Integrand size = 18, antiderivative size = 221 \[ \int (e+f x) (a+b \text {arctanh}(c+d x))^2 \, dx=\frac {a b f x}{d}+\frac {b^2 f (c+d x) \text {arctanh}(c+d x)}{d^2}+\frac {(d e-c f) (a+b \text {arctanh}(c+d x))^2}{d^2}-\frac {\left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right ) (a+b \text {arctanh}(c+d x))^2}{2 d^2 f}+\frac {(e+f x)^2 (a+b \text {arctanh}(c+d x))^2}{2 f}-\frac {2 b (d e-c f) (a+b \text {arctanh}(c+d x)) \log \left (\frac {2}{1-c-d x}\right )}{d^2}+\frac {b^2 f \log \left (1-(c+d x)^2\right )}{2 d^2}-\frac {b^2 (d e-c f) \operatorname {PolyLog}\left (2,-\frac {1+c+d x}{1-c-d x}\right )}{d^2} \]
a*b*f*x/d+b^2*f*(d*x+c)*arctanh(d*x+c)/d^2+(-c*f+d*e)*(a+b*arctanh(d*x+c)) ^2/d^2-1/2*(d^2*e^2-2*c*d*e*f+(c^2+1)*f^2)*(a+b*arctanh(d*x+c))^2/d^2/f+1/ 2*(f*x+e)^2*(a+b*arctanh(d*x+c))^2/f-2*b*(-c*f+d*e)*(a+b*arctanh(d*x+c))*l n(2/(-d*x-c+1))/d^2+1/2*b^2*f*ln(1-(d*x+c)^2)/d^2-b^2*(-c*f+d*e)*polylog(2 ,(-d*x-c-1)/(-d*x-c+1))/d^2
Time = 0.95 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.23 \[ \int (e+f x) (a+b \text {arctanh}(c+d x))^2 \, dx=\frac {2 a^2 c d e+2 a b c f-a^2 c^2 f+2 a^2 d^2 e x+2 a b d f x+a^2 d^2 f x^2+b^2 (-1+c+d x) (2 d e+f-c f+d f x) \text {arctanh}(c+d x)^2+2 b \text {arctanh}(c+d x) \left (-((c+d x) (-b f+a c f-a d (2 e+f x)))-2 b (d e-c f) \log \left (1+e^{-2 \text {arctanh}(c+d x)}\right )\right )+a b f \log (1-c-d x)-a b f \log (1+c+d x)-4 a b d e \log \left (\frac {1}{\sqrt {1-(c+d x)^2}}\right )-2 b^2 f \log \left (\frac {1}{\sqrt {1-(c+d x)^2}}\right )+4 a b c f \log \left (\frac {1}{\sqrt {1-(c+d x)^2}}\right )+2 b^2 (d e-c f) \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c+d x)}\right )}{2 d^2} \]
(2*a^2*c*d*e + 2*a*b*c*f - a^2*c^2*f + 2*a^2*d^2*e*x + 2*a*b*d*f*x + a^2*d ^2*f*x^2 + b^2*(-1 + c + d*x)*(2*d*e + f - c*f + d*f*x)*ArcTanh[c + d*x]^2 + 2*b*ArcTanh[c + d*x]*(-((c + d*x)*(-(b*f) + a*c*f - a*d*(2*e + f*x))) - 2*b*(d*e - c*f)*Log[1 + E^(-2*ArcTanh[c + d*x])]) + a*b*f*Log[1 - c - d*x ] - a*b*f*Log[1 + c + d*x] - 4*a*b*d*e*Log[1/Sqrt[1 - (c + d*x)^2]] - 2*b^ 2*f*Log[1/Sqrt[1 - (c + d*x)^2]] + 4*a*b*c*f*Log[1/Sqrt[1 - (c + d*x)^2]] + 2*b^2*(d*e - c*f)*PolyLog[2, -E^(-2*ArcTanh[c + d*x])])/(2*d^2)
Time = 0.65 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.05, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6661, 27, 6480, 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.
| \(\displaystyle \int (e+f x) (a+b \text {arctanh}(c+d x))^2 \, dx\) |
| \(\Big \downarrow \) 6661 |
| \(\displaystyle \frac {\int \frac {\left (d \left (e-\frac {c f}{d}\right )+f (c+d x)\right ) (a+b \text {arctanh}(c+d x))^2}{d}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int (d e-c f+f (c+d x)) (a+b \text {arctanh}(c+d x))^2d(c+d x)}{d^2}\) |
| \(\Big \downarrow \) 6480 |
| \(\displaystyle \frac {\frac {(f (c+d x)-c f+d e)^2 (a+b \text {arctanh}(c+d x))^2}{2 f}-\frac {b \int \left (\frac {\left (d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2+2 f (d e-c f) (c+d x)\right ) (a+b \text {arctanh}(c+d x))}{1-(c+d x)^2}-f^2 (a+b \text {arctanh}(c+d x))\right )d(c+d x)}{f}}{d^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {(f (c+d x)-c f+d e)^2 (a+b \text {arctanh}(c+d x))^2}{2 f}-\frac {b \left (\frac {\left (\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2\right ) (a+b \text {arctanh}(c+d x))^2}{2 b}-\frac {f (d e-c f) (a+b \text {arctanh}(c+d x))^2}{b}+2 f (d e-c f) \log \left (\frac {2}{-c-d x+1}\right ) (a+b \text {arctanh}(c+d x))-a f^2 (c+d x)-b f^2 (c+d x) \text {arctanh}(c+d x)+b f (d e-c f) \operatorname {PolyLog}\left (2,-\frac {c+d x+1}{-c-d x+1}\right )-\frac {1}{2} b f^2 \log \left (1-(c+d x)^2\right )\right )}{f}}{d^2}\) |
(((d*e - c*f + f*(c + d*x))^2*(a + b*ArcTanh[c + d*x])^2)/(2*f) - (b*(-(a* f^2*(c + d*x)) - b*f^2*(c + d*x)*ArcTanh[c + d*x] - (f*(d*e - c*f)*(a + b* ArcTanh[c + d*x])^2)/b + ((d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2)*(a + b*Arc Tanh[c + d*x])^2)/(2*b) + 2*f*(d*e - c*f)*(a + b*ArcTanh[c + d*x])*Log[2/( 1 - c - d*x)] - (b*f^2*Log[1 - (c + d*x)^2])/2 + b*f*(d*e - c*f)*PolyLog[2 , -((1 + c + d*x)/(1 - c - d*x))]))/f)/d^2
3.1.40.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_))^(q_.), x_S ymbol] :> Simp[(d + e*x)^(q + 1)*((a + b*ArcTanh[c*x])^p/(e*(q + 1))), x] - Simp[b*c*(p/(e*(q + 1))) Int[ExpandIntegrand[(a + b*ArcTanh[c*x])^(p - 1 ), (d + e*x)^(q + 1)/(1 - c^2*x^2), x], x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 1] && IntegerQ[q] && NeQ[q, -1]
Int[((a_.) + ArcTanh[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^( m_.), x_Symbol] :> Simp[1/d Subst[Int[((d*e - c*f)/d + f*(x/d))^m*(a + b* ArcTanh[x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IG tQ[p, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(452\) vs. \(2(217)=434\).
Time = 0.14 (sec) , antiderivative size = 453, normalized size of antiderivative = 2.05
| method | result | size |
| parts | \(a^{2} \left (\frac {1}{2} f \,x^{2}+e x \right )+\frac {b^{2} \left (\frac {\operatorname {arctanh}\left (d x +c \right )^{2} \left (d x +c \right )^{2} f}{2 d}-\frac {\operatorname {arctanh}\left (d x +c \right )^{2} c f \left (d x +c \right )}{d}+\operatorname {arctanh}\left (d x +c \right )^{2} e \left (d x +c \right )-\frac {-\operatorname {arctanh}\left (d x +c \right ) \left (d x +c \right ) f +\operatorname {arctanh}\left (d x +c \right ) \ln \left (d x +c -1\right ) c f -\operatorname {arctanh}\left (d x +c \right ) \ln \left (d x +c -1\right ) d e -\frac {\operatorname {arctanh}\left (d x +c \right ) \ln \left (d x +c -1\right ) f}{2}+\operatorname {arctanh}\left (d x +c \right ) \ln \left (d x +c +1\right ) c f -\operatorname {arctanh}\left (d x +c \right ) \ln \left (d x +c +1\right ) d e +\frac {\operatorname {arctanh}\left (d x +c \right ) \ln \left (d x +c +1\right ) f}{2}-\frac {\ln \left (d x +c -1\right ) f}{2}-\frac {\ln \left (d x +c +1\right ) f}{2}-\frac {\left (-2 c f +2 d e +f \right ) \left (\frac {\ln \left (d x +c -1\right )^{2}}{4}-\frac {\operatorname {dilog}\left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )}{2}-\frac {\ln \left (d x +c -1\right ) \ln \left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )}{2}\right )}{2}-\frac {\left (-2 c f +2 d e -f \right ) \left (-\frac {\ln \left (d x +c +1\right )^{2}}{4}+\frac {\left (\ln \left (d x +c +1\right )-\ln \left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {d x}{2}-\frac {c}{2}+\frac {1}{2}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )}{2}\right )}{2}}{d}\right )}{d}+\frac {2 a b \left (\frac {\operatorname {arctanh}\left (d x +c \right ) \left (d x +c \right )^{2} f}{2 d}-\frac {\operatorname {arctanh}\left (d x +c \right ) c f \left (d x +c \right )}{d}+\operatorname {arctanh}\left (d x +c \right ) e \left (d x +c \right )-\frac {-f \left (d x +c \right )-\frac {\left (-2 c f +2 d e +f \right ) \ln \left (d x +c -1\right )}{2}+\frac {\left (2 c f -2 d e +f \right ) \ln \left (d x +c +1\right )}{2}}{2 d}\right )}{d}\) | \(453\) |
| derivativedivides | \(\frac {-\frac {a^{2} \left (f c \left (d x +c \right )-e d \left (d x +c \right )-\frac {f \left (d x +c \right )^{2}}{2}\right )}{d}-\frac {b^{2} \left (\operatorname {arctanh}\left (d x +c \right )^{2} f c \left (d x +c \right )-\operatorname {arctanh}\left (d x +c \right )^{2} e d \left (d x +c \right )-\frac {\operatorname {arctanh}\left (d x +c \right )^{2} f \left (d x +c \right )^{2}}{2}-\operatorname {arctanh}\left (d x +c \right ) \left (d x +c \right ) f +\operatorname {arctanh}\left (d x +c \right ) \ln \left (d x +c -1\right ) c f -\operatorname {arctanh}\left (d x +c \right ) \ln \left (d x +c -1\right ) d e -\frac {\operatorname {arctanh}\left (d x +c \right ) \ln \left (d x +c -1\right ) f}{2}+\operatorname {arctanh}\left (d x +c \right ) \ln \left (d x +c +1\right ) c f -\operatorname {arctanh}\left (d x +c \right ) \ln \left (d x +c +1\right ) d e +\frac {\operatorname {arctanh}\left (d x +c \right ) \ln \left (d x +c +1\right ) f}{2}+\frac {\left (2 c f -2 d e -f \right ) \left (\frac {\ln \left (d x +c -1\right )^{2}}{4}-\frac {\operatorname {dilog}\left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )}{2}-\frac {\ln \left (d x +c -1\right ) \ln \left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )}{2}\right )}{2}+\frac {\left (2 c f -2 d e +f \right ) \left (-\frac {\ln \left (d x +c +1\right )^{2}}{4}+\frac {\left (\ln \left (d x +c +1\right )-\ln \left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {d x}{2}-\frac {c}{2}+\frac {1}{2}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )}{2}\right )}{2}-\frac {\ln \left (d x +c -1\right ) f}{2}-\frac {\ln \left (d x +c +1\right ) f}{2}\right )}{d}-\frac {2 a b \left (\operatorname {arctanh}\left (d x +c \right ) f c \left (d x +c \right )-\operatorname {arctanh}\left (d x +c \right ) e d \left (d x +c \right )-\frac {\operatorname {arctanh}\left (d x +c \right ) f \left (d x +c \right )^{2}}{2}-\frac {f \left (d x +c \right )}{2}+\frac {\left (2 c f -2 d e -f \right ) \ln \left (d x +c -1\right )}{4}-\frac {\left (-2 c f +2 d e -f \right ) \ln \left (d x +c +1\right )}{4}\right )}{d}}{d}\) | \(462\) |
| default | \(\frac {-\frac {a^{2} \left (f c \left (d x +c \right )-e d \left (d x +c \right )-\frac {f \left (d x +c \right )^{2}}{2}\right )}{d}-\frac {b^{2} \left (\operatorname {arctanh}\left (d x +c \right )^{2} f c \left (d x +c \right )-\operatorname {arctanh}\left (d x +c \right )^{2} e d \left (d x +c \right )-\frac {\operatorname {arctanh}\left (d x +c \right )^{2} f \left (d x +c \right )^{2}}{2}-\operatorname {arctanh}\left (d x +c \right ) \left (d x +c \right ) f +\operatorname {arctanh}\left (d x +c \right ) \ln \left (d x +c -1\right ) c f -\operatorname {arctanh}\left (d x +c \right ) \ln \left (d x +c -1\right ) d e -\frac {\operatorname {arctanh}\left (d x +c \right ) \ln \left (d x +c -1\right ) f}{2}+\operatorname {arctanh}\left (d x +c \right ) \ln \left (d x +c +1\right ) c f -\operatorname {arctanh}\left (d x +c \right ) \ln \left (d x +c +1\right ) d e +\frac {\operatorname {arctanh}\left (d x +c \right ) \ln \left (d x +c +1\right ) f}{2}+\frac {\left (2 c f -2 d e -f \right ) \left (\frac {\ln \left (d x +c -1\right )^{2}}{4}-\frac {\operatorname {dilog}\left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )}{2}-\frac {\ln \left (d x +c -1\right ) \ln \left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )}{2}\right )}{2}+\frac {\left (2 c f -2 d e +f \right ) \left (-\frac {\ln \left (d x +c +1\right )^{2}}{4}+\frac {\left (\ln \left (d x +c +1\right )-\ln \left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {d x}{2}-\frac {c}{2}+\frac {1}{2}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )}{2}\right )}{2}-\frac {\ln \left (d x +c -1\right ) f}{2}-\frac {\ln \left (d x +c +1\right ) f}{2}\right )}{d}-\frac {2 a b \left (\operatorname {arctanh}\left (d x +c \right ) f c \left (d x +c \right )-\operatorname {arctanh}\left (d x +c \right ) e d \left (d x +c \right )-\frac {\operatorname {arctanh}\left (d x +c \right ) f \left (d x +c \right )^{2}}{2}-\frac {f \left (d x +c \right )}{2}+\frac {\left (2 c f -2 d e -f \right ) \ln \left (d x +c -1\right )}{4}-\frac {\left (-2 c f +2 d e -f \right ) \ln \left (d x +c +1\right )}{4}\right )}{d}}{d}\) | \(462\) |
| risch | \(\frac {a b f x}{d}+\left (-\frac {b^{2} x \left (f x +2 e \right ) \ln \left (-d x -c +1\right )}{4}+\frac {b \left (2 a \,d^{2} f \,x^{2}+4 a \,d^{2} e x +\ln \left (-d x -c +1\right ) b \,c^{2} f -2 \ln \left (-d x -c +1\right ) b c d e -2 \ln \left (-d x -c +1\right ) b c f +2 \ln \left (-d x -c +1\right ) b d e +2 b d f x +\ln \left (-d x -c +1\right ) b f \right )}{4 d^{2}}\right ) \ln \left (d x +c +1\right )-\frac {e \,a^{2}}{d}-\frac {f \,a^{2}}{2 d^{2}}+\frac {b^{2} \ln \left (-d x -c -1\right ) f}{2 d^{2}}+\frac {b^{2} \operatorname {dilog}\left (-\frac {d x}{2}-\frac {c}{2}+\frac {1}{2}\right ) e}{d}-\frac {b^{2} f \ln \left (-d x -c +1\right )^{2}}{8 d^{2}}+\frac {b^{2} f \ln \left (-d x -c +1\right )}{2 d^{2}}+\frac {\ln \left (-d x -c +1\right )^{2} b^{2} e x}{4}+\frac {b^{2} f \ln \left (-d x -c +1\right )^{2} x^{2}}{8}+\frac {f \,a^{2} x^{2}}{2}+a^{2} e x -\frac {b^{2} \left (-d^{2} f \,x^{2}-2 d^{2} e x +c^{2} f -2 c d e +2 c f -2 d e +f \right ) \ln \left (d x +c +1\right )^{2}}{8 d^{2}}+\frac {e \,a^{2} c}{d}+\frac {a^{2} c f}{d^{2}}-\frac {f \,a^{2} c^{2}}{2 d^{2}}-\frac {b a f}{d^{2}}+\frac {e \ln \left (-d x -c -1\right ) a b}{d}-\frac {e \,b^{2} \ln \left (-d x -c +1\right )^{2}}{4 d}+\frac {b a c f}{d^{2}}+\frac {b^{2} \ln \left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right ) \ln \left (-d x -c +1\right ) c f}{d^{2}}-\frac {b^{2} \ln \left (-\frac {d x}{2}-\frac {c}{2}+\frac {1}{2}\right ) \ln \left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right ) c f}{d^{2}}-\frac {b \ln \left (-d x -c -1\right ) a \,c^{2} f}{2 d^{2}}+\frac {b \ln \left (-d x -c -1\right ) a c e}{d}-\frac {b \ln \left (-d x -c -1\right ) a c f}{d^{2}}+\frac {b a f \ln \left (-d x -c +1\right ) c^{2}}{2 d^{2}}-\frac {b a f \ln \left (-d x -c +1\right ) c}{d^{2}}-\frac {\ln \left (-d x -c +1\right ) a b c e}{d}+\frac {b^{2} \ln \left (-d x -c -1\right ) c f}{2 d^{2}}-\frac {b \ln \left (-d x -c -1\right ) a f}{2 d^{2}}-\frac {b^{2} f \ln \left (-d x -c +1\right ) x}{2 d}-\frac {b^{2} f \ln \left (-d x -c +1\right ) c}{2 d^{2}}+\frac {\ln \left (-d x -c +1\right ) a b e}{d}+\frac {\ln \left (-d x -c +1\right )^{2} b^{2} c e}{4 d}+\frac {b a f \ln \left (-d x -c +1\right )}{2 d^{2}}-\frac {b^{2} \operatorname {dilog}\left (-\frac {d x}{2}-\frac {c}{2}+\frac {1}{2}\right ) c f}{d^{2}}-\frac {b^{2} \ln \left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right ) \ln \left (-d x -c +1\right ) e}{d}+\frac {b^{2} \ln \left (-\frac {d x}{2}-\frac {c}{2}+\frac {1}{2}\right ) \ln \left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right ) e}{d}-\ln \left (-d x -c +1\right ) a b e x -\frac {b a f \ln \left (-d x -c +1\right ) x^{2}}{2}-\frac {b^{2} f \ln \left (-d x -c +1\right )^{2} c^{2}}{8 d^{2}}+\frac {b^{2} f \ln \left (-d x -c +1\right )^{2} c}{4 d^{2}}\) | \(903\) |
a^2*(1/2*f*x^2+e*x)+b^2/d*(1/2/d*arctanh(d*x+c)^2*(d*x+c)^2*f-1/d*arctanh( d*x+c)^2*c*f*(d*x+c)+arctanh(d*x+c)^2*e*(d*x+c)-1/d*(-arctanh(d*x+c)*(d*x+ c)*f+arctanh(d*x+c)*ln(d*x+c-1)*c*f-arctanh(d*x+c)*ln(d*x+c-1)*d*e-1/2*arc tanh(d*x+c)*ln(d*x+c-1)*f+arctanh(d*x+c)*ln(d*x+c+1)*c*f-arctanh(d*x+c)*ln (d*x+c+1)*d*e+1/2*arctanh(d*x+c)*ln(d*x+c+1)*f-1/2*ln(d*x+c-1)*f-1/2*ln(d* x+c+1)*f-1/2*(-2*c*f+2*d*e+f)*(1/4*ln(d*x+c-1)^2-1/2*dilog(1/2*d*x+1/2*c+1 /2)-1/2*ln(d*x+c-1)*ln(1/2*d*x+1/2*c+1/2))-1/2*(-2*c*f+2*d*e-f)*(-1/4*ln(d *x+c+1)^2+1/2*(ln(d*x+c+1)-ln(1/2*d*x+1/2*c+1/2))*ln(-1/2*d*x-1/2*c+1/2)-1 /2*dilog(1/2*d*x+1/2*c+1/2))))+2*a*b/d*(1/2/d*arctanh(d*x+c)*(d*x+c)^2*f-1 /d*arctanh(d*x+c)*c*f*(d*x+c)+arctanh(d*x+c)*e*(d*x+c)-1/2/d*(-f*(d*x+c)-1 /2*(-2*c*f+2*d*e+f)*ln(d*x+c-1)+1/2*(2*c*f-2*d*e+f)*ln(d*x+c+1)))
\[ \int (e+f x) (a+b \text {arctanh}(c+d x))^2 \, dx=\int { {\left (f x + e\right )} {\left (b \operatorname {artanh}\left (d x + c\right ) + a\right )}^{2} \,d x } \]
integral(a^2*f*x + a^2*e + (b^2*f*x + b^2*e)*arctanh(d*x + c)^2 + 2*(a*b*f *x + a*b*e)*arctanh(d*x + c), x)
\[ \int (e+f x) (a+b \text {arctanh}(c+d x))^2 \, dx=\int \left (a + b \operatorname {atanh}{\left (c + d x \right )}\right )^{2} \left (e + f x\right )\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 415 vs. \(2 (207) = 414\).
Time = 0.42 (sec) , antiderivative size = 415, normalized size of antiderivative = 1.88 \[ \int (e+f x) (a+b \text {arctanh}(c+d x))^2 \, dx=\frac {1}{2} \, a^{2} f x^{2} + \frac {1}{2} \, {\left (2 \, x^{2} \operatorname {artanh}\left (d x + c\right ) + d {\left (\frac {2 \, x}{d^{2}} - \frac {{\left (c^{2} + 2 \, c + 1\right )} \log \left (d x + c + 1\right )}{d^{3}} + \frac {{\left (c^{2} - 2 \, c + 1\right )} \log \left (d x + c - 1\right )}{d^{3}}\right )}\right )} a b f + a^{2} e x + \frac {{\left (2 \, {\left (d x + c\right )} \operatorname {artanh}\left (d x + c\right ) + \log \left (-{\left (d x + c\right )}^{2} + 1\right )\right )} a b e}{d} + \frac {{\left (d e - c f\right )} {\left (\log \left (d x + c + 1\right ) \log \left (-\frac {1}{2} \, d x - \frac {1}{2} \, c + \frac {1}{2}\right ) + {\rm Li}_2\left (\frac {1}{2} \, d x + \frac {1}{2} \, c + \frac {1}{2}\right )\right )} b^{2}}{d^{2}} + \frac {{\left (c f + f\right )} b^{2} \log \left (d x + c + 1\right )}{2 \, d^{2}} - \frac {{\left (c f - f\right )} b^{2} \log \left (d x + c - 1\right )}{2 \, d^{2}} + \frac {4 \, b^{2} d f x \log \left (d x + c + 1\right ) + {\left (b^{2} d^{2} f x^{2} + 2 \, b^{2} d^{2} e x - {\left (c^{2} f - 2 \, {\left (d e - f\right )} c - 2 \, d e + f\right )} b^{2}\right )} \log \left (d x + c + 1\right )^{2} + {\left (b^{2} d^{2} f x^{2} + 2 \, b^{2} d^{2} e x - {\left (c^{2} f - 2 \, {\left (d e + f\right )} c + 2 \, d e + f\right )} b^{2}\right )} \log \left (-d x - c + 1\right )^{2} - 2 \, {\left (2 \, b^{2} d f x + {\left (b^{2} d^{2} f x^{2} + 2 \, b^{2} d^{2} e x - {\left (c^{2} f - 2 \, {\left (d e - f\right )} c - 2 \, d e + f\right )} b^{2}\right )} \log \left (d x + c + 1\right )\right )} \log \left (-d x - c + 1\right )}{8 \, d^{2}} \]
1/2*a^2*f*x^2 + 1/2*(2*x^2*arctanh(d*x + c) + d*(2*x/d^2 - (c^2 + 2*c + 1) *log(d*x + c + 1)/d^3 + (c^2 - 2*c + 1)*log(d*x + c - 1)/d^3))*a*b*f + a^2 *e*x + (2*(d*x + c)*arctanh(d*x + c) + log(-(d*x + c)^2 + 1))*a*b*e/d + (d *e - c*f)*(log(d*x + c + 1)*log(-1/2*d*x - 1/2*c + 1/2) + dilog(1/2*d*x + 1/2*c + 1/2))*b^2/d^2 + 1/2*(c*f + f)*b^2*log(d*x + c + 1)/d^2 - 1/2*(c*f - f)*b^2*log(d*x + c - 1)/d^2 + 1/8*(4*b^2*d*f*x*log(d*x + c + 1) + (b^2*d ^2*f*x^2 + 2*b^2*d^2*e*x - (c^2*f - 2*(d*e - f)*c - 2*d*e + f)*b^2)*log(d* x + c + 1)^2 + (b^2*d^2*f*x^2 + 2*b^2*d^2*e*x - (c^2*f - 2*(d*e + f)*c + 2 *d*e + f)*b^2)*log(-d*x - c + 1)^2 - 2*(2*b^2*d*f*x + (b^2*d^2*f*x^2 + 2*b ^2*d^2*e*x - (c^2*f - 2*(d*e - f)*c - 2*d*e + f)*b^2)*log(d*x + c + 1))*lo g(-d*x - c + 1))/d^2
\[ \int (e+f x) (a+b \text {arctanh}(c+d x))^2 \, dx=\int { {\left (f x + e\right )} {\left (b \operatorname {artanh}\left (d x + c\right ) + a\right )}^{2} \,d x } \]
Timed out. \[ \int (e+f x) (a+b \text {arctanh}(c+d x))^2 \, dx=\int \left (e+f\,x\right )\,{\left (a+b\,\mathrm {atanh}\left (c+d\,x\right )\right )}^2 \,d x \]