Integrand size = 20, antiderivative size = 286 \[ \int x (d+c d x)^3 (a+b \text {arctanh}(c x))^2 \, dx=\frac {5 a b d^3 x}{2 c}+\frac {13 b^2 d^3 x}{10 c}+\frac {1}{4} b^2 d^3 x^2+\frac {1}{30} b^2 c d^3 x^3-\frac {13 b^2 d^3 \text {arctanh}(c x)}{10 c^2}+\frac {5 b^2 d^3 x \text {arctanh}(c x)}{2 c}+\frac {6}{5} b d^3 x^2 (a+b \text {arctanh}(c x))+\frac {1}{2} b c d^3 x^3 (a+b \text {arctanh}(c x))+\frac {1}{10} b c^2 d^3 x^4 (a+b \text {arctanh}(c x))-\frac {d^3 (1+c x)^4 (a+b \text {arctanh}(c x))^2}{4 c^2}+\frac {d^3 (1+c x)^5 (a+b \text {arctanh}(c x))^2}{5 c^2}-\frac {12 b d^3 (a+b \text {arctanh}(c x)) \log \left (\frac {2}{1-c x}\right )}{5 c^2}+\frac {3 b^2 d^3 \log \left (1-c^2 x^2\right )}{2 c^2}-\frac {6 b^2 d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{5 c^2} \]
5/2*a*b*d^3*x/c+13/10*b^2*d^3*x/c+1/4*b^2*d^3*x^2+1/30*b^2*c*d^3*x^3-13/10 *b^2*d^3*arctanh(c*x)/c^2+5/2*b^2*d^3*x*arctanh(c*x)/c+6/5*b*d^3*x^2*(a+b* arctanh(c*x))+1/2*b*c*d^3*x^3*(a+b*arctanh(c*x))+1/10*b*c^2*d^3*x^4*(a+b*a rctanh(c*x))-1/4*d^3*(c*x+1)^4*(a+b*arctanh(c*x))^2/c^2+1/5*d^3*(c*x+1)^5* (a+b*arctanh(c*x))^2/c^2-12/5*b*d^3*(a+b*arctanh(c*x))*ln(2/(-c*x+1))/c^2+ 3/2*b^2*d^3*ln(-c^2*x^2+1)/c^2-6/5*b^2*d^3*polylog(2,1-2/(-c*x+1))/c^2
Time = 1.76 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.14 \[ \int x (d+c d x)^3 (a+b \text {arctanh}(c x))^2 \, dx=\frac {d^3 \left (-18 a b-15 b^2+150 a b c x+78 b^2 c x+30 a^2 c^2 x^2+72 a b c^2 x^2+15 b^2 c^2 x^2+60 a^2 c^3 x^3+30 a b c^3 x^3+2 b^2 c^3 x^3+45 a^2 c^4 x^4+6 a b c^4 x^4+12 a^2 c^5 x^5+3 b^2 \left (-49+10 c^2 x^2+20 c^3 x^3+15 c^4 x^4+4 c^5 x^5\right ) \text {arctanh}(c x)^2+6 b \text {arctanh}(c x) \left (a c^2 x^2 \left (10+20 c x+15 c^2 x^2+4 c^3 x^3\right )+b \left (-13+25 c x+12 c^2 x^2+5 c^3 x^3+c^4 x^4\right )-24 b \log \left (1+e^{-2 \text {arctanh}(c x)}\right )\right )+75 a b \log (1-c x)-75 a b \log (1+c x)+90 b^2 \log \left (1-c^2 x^2\right )+72 a b \log \left (-1+c^2 x^2\right )+72 b^2 \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )\right )}{60 c^2} \]
(d^3*(-18*a*b - 15*b^2 + 150*a*b*c*x + 78*b^2*c*x + 30*a^2*c^2*x^2 + 72*a* b*c^2*x^2 + 15*b^2*c^2*x^2 + 60*a^2*c^3*x^3 + 30*a*b*c^3*x^3 + 2*b^2*c^3*x ^3 + 45*a^2*c^4*x^4 + 6*a*b*c^4*x^4 + 12*a^2*c^5*x^5 + 3*b^2*(-49 + 10*c^2 *x^2 + 20*c^3*x^3 + 15*c^4*x^4 + 4*c^5*x^5)*ArcTanh[c*x]^2 + 6*b*ArcTanh[c *x]*(a*c^2*x^2*(10 + 20*c*x + 15*c^2*x^2 + 4*c^3*x^3) + b*(-13 + 25*c*x + 12*c^2*x^2 + 5*c^3*x^3 + c^4*x^4) - 24*b*Log[1 + E^(-2*ArcTanh[c*x])]) + 7 5*a*b*Log[1 - c*x] - 75*a*b*Log[1 + c*x] + 90*b^2*Log[1 - c^2*x^2] + 72*a* b*Log[-1 + c^2*x^2] + 72*b^2*PolyLog[2, -E^(-2*ArcTanh[c*x])]))/(60*c^2)
Time = 0.78 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, 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.
| \(\displaystyle \int x (c d x+d)^3 (a+b \text {arctanh}(c x))^2 \, dx\) |
| \(\Big \downarrow \) 6502 |
| \(\displaystyle \int \left (\frac {(c d x+d)^4 (a+b \text {arctanh}(c x))^2}{c d}-\frac {(c d x+d)^3 (a+b \text {arctanh}(c x))^2}{c}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{10} b c^2 d^3 x^4 (a+b \text {arctanh}(c x))+\frac {d^3 (c x+1)^5 (a+b \text {arctanh}(c x))^2}{5 c^2}-\frac {d^3 (c x+1)^4 (a+b \text {arctanh}(c x))^2}{4 c^2}-\frac {12 b d^3 \log \left (\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))}{5 c^2}+\frac {1}{2} b c d^3 x^3 (a+b \text {arctanh}(c x))+\frac {6}{5} b d^3 x^2 (a+b \text {arctanh}(c x))+\frac {5 a b d^3 x}{2 c}-\frac {13 b^2 d^3 \text {arctanh}(c x)}{10 c^2}+\frac {5 b^2 d^3 x \text {arctanh}(c x)}{2 c}-\frac {6 b^2 d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{5 c^2}+\frac {3 b^2 d^3 \log \left (1-c^2 x^2\right )}{2 c^2}+\frac {1}{30} b^2 c d^3 x^3+\frac {13 b^2 d^3 x}{10 c}+\frac {1}{4} b^2 d^3 x^2\) |
(5*a*b*d^3*x)/(2*c) + (13*b^2*d^3*x)/(10*c) + (b^2*d^3*x^2)/4 + (b^2*c*d^3 *x^3)/30 - (13*b^2*d^3*ArcTanh[c*x])/(10*c^2) + (5*b^2*d^3*x*ArcTanh[c*x]) /(2*c) + (6*b*d^3*x^2*(a + b*ArcTanh[c*x]))/5 + (b*c*d^3*x^3*(a + b*ArcTan h[c*x]))/2 + (b*c^2*d^3*x^4*(a + b*ArcTanh[c*x]))/10 - (d^3*(1 + c*x)^4*(a + b*ArcTanh[c*x])^2)/(4*c^2) + (d^3*(1 + c*x)^5*(a + b*ArcTanh[c*x])^2)/( 5*c^2) - (12*b*d^3*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)])/(5*c^2) + (3*b^2 *d^3*Log[1 - c^2*x^2])/(2*c^2) - (6*b^2*d^3*PolyLog[2, 1 - 2/(1 - c*x)])/( 5*c^2)
3.1.86.3.1 Defintions of rubi rules used
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])
Time = 1.57 (sec) , antiderivative size = 377, normalized size of antiderivative = 1.32
| method | result | size |
| parts | \(d^{3} a^{2} \left (\frac {1}{5} c^{3} x^{5}+\frac {3}{4} c^{2} x^{4}+c \,x^{3}+\frac {1}{2} x^{2}\right )+\frac {d^{3} b^{2} \left (\frac {c^{5} x^{5} \operatorname {arctanh}\left (c x \right )^{2}}{5}+\frac {3 c^{4} x^{4} \operatorname {arctanh}\left (c x \right )^{2}}{4}+\operatorname {arctanh}\left (c x \right )^{2} c^{3} x^{3}+\frac {c^{2} x^{2} \operatorname {arctanh}\left (c x \right )^{2}}{2}+\frac {c^{4} x^{4} \operatorname {arctanh}\left (c x \right )}{10}+\frac {c^{3} x^{3} \operatorname {arctanh}\left (c x \right )}{2}+\frac {6 c^{2} x^{2} \operatorname {arctanh}\left (c x \right )}{5}+\frac {5 c x \,\operatorname {arctanh}\left (c x \right )}{2}+\frac {49 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )}{20}-\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )}{20}-\frac {6 \operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{5}-\frac {49 \ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{40}+\frac {49 \ln \left (c x -1\right )^{2}}{80}-\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{40}+\frac {\ln \left (c x +1\right )^{2}}{80}+\frac {c^{3} x^{3}}{30}+\frac {c^{2} x^{2}}{4}+\frac {13 c x}{10}+\frac {43 \ln \left (c x -1\right )}{20}+\frac {17 \ln \left (c x +1\right )}{20}\right )}{c^{2}}+\frac {2 d^{3} a b \left (\frac {c^{5} x^{5} \operatorname {arctanh}\left (c x \right )}{5}+\frac {3 c^{4} x^{4} \operatorname {arctanh}\left (c x \right )}{4}+c^{3} x^{3} \operatorname {arctanh}\left (c x \right )+\frac {c^{2} x^{2} \operatorname {arctanh}\left (c x \right )}{2}+\frac {c^{4} x^{4}}{20}+\frac {c^{3} x^{3}}{4}+\frac {3 c^{2} x^{2}}{5}+\frac {5 c x}{4}+\frac {49 \ln \left (c x -1\right )}{40}-\frac {\ln \left (c x +1\right )}{40}\right )}{c^{2}}\) | \(377\) |
| derivativedivides | \(\frac {d^{3} a^{2} \left (\frac {1}{5} c^{5} x^{5}+\frac {3}{4} c^{4} x^{4}+c^{3} x^{3}+\frac {1}{2} c^{2} x^{2}\right )+d^{3} b^{2} \left (\frac {c^{5} x^{5} \operatorname {arctanh}\left (c x \right )^{2}}{5}+\frac {3 c^{4} x^{4} \operatorname {arctanh}\left (c x \right )^{2}}{4}+\operatorname {arctanh}\left (c x \right )^{2} c^{3} x^{3}+\frac {c^{2} x^{2} \operatorname {arctanh}\left (c x \right )^{2}}{2}+\frac {c^{4} x^{4} \operatorname {arctanh}\left (c x \right )}{10}+\frac {c^{3} x^{3} \operatorname {arctanh}\left (c x \right )}{2}+\frac {6 c^{2} x^{2} \operatorname {arctanh}\left (c x \right )}{5}+\frac {5 c x \,\operatorname {arctanh}\left (c x \right )}{2}+\frac {49 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )}{20}-\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )}{20}-\frac {6 \operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{5}-\frac {49 \ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{40}+\frac {49 \ln \left (c x -1\right )^{2}}{80}-\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{40}+\frac {\ln \left (c x +1\right )^{2}}{80}+\frac {c^{3} x^{3}}{30}+\frac {c^{2} x^{2}}{4}+\frac {13 c x}{10}+\frac {43 \ln \left (c x -1\right )}{20}+\frac {17 \ln \left (c x +1\right )}{20}\right )+2 d^{3} a b \left (\frac {c^{5} x^{5} \operatorname {arctanh}\left (c x \right )}{5}+\frac {3 c^{4} x^{4} \operatorname {arctanh}\left (c x \right )}{4}+c^{3} x^{3} \operatorname {arctanh}\left (c x \right )+\frac {c^{2} x^{2} \operatorname {arctanh}\left (c x \right )}{2}+\frac {c^{4} x^{4}}{20}+\frac {c^{3} x^{3}}{4}+\frac {3 c^{2} x^{2}}{5}+\frac {5 c x}{4}+\frac {49 \ln \left (c x -1\right )}{40}-\frac {\ln \left (c x +1\right )}{40}\right )}{c^{2}}\) | \(380\) |
| default | \(\frac {d^{3} a^{2} \left (\frac {1}{5} c^{5} x^{5}+\frac {3}{4} c^{4} x^{4}+c^{3} x^{3}+\frac {1}{2} c^{2} x^{2}\right )+d^{3} b^{2} \left (\frac {c^{5} x^{5} \operatorname {arctanh}\left (c x \right )^{2}}{5}+\frac {3 c^{4} x^{4} \operatorname {arctanh}\left (c x \right )^{2}}{4}+\operatorname {arctanh}\left (c x \right )^{2} c^{3} x^{3}+\frac {c^{2} x^{2} \operatorname {arctanh}\left (c x \right )^{2}}{2}+\frac {c^{4} x^{4} \operatorname {arctanh}\left (c x \right )}{10}+\frac {c^{3} x^{3} \operatorname {arctanh}\left (c x \right )}{2}+\frac {6 c^{2} x^{2} \operatorname {arctanh}\left (c x \right )}{5}+\frac {5 c x \,\operatorname {arctanh}\left (c x \right )}{2}+\frac {49 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )}{20}-\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )}{20}-\frac {6 \operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{5}-\frac {49 \ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{40}+\frac {49 \ln \left (c x -1\right )^{2}}{80}-\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{40}+\frac {\ln \left (c x +1\right )^{2}}{80}+\frac {c^{3} x^{3}}{30}+\frac {c^{2} x^{2}}{4}+\frac {13 c x}{10}+\frac {43 \ln \left (c x -1\right )}{20}+\frac {17 \ln \left (c x +1\right )}{20}\right )+2 d^{3} a b \left (\frac {c^{5} x^{5} \operatorname {arctanh}\left (c x \right )}{5}+\frac {3 c^{4} x^{4} \operatorname {arctanh}\left (c x \right )}{4}+c^{3} x^{3} \operatorname {arctanh}\left (c x \right )+\frac {c^{2} x^{2} \operatorname {arctanh}\left (c x \right )}{2}+\frac {c^{4} x^{4}}{20}+\frac {c^{3} x^{3}}{4}+\frac {3 c^{2} x^{2}}{5}+\frac {5 c x}{4}+\frac {49 \ln \left (c x -1\right )}{40}-\frac {\ln \left (c x +1\right )}{40}\right )}{c^{2}}\) | \(380\) |
| risch | \(\frac {a^{2} d^{3} x^{2}}{2}+\frac {b^{2} d^{3} x^{2}}{4}+\frac {5 a b \,d^{3} x}{2 c}+\frac {13 b^{2} d^{3} x}{10 c}+\frac {b^{2} c \,d^{3} x^{3}}{30}-\frac {43 d^{3} b a}{10 c^{2}}-\frac {19 b^{2} d^{3}}{12 c^{2}}-\frac {3 d^{3} b^{2} \ln \left (-c x +1\right ) x^{2}}{5}+\frac {6 d^{3} b a \,x^{2}}{5}+\frac {a b \,c^{2} d^{3} x^{4}}{10}+\frac {a b c \,d^{3} x^{3}}{2}+\left (-\frac {d^{3} b^{2} x^{2} \left (4 c^{3} x^{3}+15 c^{2} x^{2}+20 c x +10\right ) \ln \left (-c x +1\right )}{40}-\frac {d^{3} b \left (-8 c^{5} x^{5} a -30 c^{4} x^{4} a -2 b \,c^{4} x^{4}-40 c^{3} x^{3} a -10 b \,c^{3} x^{3}-20 a \,c^{2} x^{2}-24 b \,c^{2} x^{2}-50 b c x -49 b \ln \left (-c x +1\right )\right )}{40 c^{2}}\right ) \ln \left (c x +1\right )-\frac {d^{3} b a \ln \left (-c x -1\right )}{20 c^{2}}-\frac {6 b^{2} d^{3} \ln \left (\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (-c x +1\right )}{5 c^{2}}+\frac {6 b^{2} d^{3} \ln \left (\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{5 c^{2}}+\frac {d^{3} c \,b^{2} \ln \left (-c x +1\right )^{2} x^{3}}{4}+\frac {49 d^{3} b \ln \left (-c x +1\right ) a}{20 c^{2}}+\frac {d^{3} c^{3} b^{2} \ln \left (-c x +1\right )^{2} x^{5}}{20}+\frac {3 d^{3} c^{2} b^{2} \ln \left (-c x +1\right )^{2} x^{4}}{16}-\frac {d^{3} a b \ln \left (-c x +1\right ) x^{2}}{2}-\frac {b^{2} d^{3} c \ln \left (-c x +1\right ) x^{3}}{4}-\frac {b^{2} d^{3} c^{2} \ln \left (-c x +1\right ) x^{4}}{20}-\frac {5 b^{2} d^{3} \ln \left (-c x +1\right ) x}{4 c}+\frac {d^{3} b^{2} \left (4 c^{5} x^{5}+15 c^{4} x^{4}+20 c^{3} x^{3}+10 c^{2} x^{2}-1\right ) \ln \left (c x +1\right )^{2}}{80 c^{2}}+\frac {6 b^{2} d^{3} \operatorname {dilog}\left (-\frac {c x}{2}+\frac {1}{2}\right )}{5 c^{2}}+\frac {3 d^{3} c^{2} x^{4} a^{2}}{4}+\frac {d^{3} c^{3} x^{5} a^{2}}{5}+d^{3} c \,x^{3} a^{2}+\frac {d^{3} b^{2} \ln \left (-c x +1\right )^{2} x^{2}}{8}-\frac {49 d^{3} b^{2} \ln \left (-c x +1\right )^{2}}{80 c^{2}}-\frac {49 d^{3} a^{2}}{20 c^{2}}+\frac {17 d^{3} b^{2} \ln \left (-c x -1\right )}{20 c^{2}}+\frac {43 d^{3} b^{2} \ln \left (-c x +1\right )}{20 c^{2}}-\frac {d^{3} c^{3} a b \ln \left (-c x +1\right ) x^{5}}{5}-\frac {3 d^{3} c^{2} a b \ln \left (-c x +1\right ) x^{4}}{4}-d^{3} c a b \ln \left (-c x +1\right ) x^{3}\) | \(750\) |
d^3*a^2*(1/5*c^3*x^5+3/4*c^2*x^4+c*x^3+1/2*x^2)+d^3*b^2/c^2*(1/5*c^5*x^5*a rctanh(c*x)^2+3/4*c^4*x^4*arctanh(c*x)^2+arctanh(c*x)^2*c^3*x^3+1/2*c^2*x^ 2*arctanh(c*x)^2+1/10*c^4*x^4*arctanh(c*x)+1/2*c^3*x^3*arctanh(c*x)+6/5*c^ 2*x^2*arctanh(c*x)+5/2*c*x*arctanh(c*x)+49/20*arctanh(c*x)*ln(c*x-1)-1/20* arctanh(c*x)*ln(c*x+1)-6/5*dilog(1/2*c*x+1/2)-49/40*ln(c*x-1)*ln(1/2*c*x+1 /2)+49/80*ln(c*x-1)^2-1/40*(ln(c*x+1)-ln(1/2*c*x+1/2))*ln(-1/2*c*x+1/2)+1/ 80*ln(c*x+1)^2+1/30*c^3*x^3+1/4*c^2*x^2+13/10*c*x+43/20*ln(c*x-1)+17/20*ln (c*x+1))+2*d^3*a*b/c^2*(1/5*c^5*x^5*arctanh(c*x)+3/4*c^4*x^4*arctanh(c*x)+ c^3*x^3*arctanh(c*x)+1/2*c^2*x^2*arctanh(c*x)+1/20*c^4*x^4+1/4*c^3*x^3+3/5 *c^2*x^2+5/4*c*x+49/40*ln(c*x-1)-1/40*ln(c*x+1))
\[ \int x (d+c d x)^3 (a+b \text {arctanh}(c x))^2 \, dx=\int { {\left (c d x + d\right )}^{3} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2} x \,d x } \]
integral(a^2*c^3*d^3*x^4 + 3*a^2*c^2*d^3*x^3 + 3*a^2*c*d^3*x^2 + a^2*d^3*x + (b^2*c^3*d^3*x^4 + 3*b^2*c^2*d^3*x^3 + 3*b^2*c*d^3*x^2 + b^2*d^3*x)*arc tanh(c*x)^2 + 2*(a*b*c^3*d^3*x^4 + 3*a*b*c^2*d^3*x^3 + 3*a*b*c*d^3*x^2 + a *b*d^3*x)*arctanh(c*x), x)
\[ \int x (d+c d x)^3 (a+b \text {arctanh}(c x))^2 \, dx=d^{3} \left (\int a^{2} x\, dx + \int 3 a^{2} c x^{2}\, dx + \int 3 a^{2} c^{2} x^{3}\, dx + \int a^{2} c^{3} x^{4}\, dx + \int b^{2} x \operatorname {atanh}^{2}{\left (c x \right )}\, dx + \int 2 a b x \operatorname {atanh}{\left (c x \right )}\, dx + \int 3 b^{2} c x^{2} \operatorname {atanh}^{2}{\left (c x \right )}\, dx + \int 3 b^{2} c^{2} x^{3} \operatorname {atanh}^{2}{\left (c x \right )}\, dx + \int b^{2} c^{3} x^{4} \operatorname {atanh}^{2}{\left (c x \right )}\, dx + \int 6 a b c x^{2} \operatorname {atanh}{\left (c x \right )}\, dx + \int 6 a b c^{2} x^{3} \operatorname {atanh}{\left (c x \right )}\, dx + \int 2 a b c^{3} x^{4} \operatorname {atanh}{\left (c x \right )}\, dx\right ) \]
d**3*(Integral(a**2*x, x) + Integral(3*a**2*c*x**2, x) + Integral(3*a**2*c **2*x**3, x) + Integral(a**2*c**3*x**4, x) + Integral(b**2*x*atanh(c*x)**2 , x) + Integral(2*a*b*x*atanh(c*x), x) + Integral(3*b**2*c*x**2*atanh(c*x) **2, x) + Integral(3*b**2*c**2*x**3*atanh(c*x)**2, x) + Integral(b**2*c**3 *x**4*atanh(c*x)**2, x) + Integral(6*a*b*c*x**2*atanh(c*x), x) + Integral( 6*a*b*c**2*x**3*atanh(c*x), x) + Integral(2*a*b*c**3*x**4*atanh(c*x), x))
Leaf count of result is larger than twice the leaf count of optimal. 780 vs. \(2 (255) = 510\).
Time = 0.45 (sec) , antiderivative size = 780, normalized size of antiderivative = 2.73 \[ \int x (d+c d x)^3 (a+b \text {arctanh}(c x))^2 \, dx=\frac {1}{5} \, a^{2} c^{3} d^{3} x^{5} + \frac {3}{4} \, a^{2} c^{2} d^{3} 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^{3} d^{3} + a^{2} c d^{3} x^{3} + \frac {1}{2} \, b^{2} d^{3} x^{2} \operatorname {artanh}\left (c x\right )^{2} + \frac {1}{4} \, {\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^{2} d^{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 c d^{3} + \frac {1}{2} \, a^{2} d^{3} x^{2} + \frac {1}{2} \, {\left (2 \, x^{2} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {2 \, x}{c^{2}} - \frac {\log \left (c x + 1\right )}{c^{3}} + \frac {\log \left (c x - 1\right )}{c^{3}}\right )}\right )} a b d^{3} + \frac {1}{8} \, {\left (4 \, c {\left (\frac {2 \, x}{c^{2}} - \frac {\log \left (c x + 1\right )}{c^{3}} + \frac {\log \left (c x - 1\right )}{c^{3}}\right )} \operatorname {artanh}\left (c x\right ) - \frac {2 \, {\left (\log \left (c x - 1\right ) - 2\right )} \log \left (c x + 1\right ) - \log \left (c x + 1\right )^{2} - \log \left (c x - 1\right )^{2} - 4 \, \log \left (c x - 1\right )}{c^{2}}\right )} b^{2} d^{3} + \frac {6 \, {\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^{3}}{5 \, c^{2}} + \frac {7 \, b^{2} d^{3} \log \left (c x + 1\right )}{20 \, c^{2}} + \frac {33 \, b^{2} d^{3} \log \left (c x - 1\right )}{20 \, c^{2}} + \frac {8 \, b^{2} c^{3} d^{3} x^{3} + 60 \, b^{2} c^{2} d^{3} x^{2} + 312 \, b^{2} c d^{3} x + 3 \, {\left (4 \, b^{2} c^{5} d^{3} x^{5} + 15 \, b^{2} c^{4} d^{3} x^{4} + 20 \, b^{2} c^{3} d^{3} x^{3} + 9 \, b^{2} d^{3}\right )} \log \left (c x + 1\right )^{2} + 3 \, {\left (4 \, b^{2} c^{5} d^{3} x^{5} + 15 \, b^{2} c^{4} d^{3} x^{4} + 20 \, b^{2} c^{3} d^{3} x^{3} - 39 \, b^{2} d^{3}\right )} \log \left (-c x + 1\right )^{2} + 12 \, {\left (b^{2} c^{4} d^{3} x^{4} + 5 \, b^{2} c^{3} d^{3} x^{3} + 12 \, b^{2} c^{2} d^{3} x^{2} + 15 \, b^{2} c d^{3} x\right )} \log \left (c x + 1\right ) - 6 \, {\left (2 \, b^{2} c^{4} d^{3} x^{4} + 10 \, b^{2} c^{3} d^{3} x^{3} + 24 \, b^{2} c^{2} d^{3} x^{2} + 30 \, b^{2} c d^{3} x + {\left (4 \, b^{2} c^{5} d^{3} x^{5} + 15 \, b^{2} c^{4} d^{3} x^{4} + 20 \, b^{2} c^{3} d^{3} x^{3} + 9 \, b^{2} d^{3}\right )} \log \left (c x + 1\right )\right )} \log \left (-c x + 1\right )}{240 \, c^{2}} \]
1/5*a^2*c^3*d^3*x^5 + 3/4*a^2*c^2*d^3*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^3*d^3 + a^2*c*d^3*x ^3 + 1/2*b^2*d^3*x^2*arctanh(c*x)^2 + 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^2*d^3 + ( 2*x^3*arctanh(c*x) + c*(x^2/c^2 + log(c^2*x^2 - 1)/c^4))*a*b*c*d^3 + 1/2*a ^2*d^3*x^2 + 1/2*(2*x^2*arctanh(c*x) + c*(2*x/c^2 - log(c*x + 1)/c^3 + log (c*x - 1)/c^3))*a*b*d^3 + 1/8*(4*c*(2*x/c^2 - log(c*x + 1)/c^3 + log(c*x - 1)/c^3)*arctanh(c*x) - (2*(log(c*x - 1) - 2)*log(c*x + 1) - log(c*x + 1)^ 2 - log(c*x - 1)^2 - 4*log(c*x - 1))/c^2)*b^2*d^3 + 6/5*(log(c*x + 1)*log( -1/2*c*x + 1/2) + dilog(1/2*c*x + 1/2))*b^2*d^3/c^2 + 7/20*b^2*d^3*log(c*x + 1)/c^2 + 33/20*b^2*d^3*log(c*x - 1)/c^2 + 1/240*(8*b^2*c^3*d^3*x^3 + 60 *b^2*c^2*d^3*x^2 + 312*b^2*c*d^3*x + 3*(4*b^2*c^5*d^3*x^5 + 15*b^2*c^4*d^3 *x^4 + 20*b^2*c^3*d^3*x^3 + 9*b^2*d^3)*log(c*x + 1)^2 + 3*(4*b^2*c^5*d^3*x ^5 + 15*b^2*c^4*d^3*x^4 + 20*b^2*c^3*d^3*x^3 - 39*b^2*d^3)*log(-c*x + 1)^2 + 12*(b^2*c^4*d^3*x^4 + 5*b^2*c^3*d^3*x^3 + 12*b^2*c^2*d^3*x^2 + 15*b^2*c *d^3*x)*log(c*x + 1) - 6*(2*b^2*c^4*d^3*x^4 + 10*b^2*c^3*d^3*x^3 + 24*b^2* c^2*d^3*x^2 + 30*b^2*c*d^3*x + (4*b^2*c^5*d^3*x^5 + 15*b^2*c^4*d^3*x^4 + 2 0*b^2*c^3*d^3*x^3 + 9*b^2*d^3)*log(c*x + 1))*log(-c*x + 1))/c^2
\[ \int x (d+c d x)^3 (a+b \text {arctanh}(c x))^2 \, dx=\int { {\left (c d x + d\right )}^{3} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2} x \,d x } \]
Timed out. \[ \int x (d+c d x)^3 (a+b \text {arctanh}(c x))^2 \, dx=\int x\,{\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^2\,{\left (d+c\,d\,x\right )}^3 \,d x \]