Integrand size = 40, antiderivative size = 321 \[ \int \frac {\left (a+b \cot ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2}{1-c^2 x^2} \, dx=-\frac {2 \left (a+b \cot ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2 \coth ^{-1}\left (1-\frac {2}{1+\frac {i \sqrt {1-c x}}{\sqrt {1+c x}}}\right )}{c}+\frac {i b \left (a+b \cot ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right ) \operatorname {PolyLog}\left (2,1-\frac {2 i}{i+\frac {\sqrt {1-c x}}{\sqrt {1+c x}}}\right )}{c}-\frac {i b \left (a+b \cot ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right ) \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {1-c x}}{\sqrt {1+c x} \left (i+\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{c}+\frac {b^2 \operatorname {PolyLog}\left (3,1-\frac {2 i}{i+\frac {\sqrt {1-c x}}{\sqrt {1+c x}}}\right )}{2 c}-\frac {b^2 \operatorname {PolyLog}\left (3,1-\frac {2 \sqrt {1-c x}}{\sqrt {1+c x} \left (i+\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{2 c} \]
-2*(a+b*arccot((-c*x+1)^(1/2)/(c*x+1)^(1/2)))^2*arccoth(1-2/(1+I*(-c*x+1)^ (1/2)/(c*x+1)^(1/2)))/c+I*b*(a+b*arccot((-c*x+1)^(1/2)/(c*x+1)^(1/2)))*pol ylog(2,1-2*I/(I+(-c*x+1)^(1/2)/(c*x+1)^(1/2)))/c-I*b*(a+b*arccot((-c*x+1)^ (1/2)/(c*x+1)^(1/2)))*polylog(2,1-2*(-c*x+1)^(1/2)/(I+(-c*x+1)^(1/2)/(c*x+ 1)^(1/2))/(c*x+1)^(1/2))/c+1/2*b^2*polylog(3,1-2*I/(I+(-c*x+1)^(1/2)/(c*x+ 1)^(1/2)))/c-1/2*b^2*polylog(3,1-2*(-c*x+1)^(1/2)/(I+(-c*x+1)^(1/2)/(c*x+1 )^(1/2))/(c*x+1)^(1/2))/c
\[ \int \frac {\left (a+b \cot ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2}{1-c^2 x^2} \, dx=\int \frac {\left (a+b \cot ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2}{1-c^2 x^2} \, dx \]
Time = 0.79 (sec) , antiderivative size = 323, 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, 5358, 5524, 5528, 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.
| \(\displaystyle \int \frac {\left (a+b \cot ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )^2}{1-c^2 x^2} \, dx\) |
| \(\Big \downarrow \) 7232 |
| \(\displaystyle -\frac {\int \frac {\sqrt {c x+1} \left (a+b \cot ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )^2}{\sqrt {1-c x}}d\frac {\sqrt {1-c x}}{\sqrt {c x+1}}}{c}\) |
| \(\Big \downarrow \) 5358 |
| \(\displaystyle -\frac {4 b \int \frac {\left (a+b \cot ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right ) \coth ^{-1}\left (1-\frac {2}{\frac {i \sqrt {1-c x}}{\sqrt {c x+1}}+1}\right )}{\frac {1-c x}{c x+1}+1}d\frac {\sqrt {1-c x}}{\sqrt {c x+1}}+2 \coth ^{-1}\left (1-\frac {2}{1+\frac {i \sqrt {1-c x}}{\sqrt {c x+1}}}\right ) \left (a+b \cot ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )^2}{c}\) |
| \(\Big \downarrow \) 5524 |
| \(\displaystyle -\frac {4 b \left (\frac {1}{2} \int \frac {\left (a+b \cot ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right ) \log \left (\frac {2 \sqrt {1-c x}}{\sqrt {c x+1} \left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}+i\right )}\right )}{\frac {1-c x}{c x+1}+1}d\frac {\sqrt {1-c x}}{\sqrt {c x+1}}-\frac {1}{2} \int \frac {\left (a+b \cot ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right ) \log \left (\frac {2 i}{\frac {\sqrt {1-c x}}{\sqrt {c x+1}}+i}\right )}{\frac {1-c x}{c x+1}+1}d\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )+2 \coth ^{-1}\left (1-\frac {2}{1+\frac {i \sqrt {1-c x}}{\sqrt {c x+1}}}\right ) \left (a+b \cot ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )^2}{c}\) |
| \(\Big \downarrow \) 5528 |
| \(\displaystyle -\frac {4 b \left (\frac {1}{2} \left (-\frac {1}{2} i b \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2 i}{\frac {\sqrt {1-c x}}{\sqrt {c x+1}}+i}\right )}{\frac {1-c x}{c x+1}+1}d\frac {\sqrt {1-c x}}{\sqrt {c x+1}}-\frac {1}{2} i \operatorname {PolyLog}\left (2,1-\frac {2 i}{\frac {\sqrt {1-c x}}{\sqrt {c x+1}}+i}\right ) \left (a+b \cot ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )\right )+\frac {1}{2} \left (\frac {1}{2} i b \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {1-c x}}{\sqrt {c x+1} \left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}+i\right )}\right )}{\frac {1-c x}{c x+1}+1}d\frac {\sqrt {1-c x}}{\sqrt {c x+1}}+\frac {1}{2} i \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {1-c x}}{\sqrt {c x+1} \left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}+i\right )}\right ) \left (a+b \cot ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )\right )\right )+2 \coth ^{-1}\left (1-\frac {2}{1+\frac {i \sqrt {1-c x}}{\sqrt {c x+1}}}\right ) \left (a+b \cot ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )^2}{c}\) |
| \(\Big \downarrow \) 7164 |
| \(\displaystyle -\frac {4 b \left (\frac {1}{2} \left (-\frac {1}{2} i \operatorname {PolyLog}\left (2,1-\frac {2 i}{\frac {\sqrt {1-c x}}{\sqrt {c x+1}}+i}\right ) \left (a+b \cot ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )-\frac {1}{4} b \operatorname {PolyLog}\left (3,1-\frac {2 i}{\frac {\sqrt {1-c x}}{\sqrt {c x+1}}+i}\right )\right )+\frac {1}{2} \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {1-c x}}{\sqrt {c x+1} \left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}+i\right )}\right ) \left (a+b \cot ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )+\frac {1}{4} b \operatorname {PolyLog}\left (3,1-\frac {2 \sqrt {1-c x}}{\sqrt {c x+1} \left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}+i\right )}\right )\right )\right )+2 \coth ^{-1}\left (1-\frac {2}{1+\frac {i \sqrt {1-c x}}{\sqrt {c x+1}}}\right ) \left (a+b \cot ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )^2}{c}\) |
-((2*(a + b*ArcCot[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2*ArcCoth[1 - 2/(1 + (I*S qrt[1 - c*x])/Sqrt[1 + c*x])] + 4*b*(((-1/2*I)*(a + b*ArcCot[Sqrt[1 - c*x] /Sqrt[1 + c*x]])*PolyLog[2, 1 - (2*I)/(I + Sqrt[1 - c*x]/Sqrt[1 + c*x])] - (b*PolyLog[3, 1 - (2*I)/(I + Sqrt[1 - c*x]/Sqrt[1 + c*x])])/4)/2 + ((I/2) *(a + b*ArcCot[Sqrt[1 - c*x]/Sqrt[1 + c*x]])*PolyLog[2, 1 - (2*Sqrt[1 - c* x])/(Sqrt[1 + c*x]*(I + Sqrt[1 - c*x]/Sqrt[1 + c*x]))] + (b*PolyLog[3, 1 - (2*Sqrt[1 - c*x])/(Sqrt[1 + c*x]*(I + Sqrt[1 - c*x]/Sqrt[1 + c*x]))])/4)/ 2))/c)
3.2.53.3.1 Defintions of rubi rules used
Int[((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_)/(x_), x_Symbol] :> Simp[2*(a + b*ArcCot[c*x])^p*ArcCoth[1 - 2/(1 + I*c*x)], x] + Simp[2*b*c*p Int[(a + b *ArcCot[c*x])^(p - 1)*(ArcCoth[1 - 2/(1 + I*c*x)]/(1 + c^2*x^2)), x], x] /; FreeQ[{a, b, c}, x] && IGtQ[p, 1]
Int[(ArcCoth[u_]*((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x _)^2), x_Symbol] :> Simp[1/2 Int[Log[SimplifyIntegrand[1 + 1/u, x]]*((a + b*ArcCot[c*x])^p/(d + e*x^2)), x], x] - Simp[1/2 Int[Log[SimplifyIntegra nd[1 - 1/u, x]]*((a + b*ArcCot[c*x])^p/(d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[e, c^2*d] && EqQ[u^2 - (1 - 2*(I/(I - c* x)))^2, 0]
Int[(Log[u_]*((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2 ), x_Symbol] :> Simp[I*(a + b*ArcCot[c*x])^p*(PolyLog[2, 1 - u]/(2*c*d)), x ] + Simp[b*p*(I/2) Int[(a + b*ArcCot[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[e, c^2* d] && EqQ[(1 - u)^2 - (1 - 2*(I/(I + 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]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 902 vs. \(2 (268 ) = 536\).
Time = 0.70 (sec) , antiderivative size = 903, normalized size of antiderivative = 2.81
| 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 {arccot}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{2} \ln \left (1-\frac {i+\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}}{\sqrt {\frac {-c x +1}{c x +1}+1}}\right )}{c}+\frac {2 i \operatorname {arccot}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \operatorname {polylog}\left (2, \frac {i+\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}}{\sqrt {\frac {-c x +1}{c x +1}+1}}\right )}{c}-\frac {2 \operatorname {polylog}\left (3, \frac {i+\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}}{\sqrt {\frac {-c x +1}{c x +1}+1}}\right )}{c}+\frac {\operatorname {arccot}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{2} \ln \left (1+\frac {\left (i+\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{2}}{\frac {-c x +1}{c x +1}+1}\right )}{c}-\frac {i \operatorname {arccot}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \operatorname {polylog}\left (2, -\frac {\left (i+\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{2}}{\frac {-c x +1}{c x +1}+1}\right )}{c}+\frac {\operatorname {polylog}\left (3, -\frac {\left (i+\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{2}}{\frac {-c x +1}{c x +1}+1}\right )}{2 c}-\frac {\operatorname {arccot}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{2} \ln \left (1+\frac {i+\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}}{\sqrt {\frac {-c x +1}{c x +1}+1}}\right )}{c}+\frac {2 i \operatorname {arccot}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \operatorname {polylog}\left (2, -\frac {i+\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}}{\sqrt {\frac {-c x +1}{c x +1}+1}}\right )}{c}-\frac {2 \operatorname {polylog}\left (3, -\frac {i+\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}}{\sqrt {\frac {-c x +1}{c x +1}+1}}\right )}{c}\right )-2 a b \left (-\frac {\operatorname {arccot}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \ln \left (1-\frac {i+\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}}{\sqrt {\frac {-c x +1}{c x +1}+1}}\right )}{c}+\frac {i \operatorname {polylog}\left (2, \frac {i+\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}}{\sqrt {\frac {-c x +1}{c x +1}+1}}\right )}{c}+\frac {\operatorname {arccot}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \ln \left (1+\frac {\left (i+\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{2}}{\frac {-c x +1}{c x +1}+1}\right )}{c}-\frac {i \operatorname {polylog}\left (2, -\frac {\left (i+\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{2}}{\frac {-c x +1}{c x +1}+1}\right )}{2 c}-\frac {\operatorname {arccot}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \ln \left (1+\frac {i+\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}}{\sqrt {\frac {-c x +1}{c x +1}+1}}\right )}{c}+\frac {i \operatorname {polylog}\left (2, -\frac {i+\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}}{\sqrt {\frac {-c x +1}{c x +1}+1}}\right )}{c}\right )\) | \(903\) |
| 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 {arccot}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{2} \ln \left (1-\frac {i+\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}}{\sqrt {\frac {-c x +1}{c x +1}+1}}\right )}{c}+\frac {2 i \operatorname {arccot}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \operatorname {polylog}\left (2, \frac {i+\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}}{\sqrt {\frac {-c x +1}{c x +1}+1}}\right )}{c}-\frac {2 \operatorname {polylog}\left (3, \frac {i+\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}}{\sqrt {\frac {-c x +1}{c x +1}+1}}\right )}{c}+\frac {\operatorname {arccot}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{2} \ln \left (1+\frac {\left (i+\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{2}}{\frac {-c x +1}{c x +1}+1}\right )}{c}-\frac {i \operatorname {arccot}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \operatorname {polylog}\left (2, -\frac {\left (i+\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{2}}{\frac {-c x +1}{c x +1}+1}\right )}{c}+\frac {\operatorname {polylog}\left (3, -\frac {\left (i+\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{2}}{\frac {-c x +1}{c x +1}+1}\right )}{2 c}-\frac {\operatorname {arccot}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{2} \ln \left (1+\frac {i+\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}}{\sqrt {\frac {-c x +1}{c x +1}+1}}\right )}{c}+\frac {2 i \operatorname {arccot}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \operatorname {polylog}\left (2, -\frac {i+\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}}{\sqrt {\frac {-c x +1}{c x +1}+1}}\right )}{c}-\frac {2 \operatorname {polylog}\left (3, -\frac {i+\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}}{\sqrt {\frac {-c x +1}{c x +1}+1}}\right )}{c}\right )-2 a b \left (-\frac {\operatorname {arccot}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \ln \left (1-\frac {i+\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}}{\sqrt {\frac {-c x +1}{c x +1}+1}}\right )}{c}+\frac {i \operatorname {polylog}\left (2, \frac {i+\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}}{\sqrt {\frac {-c x +1}{c x +1}+1}}\right )}{c}+\frac {\operatorname {arccot}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \ln \left (1+\frac {\left (i+\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{2}}{\frac {-c x +1}{c x +1}+1}\right )}{c}-\frac {i \operatorname {polylog}\left (2, -\frac {\left (i+\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{2}}{\frac {-c x +1}{c x +1}+1}\right )}{2 c}-\frac {\operatorname {arccot}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \ln \left (1+\frac {i+\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}}{\sqrt {\frac {-c x +1}{c x +1}+1}}\right )}{c}+\frac {i \operatorname {polylog}\left (2, -\frac {i+\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}}{\sqrt {\frac {-c x +1}{c x +1}+1}}\right )}{c}\right )\) | \(903\) |
-1/2*a^2/c*ln(c*x-1)+1/2*a^2/c*ln(c*x+1)-b^2*(-1/c*arccot((-c*x+1)^(1/2)/( c*x+1)^(1/2))^2*ln(1-(I+(-c*x+1)^(1/2)/(c*x+1)^(1/2))/((-c*x+1)/(c*x+1)+1) ^(1/2))+2*I/c*arccot((-c*x+1)^(1/2)/(c*x+1)^(1/2))*polylog(2,(I+(-c*x+1)^( 1/2)/(c*x+1)^(1/2))/((-c*x+1)/(c*x+1)+1)^(1/2))-2/c*polylog(3,(I+(-c*x+1)^ (1/2)/(c*x+1)^(1/2))/((-c*x+1)/(c*x+1)+1)^(1/2))+1/c*arccot((-c*x+1)^(1/2) /(c*x+1)^(1/2))^2*ln(1+(I+(-c*x+1)^(1/2)/(c*x+1)^(1/2))^2/((-c*x+1)/(c*x+1 )+1))-I/c*arccot((-c*x+1)^(1/2)/(c*x+1)^(1/2))*polylog(2,-(I+(-c*x+1)^(1/2 )/(c*x+1)^(1/2))^2/((-c*x+1)/(c*x+1)+1))+1/2/c*polylog(3,-(I+(-c*x+1)^(1/2 )/(c*x+1)^(1/2))^2/((-c*x+1)/(c*x+1)+1))-1/c*arccot((-c*x+1)^(1/2)/(c*x+1) ^(1/2))^2*ln(1+(I+(-c*x+1)^(1/2)/(c*x+1)^(1/2))/((-c*x+1)/(c*x+1)+1)^(1/2) )+2*I/c*arccot((-c*x+1)^(1/2)/(c*x+1)^(1/2))*polylog(2,-(I+(-c*x+1)^(1/2)/ (c*x+1)^(1/2))/((-c*x+1)/(c*x+1)+1)^(1/2))-2/c*polylog(3,-(I+(-c*x+1)^(1/2 )/(c*x+1)^(1/2))/((-c*x+1)/(c*x+1)+1)^(1/2)))-2*a*b*(-1/c*arccot((-c*x+1)^ (1/2)/(c*x+1)^(1/2))*ln(1-(I+(-c*x+1)^(1/2)/(c*x+1)^(1/2))/((-c*x+1)/(c*x+ 1)+1)^(1/2))+I/c*polylog(2,(I+(-c*x+1)^(1/2)/(c*x+1)^(1/2))/((-c*x+1)/(c*x +1)+1)^(1/2))+1/c*arccot((-c*x+1)^(1/2)/(c*x+1)^(1/2))*ln(1+(I+(-c*x+1)^(1 /2)/(c*x+1)^(1/2))^2/((-c*x+1)/(c*x+1)+1))-1/2*I/c*polylog(2,-(I+(-c*x+1)^ (1/2)/(c*x+1)^(1/2))^2/((-c*x+1)/(c*x+1)+1))-1/c*arccot((-c*x+1)^(1/2)/(c* x+1)^(1/2))*ln(1+(I+(-c*x+1)^(1/2)/(c*x+1)^(1/2))/((-c*x+1)/(c*x+1)+1)^(1/ 2))+I/c*polylog(2,-(I+(-c*x+1)^(1/2)/(c*x+1)^(1/2))/((-c*x+1)/(c*x+1)+1...
\[ \int \frac {\left (a+b \cot ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2}{1-c^2 x^2} \, dx=\int { -\frac {{\left (b \operatorname {arccot}\left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right ) + a\right )}^{2}}{c^{2} x^{2} - 1} \,d x } \]
integral(-(b^2*arccot(sqrt(-c*x + 1)/sqrt(c*x + 1))^2 + 2*a*b*arccot(sqrt( -c*x + 1)/sqrt(c*x + 1)) + a^2)/(c^2*x^2 - 1), x)
\[ \int \frac {\left (a+b \cot ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2}{1-c^2 x^2} \, dx=- \int \frac {a^{2}}{c^{2} x^{2} - 1}\, dx - \int \frac {b^{2} \operatorname {acot}^{2}{\left (\frac {\sqrt {- c x + 1}}{\sqrt {c x + 1}} \right )}}{c^{2} x^{2} - 1}\, dx - \int \frac {2 a b \operatorname {acot}{\left (\frac {\sqrt {- c x + 1}}{\sqrt {c x + 1}} \right )}}{c^{2} x^{2} - 1}\, dx \]
-Integral(a**2/(c**2*x**2 - 1), x) - Integral(b**2*acot(sqrt(-c*x + 1)/sqr t(c*x + 1))**2/(c**2*x**2 - 1), x) - Integral(2*a*b*acot(sqrt(-c*x + 1)/sq rt(c*x + 1))/(c**2*x**2 - 1), x)
\[ \int \frac {\left (a+b \cot ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2}{1-c^2 x^2} \, dx=\int { -\frac {{\left (b \operatorname {arccot}\left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right ) + a\right )}^{2}}{c^{2} x^{2} - 1} \,d x } \]
1/2*a^2*(log(c*x + 1)/c - log(c*x - 1)/c) - 1/32*(b^2*log(2)^2*log(c*x + 1 ) - b^2*log(2)^2*log(-c*x + 1) - 4*(b^2*log(c*x + 1) - b^2*log(-c*x + 1))* arctan2(sqrt(c*x + 1), sqrt(-c*x + 1))^2 - (b^2*(log(c*x + 1)/c - log(c*x - 1)/c)*log(2)^2 + 64*b^2*integrate(1/16*sqrt(c*x + 1)*sqrt(-c*x + 1)*arct an(sqrt(c*x + 1)/sqrt(-c*x + 1))*log(c*x + 1)/(c^2*x^2 - 1), x) - 64*b^2*i ntegrate(1/16*sqrt(c*x + 1)*sqrt(-c*x + 1)*arctan(sqrt(c*x + 1)/sqrt(-c*x + 1))*log(-c*x + 1)/(c^2*x^2 - 1), x) - 384*b^2*integrate(1/16*arctan(sqrt (c*x + 1)/sqrt(-c*x + 1))^2/(c^2*x^2 - 1), x) - 1024*a*b*integrate(1/16*ar ctan(sqrt(c*x + 1)/sqrt(-c*x + 1))/(c^2*x^2 - 1), x))*c)/c
\[ \int \frac {\left (a+b \cot ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2}{1-c^2 x^2} \, dx=\int { -\frac {{\left (b \operatorname {arccot}\left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right ) + a\right )}^{2}}{c^{2} x^{2} - 1} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \cot ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2}{1-c^2 x^2} \, dx=\int -\frac {{\left (a+b\,\mathrm {acot}\left (\frac {\sqrt {1-c\,x}}{\sqrt {c\,x+1}}\right )\right )}^2}{c^2\,x^2-1} \,d x \]