Integrand size = 23, antiderivative size = 168 \[ \int \frac {(a+b \text {arctanh}(c+d x))^2}{c e+d e x} \, dx=\frac {2 (a+b \text {arctanh}(c+d x))^2 \text {arctanh}\left (1-\frac {2}{1-c-d x}\right )}{d e}-\frac {b (a+b \text {arctanh}(c+d x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1-c-d x}\right )}{d e}+\frac {b (a+b \text {arctanh}(c+d x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-c-d x}\right )}{d e}+\frac {b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1-c-d x}\right )}{2 d e}-\frac {b^2 \operatorname {PolyLog}\left (3,-1+\frac {2}{1-c-d x}\right )}{2 d e} \]
-2*(a+b*arctanh(d*x+c))^2*arctanh(-1+2/(-d*x-c+1))/d/e-b*(a+b*arctanh(d*x+ c))*polylog(2,1-2/(-d*x-c+1))/d/e+b*(a+b*arctanh(d*x+c))*polylog(2,-1+2/(- d*x-c+1))/d/e+1/2*b^2*polylog(3,1-2/(-d*x-c+1))/d/e-1/2*b^2*polylog(3,-1+2 /(-d*x-c+1))/d/e
Result contains complex when optimal does not.
Time = 0.25 (sec) , antiderivative size = 424, normalized size of antiderivative = 2.52 \[ \int \frac {(a+b \text {arctanh}(c+d x))^2}{c e+d e x} \, dx=\frac {a^2 \log (c+d x)}{d e}-\frac {2 i a b \left (i \text {arctanh}(c+d x) \left (-\log \left (\frac {1}{\sqrt {1-(c+d x)^2}}\right )+\log \left (\frac {i (c+d x)}{\sqrt {1-(c+d x)^2}}\right )\right )+\frac {1}{2} \left (-\frac {1}{4} i (\pi -2 i \text {arctanh}(c+d x))^2+i \text {arctanh}(c+d x)^2+(\pi -2 i \text {arctanh}(c+d x)) \log \left (1-e^{i (\pi -2 i \text {arctanh}(c+d x))}\right )+2 i \text {arctanh}(c+d x) \log \left (1-e^{-2 \text {arctanh}(c+d x)}\right )-2 i \text {arctanh}(c+d x) \log \left (\frac {2 i (c+d x)}{\sqrt {1-(c+d x)^2}}\right )-(\pi -2 i \text {arctanh}(c+d x)) \log \left (2 \sin \left (\frac {1}{2} (\pi -2 i \text {arctanh}(c+d x))\right )\right )-i \operatorname {PolyLog}\left (2,e^{i (\pi -2 i \text {arctanh}(c+d x))}\right )-i \operatorname {PolyLog}\left (2,e^{-2 \text {arctanh}(c+d x)}\right )\right )\right )}{d e}+\frac {b^2 \left (\text {arctanh}(c+d x)^2 \log \left (1-e^{-2 \text {arctanh}(c+d x)}\right )-\text {arctanh}(c+d x)^2 \log \left (1+e^{-2 \text {arctanh}(c+d x)}\right )+\text {arctanh}(c+d x) \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c+d x)}\right )-\text {arctanh}(c+d x) \operatorname {PolyLog}\left (2,e^{-2 \text {arctanh}(c+d x)}\right )+\frac {1}{2} \operatorname {PolyLog}\left (3,-e^{-2 \text {arctanh}(c+d x)}\right )-\frac {1}{2} \operatorname {PolyLog}\left (3,e^{-2 \text {arctanh}(c+d x)}\right )\right )}{d e} \]
(a^2*Log[c + d*x])/(d*e) - ((2*I)*a*b*(I*ArcTanh[c + d*x]*(-Log[1/Sqrt[1 - (c + d*x)^2]] + Log[(I*(c + d*x))/Sqrt[1 - (c + d*x)^2]]) + ((-1/4*I)*(Pi - (2*I)*ArcTanh[c + d*x])^2 + I*ArcTanh[c + d*x]^2 + (Pi - (2*I)*ArcTanh[ c + d*x])*Log[1 - E^(I*(Pi - (2*I)*ArcTanh[c + d*x]))] + (2*I)*ArcTanh[c + d*x]*Log[1 - E^(-2*ArcTanh[c + d*x])] - (2*I)*ArcTanh[c + d*x]*Log[((2*I) *(c + d*x))/Sqrt[1 - (c + d*x)^2]] - (Pi - (2*I)*ArcTanh[c + d*x])*Log[2*S in[(Pi - (2*I)*ArcTanh[c + d*x])/2]] - I*PolyLog[2, E^(I*(Pi - (2*I)*ArcTa nh[c + d*x]))] - I*PolyLog[2, E^(-2*ArcTanh[c + d*x])])/2))/(d*e) + (b^2*( ArcTanh[c + d*x]^2*Log[1 - E^(-2*ArcTanh[c + d*x])] - ArcTanh[c + d*x]^2*L og[1 + E^(-2*ArcTanh[c + d*x])] + ArcTanh[c + d*x]*PolyLog[2, -E^(-2*ArcTa nh[c + d*x])] - ArcTanh[c + d*x]*PolyLog[2, E^(-2*ArcTanh[c + d*x])] + Pol yLog[3, -E^(-2*ArcTanh[c + d*x])]/2 - PolyLog[3, E^(-2*ArcTanh[c + d*x])]/ 2))/(d*e)
Time = 0.81 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.94, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {6657, 27, 6448, 6614, 6620, 7164}
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.
| \(\displaystyle \int \frac {(a+b \text {arctanh}(c+d x))^2}{c e+d e x} \, dx\) |
| \(\Big \downarrow \) 6657 |
| \(\displaystyle \frac {\int \frac {(a+b \text {arctanh}(c+d x))^2}{e (c+d x)}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(a+b \text {arctanh}(c+d x))^2}{c+d x}d(c+d x)}{d e}\) |
| \(\Big \downarrow \) 6448 |
| \(\displaystyle \frac {2 \text {arctanh}\left (1-\frac {2}{-c-d x+1}\right ) (a+b \text {arctanh}(c+d x))^2-4 b \int \frac {(a+b \text {arctanh}(c+d x)) \text {arctanh}\left (1-\frac {2}{-c-d x+1}\right )}{1-(c+d x)^2}d(c+d x)}{d e}\) |
| \(\Big \downarrow \) 6614 |
| \(\displaystyle \frac {2 \text {arctanh}\left (1-\frac {2}{-c-d x+1}\right ) (a+b \text {arctanh}(c+d x))^2-4 b \left (\frac {1}{2} \int \frac {(a+b \text {arctanh}(c+d x)) \log \left (2-\frac {2}{-c-d x+1}\right )}{1-(c+d x)^2}d(c+d x)-\frac {1}{2} \int \frac {(a+b \text {arctanh}(c+d x)) \log \left (\frac {2}{-c-d x+1}\right )}{1-(c+d x)^2}d(c+d x)\right )}{d e}\) |
| \(\Big \downarrow \) 6620 |
| \(\displaystyle \frac {2 \text {arctanh}\left (1-\frac {2}{-c-d x+1}\right ) (a+b \text {arctanh}(c+d x))^2-4 b \left (\frac {1}{2} \left (\frac {1}{2} \operatorname {PolyLog}\left (2,1-\frac {2}{-c-d x+1}\right ) (a+b \text {arctanh}(c+d x))-\frac {1}{2} b \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{-c-d x+1}\right )}{1-(c+d x)^2}d(c+d x)\right )+\frac {1}{2} \left (\frac {1}{2} b \int \frac {\operatorname {PolyLog}\left (2,\frac {2}{-c-d x+1}-1\right )}{1-(c+d x)^2}d(c+d x)-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{-c-d x+1}-1\right ) (a+b \text {arctanh}(c+d x))\right )\right )}{d e}\) |
| \(\Big \downarrow \) 7164 |
| \(\displaystyle \frac {2 \text {arctanh}\left (1-\frac {2}{-c-d x+1}\right ) (a+b \text {arctanh}(c+d x))^2-4 b \left (\frac {1}{2} \left (\frac {1}{2} \operatorname {PolyLog}\left (2,1-\frac {2}{-c-d x+1}\right ) (a+b \text {arctanh}(c+d x))-\frac {1}{4} b \operatorname {PolyLog}\left (3,1-\frac {2}{-c-d x+1}\right )\right )+\frac {1}{2} \left (\frac {1}{4} b \operatorname {PolyLog}\left (3,\frac {2}{-c-d x+1}-1\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{-c-d x+1}-1\right ) (a+b \text {arctanh}(c+d x))\right )\right )}{d e}\) |
(2*(a + b*ArcTanh[c + d*x])^2*ArcTanh[1 - 2/(1 - c - d*x)] - 4*b*((((a + b *ArcTanh[c + d*x])*PolyLog[2, 1 - 2/(1 - c - d*x)])/2 - (b*PolyLog[3, 1 - 2/(1 - c - d*x)])/4)/2 + (-1/2*((a + b*ArcTanh[c + d*x])*PolyLog[2, -1 + 2 /(1 - c - d*x)]) + (b*PolyLog[3, -1 + 2/(1 - c - d*x)])/4)/2))/(d*e)
3.1.18.3.1 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_)/(x_), x_Symbol] :> Simp[2*(a + b*ArcTanh[c*x])^p*ArcTanh[1 - 2/(1 - c*x)], x] - Simp[2*b*c*p Int[(a + b *ArcTanh[c*x])^(p - 1)*(ArcTanh[1 - 2/(1 - c*x)]/(1 - c^2*x^2)), x], x] /; FreeQ[{a, b, c}, x] && IGtQ[p, 1]
Int[(ArcTanh[u_]*((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*( x_)^2), x_Symbol] :> Simp[1/2 Int[Log[1 + u]*((a + b*ArcTanh[c*x])^p/(d + e*x^2)), x], x] - Simp[1/2 Int[Log[1 - u]*((a + b*ArcTanh[c*x])^p/(d + e *x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && EqQ[u^2 - (1 - 2/(1 - c*x))^2, 0]
Int[(Log[u_]*((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^ 2), x_Symbol] :> Simp[(-(a + b*ArcTanh[c*x])^p)*(PolyLog[2, 1 - u]/(2*c*d)) , x] + Simp[b*(p/2) Int[(a + b*ArcTanh[c*x])^(p - 1)*(PolyLog[2, 1 - u]/( d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && EqQ[(1 - u)^2 - (1 - 2/(1 - c*x))^2, 0]
Int[((a_.) + ArcTanh[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^( m_.), x_Symbol] :> Simp[1/d Subst[Int[(f*(x/d))^m*(a + b*ArcTanh[x])^p, x ], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[d*e - c*f, 0] && IGtQ[p, 0]
Int[(u_)*PolyLog[n_, v_], x_Symbol] :> With[{w = DerivativeDivides[v, u*v, x]}, Simp[w*PolyLog[n + 1, v], x] /; !FalseQ[w]] /; FreeQ[n, x]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 1.02 (sec) , antiderivative size = 705, normalized size of antiderivative = 4.20
| method | result | size |
| derivativedivides | \(\frac {\frac {a^{2} \ln \left (d x +c \right )}{e}+\frac {b^{2} \left (\ln \left (d x +c \right ) \operatorname {arctanh}\left (d x +c \right )^{2}-\operatorname {arctanh}\left (d x +c \right ) \operatorname {polylog}\left (2, -\frac {\left (d x +c +1\right )^{2}}{1-\left (d x +c \right )^{2}}\right )+\frac {\operatorname {polylog}\left (3, -\frac {\left (d x +c +1\right )^{2}}{1-\left (d x +c \right )^{2}}\right )}{2}-\operatorname {arctanh}\left (d x +c \right )^{2} \ln \left (\frac {\left (d x +c +1\right )^{2}}{1-\left (d x +c \right )^{2}}-1\right )+\operatorname {arctanh}\left (d x +c \right )^{2} \ln \left (1-\frac {d x +c +1}{\sqrt {1-\left (d x +c \right )^{2}}}\right )+2 \,\operatorname {arctanh}\left (d x +c \right ) \operatorname {polylog}\left (2, \frac {d x +c +1}{\sqrt {1-\left (d x +c \right )^{2}}}\right )-2 \operatorname {polylog}\left (3, \frac {d x +c +1}{\sqrt {1-\left (d x +c \right )^{2}}}\right )+\operatorname {arctanh}\left (d x +c \right )^{2} \ln \left (1+\frac {d x +c +1}{\sqrt {1-\left (d x +c \right )^{2}}}\right )+2 \,\operatorname {arctanh}\left (d x +c \right ) \operatorname {polylog}\left (2, -\frac {d x +c +1}{\sqrt {1-\left (d x +c \right )^{2}}}\right )-2 \operatorname {polylog}\left (3, -\frac {d x +c +1}{\sqrt {1-\left (d x +c \right )^{2}}}\right )+\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left (-\frac {\left (d x +c +1\right )^{2}}{\left (d x +c \right )^{2}-1}-1\right )}{1-\frac {\left (d x +c +1\right )^{2}}{\left (d x +c \right )^{2}-1}}\right ) \left (\operatorname {csgn}\left (i \left (-\frac {\left (d x +c +1\right )^{2}}{\left (d x +c \right )^{2}-1}-1\right )\right ) \operatorname {csgn}\left (\frac {i}{1-\frac {\left (d x +c +1\right )^{2}}{\left (d x +c \right )^{2}-1}}\right )-\operatorname {csgn}\left (i \left (-\frac {\left (d x +c +1\right )^{2}}{\left (d x +c \right )^{2}-1}-1\right )\right ) \operatorname {csgn}\left (\frac {i \left (-\frac {\left (d x +c +1\right )^{2}}{\left (d x +c \right )^{2}-1}-1\right )}{1-\frac {\left (d x +c +1\right )^{2}}{\left (d x +c \right )^{2}-1}}\right )-\operatorname {csgn}\left (\frac {i \left (-\frac {\left (d x +c +1\right )^{2}}{\left (d x +c \right )^{2}-1}-1\right )}{1-\frac {\left (d x +c +1\right )^{2}}{\left (d x +c \right )^{2}-1}}\right ) \operatorname {csgn}\left (\frac {i}{1-\frac {\left (d x +c +1\right )^{2}}{\left (d x +c \right )^{2}-1}}\right )+{\operatorname {csgn}\left (\frac {i \left (-\frac {\left (d x +c +1\right )^{2}}{\left (d x +c \right )^{2}-1}-1\right )}{1-\frac {\left (d x +c +1\right )^{2}}{\left (d x +c \right )^{2}-1}}\right )}^{2}\right ) \operatorname {arctanh}\left (d x +c \right )^{2}}{2}\right )}{e}+\frac {2 a b \left (\ln \left (d x +c \right ) \operatorname {arctanh}\left (d x +c \right )-\frac {\operatorname {dilog}\left (d x +c \right )}{2}-\frac {\operatorname {dilog}\left (d x +c +1\right )}{2}-\frac {\ln \left (d x +c \right ) \ln \left (d x +c +1\right )}{2}\right )}{e}}{d}\) | \(705\) |
| default | \(\frac {\frac {a^{2} \ln \left (d x +c \right )}{e}+\frac {b^{2} \left (\ln \left (d x +c \right ) \operatorname {arctanh}\left (d x +c \right )^{2}-\operatorname {arctanh}\left (d x +c \right ) \operatorname {polylog}\left (2, -\frac {\left (d x +c +1\right )^{2}}{1-\left (d x +c \right )^{2}}\right )+\frac {\operatorname {polylog}\left (3, -\frac {\left (d x +c +1\right )^{2}}{1-\left (d x +c \right )^{2}}\right )}{2}-\operatorname {arctanh}\left (d x +c \right )^{2} \ln \left (\frac {\left (d x +c +1\right )^{2}}{1-\left (d x +c \right )^{2}}-1\right )+\operatorname {arctanh}\left (d x +c \right )^{2} \ln \left (1-\frac {d x +c +1}{\sqrt {1-\left (d x +c \right )^{2}}}\right )+2 \,\operatorname {arctanh}\left (d x +c \right ) \operatorname {polylog}\left (2, \frac {d x +c +1}{\sqrt {1-\left (d x +c \right )^{2}}}\right )-2 \operatorname {polylog}\left (3, \frac {d x +c +1}{\sqrt {1-\left (d x +c \right )^{2}}}\right )+\operatorname {arctanh}\left (d x +c \right )^{2} \ln \left (1+\frac {d x +c +1}{\sqrt {1-\left (d x +c \right )^{2}}}\right )+2 \,\operatorname {arctanh}\left (d x +c \right ) \operatorname {polylog}\left (2, -\frac {d x +c +1}{\sqrt {1-\left (d x +c \right )^{2}}}\right )-2 \operatorname {polylog}\left (3, -\frac {d x +c +1}{\sqrt {1-\left (d x +c \right )^{2}}}\right )+\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left (-\frac {\left (d x +c +1\right )^{2}}{\left (d x +c \right )^{2}-1}-1\right )}{1-\frac {\left (d x +c +1\right )^{2}}{\left (d x +c \right )^{2}-1}}\right ) \left (\operatorname {csgn}\left (i \left (-\frac {\left (d x +c +1\right )^{2}}{\left (d x +c \right )^{2}-1}-1\right )\right ) \operatorname {csgn}\left (\frac {i}{1-\frac {\left (d x +c +1\right )^{2}}{\left (d x +c \right )^{2}-1}}\right )-\operatorname {csgn}\left (i \left (-\frac {\left (d x +c +1\right )^{2}}{\left (d x +c \right )^{2}-1}-1\right )\right ) \operatorname {csgn}\left (\frac {i \left (-\frac {\left (d x +c +1\right )^{2}}{\left (d x +c \right )^{2}-1}-1\right )}{1-\frac {\left (d x +c +1\right )^{2}}{\left (d x +c \right )^{2}-1}}\right )-\operatorname {csgn}\left (\frac {i \left (-\frac {\left (d x +c +1\right )^{2}}{\left (d x +c \right )^{2}-1}-1\right )}{1-\frac {\left (d x +c +1\right )^{2}}{\left (d x +c \right )^{2}-1}}\right ) \operatorname {csgn}\left (\frac {i}{1-\frac {\left (d x +c +1\right )^{2}}{\left (d x +c \right )^{2}-1}}\right )+{\operatorname {csgn}\left (\frac {i \left (-\frac {\left (d x +c +1\right )^{2}}{\left (d x +c \right )^{2}-1}-1\right )}{1-\frac {\left (d x +c +1\right )^{2}}{\left (d x +c \right )^{2}-1}}\right )}^{2}\right ) \operatorname {arctanh}\left (d x +c \right )^{2}}{2}\right )}{e}+\frac {2 a b \left (\ln \left (d x +c \right ) \operatorname {arctanh}\left (d x +c \right )-\frac {\operatorname {dilog}\left (d x +c \right )}{2}-\frac {\operatorname {dilog}\left (d x +c +1\right )}{2}-\frac {\ln \left (d x +c \right ) \ln \left (d x +c +1\right )}{2}\right )}{e}}{d}\) | \(705\) |
| parts | \(\frac {a^{2} \ln \left (d x +c \right )}{e d}+\frac {b^{2} \left (\ln \left (d x +c \right ) \operatorname {arctanh}\left (d x +c \right )^{2}-\operatorname {arctanh}\left (d x +c \right ) \operatorname {polylog}\left (2, -\frac {\left (d x +c +1\right )^{2}}{1-\left (d x +c \right )^{2}}\right )+\frac {\operatorname {polylog}\left (3, -\frac {\left (d x +c +1\right )^{2}}{1-\left (d x +c \right )^{2}}\right )}{2}-\operatorname {arctanh}\left (d x +c \right )^{2} \ln \left (\frac {\left (d x +c +1\right )^{2}}{1-\left (d x +c \right )^{2}}-1\right )+\operatorname {arctanh}\left (d x +c \right )^{2} \ln \left (1-\frac {d x +c +1}{\sqrt {1-\left (d x +c \right )^{2}}}\right )+2 \,\operatorname {arctanh}\left (d x +c \right ) \operatorname {polylog}\left (2, \frac {d x +c +1}{\sqrt {1-\left (d x +c \right )^{2}}}\right )-2 \operatorname {polylog}\left (3, \frac {d x +c +1}{\sqrt {1-\left (d x +c \right )^{2}}}\right )+\operatorname {arctanh}\left (d x +c \right )^{2} \ln \left (1+\frac {d x +c +1}{\sqrt {1-\left (d x +c \right )^{2}}}\right )+2 \,\operatorname {arctanh}\left (d x +c \right ) \operatorname {polylog}\left (2, -\frac {d x +c +1}{\sqrt {1-\left (d x +c \right )^{2}}}\right )-2 \operatorname {polylog}\left (3, -\frac {d x +c +1}{\sqrt {1-\left (d x +c \right )^{2}}}\right )+\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left (-\frac {\left (d x +c +1\right )^{2}}{\left (d x +c \right )^{2}-1}-1\right )}{1-\frac {\left (d x +c +1\right )^{2}}{\left (d x +c \right )^{2}-1}}\right ) \left (\operatorname {csgn}\left (i \left (-\frac {\left (d x +c +1\right )^{2}}{\left (d x +c \right )^{2}-1}-1\right )\right ) \operatorname {csgn}\left (\frac {i}{1-\frac {\left (d x +c +1\right )^{2}}{\left (d x +c \right )^{2}-1}}\right )-\operatorname {csgn}\left (i \left (-\frac {\left (d x +c +1\right )^{2}}{\left (d x +c \right )^{2}-1}-1\right )\right ) \operatorname {csgn}\left (\frac {i \left (-\frac {\left (d x +c +1\right )^{2}}{\left (d x +c \right )^{2}-1}-1\right )}{1-\frac {\left (d x +c +1\right )^{2}}{\left (d x +c \right )^{2}-1}}\right )-\operatorname {csgn}\left (\frac {i \left (-\frac {\left (d x +c +1\right )^{2}}{\left (d x +c \right )^{2}-1}-1\right )}{1-\frac {\left (d x +c +1\right )^{2}}{\left (d x +c \right )^{2}-1}}\right ) \operatorname {csgn}\left (\frac {i}{1-\frac {\left (d x +c +1\right )^{2}}{\left (d x +c \right )^{2}-1}}\right )+{\operatorname {csgn}\left (\frac {i \left (-\frac {\left (d x +c +1\right )^{2}}{\left (d x +c \right )^{2}-1}-1\right )}{1-\frac {\left (d x +c +1\right )^{2}}{\left (d x +c \right )^{2}-1}}\right )}^{2}\right ) \operatorname {arctanh}\left (d x +c \right )^{2}}{2}\right )}{e d}+\frac {2 a b \left (\ln \left (d x +c \right ) \operatorname {arctanh}\left (d x +c \right )-\frac {\operatorname {dilog}\left (d x +c \right )}{2}-\frac {\operatorname {dilog}\left (d x +c +1\right )}{2}-\frac {\ln \left (d x +c \right ) \ln \left (d x +c +1\right )}{2}\right )}{e d}\) | \(710\) |
1/d*(a^2/e*ln(d*x+c)+b^2/e*(ln(d*x+c)*arctanh(d*x+c)^2-arctanh(d*x+c)*poly log(2,-(d*x+c+1)^2/(1-(d*x+c)^2))+1/2*polylog(3,-(d*x+c+1)^2/(1-(d*x+c)^2) )-arctanh(d*x+c)^2*ln((d*x+c+1)^2/(1-(d*x+c)^2)-1)+arctanh(d*x+c)^2*ln(1-( d*x+c+1)/(1-(d*x+c)^2)^(1/2))+2*arctanh(d*x+c)*polylog(2,(d*x+c+1)/(1-(d*x +c)^2)^(1/2))-2*polylog(3,(d*x+c+1)/(1-(d*x+c)^2)^(1/2))+arctanh(d*x+c)^2* ln(1+(d*x+c+1)/(1-(d*x+c)^2)^(1/2))+2*arctanh(d*x+c)*polylog(2,-(d*x+c+1)/ (1-(d*x+c)^2)^(1/2))-2*polylog(3,-(d*x+c+1)/(1-(d*x+c)^2)^(1/2))+1/2*I*Pi* csgn(I*(-(d*x+c+1)^2/((d*x+c)^2-1)-1)/(1-(d*x+c+1)^2/((d*x+c)^2-1)))*(csgn (I*(-(d*x+c+1)^2/((d*x+c)^2-1)-1))*csgn(I/(1-(d*x+c+1)^2/((d*x+c)^2-1)))-c sgn(I*(-(d*x+c+1)^2/((d*x+c)^2-1)-1))*csgn(I*(-(d*x+c+1)^2/((d*x+c)^2-1)-1 )/(1-(d*x+c+1)^2/((d*x+c)^2-1)))-csgn(I*(-(d*x+c+1)^2/((d*x+c)^2-1)-1)/(1- (d*x+c+1)^2/((d*x+c)^2-1)))*csgn(I/(1-(d*x+c+1)^2/((d*x+c)^2-1)))+csgn(I*( -(d*x+c+1)^2/((d*x+c)^2-1)-1)/(1-(d*x+c+1)^2/((d*x+c)^2-1)))^2)*arctanh(d* x+c)^2)+2*a*b/e*(ln(d*x+c)*arctanh(d*x+c)-1/2*dilog(d*x+c)-1/2*dilog(d*x+c +1)-1/2*ln(d*x+c)*ln(d*x+c+1)))
\[ \int \frac {(a+b \text {arctanh}(c+d x))^2}{c e+d e x} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (d x + c\right ) + a\right )}^{2}}{d e x + c e} \,d x } \]
\[ \int \frac {(a+b \text {arctanh}(c+d x))^2}{c e+d e x} \, dx=\frac {\int \frac {a^{2}}{c + d x}\, dx + \int \frac {b^{2} \operatorname {atanh}^{2}{\left (c + d x \right )}}{c + d x}\, dx + \int \frac {2 a b \operatorname {atanh}{\left (c + d x \right )}}{c + d x}\, dx}{e} \]
(Integral(a**2/(c + d*x), x) + Integral(b**2*atanh(c + d*x)**2/(c + d*x), x) + Integral(2*a*b*atanh(c + d*x)/(c + d*x), x))/e
\[ \int \frac {(a+b \text {arctanh}(c+d x))^2}{c e+d e x} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (d x + c\right ) + a\right )}^{2}}{d e x + c e} \,d x } \]
a^2*log(d*e*x + c*e)/(d*e) + integrate(1/4*b^2*(log(d*x + c + 1) - log(-d* x - c + 1))^2/(d*e*x + c*e) + a*b*(log(d*x + c + 1) - log(-d*x - c + 1))/( d*e*x + c*e), x)
\[ \int \frac {(a+b \text {arctanh}(c+d x))^2}{c e+d e x} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (d x + c\right ) + a\right )}^{2}}{d e x + c e} \,d x } \]
Timed out. \[ \int \frac {(a+b \text {arctanh}(c+d x))^2}{c e+d e x} \, dx=\int \frac {{\left (a+b\,\mathrm {atanh}\left (c+d\,x\right )\right )}^2}{c\,e+d\,e\,x} \,d x \]