Integrand size = 40, antiderivative size = 261 \[ \int \frac {\left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3}{1-c^2 x^2} \, dx=-\frac {\left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^4}{4 b c}-\frac {\left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3 \log \left (1-e^{-2 \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{c}+\frac {3 b \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2 \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{2 c}+\frac {3 b^2 \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right ) \operatorname {PolyLog}\left (3,e^{-2 \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{2 c}+\frac {3 b^3 \operatorname {PolyLog}\left (4,e^{-2 \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{4 c} \]
-1/4*(a+b*arcsinh((-c*x+1)^(1/2)/(c*x+1)^(1/2)))^4/b/c-(a+b*arcsinh((-c*x+ 1)^(1/2)/(c*x+1)^(1/2)))^3*ln(1-1/((-c*x+1)^(1/2)/(c*x+1)^(1/2)+(1+(-c*x+1 )/(c*x+1))^(1/2))^2)/c+3/2*b*(a+b*arcsinh((-c*x+1)^(1/2)/(c*x+1)^(1/2)))^2 *polylog(2,1/((-c*x+1)^(1/2)/(c*x+1)^(1/2)+(1+(-c*x+1)/(c*x+1))^(1/2))^2)/ c+3/2*b^2*(a+b*arcsinh((-c*x+1)^(1/2)/(c*x+1)^(1/2)))*polylog(3,1/((-c*x+1 )^(1/2)/(c*x+1)^(1/2)+(1+(-c*x+1)/(c*x+1))^(1/2))^2)/c+3/4*b^3*polylog(4,1 /((-c*x+1)^(1/2)/(c*x+1)^(1/2)+(1+(-c*x+1)/(c*x+1))^(1/2))^2)/c
Time = 0.05 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.93 \[ \int \frac {\left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3}{1-c^2 x^2} \, dx=\frac {\frac {\left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^4}{b}-4 \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3 \log \left (1-e^{2 \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )-6 b \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2 \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )+6 b^2 \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right ) \operatorname {PolyLog}\left (3,e^{2 \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )-3 b^3 \operatorname {PolyLog}\left (4,e^{2 \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{4 c} \]
((a + b*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^4/b - 4*(a + b*ArcSinh[Sqrt[ 1 - c*x]/Sqrt[1 + c*x]])^3*Log[1 - E^(2*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x ]])] - 6*b*(a + b*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2*PolyLog[2, 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[3, E^(2*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])] - 3*b^3*PolyLog[4, E^(2*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])])/(4*c)
Result contains complex when optimal does not.
Time = 0.99 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.12, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.275, Rules used = {7232, 6190, 25, 3042, 26, 4201, 2620, 3011, 7163, 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.
| \(\displaystyle \int \frac {\left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )^3}{1-c^2 x^2} \, dx\) |
| \(\Big \downarrow \) 7232 |
| \(\displaystyle -\frac {\int \frac {\sqrt {c x+1} \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )^3}{\sqrt {1-c x}}d\frac {\sqrt {1-c x}}{\sqrt {c x+1}}}{c}\) |
| \(\Big \downarrow \) 6190 |
| \(\displaystyle -\frac {\int -\frac {(1-c x)^{3/2} \coth \left (\frac {a}{b}-\frac {a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )}{b}\right )}{(c x+1)^{3/2}}d\left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{b c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {(1-c x)^{3/2} \coth \left (\frac {a}{b}-\frac {a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )}{b}\right )}{(c x+1)^{3/2}}d\left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{b c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int -\frac {i (1-c x)^{3/2} \tan \left (\frac {i a}{b}-\frac {i \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{b}+\frac {\pi }{2}\right )}{(c x+1)^{3/2}}d\left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{b c}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {i \int \frac {(1-c x)^{3/2} \tan \left (\frac {1}{2} \left (\frac {2 i a}{b}+\pi \right )-\frac {i \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{b}\right )}{(c x+1)^{3/2}}d\left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{b c}\) |
| \(\Big \downarrow \) 4201 |
| \(\displaystyle -\frac {i \left (2 i \int \frac {\exp \left (\frac {2 a}{b}-\frac {2 \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{b}-i \pi \right ) (1-c x)^{3/2}}{\left (1+\exp \left (\frac {2 a}{b}-\frac {2 \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{b}-i \pi \right )\right ) (c x+1)^{3/2}}d\left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )-\frac {i (1-c x)^2}{4 (c x+1)^2}\right )}{b c}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle -\frac {i \left (2 i \left (\frac {3}{2} b \int \frac {(1-c x) \log \left (1+\exp \left (\frac {2 a}{b}-\frac {2 \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{b}-i \pi \right )\right )}{c x+1}d\left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )-\frac {b (1-c x)^{3/2} \log \left (1+\exp \left (-\frac {2 \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{b}+\frac {2 a}{b}-i \pi \right )\right )}{2 (c x+1)^{3/2}}\right )-\frac {i (1-c x)^2}{4 (c x+1)^2}\right )}{b c}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle -\frac {i \left (2 i \left (\frac {3}{2} b \left (\frac {b (1-c x) \operatorname {PolyLog}\left (2,-\exp \left (\frac {2 a}{b}-\frac {2 \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{b}-i \pi \right )\right )}{2 (c x+1)}-b \int \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right ) \operatorname {PolyLog}\left (2,-e^{\frac {2 a}{b}-\frac {2 \sqrt {1-c x}}{b \sqrt {c x+1}}-i \pi }\right )d\left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )\right )-\frac {b (1-c x)^{3/2} \log \left (1+\exp \left (-\frac {2 \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{b}+\frac {2 a}{b}-i \pi \right )\right )}{2 (c x+1)^{3/2}}\right )-\frac {i (1-c x)^2}{4 (c x+1)^2}\right )}{b c}\) |
| \(\Big \downarrow \) 7163 |
| \(\displaystyle -\frac {i \left (2 i \left (\frac {3}{2} b \left (\frac {b (1-c x) \operatorname {PolyLog}\left (2,-\exp \left (\frac {2 a}{b}-\frac {2 \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{b}-i \pi \right )\right )}{2 (c x+1)}-b \left (\frac {1}{2} b \int \operatorname {PolyLog}\left (3,-\exp \left (\frac {2 a}{b}-\frac {2 \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{b}-i \pi \right )\right )d\left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )-\frac {1}{2} b \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right ) \operatorname {PolyLog}\left (3,-e^{\frac {2 a}{b}-\frac {2 \sqrt {1-c x}}{b \sqrt {c x+1}}-i \pi }\right )\right )\right )-\frac {b (1-c x)^{3/2} \log \left (1+\exp \left (-\frac {2 \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{b}+\frac {2 a}{b}-i \pi \right )\right )}{2 (c x+1)^{3/2}}\right )-\frac {i (1-c x)^2}{4 (c x+1)^2}\right )}{b c}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle -\frac {i \left (2 i \left (\frac {3}{2} b \left (\frac {b (1-c x) \operatorname {PolyLog}\left (2,-\exp \left (\frac {2 a}{b}-\frac {2 \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{b}-i \pi \right )\right )}{2 (c x+1)}-b \left (-\frac {1}{4} b^2 \int \frac {\sqrt {c x+1} \operatorname {PolyLog}\left (3,-a-b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{\sqrt {1-c x}}d\exp \left (\frac {2 a}{b}-\frac {2 \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{b}-i \pi \right )-\frac {1}{2} b \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right ) \operatorname {PolyLog}\left (3,-e^{\frac {2 a}{b}-\frac {2 \sqrt {1-c x}}{b \sqrt {c x+1}}-i \pi }\right )\right )\right )-\frac {b (1-c x)^{3/2} \log \left (1+\exp \left (-\frac {2 \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{b}+\frac {2 a}{b}-i \pi \right )\right )}{2 (c x+1)^{3/2}}\right )-\frac {i (1-c x)^2}{4 (c x+1)^2}\right )}{b c}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle -\frac {i \left (2 i \left (\frac {3}{2} b \left (\frac {b (1-c x) \operatorname {PolyLog}\left (2,-\exp \left (\frac {2 a}{b}-\frac {2 \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{b}-i \pi \right )\right )}{2 (c x+1)}-b \left (-\frac {1}{4} b^2 \operatorname {PolyLog}\left (4,-a-b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )-\frac {1}{2} b \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right ) \operatorname {PolyLog}\left (3,-e^{\frac {2 a}{b}-\frac {2 \sqrt {1-c x}}{b \sqrt {c x+1}}-i \pi }\right )\right )\right )-\frac {b (1-c x)^{3/2} \log \left (1+\exp \left (-\frac {2 \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{b}+\frac {2 a}{b}-i \pi \right )\right )}{2 (c x+1)^{3/2}}\right )-\frac {i (1-c x)^2}{4 (c x+1)^2}\right )}{b c}\) |
((-I)*(((-1/4*I)*(1 - c*x)^2)/(1 + c*x)^2 + (2*I)*(-1/2*(b*(1 - c*x)^(3/2) *Log[1 + E^((2*a)/b - I*Pi - (2*(a + b*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x] ]))/b)])/(1 + c*x)^(3/2) + (3*b*((b*(1 - c*x)*PolyLog[2, -E^((2*a)/b - I*P i - (2*(a + b*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x]]))/b)])/(2*(1 + c*x)) - b*(-1/2*(b*(a + b*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])*PolyLog[3, -E^((2* a)/b - I*Pi - (2*Sqrt[1 - c*x])/(b*Sqrt[1 + c*x]))]) - (b^2*PolyLog[4, -a - b*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x]]])/4)))/2)))/(b*c)
3.4.43.3.1 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[((e_.) + (f_.)*(x_))^(m_.)*PolyLog[n_, (d_.)*((F_)^((c_.)*((a_.) + (b_. )*(x_))))^(p_.)], x_Symbol] :> Simp[(e + f*x)^m*(PolyLog[n + 1, d*(F^(c*(a + b*x)))^p]/(b*c*p*Log[F])), x] - Simp[f*(m/(b*c*p*Log[F])) Int[(e + f*x) ^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c , d, e, f, n, p}, x] && GtQ[m, 0]
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. \(1112\) vs. \(2(285)=570\).
Time = 1.40 (sec) , antiderivative size = 1113, normalized size of antiderivative = 4.26
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1113\) |
| parts | \(\text {Expression too large to display}\) | \(1113\) |
1/2*a^3/c*ln(c*x+1)-1/2*a^3/c*ln(c*x-1)-b^3*(-1/4/c*arcsinh((-c*x+1)^(1/2) /(c*x+1)^(1/2))^4+1/c*arcsinh((-c*x+1)^(1/2)/(c*x+1)^(1/2))^3*ln((-c*x+1)^ (1/2)/(c*x+1)^(1/2)+(1+(-c*x+1)/(c*x+1))^(1/2)+1)+3/c*arcsinh((-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))^(1/2))-6/c*arcsinh((-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))^(1/2))+6/c*polylog(4,-(-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))^3*ln(1-(-c*x+1)^(1/2)/(c*x+1)^(1/2)-(1+(-c*x+1)/(c*x+1))^( 1/2))+3/c*arcsinh((-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))^(1/2))-6/c*arcsinh((-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/ 2))+6/c*polylog(4,(-c*x+1)^(1/2)/(c*x+1)^(1/2)+(1+(-c*x+1)/(c*x+1))^(1/2)) )-3*a*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)...
\[ \int \frac {\left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3}{1-c^2 x^2} \, dx=\int { -\frac {{\left (b \operatorname {arsinh}\left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right ) + a\right )}^{3}}{c^{2} x^{2} - 1} \,d x } \]
integral(-(b^3*arcsinh(sqrt(-c*x + 1)/sqrt(c*x + 1))^3 + 3*a*b^2*arcsinh(s qrt(-c*x + 1)/sqrt(c*x + 1))^2 + 3*a^2*b*arcsinh(sqrt(-c*x + 1)/sqrt(c*x + 1)) + a^3)/(c^2*x^2 - 1), x)
\[ \int \frac {\left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3}{1-c^2 x^2} \, dx=- \int \frac {a^{3}}{c^{2} x^{2} - 1}\, dx - \int \frac {b^{3} \operatorname {asinh}^{3}{\left (\frac {\sqrt {- c x + 1}}{\sqrt {c x + 1}} \right )}}{c^{2} x^{2} - 1}\, dx - \int \frac {3 a b^{2} \operatorname {asinh}^{2}{\left (\frac {\sqrt {- c x + 1}}{\sqrt {c x + 1}} \right )}}{c^{2} x^{2} - 1}\, dx - \int \frac {3 a^{2} b \operatorname {asinh}{\left (\frac {\sqrt {- c x + 1}}{\sqrt {c x + 1}} \right )}}{c^{2} x^{2} - 1}\, dx \]
-Integral(a**3/(c**2*x**2 - 1), x) - Integral(b**3*asinh(sqrt(-c*x + 1)/sq rt(c*x + 1))**3/(c**2*x**2 - 1), x) - Integral(3*a*b**2*asinh(sqrt(-c*x + 1)/sqrt(c*x + 1))**2/(c**2*x**2 - 1), x) - Integral(3*a**2*b*asinh(sqrt(-c *x + 1)/sqrt(c*x + 1))/(c**2*x**2 - 1), x)
\[ \int \frac {\left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3}{1-c^2 x^2} \, dx=\int { -\frac {{\left (b \operatorname {arsinh}\left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right ) + a\right )}^{3}}{c^{2} x^{2} - 1} \,d x } \]
1/2*a^3*(log(c*x + 1)/c - log(c*x - 1)/c) + 1/2*(b^3*log(c*x + 1) - b^3*lo g(-c*x + 1))*log(sqrt(2) + sqrt(-c*x + 1))^3/c + integrate(1/8*((sqrt(2)*b ^3 + sqrt(-c*x + 1)*b^3)*log(c*x + 1)^3 - 6*(sqrt(2)*a*b^2 + sqrt(-c*x + 1 )*a*b^2)*log(c*x + 1)^2 - 6*(4*sqrt(2)*a*b^2 - 2*(sqrt(2)*b^3 + sqrt(-c*x + 1)*b^3)*log(c*x + 1) + (4*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))*log(sqrt(2) + sqrt(-c*x + 1))^2 + 12*(sqrt(2)*a^2*b + sqrt(-c*x + 1)*a^2*b)*log(c*x + 1) - 6*(4*sqrt(2)*a^2* b + 4*sqrt(-c*x + 1)*a^2*b + (sqrt(2)*b^3 + sqrt(-c*x + 1)*b^3)*log(c*x + 1)^2 - 4*(sqrt(2)*a*b^2 + sqrt(-c*x + 1)*a*b^2)*log(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)
\[ \int \frac {\left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3}{1-c^2 x^2} \, dx=\int { -\frac {{\left (b \operatorname {arsinh}\left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right ) + a\right )}^{3}}{c^{2} x^{2} - 1} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3}{1-c^2 x^2} \, dx=\int -\frac {{\left (a+b\,\mathrm {asinh}\left (\frac {\sqrt {1-c\,x}}{\sqrt {c\,x+1}}\right )\right )}^3}{c^2\,x^2-1} \,d x \]