Integrand size = 22, antiderivative size = 356 \[ \int x^3 (d+c d x)^2 (a+b \text {arctanh}(c x))^2 \, dx=\frac {5 a b d^2 x}{6 c^3}+\frac {3 b^2 d^2 x}{5 c^3}+\frac {31 b^2 d^2 x^2}{180 c^2}+\frac {b^2 d^2 x^3}{15 c}+\frac {1}{60} b^2 d^2 x^4-\frac {3 b^2 d^2 \text {arctanh}(c x)}{5 c^4}+\frac {5 b^2 d^2 x \text {arctanh}(c x)}{6 c^3}+\frac {2 b d^2 x^2 (a+b \text {arctanh}(c x))}{5 c^2}+\frac {5 b d^2 x^3 (a+b \text {arctanh}(c x))}{18 c}+\frac {1}{5} b d^2 x^4 (a+b \text {arctanh}(c x))+\frac {1}{15} b c d^2 x^5 (a+b \text {arctanh}(c x))-\frac {d^2 (a+b \text {arctanh}(c x))^2}{60 c^4}+\frac {1}{4} d^2 x^4 (a+b \text {arctanh}(c x))^2+\frac {2}{5} c d^2 x^5 (a+b \text {arctanh}(c x))^2+\frac {1}{6} c^2 d^2 x^6 (a+b \text {arctanh}(c x))^2-\frac {4 b d^2 (a+b \text {arctanh}(c x)) \log \left (\frac {2}{1-c x}\right )}{5 c^4}+\frac {53 b^2 d^2 \log \left (1-c^2 x^2\right )}{90 c^4}-\frac {2 b^2 d^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{5 c^4} \]
5/6*a*b*d^2*x/c^3+3/5*b^2*d^2*x/c^3+31/180*b^2*d^2*x^2/c^2+1/15*b^2*d^2*x^ 3/c+1/60*b^2*d^2*x^4-3/5*b^2*d^2*arctanh(c*x)/c^4+5/6*b^2*d^2*x*arctanh(c* x)/c^3+2/5*b*d^2*x^2*(a+b*arctanh(c*x))/c^2+5/18*b*d^2*x^3*(a+b*arctanh(c* x))/c+1/5*b*d^2*x^4*(a+b*arctanh(c*x))+1/15*b*c*d^2*x^5*(a+b*arctanh(c*x)) -1/60*d^2*(a+b*arctanh(c*x))^2/c^4+1/4*d^2*x^4*(a+b*arctanh(c*x))^2+2/5*c* d^2*x^5*(a+b*arctanh(c*x))^2+1/6*c^2*d^2*x^6*(a+b*arctanh(c*x))^2-4/5*b*d^ 2*(a+b*arctanh(c*x))*ln(2/(-c*x+1))/c^4+53/90*b^2*d^2*ln(-c^2*x^2+1)/c^4-2 /5*b^2*d^2*polylog(2,1-2/(-c*x+1))/c^4
Time = 0.73 (sec) , antiderivative size = 329, normalized size of antiderivative = 0.92 \[ \int x^3 (d+c d x)^2 (a+b \text {arctanh}(c x))^2 \, dx=\frac {d^2 \left (-108 a b-34 b^2+150 a b c x+108 b^2 c x+72 a b c^2 x^2+31 b^2 c^2 x^2+50 a b c^3 x^3+12 b^2 c^3 x^3+45 a^2 c^4 x^4+36 a b c^4 x^4+3 b^2 c^4 x^4+72 a^2 c^5 x^5+12 a b c^5 x^5+30 a^2 c^6 x^6+3 b^2 \left (-49+15 c^4 x^4+24 c^5 x^5+10 c^6 x^6\right ) \text {arctanh}(c x)^2+2 b \text {arctanh}(c x) \left (3 a c^4 x^4 \left (15+24 c x+10 c^2 x^2\right )+b \left (-54+75 c x+36 c^2 x^2+25 c^3 x^3+18 c^4 x^4+6 c^5 x^5\right )-72 b \log \left (1+e^{-2 \text {arctanh}(c x)}\right )\right )+75 a b \log (1-c x)-75 a b \log (1+c x)+106 b^2 \log \left (1-c^2 x^2\right )+72 a b \log \left (-1+c^2 x^2\right )+72 b^2 \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )\right )}{180 c^4} \]
(d^2*(-108*a*b - 34*b^2 + 150*a*b*c*x + 108*b^2*c*x + 72*a*b*c^2*x^2 + 31* b^2*c^2*x^2 + 50*a*b*c^3*x^3 + 12*b^2*c^3*x^3 + 45*a^2*c^4*x^4 + 36*a*b*c^ 4*x^4 + 3*b^2*c^4*x^4 + 72*a^2*c^5*x^5 + 12*a*b*c^5*x^5 + 30*a^2*c^6*x^6 + 3*b^2*(-49 + 15*c^4*x^4 + 24*c^5*x^5 + 10*c^6*x^6)*ArcTanh[c*x]^2 + 2*b*A rcTanh[c*x]*(3*a*c^4*x^4*(15 + 24*c*x + 10*c^2*x^2) + b*(-54 + 75*c*x + 36 *c^2*x^2 + 25*c^3*x^3 + 18*c^4*x^4 + 6*c^5*x^5) - 72*b*Log[1 + E^(-2*ArcTa nh[c*x])]) + 75*a*b*Log[1 - c*x] - 75*a*b*Log[1 + c*x] + 106*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] )]))/(180*c^4)
Time = 1.22 (sec) , antiderivative size = 356, 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.
| \(\displaystyle \int x^3 (c d x+d)^2 (a+b \text {arctanh}(c x))^2 \, dx\) |
| \(\Big \downarrow \) 6502 |
| \(\displaystyle \int \left (c^2 d^2 x^5 (a+b \text {arctanh}(c x))^2+2 c d^2 x^4 (a+b \text {arctanh}(c x))^2+d^2 x^3 (a+b \text {arctanh}(c x))^2\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {d^2 (a+b \text {arctanh}(c x))^2}{60 c^4}-\frac {4 b d^2 \log \left (\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))}{5 c^4}+\frac {1}{6} c^2 d^2 x^6 (a+b \text {arctanh}(c x))^2+\frac {2 b d^2 x^2 (a+b \text {arctanh}(c x))}{5 c^2}+\frac {2}{5} c d^2 x^5 (a+b \text {arctanh}(c x))^2+\frac {1}{15} b c d^2 x^5 (a+b \text {arctanh}(c x))+\frac {1}{4} d^2 x^4 (a+b \text {arctanh}(c x))^2+\frac {1}{5} b d^2 x^4 (a+b \text {arctanh}(c x))+\frac {5 b d^2 x^3 (a+b \text {arctanh}(c x))}{18 c}+\frac {5 a b d^2 x}{6 c^3}-\frac {3 b^2 d^2 \text {arctanh}(c x)}{5 c^4}+\frac {5 b^2 d^2 x \text {arctanh}(c x)}{6 c^3}-\frac {2 b^2 d^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{5 c^4}+\frac {3 b^2 d^2 x}{5 c^3}+\frac {31 b^2 d^2 x^2}{180 c^2}+\frac {53 b^2 d^2 \log \left (1-c^2 x^2\right )}{90 c^4}+\frac {b^2 d^2 x^3}{15 c}+\frac {1}{60} b^2 d^2 x^4\) |
(5*a*b*d^2*x)/(6*c^3) + (3*b^2*d^2*x)/(5*c^3) + (31*b^2*d^2*x^2)/(180*c^2) + (b^2*d^2*x^3)/(15*c) + (b^2*d^2*x^4)/60 - (3*b^2*d^2*ArcTanh[c*x])/(5*c ^4) + (5*b^2*d^2*x*ArcTanh[c*x])/(6*c^3) + (2*b*d^2*x^2*(a + b*ArcTanh[c*x ]))/(5*c^2) + (5*b*d^2*x^3*(a + b*ArcTanh[c*x]))/(18*c) + (b*d^2*x^4*(a + b*ArcTanh[c*x]))/5 + (b*c*d^2*x^5*(a + b*ArcTanh[c*x]))/15 - (d^2*(a + b*A rcTanh[c*x])^2)/(60*c^4) + (d^2*x^4*(a + b*ArcTanh[c*x])^2)/4 + (2*c*d^2*x ^5*(a + b*ArcTanh[c*x])^2)/5 + (c^2*d^2*x^6*(a + b*ArcTanh[c*x])^2)/6 - (4 *b*d^2*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)])/(5*c^4) + (53*b^2*d^2*Log[1 - c^2*x^2])/(90*c^4) - (2*b^2*d^2*PolyLog[2, 1 - 2/(1 - c*x)])/(5*c^4)
3.1.76.3.1 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 = 1.83 (sec) , antiderivative size = 374, normalized size of antiderivative = 1.05
| method | result | size |
| parts | \(d^{2} a^{2} \left (\frac {1}{6} c^{2} x^{6}+\frac {2}{5} c \,x^{5}+\frac {1}{4} x^{4}\right )+\frac {d^{2} b^{2} \left (\frac {c^{6} x^{6} \operatorname {arctanh}\left (c x \right )^{2}}{6}+\frac {2 c^{5} x^{5} \operatorname {arctanh}\left (c x \right )^{2}}{5}+\frac {c^{4} x^{4} \operatorname {arctanh}\left (c x \right )^{2}}{4}+\frac {c^{5} x^{5} \operatorname {arctanh}\left (c x \right )}{15}+\frac {c^{4} x^{4} \operatorname {arctanh}\left (c x \right )}{5}+\frac {5 c^{3} x^{3} \operatorname {arctanh}\left (c x \right )}{18}+\frac {2 c^{2} x^{2} \operatorname {arctanh}\left (c x \right )}{5}+\frac {5 c x \,\operatorname {arctanh}\left (c x \right )}{6}+\frac {49 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )}{60}-\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )}{60}-\frac {2 \operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{5}-\frac {49 \ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{120}+\frac {49 \ln \left (c x -1\right )^{2}}{240}-\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{120}+\frac {\ln \left (c x +1\right )^{2}}{240}+\frac {c^{4} x^{4}}{60}+\frac {c^{3} x^{3}}{15}+\frac {31 c^{2} x^{2}}{180}+\frac {3 c x}{5}+\frac {8 \ln \left (c x -1\right )}{9}+\frac {13 \ln \left (c x +1\right )}{45}\right )}{c^{4}}+\frac {2 a b \,d^{2} \left (\frac {c^{6} x^{6} \operatorname {arctanh}\left (c x \right )}{6}+\frac {2 c^{5} x^{5} \operatorname {arctanh}\left (c x \right )}{5}+\frac {c^{4} x^{4} \operatorname {arctanh}\left (c x \right )}{4}+\frac {c^{5} x^{5}}{30}+\frac {c^{4} x^{4}}{10}+\frac {5 c^{3} x^{3}}{36}+\frac {c^{2} x^{2}}{5}+\frac {5 c x}{12}+\frac {49 \ln \left (c x -1\right )}{120}-\frac {\ln \left (c x +1\right )}{120}\right )}{c^{4}}\) | \(374\) |
| derivativedivides | \(\frac {d^{2} a^{2} \left (\frac {1}{6} c^{6} x^{6}+\frac {2}{5} c^{5} x^{5}+\frac {1}{4} c^{4} x^{4}\right )+d^{2} b^{2} \left (\frac {c^{6} x^{6} \operatorname {arctanh}\left (c x \right )^{2}}{6}+\frac {2 c^{5} x^{5} \operatorname {arctanh}\left (c x \right )^{2}}{5}+\frac {c^{4} x^{4} \operatorname {arctanh}\left (c x \right )^{2}}{4}+\frac {c^{5} x^{5} \operatorname {arctanh}\left (c x \right )}{15}+\frac {c^{4} x^{4} \operatorname {arctanh}\left (c x \right )}{5}+\frac {5 c^{3} x^{3} \operatorname {arctanh}\left (c x \right )}{18}+\frac {2 c^{2} x^{2} \operatorname {arctanh}\left (c x \right )}{5}+\frac {5 c x \,\operatorname {arctanh}\left (c x \right )}{6}+\frac {49 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )}{60}-\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )}{60}-\frac {2 \operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{5}-\frac {49 \ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{120}+\frac {49 \ln \left (c x -1\right )^{2}}{240}-\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{120}+\frac {\ln \left (c x +1\right )^{2}}{240}+\frac {c^{4} x^{4}}{60}+\frac {c^{3} x^{3}}{15}+\frac {31 c^{2} x^{2}}{180}+\frac {3 c x}{5}+\frac {8 \ln \left (c x -1\right )}{9}+\frac {13 \ln \left (c x +1\right )}{45}\right )+2 a b \,d^{2} \left (\frac {c^{6} x^{6} \operatorname {arctanh}\left (c x \right )}{6}+\frac {2 c^{5} x^{5} \operatorname {arctanh}\left (c x \right )}{5}+\frac {c^{4} x^{4} \operatorname {arctanh}\left (c x \right )}{4}+\frac {c^{5} x^{5}}{30}+\frac {c^{4} x^{4}}{10}+\frac {5 c^{3} x^{3}}{36}+\frac {c^{2} x^{2}}{5}+\frac {5 c x}{12}+\frac {49 \ln \left (c x -1\right )}{120}-\frac {\ln \left (c x +1\right )}{120}\right )}{c^{4}}\) | \(377\) |
| default | \(\frac {d^{2} a^{2} \left (\frac {1}{6} c^{6} x^{6}+\frac {2}{5} c^{5} x^{5}+\frac {1}{4} c^{4} x^{4}\right )+d^{2} b^{2} \left (\frac {c^{6} x^{6} \operatorname {arctanh}\left (c x \right )^{2}}{6}+\frac {2 c^{5} x^{5} \operatorname {arctanh}\left (c x \right )^{2}}{5}+\frac {c^{4} x^{4} \operatorname {arctanh}\left (c x \right )^{2}}{4}+\frac {c^{5} x^{5} \operatorname {arctanh}\left (c x \right )}{15}+\frac {c^{4} x^{4} \operatorname {arctanh}\left (c x \right )}{5}+\frac {5 c^{3} x^{3} \operatorname {arctanh}\left (c x \right )}{18}+\frac {2 c^{2} x^{2} \operatorname {arctanh}\left (c x \right )}{5}+\frac {5 c x \,\operatorname {arctanh}\left (c x \right )}{6}+\frac {49 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )}{60}-\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )}{60}-\frac {2 \operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{5}-\frac {49 \ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{120}+\frac {49 \ln \left (c x -1\right )^{2}}{240}-\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{120}+\frac {\ln \left (c x +1\right )^{2}}{240}+\frac {c^{4} x^{4}}{60}+\frac {c^{3} x^{3}}{15}+\frac {31 c^{2} x^{2}}{180}+\frac {3 c x}{5}+\frac {8 \ln \left (c x -1\right )}{9}+\frac {13 \ln \left (c x +1\right )}{45}\right )+2 a b \,d^{2} \left (\frac {c^{6} x^{6} \operatorname {arctanh}\left (c x \right )}{6}+\frac {2 c^{5} x^{5} \operatorname {arctanh}\left (c x \right )}{5}+\frac {c^{4} x^{4} \operatorname {arctanh}\left (c x \right )}{4}+\frac {c^{5} x^{5}}{30}+\frac {c^{4} x^{4}}{10}+\frac {5 c^{3} x^{3}}{36}+\frac {c^{2} x^{2}}{5}+\frac {5 c x}{12}+\frac {49 \ln \left (c x -1\right )}{120}-\frac {\ln \left (c x +1\right )}{120}\right )}{c^{4}}\) | \(377\) |
| risch | \(\frac {b^{2} d^{2} x^{4}}{60}+\frac {5 a b \,d^{2} x}{6 c^{3}}+\frac {3 b^{2} d^{2} x}{5 c^{3}}+\frac {31 b^{2} d^{2} x^{2}}{180 c^{2}}+\frac {b^{2} d^{2} x^{3}}{15 c}-\frac {2 d^{2} c a b \ln \left (-c x +1\right ) x^{5}}{5}-\frac {d^{2} c^{2} a b \ln \left (-c x +1\right ) x^{6}}{6}+\frac {8 d^{2} b^{2} \ln \left (-c x +1\right )}{9 c^{4}}+\frac {2 d^{2} b^{2} \operatorname {dilog}\left (-\frac {c x}{2}+\frac {1}{2}\right )}{5 c^{4}}+\frac {d^{2} b \,x^{4} a}{5}+\frac {2 d^{2} c \,x^{5} a^{2}}{5}+\frac {d^{2} c^{2} x^{6} a^{2}}{6}-\frac {49 d^{2} b^{2} \ln \left (-c x +1\right )^{2}}{240 c^{4}}+\frac {d^{2} b^{2} \ln \left (-c x +1\right )^{2} x^{4}}{16}-\frac {d^{2} b^{2} \ln \left (-c x +1\right ) x^{4}}{10}-\frac {49 d^{2} a^{2}}{60 c^{4}}+\frac {d^{2} x^{4} a^{2}}{4}+\left (-\frac {d^{2} b^{2} x^{4} \left (10 c^{2} x^{2}+24 c x +15\right ) \ln \left (-c x +1\right )}{120}-\frac {d^{2} b \left (-60 a \,c^{6} x^{6}-144 c^{5} x^{5} a -12 b \,c^{5} x^{5}-90 c^{4} x^{4} a -36 b \,c^{4} x^{4}-50 b \,c^{3} x^{3}-72 b \,c^{2} x^{2}-150 b c x -147 b \ln \left (-c x +1\right )\right )}{360 c^{4}}\right ) \ln \left (c x +1\right )-\frac {d^{2} b a \ln \left (-c x -1\right )}{60 c^{4}}-\frac {2 d^{2} b^{2} \ln \left (\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (-c x +1\right )}{5 c^{4}}+\frac {2 d^{2} b^{2} \ln \left (\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{5 c^{4}}+\frac {49 d^{2} b \ln \left (-c x +1\right ) a}{60 c^{4}}+\frac {d^{2} c \,b^{2} \ln \left (-c x +1\right )^{2} x^{5}}{10}+\frac {d^{2} c^{2} b^{2} \ln \left (-c x +1\right )^{2} x^{6}}{24}-\frac {5 d^{2} b^{2} \ln \left (-c x +1\right ) x^{3}}{36 c}-\frac {d^{2} b^{2} \ln \left (-c x +1\right ) x^{2}}{5 c^{2}}-\frac {5 d^{2} b^{2} \ln \left (-c x +1\right ) x}{12 c^{3}}-\frac {d^{2} b^{2} c \ln \left (-c x +1\right ) x^{5}}{30}+\frac {2 d^{2} b a \,x^{2}}{5 c^{2}}+\frac {5 d^{2} b a \,x^{3}}{18 c}+\frac {d^{2} b c \,x^{5} a}{15}+\frac {d^{2} b^{2} \left (10 c^{6} x^{6}+24 c^{5} x^{5}+15 c^{4} x^{4}-1\right ) \ln \left (c x +1\right )^{2}}{240 c^{4}}-\frac {d^{2} a b \ln \left (-c x +1\right ) x^{4}}{4}-\frac {16 d^{2} b a}{9 c^{4}}+\frac {13 d^{2} b^{2} \ln \left (-c x -1\right )}{45 c^{4}}-\frac {77 d^{2} b^{2}}{90 c^{4}}\) | \(728\) |
d^2*a^2*(1/6*c^2*x^6+2/5*c*x^5+1/4*x^4)+d^2*b^2/c^4*(1/6*c^6*x^6*arctanh(c *x)^2+2/5*c^5*x^5*arctanh(c*x)^2+1/4*c^4*x^4*arctanh(c*x)^2+1/15*c^5*x^5*a rctanh(c*x)+1/5*c^4*x^4*arctanh(c*x)+5/18*c^3*x^3*arctanh(c*x)+2/5*c^2*x^2 *arctanh(c*x)+5/6*c*x*arctanh(c*x)+49/60*arctanh(c*x)*ln(c*x-1)-1/60*arcta nh(c*x)*ln(c*x+1)-2/5*dilog(1/2*c*x+1/2)-49/120*ln(c*x-1)*ln(1/2*c*x+1/2)+ 49/240*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)+1/24 0*ln(c*x+1)^2+1/60*c^4*x^4+1/15*c^3*x^3+31/180*c^2*x^2+3/5*c*x+8/9*ln(c*x- 1)+13/45*ln(c*x+1))+2*a*b*d^2/c^4*(1/6*c^6*x^6*arctanh(c*x)+2/5*c^5*x^5*ar ctanh(c*x)+1/4*c^4*x^4*arctanh(c*x)+1/30*c^5*x^5+1/10*c^4*x^4+5/36*c^3*x^3 +1/5*c^2*x^2+5/12*c*x+49/120*ln(c*x-1)-1/120*ln(c*x+1))
\[ \int x^3 (d+c d x)^2 (a+b \text {arctanh}(c x))^2 \, dx=\int { {\left (c d x + d\right )}^{2} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2} x^{3} \,d x } \]
integral(a^2*c^2*d^2*x^5 + 2*a^2*c*d^2*x^4 + a^2*d^2*x^3 + (b^2*c^2*d^2*x^ 5 + 2*b^2*c*d^2*x^4 + b^2*d^2*x^3)*arctanh(c*x)^2 + 2*(a*b*c^2*d^2*x^5 + 2 *a*b*c*d^2*x^4 + a*b*d^2*x^3)*arctanh(c*x), x)
\[ \int x^3 (d+c d x)^2 (a+b \text {arctanh}(c x))^2 \, dx=d^{2} \left (\int a^{2} x^{3}\, dx + \int 2 a^{2} c x^{4}\, dx + \int a^{2} c^{2} x^{5}\, dx + \int b^{2} x^{3} \operatorname {atanh}^{2}{\left (c x \right )}\, dx + \int 2 a b x^{3} \operatorname {atanh}{\left (c x \right )}\, dx + \int 2 b^{2} c x^{4} \operatorname {atanh}^{2}{\left (c x \right )}\, dx + \int b^{2} c^{2} x^{5} \operatorname {atanh}^{2}{\left (c x \right )}\, dx + \int 4 a b c x^{4} \operatorname {atanh}{\left (c x \right )}\, dx + \int 2 a b c^{2} x^{5} \operatorname {atanh}{\left (c x \right )}\, dx\right ) \]
d**2*(Integral(a**2*x**3, x) + Integral(2*a**2*c*x**4, x) + Integral(a**2* c**2*x**5, x) + Integral(b**2*x**3*atanh(c*x)**2, x) + Integral(2*a*b*x**3 *atanh(c*x), x) + Integral(2*b**2*c*x**4*atanh(c*x)**2, x) + Integral(b**2 *c**2*x**5*atanh(c*x)**2, x) + Integral(4*a*b*c*x**4*atanh(c*x), x) + Inte gral(2*a*b*c**2*x**5*atanh(c*x), x))
Leaf count of result is larger than twice the leaf count of optimal. 766 vs. \(2 (317) = 634\).
Time = 0.45 (sec) , antiderivative size = 766, normalized size of antiderivative = 2.15 \[ \int x^3 (d+c d x)^2 (a+b \text {arctanh}(c x))^2 \, dx=\frac {1}{6} \, a^{2} c^{2} d^{2} x^{6} + \frac {2}{5} \, a^{2} c d^{2} x^{5} + \frac {1}{4} \, b^{2} d^{2} x^{4} \operatorname {artanh}\left (c x\right )^{2} + \frac {1}{4} \, a^{2} d^{2} x^{4} + \frac {1}{90} \, {\left (30 \, x^{6} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {2 \, {\left (3 \, c^{4} x^{5} + 5 \, c^{2} x^{3} + 15 \, x\right )}}{c^{6}} - \frac {15 \, \log \left (c x + 1\right )}{c^{7}} + \frac {15 \, \log \left (c x - 1\right )}{c^{7}}\right )}\right )} a b c^{2} d^{2} + \frac {1}{5} \, {\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 d^{2} + \frac {1}{12} \, {\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 d^{2} + \frac {1}{48} \, {\left (4 \, 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 )} \operatorname {artanh}\left (c x\right ) + \frac {4 \, c^{2} x^{2} - 2 \, {\left (3 \, \log \left (c x - 1\right ) - 8\right )} \log \left (c x + 1\right ) + 3 \, \log \left (c x + 1\right )^{2} + 3 \, \log \left (c x - 1\right )^{2} + 16 \, \log \left (c x - 1\right )}{c^{4}}\right )} b^{2} d^{2} + \frac {2 \, {\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}}{5 \, c^{4}} - \frac {2 \, b^{2} d^{2} \log \left (c x + 1\right )}{45 \, c^{4}} + \frac {5 \, b^{2} d^{2} \log \left (c x - 1\right )}{9 \, c^{4}} + \frac {6 \, b^{2} c^{4} d^{2} x^{4} + 24 \, b^{2} c^{3} d^{2} x^{3} + 32 \, b^{2} c^{2} d^{2} x^{2} + 216 \, b^{2} c d^{2} x + 3 \, {\left (5 \, b^{2} c^{6} d^{2} x^{6} + 12 \, b^{2} c^{5} d^{2} x^{5} + 7 \, b^{2} d^{2}\right )} \log \left (c x + 1\right )^{2} + 3 \, {\left (5 \, b^{2} c^{6} d^{2} x^{6} + 12 \, b^{2} c^{5} d^{2} x^{5} - 17 \, b^{2} d^{2}\right )} \log \left (-c x + 1\right )^{2} + 4 \, {\left (3 \, b^{2} c^{5} d^{2} x^{5} + 9 \, b^{2} c^{4} d^{2} x^{4} + 5 \, b^{2} c^{3} d^{2} x^{3} + 18 \, b^{2} c^{2} d^{2} x^{2} + 15 \, b^{2} c d^{2} x\right )} \log \left (c x + 1\right ) - 2 \, {\left (6 \, b^{2} c^{5} d^{2} x^{5} + 18 \, b^{2} c^{4} d^{2} x^{4} + 10 \, b^{2} c^{3} d^{2} x^{3} + 36 \, b^{2} c^{2} d^{2} x^{2} + 30 \, b^{2} c d^{2} x + 3 \, {\left (5 \, b^{2} c^{6} d^{2} x^{6} + 12 \, b^{2} c^{5} d^{2} x^{5} + 7 \, b^{2} d^{2}\right )} \log \left (c x + 1\right )\right )} \log \left (-c x + 1\right )}{360 \, c^{4}} \]
1/6*a^2*c^2*d^2*x^6 + 2/5*a^2*c*d^2*x^5 + 1/4*b^2*d^2*x^4*arctanh(c*x)^2 + 1/4*a^2*d^2*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^2*d^2 + 1 /5*(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*d^2 + 1/12*(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*d^2 + 1/48*(4*c*(2*(c^2*x^3 + 3*x )/c^4 - 3*log(c*x + 1)/c^5 + 3*log(c*x - 1)/c^5)*arctanh(c*x) + (4*c^2*x^2 - 2*(3*log(c*x - 1) - 8)*log(c*x + 1) + 3*log(c*x + 1)^2 + 3*log(c*x - 1) ^2 + 16*log(c*x - 1))/c^4)*b^2*d^2 + 2/5*(log(c*x + 1)*log(-1/2*c*x + 1/2) + dilog(1/2*c*x + 1/2))*b^2*d^2/c^4 - 2/45*b^2*d^2*log(c*x + 1)/c^4 + 5/9 *b^2*d^2*log(c*x - 1)/c^4 + 1/360*(6*b^2*c^4*d^2*x^4 + 24*b^2*c^3*d^2*x^3 + 32*b^2*c^2*d^2*x^2 + 216*b^2*c*d^2*x + 3*(5*b^2*c^6*d^2*x^6 + 12*b^2*c^5 *d^2*x^5 + 7*b^2*d^2)*log(c*x + 1)^2 + 3*(5*b^2*c^6*d^2*x^6 + 12*b^2*c^5*d ^2*x^5 - 17*b^2*d^2)*log(-c*x + 1)^2 + 4*(3*b^2*c^5*d^2*x^5 + 9*b^2*c^4*d^ 2*x^4 + 5*b^2*c^3*d^2*x^3 + 18*b^2*c^2*d^2*x^2 + 15*b^2*c*d^2*x)*log(c*x + 1) - 2*(6*b^2*c^5*d^2*x^5 + 18*b^2*c^4*d^2*x^4 + 10*b^2*c^3*d^2*x^3 + 36* b^2*c^2*d^2*x^2 + 30*b^2*c*d^2*x + 3*(5*b^2*c^6*d^2*x^6 + 12*b^2*c^5*d^2*x ^5 + 7*b^2*d^2)*log(c*x + 1))*log(-c*x + 1))/c^4
Leaf count of result is larger than twice the leaf count of optimal. 1135 vs. \(2 (317) = 634\).
Time = 2.37 (sec) , antiderivative size = 1135, normalized size of antiderivative = 3.19 \[ \int x^3 (d+c d x)^2 (a+b \text {arctanh}(c x))^2 \, dx=\text {Too large to display} \]
1/63*(84*((c*x + 1)^5*b^2*d^2/(c*x - 1)^5 + (c*x + 1)^4*b^2*d^2/(c*x - 1)^ 4 + (c*x + 1)^3*b^2*d^2/(c*x - 1)^3)*log(-(c*x + 1)/(c*x - 1))^2/((c*x + 1 )^8*c^7/(c*x - 1)^8 - 8*(c*x + 1)^7*c^7/(c*x - 1)^7 + 28*(c*x + 1)^6*c^7/( c*x - 1)^6 - 56*(c*x + 1)^5*c^7/(c*x - 1)^5 + 70*(c*x + 1)^4*c^7/(c*x - 1) ^4 - 56*(c*x + 1)^3*c^7/(c*x - 1)^3 + 28*(c*x + 1)^2*c^7/(c*x - 1)^2 - 8*( c*x + 1)*c^7/(c*x - 1) + c^7) + 2*(168*(c*x + 1)^5*a*b*d^2/(c*x - 1)^5 + 1 68*(c*x + 1)^4*a*b*d^2/(c*x - 1)^4 + 168*(c*x + 1)^3*a*b*d^2/(c*x - 1)^3 + 28*(c*x + 1)^5*b^2*d^2/(c*x - 1)^5 - 35*(c*x + 1)^4*b^2*d^2/(c*x - 1)^4 + 28*(c*x + 1)^3*b^2*d^2/(c*x - 1)^3 - 28*(c*x + 1)^2*b^2*d^2/(c*x - 1)^2 + 8*(c*x + 1)*b^2*d^2/(c*x - 1) - b^2*d^2)*log(-(c*x + 1)/(c*x - 1))/((c*x + 1)^8*c^7/(c*x - 1)^8 - 8*(c*x + 1)^7*c^7/(c*x - 1)^7 + 28*(c*x + 1)^6*c^ 7/(c*x - 1)^6 - 56*(c*x + 1)^5*c^7/(c*x - 1)^5 + 70*(c*x + 1)^4*c^7/(c*x - 1)^4 - 56*(c*x + 1)^3*c^7/(c*x - 1)^3 + 28*(c*x + 1)^2*c^7/(c*x - 1)^2 - 8*(c*x + 1)*c^7/(c*x - 1) + c^7) + (336*(c*x + 1)^5*a^2*d^2/(c*x - 1)^5 + 336*(c*x + 1)^4*a^2*d^2/(c*x - 1)^4 + 336*(c*x + 1)^3*a^2*d^2/(c*x - 1)^3 + 112*(c*x + 1)^5*a*b*d^2/(c*x - 1)^5 - 140*(c*x + 1)^4*a*b*d^2/(c*x - 1)^ 4 + 112*(c*x + 1)^3*a*b*d^2/(c*x - 1)^3 - 112*(c*x + 1)^2*a*b*d^2/(c*x - 1 )^2 + 32*(c*x + 1)*a*b*d^2/(c*x - 1) - 4*a*b*d^2 - 2*(c*x + 1)^7*b^2*d^2/( c*x - 1)^7 + 15*(c*x + 1)^6*b^2*d^2/(c*x - 1)^6 - 30*(c*x + 1)^5*b^2*d^2/( c*x - 1)^5 + 34*(c*x + 1)^4*b^2*d^2/(c*x - 1)^4 - 30*(c*x + 1)^3*b^2*d^...
Timed out. \[ \int x^3 (d+c d x)^2 (a+b \text {arctanh}(c x))^2 \, dx=\int x^3\,{\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^2\,{\left (d+c\,d\,x\right )}^2 \,d x \]