Integrand size = 40, antiderivative size = 268 \[ \int \frac {\left (a+b \text {arctanh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2}{1-c^2 x^2} \, dx=-\frac {2 \left (a+b \text {arctanh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2 \text {arctanh}\left (1-\frac {2}{1-\frac {\sqrt {1-c x}}{\sqrt {1+c x}}}\right )}{c}+\frac {b \left (a+b \text {arctanh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{1-\frac {\sqrt {1-c x}}{\sqrt {1+c x}}}\right )}{c}-\frac {b \left (a+b \text {arctanh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right ) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-\frac {\sqrt {1-c x}}{\sqrt {1+c x}}}\right )}{c}-\frac {b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1-\frac {\sqrt {1-c x}}{\sqrt {1+c x}}}\right )}{2 c}+\frac {b^2 \operatorname {PolyLog}\left (3,-1+\frac {2}{1-\frac {\sqrt {1-c x}}{\sqrt {1+c x}}}\right )}{2 c} \]
2*arctanh(-1+2/(1-(-c*x+1)^(1/2)/(c*x+1)^(1/2)))*(a+b*arctanh((-c*x+1)^(1/ 2)/(c*x+1)^(1/2)))^2/c+b*(a+b*arctanh((-c*x+1)^(1/2)/(c*x+1)^(1/2)))*polyl og(2,1-2/(1-(-c*x+1)^(1/2)/(c*x+1)^(1/2)))/c-b*(a+b*arctanh((-c*x+1)^(1/2) /(c*x+1)^(1/2)))*polylog(2,-1+2/(1-(-c*x+1)^(1/2)/(c*x+1)^(1/2)))/c-1/2*b^ 2*polylog(3,1-2/(1-(-c*x+1)^(1/2)/(c*x+1)^(1/2)))/c+1/2*b^2*polylog(3,-1+2 /(1-(-c*x+1)^(1/2)/(c*x+1)^(1/2)))/c
\[ \int \frac {\left (a+b \text {arctanh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2}{1-c^2 x^2} \, dx=\int \frac {\left (a+b \text {arctanh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2}{1-c^2 x^2} \, dx \]
Time = 0.86 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.01, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {7232, 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.
| \(\displaystyle \int \frac {\left (a+b \text {arctanh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )^2}{1-c^2 x^2} \, dx\) |
| \(\Big \downarrow \) 7232 |
| \(\displaystyle -\frac {\int \frac {\sqrt {c x+1} \left (a+b \text {arctanh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )^2}{\sqrt {1-c x}}d\frac {\sqrt {1-c x}}{\sqrt {c x+1}}}{c}\) |
| \(\Big \downarrow \) 6448 |
| \(\displaystyle -\frac {2 \text {arctanh}\left (1-\frac {2}{1-\frac {\sqrt {1-c x}}{\sqrt {c x+1}}}\right ) \left (a+b \text {arctanh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )^2-4 b \int \frac {\left (a+b \text {arctanh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right ) \text {arctanh}\left (1-\frac {2}{1-\frac {\sqrt {1-c x}}{\sqrt {c x+1}}}\right )}{1-\frac {1-c x}{c x+1}}d\frac {\sqrt {1-c x}}{\sqrt {c x+1}}}{c}\) |
| \(\Big \downarrow \) 6614 |
| \(\displaystyle -\frac {2 \text {arctanh}\left (1-\frac {2}{1-\frac {\sqrt {1-c x}}{\sqrt {c x+1}}}\right ) \left (a+b \text {arctanh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )^2-4 b \left (\frac {1}{2} \int \frac {\left (a+b \text {arctanh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right ) \log \left (2-\frac {2}{1-\frac {\sqrt {1-c x}}{\sqrt {c x+1}}}\right )}{1-\frac {1-c x}{c x+1}}d\frac {\sqrt {1-c x}}{\sqrt {c x+1}}-\frac {1}{2} \int \frac {\left (a+b \text {arctanh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right ) \log \left (\frac {2}{1-\frac {\sqrt {1-c x}}{\sqrt {c x+1}}}\right )}{1-\frac {1-c x}{c x+1}}d\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )}{c}\) |
| \(\Big \downarrow \) 6620 |
| \(\displaystyle -\frac {2 \text {arctanh}\left (1-\frac {2}{1-\frac {\sqrt {1-c x}}{\sqrt {c x+1}}}\right ) \left (a+b \text {arctanh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )^2-4 b \left (\frac {1}{2} \left (\frac {1}{2} \operatorname {PolyLog}\left (2,1-\frac {2}{1-\frac {\sqrt {1-c x}}{\sqrt {c x+1}}}\right ) \left (a+b \text {arctanh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )-\frac {1}{2} b \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-\frac {\sqrt {1-c x}}{\sqrt {c x+1}}}\right )}{1-\frac {1-c x}{c x+1}}d\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )+\frac {1}{2} \left (\frac {1}{2} b \int \frac {\operatorname {PolyLog}\left (2,\frac {2}{1-\frac {\sqrt {1-c x}}{\sqrt {c x+1}}}-1\right )}{1-\frac {1-c x}{c x+1}}d\frac {\sqrt {1-c x}}{\sqrt {c x+1}}-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{1-\frac {\sqrt {1-c x}}{\sqrt {c x+1}}}-1\right ) \left (a+b \text {arctanh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )\right )\right )}{c}\) |
| \(\Big \downarrow \) 7164 |
| \(\displaystyle -\frac {2 \text {arctanh}\left (1-\frac {2}{1-\frac {\sqrt {1-c x}}{\sqrt {c x+1}}}\right ) \left (a+b \text {arctanh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )^2-4 b \left (\frac {1}{2} \left (\frac {1}{2} \operatorname {PolyLog}\left (2,1-\frac {2}{1-\frac {\sqrt {1-c x}}{\sqrt {c x+1}}}\right ) \left (a+b \text {arctanh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )-\frac {1}{4} b \operatorname {PolyLog}\left (3,1-\frac {2}{1-\frac {\sqrt {1-c x}}{\sqrt {c x+1}}}\right )\right )+\frac {1}{2} \left (\frac {1}{4} b \operatorname {PolyLog}\left (3,\frac {2}{1-\frac {\sqrt {1-c x}}{\sqrt {c x+1}}}-1\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{1-\frac {\sqrt {1-c x}}{\sqrt {c x+1}}}-1\right ) \left (a+b \text {arctanh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )\right )\right )}{c}\) |
-((2*(a + b*ArcTanh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2*ArcTanh[1 - 2/(1 - Sqr t[1 - c*x]/Sqrt[1 + c*x])] - 4*b*((((a + b*ArcTanh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])*PolyLog[2, 1 - 2/(1 - Sqrt[1 - c*x]/Sqrt[1 + c*x])])/2 - (b*PolyLog [3, 1 - 2/(1 - Sqrt[1 - c*x]/Sqrt[1 + c*x])])/4)/2 + (-1/2*((a + b*ArcTanh [Sqrt[1 - c*x]/Sqrt[1 + c*x]])*PolyLog[2, -1 + 2/(1 - Sqrt[1 - c*x]/Sqrt[1 + c*x])]) + (b*PolyLog[3, -1 + 2/(1 - Sqrt[1 - c*x]/Sqrt[1 + c*x])])/4)/2 ))/c)
3.1.32.3.1 Defintions of rubi rules used
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[(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]
Int[((a_.) + (b_.)*(F_)[((c_.)*Sqrt[(d_.) + (e_.)*(x_)])/Sqrt[(f_.) + (g_.) *(x_)]])^(n_.)/((A_.) + (C_.)*(x_)^2), x_Symbol] :> Simp[2*e*(g/(C*(e*f - d *g))) Subst[Int[(a + b*F[c*x])^n/x, x], x, Sqrt[d + e*x]/Sqrt[f + g*x]], x] /; FreeQ[{a, b, c, d, e, f, g, A, C, F}, x] && EqQ[C*d*f - A*e*g, 0] && EqQ[e*f + d*g, 0] && IGtQ[n, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(892\) vs. \(2(232)=464\).
Time = 0.26 (sec) , antiderivative size = 893, normalized size of antiderivative = 3.33
| method | result | size |
| default | \(-\frac {a^{2} \ln \left (c x -1\right )}{2 c}+\frac {a^{2} \ln \left (c x +1\right )}{2 c}-b^{2} \left (\frac {\operatorname {arctanh}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{2} \ln \left (1-\frac {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}{\sqrt {-\frac {-c x +1}{c x +1}+1}}\right )}{c}+\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \operatorname {polylog}\left (2, \frac {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}{\sqrt {-\frac {-c x +1}{c x +1}+1}}\right )}{c}-\frac {2 \operatorname {polylog}\left (3, \frac {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}{\sqrt {-\frac {-c x +1}{c x +1}+1}}\right )}{c}+\frac {\operatorname {arctanh}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{2} \ln \left (1+\frac {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}{\sqrt {-\frac {-c x +1}{c x +1}+1}}\right )}{c}+\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \operatorname {polylog}\left (2, -\frac {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}{\sqrt {-\frac {-c x +1}{c x +1}+1}}\right )}{c}-\frac {2 \operatorname {polylog}\left (3, -\frac {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}{\sqrt {-\frac {-c x +1}{c x +1}+1}}\right )}{c}-\frac {\operatorname {arctanh}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{2} \ln \left (1+\frac {\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1\right )^{2}}{-\frac {-c x +1}{c x +1}+1}\right )}{c}-\frac {\operatorname {arctanh}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \operatorname {polylog}\left (2, -\frac {\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1\right )^{2}}{-\frac {-c x +1}{c x +1}+1}\right )}{c}+\frac {\operatorname {polylog}\left (3, -\frac {\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1\right )^{2}}{-\frac {-c x +1}{c x +1}+1}\right )}{2 c}\right )-2 a b \left (\frac {\operatorname {arctanh}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \ln \left (1-\frac {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}{\sqrt {-\frac {-c x +1}{c x +1}+1}}\right )}{c}+\frac {\operatorname {polylog}\left (2, \frac {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}{\sqrt {-\frac {-c x +1}{c x +1}+1}}\right )}{c}+\frac {\operatorname {arctanh}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \ln \left (1+\frac {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}{\sqrt {-\frac {-c x +1}{c x +1}+1}}\right )}{c}+\frac {\operatorname {polylog}\left (2, -\frac {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}{\sqrt {-\frac {-c x +1}{c x +1}+1}}\right )}{c}-\frac {\operatorname {arctanh}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \ln \left (1+\frac {\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1\right )^{2}}{-\frac {-c x +1}{c x +1}+1}\right )}{c}-\frac {\operatorname {polylog}\left (2, -\frac {\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1\right )^{2}}{-\frac {-c x +1}{c x +1}+1}\right )}{2 c}\right )\) | \(893\) |
| parts | \(-\frac {a^{2} \ln \left (c x -1\right )}{2 c}+\frac {a^{2} \ln \left (c x +1\right )}{2 c}-b^{2} \left (\frac {\operatorname {arctanh}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{2} \ln \left (1-\frac {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}{\sqrt {-\frac {-c x +1}{c x +1}+1}}\right )}{c}+\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \operatorname {polylog}\left (2, \frac {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}{\sqrt {-\frac {-c x +1}{c x +1}+1}}\right )}{c}-\frac {2 \operatorname {polylog}\left (3, \frac {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}{\sqrt {-\frac {-c x +1}{c x +1}+1}}\right )}{c}+\frac {\operatorname {arctanh}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{2} \ln \left (1+\frac {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}{\sqrt {-\frac {-c x +1}{c x +1}+1}}\right )}{c}+\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \operatorname {polylog}\left (2, -\frac {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}{\sqrt {-\frac {-c x +1}{c x +1}+1}}\right )}{c}-\frac {2 \operatorname {polylog}\left (3, -\frac {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}{\sqrt {-\frac {-c x +1}{c x +1}+1}}\right )}{c}-\frac {\operatorname {arctanh}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{2} \ln \left (1+\frac {\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1\right )^{2}}{-\frac {-c x +1}{c x +1}+1}\right )}{c}-\frac {\operatorname {arctanh}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \operatorname {polylog}\left (2, -\frac {\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1\right )^{2}}{-\frac {-c x +1}{c x +1}+1}\right )}{c}+\frac {\operatorname {polylog}\left (3, -\frac {\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1\right )^{2}}{-\frac {-c x +1}{c x +1}+1}\right )}{2 c}\right )-2 a b \left (\frac {\operatorname {arctanh}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \ln \left (1-\frac {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}{\sqrt {-\frac {-c x +1}{c x +1}+1}}\right )}{c}+\frac {\operatorname {polylog}\left (2, \frac {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}{\sqrt {-\frac {-c x +1}{c x +1}+1}}\right )}{c}+\frac {\operatorname {arctanh}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \ln \left (1+\frac {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}{\sqrt {-\frac {-c x +1}{c x +1}+1}}\right )}{c}+\frac {\operatorname {polylog}\left (2, -\frac {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}{\sqrt {-\frac {-c x +1}{c x +1}+1}}\right )}{c}-\frac {\operatorname {arctanh}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \ln \left (1+\frac {\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1\right )^{2}}{-\frac {-c x +1}{c x +1}+1}\right )}{c}-\frac {\operatorname {polylog}\left (2, -\frac {\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1\right )^{2}}{-\frac {-c x +1}{c x +1}+1}\right )}{2 c}\right )\) | \(893\) |
-1/2*a^2/c*ln(c*x-1)+1/2*a^2/c*ln(c*x+1)-b^2*(1/c*arctanh((-c*x+1)^(1/2)/( c*x+1)^(1/2))^2*ln(1-((-c*x+1)^(1/2)/(c*x+1)^(1/2)+1)/(-(-c*x+1)/(c*x+1)+1 )^(1/2))+2/c*arctanh((-c*x+1)^(1/2)/(c*x+1)^(1/2))*polylog(2,((-c*x+1)^(1/ 2)/(c*x+1)^(1/2)+1)/(-(-c*x+1)/(c*x+1)+1)^(1/2))-2/c*polylog(3,((-c*x+1)^( 1/2)/(c*x+1)^(1/2)+1)/(-(-c*x+1)/(c*x+1)+1)^(1/2))+1/c*arctanh((-c*x+1)^(1 /2)/(c*x+1)^(1/2))^2*ln(1+((-c*x+1)^(1/2)/(c*x+1)^(1/2)+1)/(-(-c*x+1)/(c*x +1)+1)^(1/2))+2/c*arctanh((-c*x+1)^(1/2)/(c*x+1)^(1/2))*polylog(2,-((-c*x+ 1)^(1/2)/(c*x+1)^(1/2)+1)/(-(-c*x+1)/(c*x+1)+1)^(1/2))-2/c*polylog(3,-((-c *x+1)^(1/2)/(c*x+1)^(1/2)+1)/(-(-c*x+1)/(c*x+1)+1)^(1/2))-1/c*arctanh((-c* x+1)^(1/2)/(c*x+1)^(1/2))^2*ln(1+((-c*x+1)^(1/2)/(c*x+1)^(1/2)+1)^2/(-(-c* x+1)/(c*x+1)+1))-1/c*arctanh((-c*x+1)^(1/2)/(c*x+1)^(1/2))*polylog(2,-((-c *x+1)^(1/2)/(c*x+1)^(1/2)+1)^2/(-(-c*x+1)/(c*x+1)+1))+1/2/c*polylog(3,-((- c*x+1)^(1/2)/(c*x+1)^(1/2)+1)^2/(-(-c*x+1)/(c*x+1)+1)))-2*a*b*(1/c*arctanh ((-c*x+1)^(1/2)/(c*x+1)^(1/2))*ln(1-((-c*x+1)^(1/2)/(c*x+1)^(1/2)+1)/(-(-c *x+1)/(c*x+1)+1)^(1/2))+1/c*polylog(2,((-c*x+1)^(1/2)/(c*x+1)^(1/2)+1)/(-( -c*x+1)/(c*x+1)+1)^(1/2))+1/c*arctanh((-c*x+1)^(1/2)/(c*x+1)^(1/2))*ln(1+( (-c*x+1)^(1/2)/(c*x+1)^(1/2)+1)/(-(-c*x+1)/(c*x+1)+1)^(1/2))+1/c*polylog(2 ,-((-c*x+1)^(1/2)/(c*x+1)^(1/2)+1)/(-(-c*x+1)/(c*x+1)+1)^(1/2))-1/c*arctan h((-c*x+1)^(1/2)/(c*x+1)^(1/2))*ln(1+((-c*x+1)^(1/2)/(c*x+1)^(1/2)+1)^2/(- (-c*x+1)/(c*x+1)+1))-1/2/c*polylog(2,-((-c*x+1)^(1/2)/(c*x+1)^(1/2)+1)^...
\[ \int \frac {\left (a+b \text {arctanh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2}{1-c^2 x^2} \, dx=\int { -\frac {{\left (b \operatorname {artanh}\left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right ) + a\right )}^{2}}{c^{2} x^{2} - 1} \,d x } \]
integral(-(b^2*arctanh(sqrt(-c*x + 1)/sqrt(c*x + 1))^2 + 2*a*b*arctanh(sqr t(-c*x + 1)/sqrt(c*x + 1)) + a^2)/(c^2*x^2 - 1), x)
\[ \int \frac {\left (a+b \text {arctanh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2}{1-c^2 x^2} \, dx=- \int \frac {a^{2}}{c^{2} x^{2} - 1}\, dx - \int \frac {b^{2} \operatorname {atanh}^{2}{\left (\frac {\sqrt {- c x + 1}}{\sqrt {c x + 1}} \right )}}{c^{2} x^{2} - 1}\, dx - \int \frac {2 a b \operatorname {atanh}{\left (\frac {\sqrt {- c x + 1}}{\sqrt {c x + 1}} \right )}}{c^{2} x^{2} - 1}\, dx \]
-Integral(a**2/(c**2*x**2 - 1), x) - Integral(b**2*atanh(sqrt(-c*x + 1)/sq rt(c*x + 1))**2/(c**2*x**2 - 1), x) - Integral(2*a*b*atanh(sqrt(-c*x + 1)/ sqrt(c*x + 1))/(c**2*x**2 - 1), x)
\[ \int \frac {\left (a+b \text {arctanh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2}{1-c^2 x^2} \, dx=\int { -\frac {{\left (b \operatorname {artanh}\left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right ) + a\right )}^{2}}{c^{2} x^{2} - 1} \,d x } \]
1/2*a^2*(log(c*x + 1)/c - log(c*x - 1)/c) + 1/8*(b^2*log(c*x + 1) - b^2*lo g(-c*x + 1))*log(sqrt(c*x + 1) - sqrt(-c*x + 1))^2/c + integrate(-1/8*(2*( sqrt(c*x + 1)*b^2 - sqrt(-c*x + 1)*b^2)*log(sqrt(c*x + 1) + sqrt(-c*x + 1) )^2 + 8*(sqrt(c*x + 1)*a*b - sqrt(-c*x + 1)*a*b)*log(sqrt(c*x + 1) + sqrt( -c*x + 1)) - (4*(sqrt(c*x + 1)*b^2 - sqrt(-c*x + 1)*b^2)*log(sqrt(c*x + 1) + sqrt(-c*x + 1)) + (8*a*b - (b^2*c*x - b^2)*log(c*x + 1) + (b^2*c*x - b^ 2)*log(-c*x + 1))*sqrt(c*x + 1) - (8*a*b - (b^2*c*x + b^2)*log(c*x + 1) + (b^2*c*x + b^2)*log(-c*x + 1))*sqrt(-c*x + 1))*log(sqrt(c*x + 1) - sqrt(-c *x + 1)))/((c^2*x^2 - 1)*sqrt(c*x + 1) - (c^2*x^2 - 1)*sqrt(-c*x + 1)), x)
\[ \int \frac {\left (a+b \text {arctanh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2}{1-c^2 x^2} \, dx=\int { -\frac {{\left (b \operatorname {artanh}\left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right ) + a\right )}^{2}}{c^{2} x^{2} - 1} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \text {arctanh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2}{1-c^2 x^2} \, dx=\int -\frac {{\left (a+b\,\mathrm {atanh}\left (\frac {\sqrt {1-c\,x}}{\sqrt {c\,x+1}}\right )\right )}^2}{c^2\,x^2-1} \,d x \]