname {arctanh}\left (c x \right )^{2} c^{5} x^{5}}{5}+\frac {3 \operatorname {arctanh}\left (c x \right )^{2} c^{4} x^{4}}{4}+\frac {\operatorname {arctanh}\left (c x \right )^{2} c^{3} x^{3}}{3}+\frac {\operatorname {arctanh}\left (c x \right ) c^{5} x^{5}}{15}+\frac {3 \,\operatorname {arctanh}\left (c x \right ) c^{4} x^{4}}{10}+\frac {11 \,\operatorname {arctanh}\left (c x \right ) c^{3} x^{3}}{18}+\frac {14 \,\operatorname {arctanh}\left (c x \right ) c^{2} x^{2}}{15}+\frac {11 \,\operatorname {arctanh}\left (c x \right ) c x}{6}+\frac {37 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )}{20}+\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )}{60}+\frac {37 \ln \left (c x -1\right )^{2}}{80}-\frac {14 \operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{15}-\frac {37 \ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{40}-\frac {\ln \left (c x +1\right )^{2}}{240}+\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 )}{120}+\frac {c^{4} x^{4}}{60}+\frac {x^{3} c^{3}}{10}+\frac {61 c^{2} x^{2}}{180}+\frac {37 c x}{30}+\frac {337 \ln \left (c x -1\right )}{180}+\frac {23 \ln \left (c x +1\right )}{36}\right )+2 d^{3} a b \left (\frac {\operatorname {arctanh}\left (c x \right ) c^{6} x^{6}}{6}+\frac {3 \,\operatorname {arctanh}\left (c x \right ) c^{5} x^{5}}{5}+\frac {3 \,\operatorname {arctanh}\left (c x \right ) c^{4} x^{4}}{4}+\frac {\operatorname {arctanh}\left (c x \right ) c^{3} x^{3}}{3}+\frac {c^{5} x^{5}}{30}+\frac {3 c^{4} x^{4}}{20}+\frac {11 x^{3} c^{3}}{36}+\frac {7 c^{2} x^{2}}{15}+\frac {11 c x}{12}+\frac {37 \ln \left (c x -1\right )}{40}+\frac {\ln \left (c x +1\right )}{120}\right )}{c^{3}}\)	\(411\)
default	\(\frac {d^{3} a^{2} \left (\frac {1}{6} c^{6} x^{6}+\frac {3}{5} c^{5} x^{5}+\frac {3}{4} c^{4} x^{4}+\frac {1}{3} x^{3} c^{3}\right )+d^{3} b^{2} \left (\frac {\operatorname {arctanh}\left (c x \right )^{2} c^{6} x^{6}}{6}+\frac {3 \operatorname {arctanh}\left (c x \right )^{2} c^{5} x^{5}}{5}+\frac {3 \operatorname {arctanh}\left (c x \right )^{2} c^{4} x^{4}}{4}+\frac {\operatorname {arctanh}\left (c x \right )^{2} c^{3} x^{3}}{3}+\frac {\operatorname {arctanh}\left (c x \right ) c^{5} x^{5}}{15}+\frac {3 \,\operatorname {arctanh}\left (c x \right ) c^{4} x^{4}}{10}+\frac {11 \,\operatorname {arctanh}\left (c x \right ) c^{3} x^{3}}{18}+\frac {14 \,\operatorname {arctanh}\left (c x \right ) c^{2} x^{2}}{15}+\frac {11 \,\operatorname {arctanh}\left (c x \right ) c x}{6}+\frac {37 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )}{20}+\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )}{60}+\frac {37 \ln \left (c x -1\right )^{2}}{80}-\frac {14 \operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{15}-\frac {37 \ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{40}-\frac {\ln \left (c x +1\right )^{2}}{240}+\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 )}{120}+\frac {c^{4} x^{4}}{60}+\frac {x^{3} c^{3}}{10}+\frac {61 c^{2} x^{2}}{180}+\frac {37 c x}{30}+\frac {337 \ln \left (c x -1\right )}{180}+\frac {23 \ln \left (c x +1\right )}{36}\right )+2 d^{3} a b \left (\frac {\operatorname {arctanh}\left (c x \right ) c^{6} x^{6}}{6}+\frac {3 \,\operatorname {arctanh}\left (c x \right ) c^{5} x^{5}}{5}+\frac {3 \,\operatorname {arctanh}\left (c x \right ) c^{4} x^{4}}{4}+\frac {\operatorname {arctanh}\left (c x \right ) c^{3} x^{3}}{3}+\frac {c^{5} x^{5}}{30}+\frac {3 c^{4} x^{4}}{20}+\frac {11 x^{3} c^{3}}{36}+\frac {7 c^{2} x^{2}}{15}+\frac {11 c x}{12}+\frac {37 \ln \left (c x -1\right )}{40}+\frac {\ln \left (c x +1\right )}{120}\right )}{c^{3}}\)	\(411\)
risch	\(\frac {11 a b \,d^{3} x}{6 c^{2}}-\frac {d^{3} a b \ln \left (-c x +1\right ) x^{3}}{3}+\frac {d^{3} c^{3} b^{2} \ln \left (-c x +1\right )^{2} x^{6}}{24}+\frac {3 d^{3} c^{2} b^{2} \ln \left (-c x +1\right )^{2} x^{5}}{20}+\frac {3 d^{3} c \,b^{2} \ln \left (-c x +1\right )^{2} x^{4}}{16}+\frac {d^{3} b \,c^{2} a \,x^{5}}{15}+\frac {3 d^{3} b c a \,x^{4}}{10}+\frac {14 d^{3} b a \,x^{2}}{15 c}-\frac {11 d^{3} b^{2} \ln \left (-c x +1\right ) x}{12 c^{2}}-\frac {7 d^{3} b^{2} \ln \left (-c x +1\right ) x^{2}}{15 c}-\frac {d^{3} b^{2} c^{2} \ln \left (-c x +1\right ) x^{5}}{30}-\frac {3 d^{3} b^{2} c \ln \left (-c x +1\right ) x^{4}}{20}+\frac {11 d^{3} b \,x^{3} a}{18}-\frac {11 d^{3} b^{2} \ln \left (-c x +1\right ) x^{3}}{36}+\frac {d^{3} c^{3} x^{6} a^{2}}{6}+\frac {3 d^{3} c^{2} x^{5} a^{2}}{5}+\frac {3 d^{3} c \,x^{4} a^{2}}{4}+\frac {d^{3} b^{2} \ln \left (-c x +1\right )^{2} x^{3}}{12}-\frac {37 d^{3} b^{2} \ln \left (-c x +1\right )^{2}}{80 c^{3}}+\frac {14 d^{3} b^{2} \operatorname {dilog}\left (-\frac {c x}{2}+\frac {1}{2}\right )}{15 c^{3}}+\frac {23 d^{3} b^{2} \ln \left (-c x -1\right )}{36 c^{3}}+\frac {37 b^{2} d^{3} x}{30 c^{2}}+\frac {61 b^{2} d^{3} x^{2}}{180 c}+\frac {b^{2} c \,d^{3} x^{4}}{60}+\frac {337 b^{2} d^{3} \ln \left (-c x +1\right )}{180 c^{3}}+\frac {d^{3} b^{2} \left (10 c^{6} x^{6}+36 c^{5} x^{5}+45 c^{4} x^{4}+20 x^{3} c^{3}+1\right ) \ln \left (c x +1\right )^{2}}{240 c^{3}}-\frac {76 d^{3} b^{2}}{45 c^{3}}-\frac {337 d^{3} b a}{90 c^{3}}-\frac {3 d^{3} c a b \ln \left (-c x +1\right ) x^{4}}{4}-\frac {d^{3} c^{3} a b \ln \left (-c x +1\right ) x^{6}}{6}-\frac {3 d^{3} c^{2} a b \ln \left (-c x +1\right ) x^{5}}{5}+\frac {b^{2} d^{3} x^{3}}{10}+\left (-\frac {d^{3} b^{2} x^{3} \left (10 x^{3} c^{3}+36 c^{2} x^{2}+45 c x +20\right ) \ln \left (-c x +1\right )}{120}+\frac {d^{3} b \left (60 a \,c^{6} x^{6}+216 a \,c^{5} x^{5}+12 b \,c^{5} x^{5}+270 a \,c^{4} x^{4}+54 b \,c^{4} x^{4}+120 a \,c^{3} x^{3}+110 b \,c^{3} x^{3}+168 b \,c^{2} x^{2}+330 b c x +333 b \ln \left (-c x +1\right )\right )}{360 c^{3}}\right ) \ln \left (c x +1\right )-\frac {37 d^{3} a^{2}}{20 c^{3}}+\frac {d^{3} x^{3} a^{2}}{3}+\frac {37 d^{3} a b \ln \left (-c x +1\right )}{20 c^{3}}-\frac {14 d^{3} b^{2} \ln \left (\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (-c x +1\right )}{15 c^{3}}+\frac {14 d^{3} 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 {d^{3} b \ln \left (-c x -1\right ) a}{60 c^{3}}\)	\(808\)

Input:

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

Output:

d^3*a^2*(1/6*c^3*x^6+3/5*c^2*x^5+3/4*c*x^4+1/3*x^3)+d^3*b^2/c^3*(1/6*arcta 
nh(c*x)^2*c^6*x^6+3/5*arctanh(c*x)^2*c^5*x^5+3/4*arctanh(c*x)^2*c^4*x^4+1/ 
3*arctanh(c*x)^2*c^3*x^3+1/15*arctanh(c*x)*c^5*x^5+3/10*arctanh(c*x)*c^4*x 
^4+11/18*arctanh(c*x)*c^3*x^3+14/15*arctanh(c*x)*c^2*x^2+11/6*arctanh(c*x) 
*c*x+37/20*arctanh(c*x)*ln(c*x-1)+1/60*arctanh(c*x)*ln(c*x+1)+37/80*ln(c*x 
-1)^2-14/15*dilog(1/2*c*x+1/2)-37/40*ln(c*x-1)*ln(1/2*c*x+1/2)-1/240*ln(c* 
x+1)^2+1/120*(ln(c*x+1)-ln(1/2*c*x+1/2))*ln(-1/2*c*x+1/2)+1/60*c^4*x^4+1/1 
0*x^3*c^3+61/180*c^2*x^2+37/30*c*x+337/180*ln(c*x-1)+23/36*ln(c*x+1))+2*d^ 
3*a*b/c^3*(1/6*arctanh(c*x)*c^6*x^6+3/5*arctanh(c*x)*c^5*x^5+3/4*arctanh(c 
*x)*c^4*x^4+1/3*arctanh(c*x)*c^3*x^3+1/30*c^5*x^5+3/20*c^4*x^4+11/36*x^3*c 
^3+7/15*c^2*x^2+11/12*c*x+37/40*ln(c*x-1)+1/120*ln(c*x+1))
 

Fricas [F]

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

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

Output:

integral(a^2*c^3*d^3*x^5 + 3*a^2*c^2*d^3*x^4 + 3*a^2*c*d^3*x^3 + a^2*d^3*x 
^2 + (b^2*c^3*d^3*x^5 + 3*b^2*c^2*d^3*x^4 + 3*b^2*c*d^3*x^3 + b^2*d^3*x^2) 
*arctanh(c*x)^2 + 2*(a*b*c^3*d^3*x^5 + 3*a*b*c^2*d^3*x^4 + 3*a*b*c*d^3*x^3 
 + a*b*d^3*x^2)*arctanh(c*x), x)
 

Sympy [F]

\[ \int x^2 (d+c d x)^3 (a+b \text {arctanh}(c x))^2 \, dx=d^{3} \left (\int a^{2} x^{2}\, dx + \int 3 a^{2} c x^{3}\, dx + \int 3 a^{2} c^{2} x^{4}\, dx + \int a^{2} c^{3} x^{5}\, dx + \int b^{2} x^{2} \operatorname {atanh}^{2}{\left (c x \right )}\, dx + \int 2 a b x^{2} \operatorname {atanh}{\left (c x \right )}\, dx + \int 3 b^{2} c x^{3} \operatorname {atanh}^{2}{\left (c x \right )}\, dx + \int 3 b^{2} c^{2} x^{4} \operatorname {atanh}^{2}{\left (c x \right )}\, dx + \int b^{2} c^{3} x^{5} \operatorname {atanh}^{2}{\left (c x \right )}\, dx + \int 6 a b c x^{3} \operatorname {atanh}{\left (c x \right )}\, dx + \int 6 a b c^{2} x^{4} \operatorname {atanh}{\left (c x \right )}\, dx + \int 2 a b c^{3} x^{5} \operatorname {atanh}{\left (c x \right )}\, dx\right ) \] Input:

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

Output:

d**3*(Integral(a**2*x**2, x) + Integral(3*a**2*c*x**3, x) + Integral(3*a** 
2*c**2*x**4, x) + Integral(a**2*c**3*x**5, x) + Integral(b**2*x**2*atanh(c 
*x)**2, x) + Integral(2*a*b*x**2*atanh(c*x), x) + Integral(3*b**2*c*x**3*a 
tanh(c*x)**2, x) + Integral(3*b**2*c**2*x**4*atanh(c*x)**2, x) + Integral( 
b**2*c**3*x**5*atanh(c*x)**2, x) + Integral(6*a*b*c*x**3*atanh(c*x), x) + 
Integral(6*a*b*c**2*x**4*atanh(c*x), x) + Integral(2*a*b*c**3*x**5*atanh(c 
*x), x))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 775 vs. \(2 (336) = 672\).

Time = 0.30 (sec) , antiderivative size = 775, normalized size of antiderivative = 2.06 \[ \int x^2 (d+c d x)^3 (a+b \text {arctanh}(c x))^2 \, dx =\text {Too large to display} \] Input:

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

Output:

1/6*a^2*c^3*d^3*x^6 + 3/5*a^2*c^2*d^3*x^5 + 3/4*a^2*c*d^3*x^4 + 1/90*(30*x 
^6*arctanh(c*x) + c*(2*(3*c^4*x^5 + 5*c^2*x^3 + 15*x)/c^6 - 15*log(c*x + 1 
)/c^7 + 15*log(c*x - 1)/c^7))*a*b*c^3*d^3 + 3/10*(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^3 + 1/3*a^2*d^3 
*x^3 + 1/4*(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^3 + 1/3*(2*x^3*arctanh(c*x) + c*(x^2/c 
^2 + log(c^2*x^2 - 1)/c^4))*a*b*d^3 + 14/15*(log(c*x + 1)*log(-1/2*c*x + 1 
/2) + dilog(1/2*c*x + 1/2))*b^2*d^3/c^3 + 23/36*b^2*d^3*log(c*x + 1)/c^3 + 
 337/180*b^2*d^3*log(c*x - 1)/c^3 + 1/720*(12*b^2*c^4*d^3*x^4 + 72*b^2*c^3 
*d^3*x^3 + 244*b^2*c^2*d^3*x^2 + 888*b^2*c*d^3*x + 3*(10*b^2*c^6*d^3*x^6 + 
 36*b^2*c^5*d^3*x^5 + 45*b^2*c^4*d^3*x^4 + 20*b^2*c^3*d^3*x^3 + b^2*d^3)*l 
og(c*x + 1)^2 + 3*(10*b^2*c^6*d^3*x^6 + 36*b^2*c^5*d^3*x^5 + 45*b^2*c^4*d^ 
3*x^4 + 20*b^2*c^3*d^3*x^3 - 111*b^2*d^3)*log(-c*x + 1)^2 + 4*(6*b^2*c^5*d 
^3*x^5 + 27*b^2*c^4*d^3*x^4 + 55*b^2*c^3*d^3*x^3 + 84*b^2*c^2*d^3*x^2 + 16 
5*b^2*c*d^3*x)*log(c*x + 1) - 2*(12*b^2*c^5*d^3*x^5 + 54*b^2*c^4*d^3*x^4 + 
 110*b^2*c^3*d^3*x^3 + 168*b^2*c^2*d^3*x^2 + 330*b^2*c*d^3*x + 3*(10*b^2*c 
^6*d^3*x^6 + 36*b^2*c^5*d^3*x^5 + 45*b^2*c^4*d^3*x^4 + 20*b^2*c^3*d^3*x^3 
+ b^2*d^3)*log(c*x + 1))*log(-c*x + 1))/c^3
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int x^2 (d+c d x)^3 (a+b \text {arctanh}(c x))^2 \, dx=\frac {d^{3} \left (336 \,\mathrm {log}\left (c^{2} x -c \right ) a b +108 a^{2} c^{5} x^{5}+222 b^{2} c x +6 \mathit {atanh} \left (c x \right ) a b +61 b^{2} c^{2} x^{2}+30 \mathit {atanh} \left (c x \right )^{2} b^{2} c^{6} x^{6}+12 \mathit {atanh} \left (c x \right ) b^{2} c^{5} x^{5}+12 a b \,c^{5} x^{5}+110 \mathit {atanh} \left (c x \right ) b^{2} c^{3} x^{3}+330 \mathit {atanh} \left (c x \right ) b^{2} c x +110 a b \,c^{3} x^{3}+330 a b c x +18 b^{2} c^{3} x^{3}+108 \mathit {atanh} \left (c x \right )^{2} b^{2} c^{5} x^{5}-168 \mathit {atanh} \left (c x \right )^{2} b^{2} c x +54 \mathit {atanh} \left (c x \right ) b^{2} c^{4} x^{4}+168 \mathit {atanh} \left (c x \right ) b^{2} c^{2} x^{2}+54 a b \,c^{4} x^{4}+168 a b \,c^{2} x^{2}+135 \mathit {atanh} \left (c x \right )^{2} b^{2} c^{4} x^{4}+60 \mathit {atanh} \left (c x \right )^{2} b^{2} c^{3} x^{3}+30 a^{2} c^{6} x^{6}+3 b^{2} c^{4} x^{4}+60 a^{2} c^{3} x^{3}+270 \mathit {atanh} \left (c x \right ) a b \,c^{4} x^{4}+120 \mathit {atanh} \left (c x \right ) a b \,c^{3} x^{3}-165 \mathit {atanh} \left (c x \right )^{2} b^{2}+230 \mathit {atanh} \left (c x \right ) b^{2}+452 \,\mathrm {log}\left (c^{2} x -c \right ) b^{2}+168 \left (\int \mathit {atanh} \left (c x \right )^{2}d x \right ) b^{2} c +135 a^{2} c^{4} x^{4}+60 \mathit {atanh} \left (c x \right ) a b \,c^{6} x^{6}+216 \mathit {atanh} \left (c x \right ) a b \,c^{5} x^{5}\right )}{180 c^{3}} \] Input:

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

Output:

(d**3*(30*atanh(c*x)**2*b**2*c**6*x**6 + 108*atanh(c*x)**2*b**2*c**5*x**5 
+ 135*atanh(c*x)**2*b**2*c**4*x**4 + 60*atanh(c*x)**2*b**2*c**3*x**3 - 168 
*atanh(c*x)**2*b**2*c*x - 165*atanh(c*x)**2*b**2 + 60*atanh(c*x)*a*b*c**6* 
x**6 + 216*atanh(c*x)*a*b*c**5*x**5 + 270*atanh(c*x)*a*b*c**4*x**4 + 120*a 
tanh(c*x)*a*b*c**3*x**3 + 6*atanh(c*x)*a*b + 12*atanh(c*x)*b**2*c**5*x**5 
+ 54*atanh(c*x)*b**2*c**4*x**4 + 110*atanh(c*x)*b**2*c**3*x**3 + 168*atanh 
(c*x)*b**2*c**2*x**2 + 330*atanh(c*x)*b**2*c*x + 230*atanh(c*x)*b**2 + 168 
*int(atanh(c*x)**2,x)*b**2*c + 336*log(c**2*x - c)*a*b + 452*log(c**2*x - 
c)*b**2 + 30*a**2*c**6*x**6 + 108*a**2*c**5*x**5 + 135*a**2*c**4*x**4 + 60 
*a**2*c**3*x**3 + 12*a*b*c**5*x**5 + 54*a*b*c**4*x**4 + 110*a*b*c**3*x**3 
+ 168*a*b*c**2*x**2 + 330*a*b*c*x + 3*b**2*c**4*x**4 + 18*b**2*c**3*x**3 + 
 61*b**2*c**2*x**2 + 222*b**2*c*x))/(180*c**3)