Integrand size = 40, antiderivative size = 194 \[ \int \frac {\left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2}{1-c^2 x^2} \, dx=-\frac {\left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3}{3 b c}-\frac {\left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2 \log \left (1-e^{-2 \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{c}+\frac {b \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right ) \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{c}+\frac {b^2 \operatorname {PolyLog}\left (3,e^{-2 \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{2 c} \]
-1/3*(a+b*arcsinh((-c*x+1)^(1/2)/(c*x+1)^(1/2)))^3/b/c-(a+b*arcsinh((-c*x+ 1)^(1/2)/(c*x+1)^(1/2)))^2*ln(1-1/((-c*x+1)^(1/2)/(c*x+1)^(1/2)+(1+(-c*x+1 )/(c*x+1))^(1/2))^2)/c+b*(a+b*arcsinh((-c*x+1)^(1/2)/(c*x+1)^(1/2)))*polyl og(2,1/((-c*x+1)^(1/2)/(c*x+1)^(1/2)+(1+(-c*x+1)/(c*x+1))^(1/2))^2)/c+1/2* b^2*polylog(3,1/((-c*x+1)^(1/2)/(c*x+1)^(1/2)+(1+(-c*x+1)/(c*x+1))^(1/2))^ 2)/c
Time = 0.04 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.96 \[ \int \frac {\left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2}{1-c^2 x^2} \, dx=\frac {2 \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2 \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )-3 b \log \left (1-e^{2 \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )\right )-6 b^2 \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right ) \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )+3 b^3 \operatorname {PolyLog}\left (3,e^{2 \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{6 b c} \]
(2*(a + b*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2*(a + b*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x]] - 3*b*Log[1 - E^(2*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x] ])]) - 6*b^2*(a + b*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])*PolyLog[2, E^(2* ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])] + 3*b^3*PolyLog[3, E^(2*ArcSinh[Sqr t[1 - c*x]/Sqrt[1 + c*x]])])/(6*b*c)
Result contains complex when optimal does not.
Time = 0.84 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.12, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {7232, 6190, 25, 3042, 26, 4201, 2620, 3011, 2720, 7143}
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.
| \(\displaystyle \int \frac {\left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )^2}{1-c^2 x^2} \, dx\) |
| \(\Big \downarrow \) 7232 |
| \(\displaystyle -\frac {\int \frac {\sqrt {c x+1} \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )^2}{\sqrt {1-c x}}d\frac {\sqrt {1-c x}}{\sqrt {c x+1}}}{c}\) |
| \(\Big \downarrow \) 6190 |
| \(\displaystyle -\frac {\int -\frac {(1-c x) \coth \left (\frac {a}{b}-\frac {a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )}{b}\right )}{c x+1}d\left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{b c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {(1-c x) \coth \left (\frac {a}{b}-\frac {a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )}{b}\right )}{c x+1}d\left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{b c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int -\frac {i (1-c x) \tan \left (\frac {i a}{b}-\frac {i \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{b}+\frac {\pi }{2}\right )}{c x+1}d\left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{b c}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {i \int \frac {(1-c x) \tan \left (\frac {1}{2} \left (\frac {2 i a}{b}+\pi \right )-\frac {i \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{b}\right )}{c x+1}d\left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{b c}\) |
| \(\Big \downarrow \) 4201 |
| \(\displaystyle -\frac {i \left (2 i \int \frac {\exp \left (\frac {2 a}{b}-\frac {2 \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{b}-i \pi \right ) (1-c x)}{\left (1+\exp \left (\frac {2 a}{b}-\frac {2 \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{b}-i \pi \right )\right ) (c x+1)}d\left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )-\frac {i (1-c x)^{3/2}}{3 (c x+1)^{3/2}}\right )}{b c}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle -\frac {i \left (2 i \left (b \int \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right ) \log \left (1+e^{\frac {2 a}{b}-\frac {2 \sqrt {1-c x}}{b \sqrt {c x+1}}-i \pi }\right )d\left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )-\frac {b (1-c x) \log \left (1+\exp \left (-\frac {2 \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{b}+\frac {2 a}{b}-i \pi \right )\right )}{2 (c x+1)}\right )-\frac {i (1-c x)^{3/2}}{3 (c x+1)^{3/2}}\right )}{b c}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle -\frac {i \left (2 i \left (b \left (\frac {1}{2} b \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right ) \operatorname {PolyLog}\left (2,-e^{\frac {2 a}{b}-\frac {2 \sqrt {1-c x}}{b \sqrt {c x+1}}-i \pi }\right )-\frac {1}{2} b \int \operatorname {PolyLog}\left (2,-\exp \left (\frac {2 a}{b}-\frac {2 \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{b}-i \pi \right )\right )d\left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )\right )-\frac {b (1-c x) \log \left (1+\exp \left (-\frac {2 \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{b}+\frac {2 a}{b}-i \pi \right )\right )}{2 (c x+1)}\right )-\frac {i (1-c x)^{3/2}}{3 (c x+1)^{3/2}}\right )}{b c}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle -\frac {i \left (2 i \left (b \left (\frac {1}{4} b^2 \int \frac {\sqrt {c x+1} \operatorname {PolyLog}\left (2,-a-b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{\sqrt {1-c x}}d\exp \left (\frac {2 a}{b}-\frac {2 \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{b}-i \pi \right )+\frac {1}{2} b \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right ) \operatorname {PolyLog}\left (2,-e^{\frac {2 a}{b}-\frac {2 \sqrt {1-c x}}{b \sqrt {c x+1}}-i \pi }\right )\right )-\frac {b (1-c x) \log \left (1+\exp \left (-\frac {2 \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{b}+\frac {2 a}{b}-i \pi \right )\right )}{2 (c x+1)}\right )-\frac {i (1-c x)^{3/2}}{3 (c x+1)^{3/2}}\right )}{b c}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle -\frac {i \left (2 i \left (b \left (\frac {1}{4} b^2 \operatorname {PolyLog}\left (3,-a-b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )+\frac {1}{2} b \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right ) \operatorname {PolyLog}\left (2,-e^{\frac {2 a}{b}-\frac {2 \sqrt {1-c x}}{b \sqrt {c x+1}}-i \pi }\right )\right )-\frac {b (1-c x) \log \left (1+\exp \left (-\frac {2 \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{b}+\frac {2 a}{b}-i \pi \right )\right )}{2 (c x+1)}\right )-\frac {i (1-c x)^{3/2}}{3 (c x+1)^{3/2}}\right )}{b c}\) |
((-I)*(((-1/3*I)*(1 - c*x)^(3/2))/(1 + c*x)^(3/2) + (2*I)*(-1/2*(b*(1 - c* x)*Log[1 + E^((2*a)/b - I*Pi - (2*(a + b*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c* x]]))/b)])/(1 + c*x) + b*((b*(a + b*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])* PolyLog[2, -E^((2*a)/b - I*Pi - (2*Sqrt[1 - c*x])/(b*Sqrt[1 + c*x]))])/2 + (b^2*PolyLog[3, -a - b*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x]]])/4))))/(b*c)
3.4.44.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ ((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp [((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si mp[d*(m/(b*f*g*n*Log[F])) Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x )))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) *(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F])) Int[(f + g*x)^( m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e , f, g, n}, x] && GtQ[m, 0]
Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (Complex[0, fz_])*(f_.)*(x_)], x _Symbol] :> Simp[(-I)*((c + d*x)^(m + 1)/(d*(m + 1))), x] + Simp[2*I Int[ (c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*fz*x)))), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Simp[1/b Subst[Int[x^n*Coth[-a/b + x/b], x], x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a , b, c}, x] && IGtQ[n, 0]
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d , e, n, p}, x] && EqQ[b*d, a*e]
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. \(623\) vs. \(2(214)=428\).
Time = 0.40 (sec) , antiderivative size = 624, normalized size of antiderivative = 3.22
| method | result | size |
| default | \(\frac {a^{2} \ln \left (c x +1\right )}{2 c}-\frac {a^{2} \ln \left (c x -1\right )}{2 c}-b^{2} \left (-\frac {\operatorname {arcsinh}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{3}}{3 c}+\frac {\operatorname {arcsinh}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{2} \ln \left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+\sqrt {1+\frac {-c x +1}{c x +1}}+1\right )}{c}+\frac {2 \,\operatorname {arcsinh}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \operatorname {polylog}\left (2, -\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-\sqrt {1+\frac {-c x +1}{c x +1}}\right )}{c}-\frac {2 \operatorname {polylog}\left (3, -\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-\sqrt {1+\frac {-c x +1}{c x +1}}\right )}{c}+\frac {\operatorname {arcsinh}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{2} \ln \left (1-\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-\sqrt {1+\frac {-c x +1}{c x +1}}\right )}{c}+\frac {2 \,\operatorname {arcsinh}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \operatorname {polylog}\left (2, \frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+\sqrt {1+\frac {-c x +1}{c x +1}}\right )}{c}-\frac {2 \operatorname {polylog}\left (3, \frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+\sqrt {1+\frac {-c x +1}{c x +1}}\right )}{c}\right )-2 a b \left (-\frac {\operatorname {arcsinh}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{2}}{2 c}+\frac {\operatorname {arcsinh}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \ln \left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+\sqrt {1+\frac {-c x +1}{c x +1}}+1\right )}{c}+\frac {\operatorname {polylog}\left (2, -\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-\sqrt {1+\frac {-c x +1}{c x +1}}\right )}{c}+\frac {\operatorname {arcsinh}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \ln \left (1-\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-\sqrt {1+\frac {-c x +1}{c x +1}}\right )}{c}+\frac {\operatorname {polylog}\left (2, \frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+\sqrt {1+\frac {-c x +1}{c x +1}}\right )}{c}\right )\) | \(624\) |
| parts | \(\frac {a^{2} \ln \left (c x +1\right )}{2 c}-\frac {a^{2} \ln \left (c x -1\right )}{2 c}-b^{2} \left (-\frac {\operatorname {arcsinh}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{3}}{3 c}+\frac {\operatorname {arcsinh}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{2} \ln \left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+\sqrt {1+\frac {-c x +1}{c x +1}}+1\right )}{c}+\frac {2 \,\operatorname {arcsinh}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \operatorname {polylog}\left (2, -\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-\sqrt {1+\frac {-c x +1}{c x +1}}\right )}{c}-\frac {2 \operatorname {polylog}\left (3, -\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-\sqrt {1+\frac {-c x +1}{c x +1}}\right )}{c}+\frac {\operatorname {arcsinh}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{2} \ln \left (1-\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-\sqrt {1+\frac {-c x +1}{c x +1}}\right )}{c}+\frac {2 \,\operatorname {arcsinh}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \operatorname {polylog}\left (2, \frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+\sqrt {1+\frac {-c x +1}{c x +1}}\right )}{c}-\frac {2 \operatorname {polylog}\left (3, \frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+\sqrt {1+\frac {-c x +1}{c x +1}}\right )}{c}\right )-2 a b \left (-\frac {\operatorname {arcsinh}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{2}}{2 c}+\frac {\operatorname {arcsinh}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \ln \left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+\sqrt {1+\frac {-c x +1}{c x +1}}+1\right )}{c}+\frac {\operatorname {polylog}\left (2, -\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-\sqrt {1+\frac {-c x +1}{c x +1}}\right )}{c}+\frac {\operatorname {arcsinh}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \ln \left (1-\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-\sqrt {1+\frac {-c x +1}{c x +1}}\right )}{c}+\frac {\operatorname {polylog}\left (2, \frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+\sqrt {1+\frac {-c x +1}{c x +1}}\right )}{c}\right )\) | \(624\) |
1/2*a^2/c*ln(c*x+1)-1/2*a^2/c*ln(c*x-1)-b^2*(-1/3/c*arcsinh((-c*x+1)^(1/2) /(c*x+1)^(1/2))^3+1/c*arcsinh((-c*x+1)^(1/2)/(c*x+1)^(1/2))^2*ln((-c*x+1)^ (1/2)/(c*x+1)^(1/2)+(1+(-c*x+1)/(c*x+1))^(1/2)+1)+2/c*arcsinh((-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))^(1/2))-2/c*polylog(3,-(-c*x+1)^(1/2)/(c*x+1)^(1/2)-(1+(-c*x+1)/(c*x+1 ))^(1/2))+1/c*arcsinh((-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))^(1/2))+2/c*arcsinh((-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))^(1/2 ))-2/c*polylog(3,(-c*x+1)^(1/2)/(c*x+1)^(1/2)+(1+(-c*x+1)/(c*x+1))^(1/2))) -2*a*b*(-1/2/c*arcsinh((-c*x+1)^(1/2)/(c*x+1)^(1/2))^2+1/c*arcsinh((-c*x+1 )^(1/2)/(c*x+1)^(1/2))*ln((-c*x+1)^(1/2)/(c*x+1)^(1/2)+(1+(-c*x+1)/(c*x+1) )^(1/2)+1)+1/c*polylog(2,-(-c*x+1)^(1/2)/(c*x+1)^(1/2)-(1+(-c*x+1)/(c*x+1) )^(1/2))+1/c*arcsinh((-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))^(1/2))+1/c*polylog(2,(-c*x+1)^(1/2)/(c*x+1 )^(1/2)+(1+(-c*x+1)/(c*x+1))^(1/2)))
\[ \int \frac {\left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2}{1-c^2 x^2} \, dx=\int { -\frac {{\left (b \operatorname {arsinh}\left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right ) + a\right )}^{2}}{c^{2} x^{2} - 1} \,d x } \]
integral(-(b^2*arcsinh(sqrt(-c*x + 1)/sqrt(c*x + 1))^2 + 2*a*b*arcsinh(sqr t(-c*x + 1)/sqrt(c*x + 1)) + a^2)/(c^2*x^2 - 1), x)
\[ \int \frac {\left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2}{1-c^2 x^2} \, dx=- \int \frac {a^{2}}{c^{2} x^{2} - 1}\, dx - \int \frac {b^{2} \operatorname {asinh}^{2}{\left (\frac {\sqrt {- c x + 1}}{\sqrt {c x + 1}} \right )}}{c^{2} x^{2} - 1}\, dx - \int \frac {2 a b \operatorname {asinh}{\left (\frac {\sqrt {- c x + 1}}{\sqrt {c x + 1}} \right )}}{c^{2} x^{2} - 1}\, dx \]
-Integral(a**2/(c**2*x**2 - 1), x) - Integral(b**2*asinh(sqrt(-c*x + 1)/sq rt(c*x + 1))**2/(c**2*x**2 - 1), x) - Integral(2*a*b*asinh(sqrt(-c*x + 1)/ sqrt(c*x + 1))/(c**2*x**2 - 1), x)
\[ \int \frac {\left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2}{1-c^2 x^2} \, dx=\int { -\frac {{\left (b \operatorname {arsinh}\left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right ) + a\right )}^{2}}{c^{2} x^{2} - 1} \,d x } \]
1/2*a^2*(log(c*x + 1)/c - log(c*x - 1)/c) + 1/2*(b^2*log(c*x + 1) - b^2*lo g(-c*x + 1))*log(sqrt(2) + sqrt(-c*x + 1))^2/c + integrate(-1/4*((sqrt(2)* b^2 + sqrt(-c*x + 1)*b^2)*log(c*x + 1)^2 - 4*(sqrt(2)*a*b + sqrt(-c*x + 1) *a*b)*log(c*x + 1) + 2*(4*sqrt(2)*a*b - 2*(sqrt(2)*b^2 + sqrt(-c*x + 1)*b^ 2)*log(c*x + 1) + (4*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(2) + sqrt(-c*x + 1)))/(sqrt(2)*c^2 *x^2 + (c^2*x^2 - 1)*sqrt(-c*x + 1) - sqrt(2)), x)
\[ \int \frac {\left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2}{1-c^2 x^2} \, dx=\int { -\frac {{\left (b \operatorname {arsinh}\left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right ) + a\right )}^{2}}{c^{2} x^{2} - 1} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2}{1-c^2 x^2} \, dx=\int -\frac {{\left (a+b\,\mathrm {asinh}\left (\frac {\sqrt {1-c\,x}}{\sqrt {c\,x+1}}\right )\right )}^2}{c^2\,x^2-1} \,d x \]