Integrand size = 40, antiderivative size = 409 \[ \int \frac {\left (a+b \text {arctanh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3}{1-c^2 x^2} \, dx=-\frac {2 \left (a+b \text {arctanh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3 \text {arctanh}\left (1-\frac {2}{1-\frac {\sqrt {1-c x}}{\sqrt {1+c x}}}\right )}{c}+\frac {3 b \left (a+b \text {arctanh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-\frac {\sqrt {1-c x}}{\sqrt {1+c x}}}\right )}{2 c}-\frac {3 b \left (a+b \text {arctanh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-\frac {\sqrt {1-c x}}{\sqrt {1+c x}}}\right )}{2 c}-\frac {3 b^2 \left (a+b \text {arctanh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right ) \operatorname {PolyLog}\left (3,1-\frac {2}{1-\frac {\sqrt {1-c x}}{\sqrt {1+c x}}}\right )}{2 c}+\frac {3 b^2 \left (a+b \text {arctanh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right ) \operatorname {PolyLog}\left (3,-1+\frac {2}{1-\frac {\sqrt {1-c x}}{\sqrt {1+c x}}}\right )}{2 c}+\frac {3 b^3 \operatorname {PolyLog}\left (4,1-\frac {2}{1-\frac {\sqrt {1-c x}}{\sqrt {1+c x}}}\right )}{4 c}-\frac {3 b^3 \operatorname {PolyLog}\left (4,-1+\frac {2}{1-\frac {\sqrt {1-c x}}{\sqrt {1+c x}}}\right )}{4 c} \]
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)))^3/c+3/2*b*(a+b*arctanh((-c*x+1)^(1/2)/(c*x+1)^(1/2)))^2 *polylog(2,1-2/(1-(-c*x+1)^(1/2)/(c*x+1)^(1/2)))/c-3/2*b*(a+b*arctanh((-c* x+1)^(1/2)/(c*x+1)^(1/2)))^2*polylog(2,-1+2/(1-(-c*x+1)^(1/2)/(c*x+1)^(1/2 )))/c-3/2*b^2*(a+b*arctanh((-c*x+1)^(1/2)/(c*x+1)^(1/2)))*polylog(3,1-2/(1 -(-c*x+1)^(1/2)/(c*x+1)^(1/2)))/c+3/2*b^2*(a+b*arctanh((-c*x+1)^(1/2)/(c*x +1)^(1/2)))*polylog(3,-1+2/(1-(-c*x+1)^(1/2)/(c*x+1)^(1/2)))/c+3/4*b^3*pol ylog(4,1-2/(1-(-c*x+1)^(1/2)/(c*x+1)^(1/2)))/c-3/4*b^3*polylog(4,-1+2/(1-( -c*x+1)^(1/2)/(c*x+1)^(1/2)))/c
\[ \int \frac {\left (a+b \text {arctanh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3}{1-c^2 x^2} \, dx=\int \frac {\left (a+b \text {arctanh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3}{1-c^2 x^2} \, dx \]
Time = 1.21 (sec) , antiderivative size = 402, normalized size of antiderivative = 0.98, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {7232, 6448, 6614, 6620, 6624, 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.
| \(\displaystyle \int \frac {\left (a+b \text {arctanh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )^3}{1-c^2 x^2} \, dx\) |
| \(\Big \downarrow \) 7232 |
| \(\displaystyle -\frac {\int \frac {\sqrt {c x+1} \left (a+b \text {arctanh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )^3}{\sqrt {1-c x}}d\frac {\sqrt {1-c x}}{\sqrt {c x+1}}}{c}\) |
| \(\Big \downarrow \) 6448 |
| \(\displaystyle -\frac {2 \text {arctanh}\left (1-\frac {2}{1-\frac {\sqrt {1-c x}}{\sqrt {c x+1}}}\right ) \left (a+b \text {arctanh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )^3-6 b \int \frac {\left (a+b \text {arctanh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )^2 \text {arctanh}\left (1-\frac {2}{1-\frac {\sqrt {1-c x}}{\sqrt {c x+1}}}\right )}{1-\frac {1-c x}{c x+1}}d\frac {\sqrt {1-c x}}{\sqrt {c x+1}}}{c}\) |
| \(\Big \downarrow \) 6614 |
| \(\displaystyle -\frac {2 \text {arctanh}\left (1-\frac {2}{1-\frac {\sqrt {1-c x}}{\sqrt {c x+1}}}\right ) \left (a+b \text {arctanh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )^3-6 b \left (\frac {1}{2} \int \frac {\left (a+b \text {arctanh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )^2 \log \left (2-\frac {2}{1-\frac {\sqrt {1-c x}}{\sqrt {c x+1}}}\right )}{1-\frac {1-c x}{c x+1}}d\frac {\sqrt {1-c x}}{\sqrt {c x+1}}-\frac {1}{2} \int \frac {\left (a+b \text {arctanh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )^2 \log \left (\frac {2}{1-\frac {\sqrt {1-c x}}{\sqrt {c x+1}}}\right )}{1-\frac {1-c x}{c x+1}}d\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )}{c}\) |
| \(\Big \downarrow \) 6620 |
| \(\displaystyle -\frac {2 \text {arctanh}\left (1-\frac {2}{1-\frac {\sqrt {1-c x}}{\sqrt {c x+1}}}\right ) \left (a+b \text {arctanh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )^3-6 b \left (\frac {1}{2} \left (\frac {1}{2} \operatorname {PolyLog}\left (2,1-\frac {2}{1-\frac {\sqrt {1-c x}}{\sqrt {c x+1}}}\right ) \left (a+b \text {arctanh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )^2-b \int \frac {\left (a+b \text {arctanh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{1-\frac {\sqrt {1-c x}}{\sqrt {c x+1}}}\right )}{1-\frac {1-c x}{c x+1}}d\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )+\frac {1}{2} \left (b \int \frac {\left (a+b \text {arctanh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right ) \operatorname {PolyLog}\left (2,\frac {2}{1-\frac {\sqrt {1-c x}}{\sqrt {c x+1}}}-1\right )}{1-\frac {1-c x}{c x+1}}d\frac {\sqrt {1-c x}}{\sqrt {c x+1}}-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{1-\frac {\sqrt {1-c x}}{\sqrt {c x+1}}}-1\right ) \left (a+b \text {arctanh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )^2\right )\right )}{c}\) |
| \(\Big \downarrow \) 6624 |
| \(\displaystyle -\frac {2 \text {arctanh}\left (1-\frac {2}{1-\frac {\sqrt {1-c x}}{\sqrt {c x+1}}}\right ) \left (a+b \text {arctanh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )^3-6 b \left (\frac {1}{2} \left (\frac {1}{2} \operatorname {PolyLog}\left (2,1-\frac {2}{1-\frac {\sqrt {1-c x}}{\sqrt {c x+1}}}\right ) \left (a+b \text {arctanh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )^2-b \left (\frac {1}{2} \operatorname {PolyLog}\left (3,1-\frac {2}{1-\frac {\sqrt {1-c x}}{\sqrt {c x+1}}}\right ) \left (a+b \text {arctanh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )-\frac {1}{2} b \int \frac {\operatorname {PolyLog}\left (3,1-\frac {2}{1-\frac {\sqrt {1-c x}}{\sqrt {c x+1}}}\right )}{1-\frac {1-c x}{c x+1}}d\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )+\frac {1}{2} \left (b \left (\frac {1}{2} \operatorname {PolyLog}\left (3,\frac {2}{1-\frac {\sqrt {1-c x}}{\sqrt {c x+1}}}-1\right ) \left (a+b \text {arctanh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )-\frac {1}{2} b \int \frac {\operatorname {PolyLog}\left (3,\frac {2}{1-\frac {\sqrt {1-c x}}{\sqrt {c x+1}}}-1\right )}{1-\frac {1-c x}{c x+1}}d\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{1-\frac {\sqrt {1-c x}}{\sqrt {c x+1}}}-1\right ) \left (a+b \text {arctanh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )^2\right )\right )}{c}\) |
| \(\Big \downarrow \) 7164 |
| \(\displaystyle -\frac {2 \text {arctanh}\left (1-\frac {2}{1-\frac {\sqrt {1-c x}}{\sqrt {c x+1}}}\right ) \left (a+b \text {arctanh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )^3-6 b \left (\frac {1}{2} \left (\frac {1}{2} \operatorname {PolyLog}\left (2,1-\frac {2}{1-\frac {\sqrt {1-c x}}{\sqrt {c x+1}}}\right ) \left (a+b \text {arctanh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )^2-b \left (\frac {1}{2} \operatorname {PolyLog}\left (3,1-\frac {2}{1-\frac {\sqrt {1-c x}}{\sqrt {c x+1}}}\right ) \left (a+b \text {arctanh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )-\frac {1}{4} b \operatorname {PolyLog}\left (4,1-\frac {2}{1-\frac {\sqrt {1-c x}}{\sqrt {c x+1}}}\right )\right )\right )+\frac {1}{2} \left (b \left (\frac {1}{2} \operatorname {PolyLog}\left (3,\frac {2}{1-\frac {\sqrt {1-c x}}{\sqrt {c x+1}}}-1\right ) \left (a+b \text {arctanh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )-\frac {1}{4} b \operatorname {PolyLog}\left (4,\frac {2}{1-\frac {\sqrt {1-c x}}{\sqrt {c x+1}}}-1\right )\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{1-\frac {\sqrt {1-c x}}{\sqrt {c x+1}}}-1\right ) \left (a+b \text {arctanh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )^2\right )\right )}{c}\) |
-((2*(a + b*ArcTanh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^3*ArcTanh[1 - 2/(1 - Sqr t[1 - c*x]/Sqrt[1 + c*x])] - 6*b*((((a + b*ArcTanh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2*PolyLog[2, 1 - 2/(1 - Sqrt[1 - c*x]/Sqrt[1 + c*x])])/2 - b*(((a + b*ArcTanh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])*PolyLog[3, 1 - 2/(1 - Sqrt[1 - c* x]/Sqrt[1 + c*x])])/2 - (b*PolyLog[4, 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]])^2*PolyLo g[2, -1 + 2/(1 - Sqrt[1 - c*x]/Sqrt[1 + c*x])]) + b*(((a + b*ArcTanh[Sqrt[ 1 - c*x]/Sqrt[1 + c*x]])*PolyLog[3, -1 + 2/(1 - Sqrt[1 - c*x]/Sqrt[1 + c*x ])])/2 - (b*PolyLog[4, -1 + 2/(1 - Sqrt[1 - c*x]/Sqrt[1 + c*x])])/4))/2))/ c)
3.1.31.3.1 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[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*PolyLog[k_, u_])/((d_) + (e_ .)*(x_)^2), x_Symbol] :> Simp[(a + b*ArcTanh[c*x])^p*(PolyLog[k + 1, u]/(2* c*d)), x] - Simp[b*(p/2) Int[(a + b*ArcTanh[c*x])^(p - 1)*(PolyLog[k + 1, u]/(d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e, k}, x] && IGtQ[p, 0] && E qQ[c^2*d + e, 0] && EqQ[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. \(1622\) vs. \(2(349)=698\).
Time = 0.55 (sec) , antiderivative size = 1623, normalized size of antiderivative = 3.97
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1623\) |
| parts | \(\text {Expression too large to display}\) | \(1623\) |
-1/2*a^3/c*ln(c*x-1)+1/2*a^3/c*ln(c*x+1)-b^3*(1/c*arctanh((-c*x+1)^(1/2)/( c*x+1)^(1/2))^3*ln(1-((-c*x+1)^(1/2)/(c*x+1)^(1/2)+1)/(-(-c*x+1)/(c*x+1)+1 )^(1/2))+3/c*arctanh((-c*x+1)^(1/2)/(c*x+1)^(1/2))^2*polylog(2,((-c*x+1)^( 1/2)/(c*x+1)^(1/2)+1)/(-(-c*x+1)/(c*x+1)+1)^(1/2))-6/c*arctanh((-c*x+1)^(1 /2)/(c*x+1)^(1/2))*polylog(3,((-c*x+1)^(1/2)/(c*x+1)^(1/2)+1)/(-(-c*x+1)/( c*x+1)+1)^(1/2))+6/c*polylog(4,((-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))^3*ln(1+((-c*x +1)^(1/2)/(c*x+1)^(1/2)+1)/(-(-c*x+1)/(c*x+1)+1)^(1/2))+3/c*arctanh((-c*x+ 1)^(1/2)/(c*x+1)^(1/2))^2*polylog(2,-((-c*x+1)^(1/2)/(c*x+1)^(1/2)+1)/(-(- c*x+1)/(c*x+1)+1)^(1/2))-6/c*arctanh((-c*x+1)^(1/2)/(c*x+1)^(1/2))*polylog (3,-((-c*x+1)^(1/2)/(c*x+1)^(1/2)+1)/(-(-c*x+1)/(c*x+1)+1)^(1/2))+6/c*poly log(4,-((-c*x+1)^(1/2)/(c*x+1)^(1/2)+1)/(-(-c*x+1)/(c*x+1)+1)^(1/2))-1/c*a rctanh((-c*x+1)^(1/2)/(c*x+1)^(1/2))^3*ln(1+((-c*x+1)^(1/2)/(c*x+1)^(1/2)+ 1)^2/(-(-c*x+1)/(c*x+1)+1))-3/2/c*arctanh((-c*x+1)^(1/2)/(c*x+1)^(1/2))^2* polylog(2,-((-c*x+1)^(1/2)/(c*x+1)^(1/2)+1)^2/(-(-c*x+1)/(c*x+1)+1))+3/2/c *arctanh((-c*x+1)^(1/2)/(c*x+1)^(1/2))*polylog(3,-((-c*x+1)^(1/2)/(c*x+1)^ (1/2)+1)^2/(-(-c*x+1)/(c*x+1)+1))-3/4/c*polylog(4,-((-c*x+1)^(1/2)/(c*x+1) ^(1/2)+1)^2/(-(-c*x+1)/(c*x+1)+1)))-3*a*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)^(...
\[ \int \frac {\left (a+b \text {arctanh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3}{1-c^2 x^2} \, dx=\int { -\frac {{\left (b \operatorname {artanh}\left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right ) + a\right )}^{3}}{c^{2} x^{2} - 1} \,d x } \]
integral(-(b^3*arctanh(sqrt(-c*x + 1)/sqrt(c*x + 1))^3 + 3*a*b^2*arctanh(s qrt(-c*x + 1)/sqrt(c*x + 1))^2 + 3*a^2*b*arctanh(sqrt(-c*x + 1)/sqrt(c*x + 1)) + a^3)/(c^2*x^2 - 1), x)
\[ \int \frac {\left (a+b \text {arctanh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3}{1-c^2 x^2} \, dx=- \int \frac {a^{3}}{c^{2} x^{2} - 1}\, dx - \int \frac {b^{3} \operatorname {atanh}^{3}{\left (\frac {\sqrt {- c x + 1}}{\sqrt {c x + 1}} \right )}}{c^{2} x^{2} - 1}\, dx - \int \frac {3 a b^{2} \operatorname {atanh}^{2}{\left (\frac {\sqrt {- c x + 1}}{\sqrt {c x + 1}} \right )}}{c^{2} x^{2} - 1}\, dx - \int \frac {3 a^{2} b \operatorname {atanh}{\left (\frac {\sqrt {- c x + 1}}{\sqrt {c x + 1}} \right )}}{c^{2} x^{2} - 1}\, dx \]
-Integral(a**3/(c**2*x**2 - 1), x) - Integral(b**3*atanh(sqrt(-c*x + 1)/sq rt(c*x + 1))**3/(c**2*x**2 - 1), x) - Integral(3*a*b**2*atanh(sqrt(-c*x + 1)/sqrt(c*x + 1))**2/(c**2*x**2 - 1), x) - Integral(3*a**2*b*atanh(sqrt(-c *x + 1)/sqrt(c*x + 1))/(c**2*x**2 - 1), x)
\[ \int \frac {\left (a+b \text {arctanh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3}{1-c^2 x^2} \, dx=\int { -\frac {{\left (b \operatorname {artanh}\left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right ) + a\right )}^{3}}{c^{2} x^{2} - 1} \,d x } \]
1/2*a^3*(log(c*x + 1)/c - log(c*x - 1)/c) - 1/16*(b^3*log(c*x + 1) - b^3*l og(-c*x + 1))*log(sqrt(c*x + 1) - sqrt(-c*x + 1))^3/c - integrate(1/32*(4* (sqrt(c*x + 1)*b^3 - sqrt(-c*x + 1)*b^3)*log(sqrt(c*x + 1) + sqrt(-c*x + 1 ))^3 + 24*(sqrt(c*x + 1)*a*b^2 - sqrt(-c*x + 1)*a*b^2)*log(sqrt(c*x + 1) + sqrt(-c*x + 1))^2 + 3*(4*(sqrt(c*x + 1)*b^3 - sqrt(-c*x + 1)*b^3)*log(sqr t(c*x + 1) + sqrt(-c*x + 1)) + (8*a*b^2 - (b^3*c*x - b^3)*log(c*x + 1) + ( b^3*c*x - b^3)*log(-c*x + 1))*sqrt(c*x + 1) - (8*a*b^2 - (b^3*c*x + b^3)*l og(c*x + 1) + (b^3*c*x + b^3)*log(-c*x + 1))*sqrt(-c*x + 1))*log(sqrt(c*x + 1) - sqrt(-c*x + 1))^2 + 48*(sqrt(c*x + 1)*a^2*b - sqrt(-c*x + 1)*a^2*b) *log(sqrt(c*x + 1) + sqrt(-c*x + 1)) - 12*(4*sqrt(c*x + 1)*a^2*b - 4*sqrt( -c*x + 1)*a^2*b + (sqrt(c*x + 1)*b^3 - sqrt(-c*x + 1)*b^3)*log(sqrt(c*x + 1) + sqrt(-c*x + 1))^2 + 4*(sqrt(c*x + 1)*a*b^2 - sqrt(-c*x + 1)*a*b^2)*lo g(sqrt(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)
\[ \int \frac {\left (a+b \text {arctanh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3}{1-c^2 x^2} \, dx=\int { -\frac {{\left (b \operatorname {artanh}\left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right ) + a\right )}^{3}}{c^{2} x^{2} - 1} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \text {arctanh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3}{1-c^2 x^2} \, dx=\int -\frac {{\left (a+b\,\mathrm {atanh}\left (\frac {\sqrt {1-c\,x}}{\sqrt {c\,x+1}}\right )\right )}^3}{c^2\,x^2-1} \,d x \]