Integrand size = 20, antiderivative size = 188 \[ \int \frac {x (a+b \text {arctanh}(c x))^2}{(d+c d x)^2} \, dx=\frac {b^2}{2 c^2 d^2 (1+c x)}-\frac {b^2 \text {arctanh}(c x)}{2 c^2 d^2}+\frac {b (a+b \text {arctanh}(c x))}{c^2 d^2 (1+c x)}-\frac {(a+b \text {arctanh}(c x))^2}{2 c^2 d^2}+\frac {(a+b \text {arctanh}(c x))^2}{c^2 d^2 (1+c x)}-\frac {(a+b \text {arctanh}(c x))^2 \log \left (\frac {2}{1+c x}\right )}{c^2 d^2}+\frac {b (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+c x}\right )}{c^2 d^2}+\frac {b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1+c x}\right )}{2 c^2 d^2} \]
1/2*b^2/c^2/d^2/(c*x+1)-1/2*b^2*arctanh(c*x)/c^2/d^2+b*(a+b*arctanh(c*x))/ c^2/d^2/(c*x+1)-1/2*(a+b*arctanh(c*x))^2/c^2/d^2+(a+b*arctanh(c*x))^2/c^2/ d^2/(c*x+1)-(a+b*arctanh(c*x))^2*ln(2/(c*x+1))/c^2/d^2+b*(a+b*arctanh(c*x) )*polylog(2,1-2/(c*x+1))/c^2/d^2+1/2*b^2*polylog(3,1-2/(c*x+1))/c^2/d^2
Time = 0.68 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.24 \[ \int \frac {x (a+b \text {arctanh}(c x))^2}{(d+c d x)^2} \, dx=\frac {\frac {4 a^2}{1+c x}+4 a^2 \log (1+c x)+2 a b \left (\cosh (2 \text {arctanh}(c x))+2 \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )+2 \text {arctanh}(c x) \left (\cosh (2 \text {arctanh}(c x))-2 \log \left (1+e^{-2 \text {arctanh}(c x)}\right )-\sinh (2 \text {arctanh}(c x))\right )-\sinh (2 \text {arctanh}(c x))\right )+b^2 \left (\cosh (2 \text {arctanh}(c x))+2 \text {arctanh}(c x) \cosh (2 \text {arctanh}(c x))+2 \text {arctanh}(c x)^2 \cosh (2 \text {arctanh}(c x))-4 \text {arctanh}(c x)^2 \log \left (1+e^{-2 \text {arctanh}(c x)}\right )+4 \text {arctanh}(c x) \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )+2 \operatorname {PolyLog}\left (3,-e^{-2 \text {arctanh}(c x)}\right )-\sinh (2 \text {arctanh}(c x))-2 \text {arctanh}(c x) \sinh (2 \text {arctanh}(c x))-2 \text {arctanh}(c x)^2 \sinh (2 \text {arctanh}(c x))\right )}{4 c^2 d^2} \]
((4*a^2)/(1 + c*x) + 4*a^2*Log[1 + c*x] + 2*a*b*(Cosh[2*ArcTanh[c*x]] + 2* PolyLog[2, -E^(-2*ArcTanh[c*x])] + 2*ArcTanh[c*x]*(Cosh[2*ArcTanh[c*x]] - 2*Log[1 + E^(-2*ArcTanh[c*x])] - Sinh[2*ArcTanh[c*x]]) - Sinh[2*ArcTanh[c* x]]) + b^2*(Cosh[2*ArcTanh[c*x]] + 2*ArcTanh[c*x]*Cosh[2*ArcTanh[c*x]] + 2 *ArcTanh[c*x]^2*Cosh[2*ArcTanh[c*x]] - 4*ArcTanh[c*x]^2*Log[1 + E^(-2*ArcT anh[c*x])] + 4*ArcTanh[c*x]*PolyLog[2, -E^(-2*ArcTanh[c*x])] + 2*PolyLog[3 , -E^(-2*ArcTanh[c*x])] - Sinh[2*ArcTanh[c*x]] - 2*ArcTanh[c*x]*Sinh[2*Arc Tanh[c*x]] - 2*ArcTanh[c*x]^2*Sinh[2*ArcTanh[c*x]]))/(4*c^2*d^2)
Time = 0.59 (sec) , antiderivative size = 188, 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.
| \(\displaystyle \int \frac {x (a+b \text {arctanh}(c x))^2}{(c d x+d)^2} \, dx\) |
| \(\Big \downarrow \) 6502 |
| \(\displaystyle \int \left (\frac {(a+b \text {arctanh}(c x))^2}{c d^2 (c x+1)}-\frac {(a+b \text {arctanh}(c x))^2}{c d^2 (c x+1)^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {b \operatorname {PolyLog}\left (2,1-\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))}{c^2 d^2}+\frac {b (a+b \text {arctanh}(c x))}{c^2 d^2 (c x+1)}+\frac {(a+b \text {arctanh}(c x))^2}{c^2 d^2 (c x+1)}-\frac {(a+b \text {arctanh}(c x))^2}{2 c^2 d^2}-\frac {\log \left (\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))^2}{c^2 d^2}-\frac {b^2 \text {arctanh}(c x)}{2 c^2 d^2}+\frac {b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{c x+1}\right )}{2 c^2 d^2}+\frac {b^2}{2 c^2 d^2 (c x+1)}\) |
b^2/(2*c^2*d^2*(1 + c*x)) - (b^2*ArcTanh[c*x])/(2*c^2*d^2) + (b*(a + b*Arc Tanh[c*x]))/(c^2*d^2*(1 + c*x)) - (a + b*ArcTanh[c*x])^2/(2*c^2*d^2) + (a + b*ArcTanh[c*x])^2/(c^2*d^2*(1 + c*x)) - ((a + b*ArcTanh[c*x])^2*Log[2/(1 + c*x)])/(c^2*d^2) + (b*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 + c*x)]) /(c^2*d^2) + (b^2*PolyLog[3, 1 - 2/(1 + c*x)])/(2*c^2*d^2)
3.2.6.3.1 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])
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 2.35 (sec) , antiderivative size = 729, normalized size of antiderivative = 3.88
| method | result | size |
| derivativedivides | \(\frac {\frac {a^{2} \left (\ln \left (c x +1\right )+\frac {1}{c x +1}\right )}{d^{2}}+\frac {b^{2} \left (\operatorname {arctanh}\left (c x \right )^{2} \ln \left (c x +1\right )+\frac {\operatorname {arctanh}\left (c x \right )^{2}}{c x +1}-2 \operatorname {arctanh}\left (c x \right )^{2} \ln \left (\frac {c x +1}{\sqrt {-c^{2} x^{2}+1}}\right )+\frac {2 \operatorname {arctanh}\left (c x \right )^{3}}{3}-\frac {\operatorname {arctanh}\left (c x \right ) \left (c x -1\right )}{2 \left (c x +1\right )}-\frac {c x -1}{4 \left (c x +1\right )}-\frac {\left (-i \pi \,\operatorname {csgn}\left (\frac {i}{1-\frac {\left (c x +1\right )^{2}}{c^{2} x^{2}-1}}\right ) \operatorname {csgn}\left (\frac {i \left (c x +1\right )^{2}}{c^{2} x^{2}-1}\right ) \operatorname {csgn}\left (\frac {i \left (c x +1\right )^{2}}{\left (c^{2} x^{2}-1\right ) \left (1-\frac {\left (c x +1\right )^{2}}{c^{2} x^{2}-1}\right )}\right )+i \pi \,\operatorname {csgn}\left (\frac {i}{1-\frac {\left (c x +1\right )^{2}}{c^{2} x^{2}-1}}\right ) \operatorname {csgn}\left (\frac {i \left (c x +1\right )^{2}}{\left (c^{2} x^{2}-1\right ) \left (1-\frac {\left (c x +1\right )^{2}}{c^{2} x^{2}-1}\right )}\right )^{2}+i \pi {\operatorname {csgn}\left (\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )}^{2} \operatorname {csgn}\left (\frac {i \left (c x +1\right )^{2}}{c^{2} x^{2}-1}\right )+2 i \pi \,\operatorname {csgn}\left (\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right ) \operatorname {csgn}\left (\frac {i \left (c x +1\right )^{2}}{c^{2} x^{2}-1}\right )^{2}+i \pi \operatorname {csgn}\left (\frac {i \left (c x +1\right )^{2}}{c^{2} x^{2}-1}\right )^{3}-i \pi \,\operatorname {csgn}\left (\frac {i \left (c x +1\right )^{2}}{c^{2} x^{2}-1}\right ) \operatorname {csgn}\left (\frac {i \left (c x +1\right )^{2}}{\left (c^{2} x^{2}-1\right ) \left (1-\frac {\left (c x +1\right )^{2}}{c^{2} x^{2}-1}\right )}\right )^{2}+1+i \pi \operatorname {csgn}\left (\frac {i \left (c x +1\right )^{2}}{\left (c^{2} x^{2}-1\right ) \left (1-\frac {\left (c x +1\right )^{2}}{c^{2} x^{2}-1}\right )}\right )^{3}+2 \ln \left (2\right )\right ) \operatorname {arctanh}\left (c x \right )^{2}}{2}-\operatorname {arctanh}\left (c x \right ) \operatorname {polylog}\left (2, -\frac {\left (c x +1\right )^{2}}{-c^{2} x^{2}+1}\right )+\frac {\operatorname {polylog}\left (3, -\frac {\left (c x +1\right )^{2}}{-c^{2} x^{2}+1}\right )}{2}\right )}{d^{2}}+\frac {2 a b \left (\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )+\frac {\operatorname {arctanh}\left (c x \right )}{c x +1}+\frac {\ln \left (c x -1\right )}{4}+\frac {1}{2 c x +2}-\frac {\ln \left (c x +1\right )}{4}+\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 )}{2}-\frac {\operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{2}-\frac {\ln \left (c x +1\right )^{2}}{4}\right )}{d^{2}}}{c^{2}}\) | \(729\) |
| default | \(\frac {\frac {a^{2} \left (\ln \left (c x +1\right )+\frac {1}{c x +1}\right )}{d^{2}}+\frac {b^{2} \left (\operatorname {arctanh}\left (c x \right )^{2} \ln \left (c x +1\right )+\frac {\operatorname {arctanh}\left (c x \right )^{2}}{c x +1}-2 \operatorname {arctanh}\left (c x \right )^{2} \ln \left (\frac {c x +1}{\sqrt {-c^{2} x^{2}+1}}\right )+\frac {2 \operatorname {arctanh}\left (c x \right )^{3}}{3}-\frac {\operatorname {arctanh}\left (c x \right ) \left (c x -1\right )}{2 \left (c x +1\right )}-\frac {c x -1}{4 \left (c x +1\right )}-\frac {\left (-i \pi \,\operatorname {csgn}\left (\frac {i}{1-\frac {\left (c x +1\right )^{2}}{c^{2} x^{2}-1}}\right ) \operatorname {csgn}\left (\frac {i \left (c x +1\right )^{2}}{c^{2} x^{2}-1}\right ) \operatorname {csgn}\left (\frac {i \left (c x +1\right )^{2}}{\left (c^{2} x^{2}-1\right ) \left (1-\frac {\left (c x +1\right )^{2}}{c^{2} x^{2}-1}\right )}\right )+i \pi \,\operatorname {csgn}\left (\frac {i}{1-\frac {\left (c x +1\right )^{2}}{c^{2} x^{2}-1}}\right ) \operatorname {csgn}\left (\frac {i \left (c x +1\right )^{2}}{\left (c^{2} x^{2}-1\right ) \left (1-\frac {\left (c x +1\right )^{2}}{c^{2} x^{2}-1}\right )}\right )^{2}+i \pi {\operatorname {csgn}\left (\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )}^{2} \operatorname {csgn}\left (\frac {i \left (c x +1\right )^{2}}{c^{2} x^{2}-1}\right )+2 i \pi \,\operatorname {csgn}\left (\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right ) \operatorname {csgn}\left (\frac {i \left (c x +1\right )^{2}}{c^{2} x^{2}-1}\right )^{2}+i \pi \operatorname {csgn}\left (\frac {i \left (c x +1\right )^{2}}{c^{2} x^{2}-1}\right )^{3}-i \pi \,\operatorname {csgn}\left (\frac {i \left (c x +1\right )^{2}}{c^{2} x^{2}-1}\right ) \operatorname {csgn}\left (\frac {i \left (c x +1\right )^{2}}{\left (c^{2} x^{2}-1\right ) \left (1-\frac {\left (c x +1\right )^{2}}{c^{2} x^{2}-1}\right )}\right )^{2}+1+i \pi \operatorname {csgn}\left (\frac {i \left (c x +1\right )^{2}}{\left (c^{2} x^{2}-1\right ) \left (1-\frac {\left (c x +1\right )^{2}}{c^{2} x^{2}-1}\right )}\right )^{3}+2 \ln \left (2\right )\right ) \operatorname {arctanh}\left (c x \right )^{2}}{2}-\operatorname {arctanh}\left (c x \right ) \operatorname {polylog}\left (2, -\frac {\left (c x +1\right )^{2}}{-c^{2} x^{2}+1}\right )+\frac {\operatorname {polylog}\left (3, -\frac {\left (c x +1\right )^{2}}{-c^{2} x^{2}+1}\right )}{2}\right )}{d^{2}}+\frac {2 a b \left (\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )+\frac {\operatorname {arctanh}\left (c x \right )}{c x +1}+\frac {\ln \left (c x -1\right )}{4}+\frac {1}{2 c x +2}-\frac {\ln \left (c x +1\right )}{4}+\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 )}{2}-\frac {\operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{2}-\frac {\ln \left (c x +1\right )^{2}}{4}\right )}{d^{2}}}{c^{2}}\) | \(729\) |
| parts | \(\frac {a^{2} \left (\frac {\ln \left (c x +1\right )}{c^{2}}+\frac {1}{c^{2} \left (c x +1\right )}\right )}{d^{2}}+\frac {b^{2} \left (\operatorname {arctanh}\left (c x \right )^{2} \ln \left (c x +1\right )+\frac {\operatorname {arctanh}\left (c x \right )^{2}}{c x +1}-2 \operatorname {arctanh}\left (c x \right )^{2} \ln \left (\frac {c x +1}{\sqrt {-c^{2} x^{2}+1}}\right )+\frac {2 \operatorname {arctanh}\left (c x \right )^{3}}{3}-\frac {\operatorname {arctanh}\left (c x \right ) \left (c x -1\right )}{2 \left (c x +1\right )}-\frac {c x -1}{4 \left (c x +1\right )}-\frac {\left (-i \pi \,\operatorname {csgn}\left (\frac {i}{1-\frac {\left (c x +1\right )^{2}}{c^{2} x^{2}-1}}\right ) \operatorname {csgn}\left (\frac {i \left (c x +1\right )^{2}}{c^{2} x^{2}-1}\right ) \operatorname {csgn}\left (\frac {i \left (c x +1\right )^{2}}{\left (c^{2} x^{2}-1\right ) \left (1-\frac {\left (c x +1\right )^{2}}{c^{2} x^{2}-1}\right )}\right )+i \pi \,\operatorname {csgn}\left (\frac {i}{1-\frac {\left (c x +1\right )^{2}}{c^{2} x^{2}-1}}\right ) \operatorname {csgn}\left (\frac {i \left (c x +1\right )^{2}}{\left (c^{2} x^{2}-1\right ) \left (1-\frac {\left (c x +1\right )^{2}}{c^{2} x^{2}-1}\right )}\right )^{2}+i \pi {\operatorname {csgn}\left (\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )}^{2} \operatorname {csgn}\left (\frac {i \left (c x +1\right )^{2}}{c^{2} x^{2}-1}\right )+2 i \pi \,\operatorname {csgn}\left (\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right ) \operatorname {csgn}\left (\frac {i \left (c x +1\right )^{2}}{c^{2} x^{2}-1}\right )^{2}+i \pi \operatorname {csgn}\left (\frac {i \left (c x +1\right )^{2}}{c^{2} x^{2}-1}\right )^{3}-i \pi \,\operatorname {csgn}\left (\frac {i \left (c x +1\right )^{2}}{c^{2} x^{2}-1}\right ) \operatorname {csgn}\left (\frac {i \left (c x +1\right )^{2}}{\left (c^{2} x^{2}-1\right ) \left (1-\frac {\left (c x +1\right )^{2}}{c^{2} x^{2}-1}\right )}\right )^{2}+1+i \pi \operatorname {csgn}\left (\frac {i \left (c x +1\right )^{2}}{\left (c^{2} x^{2}-1\right ) \left (1-\frac {\left (c x +1\right )^{2}}{c^{2} x^{2}-1}\right )}\right )^{3}+2 \ln \left (2\right )\right ) \operatorname {arctanh}\left (c x \right )^{2}}{2}-\operatorname {arctanh}\left (c x \right ) \operatorname {polylog}\left (2, -\frac {\left (c x +1\right )^{2}}{-c^{2} x^{2}+1}\right )+\frac {\operatorname {polylog}\left (3, -\frac {\left (c x +1\right )^{2}}{-c^{2} x^{2}+1}\right )}{2}\right )}{d^{2} c^{2}}+\frac {2 a b \left (\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )+\frac {\operatorname {arctanh}\left (c x \right )}{c x +1}+\frac {\ln \left (c x -1\right )}{4}+\frac {1}{2 c x +2}-\frac {\ln \left (c x +1\right )}{4}+\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 )}{2}-\frac {\operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{2}-\frac {\ln \left (c x +1\right )^{2}}{4}\right )}{d^{2} c^{2}}\) | \(739\) |
1/c^2*(a^2/d^2*(ln(c*x+1)+1/(c*x+1))+b^2/d^2*(arctanh(c*x)^2*ln(c*x+1)+1/( c*x+1)*arctanh(c*x)^2-2*arctanh(c*x)^2*ln((c*x+1)/(-c^2*x^2+1)^(1/2))+2/3* arctanh(c*x)^3-1/2*arctanh(c*x)*(c*x-1)/(c*x+1)-1/4*(c*x-1)/(c*x+1)-1/2*(- I*Pi*csgn(I/(1-(c*x+1)^2/(c^2*x^2-1)))*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)))+I*Pi*csgn(I/(1-(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+ 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))+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+I* Pi*csgn(I*(c*x+1)^2/(c^2*x^2-1))^3-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+1+I*Pi*csgn(I*(c*x+1 )^2/(c^2*x^2-1)/(1-(c*x+1)^2/(c^2*x^2-1)))^3+2*ln(2))*arctanh(c*x)^2-arcta nh(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*a*b/d^2*(arctanh(c*x)*ln(c*x+1)+1/(c*x+1)*arctanh(c*x)+1/4*ln(c *x-1)+1/2/(c*x+1)-1/4*ln(c*x+1)+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)-1/4*ln(c*x+1)^2))
\[ \int \frac {x (a+b \text {arctanh}(c x))^2}{(d+c d x)^2} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2} x}{{\left (c d x + d\right )}^{2}} \,d x } \]
integral((b^2*x*arctanh(c*x)^2 + 2*a*b*x*arctanh(c*x) + a^2*x)/(c^2*d^2*x^ 2 + 2*c*d^2*x + d^2), x)
\[ \int \frac {x (a+b \text {arctanh}(c x))^2}{(d+c d x)^2} \, dx=\frac {\int \frac {a^{2} x}{c^{2} x^{2} + 2 c x + 1}\, dx + \int \frac {b^{2} x \operatorname {atanh}^{2}{\left (c x \right )}}{c^{2} x^{2} + 2 c x + 1}\, dx + \int \frac {2 a b x \operatorname {atanh}{\left (c x \right )}}{c^{2} x^{2} + 2 c x + 1}\, dx}{d^{2}} \]
(Integral(a**2*x/(c**2*x**2 + 2*c*x + 1), x) + Integral(b**2*x*atanh(c*x)* *2/(c**2*x**2 + 2*c*x + 1), x) + Integral(2*a*b*x*atanh(c*x)/(c**2*x**2 + 2*c*x + 1), x))/d**2
\[ \int \frac {x (a+b \text {arctanh}(c x))^2}{(d+c d x)^2} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2} x}{{\left (c d x + d\right )}^{2}} \,d x } \]
a^2*(1/(c^3*d^2*x + c^2*d^2) + log(c*x + 1)/(c^2*d^2)) + 1/4*(b^2 + (b^2*c *x + b^2)*log(c*x + 1))*log(-c*x + 1)^2/(c^3*d^2*x + c^2*d^2) - integrate( -1/4*((b^2*c^2*x^2 - b^2*c*x)*log(c*x + 1)^2 + 4*(a*b*c^2*x^2 - a*b*c*x)*l og(c*x + 1) - 2*(2*a*b*c^2*x^2 + b^2 - (2*a*b*c - b^2*c)*x + (2*b^2*c^2*x^ 2 + b^2*c*x + b^2)*log(c*x + 1))*log(-c*x + 1))/(c^4*d^2*x^3 + c^3*d^2*x^2 - c^2*d^2*x - c*d^2), x)
\[ \int \frac {x (a+b \text {arctanh}(c x))^2}{(d+c d x)^2} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2} x}{{\left (c d x + d\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {x (a+b \text {arctanh}(c x))^2}{(d+c d x)^2} \, dx=\int \frac {x\,{\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^2}{{\left (d+c\,d\,x\right )}^2} \,d x \]