Integrand size = 14, antiderivative size = 112 \[ \int \frac {1}{x^3 \left (a+b \text {sech}^{-1}(c x)\right )^3} \, dx=\frac {c^2 \cosh \left (2 \text {sech}^{-1}(c x)\right )}{2 b^2 \left (a+b \text {sech}^{-1}(c x)\right )}+\frac {c^2 \text {Chi}\left (\frac {2 a}{b}+2 \text {sech}^{-1}(c x)\right ) \sinh \left (\frac {2 a}{b}\right )}{b^3}+\frac {c^2 \sinh \left (2 \text {sech}^{-1}(c x)\right )}{4 b \left (a+b \text {sech}^{-1}(c x)\right )^2}-\frac {c^2 \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \text {sech}^{-1}(c x)\right )}{b^3} \]
1/2*c^2*cosh(2*arcsech(c*x))/b^2/(a+b*arcsech(c*x))-c^2*cosh(2*a/b)*Shi(2* a/b+2*arcsech(c*x))/b^3+c^2*Chi(2*a/b+2*arcsech(c*x))*sinh(2*a/b)/b^3+1/4* c^2*sinh(2*arcsech(c*x))/b/(a+b*arcsech(c*x))^2
Time = 0.47 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.09 \[ \int \frac {1}{x^3 \left (a+b \text {sech}^{-1}(c x)\right )^3} \, dx=\frac {\frac {b^2 \sqrt {\frac {1-c x}{1+c x}} (1+c x)}{x^2 \left (a+b \text {sech}^{-1}(c x)\right )^2}+\frac {b \left (2-c^2 x^2\right )}{x^2 \left (a+b \text {sech}^{-1}(c x)\right )}+2 c^2 \left (\text {Chi}\left (2 \left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )\right ) \sinh \left (\frac {2 a}{b}\right )-\cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )\right )\right )}{2 b^3} \]
((b^2*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x))/(x^2*(a + b*ArcSech[c*x])^2) + (b*(2 - c^2*x^2))/(x^2*(a + b*ArcSech[c*x])) + 2*c^2*(CoshIntegral[2*(a/b + ArcSech[c*x])]*Sinh[(2*a)/b] - Cosh[(2*a)/b]*SinhIntegral[2*(a/b + ArcSe ch[c*x])]))/(2*b^3)
Result contains complex when optimal does not.
Time = 0.77 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.16, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.214, Rules used = {6839, 5971, 27, 3042, 26, 3778, 3042, 3778, 26, 3042, 26, 3784, 26, 3042, 26, 3779, 3782}
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.
| \(\displaystyle \int \frac {1}{x^3 \left (a+b \text {sech}^{-1}(c x)\right )^3} \, dx\) |
| \(\Big \downarrow \) 6839 |
| \(\displaystyle -c^2 \int \frac {\sqrt {\frac {1-c x}{c x+1}} (c x+1)}{c^2 x^2 \left (a+b \text {sech}^{-1}(c x)\right )^3}d\text {sech}^{-1}(c x)\) |
| \(\Big \downarrow \) 5971 |
| \(\displaystyle -c^2 \int \frac {\sinh \left (2 \text {sech}^{-1}(c x)\right )}{2 \left (a+b \text {sech}^{-1}(c x)\right )^3}d\text {sech}^{-1}(c x)\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{2} c^2 \int \frac {\sinh \left (2 \text {sech}^{-1}(c x)\right )}{\left (a+b \text {sech}^{-1}(c x)\right )^3}d\text {sech}^{-1}(c x)\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {1}{2} c^2 \int -\frac {i \sin \left (2 i \text {sech}^{-1}(c x)\right )}{\left (a+b \text {sech}^{-1}(c x)\right )^3}d\text {sech}^{-1}(c x)\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {1}{2} i c^2 \int \frac {\sin \left (2 i \text {sech}^{-1}(c x)\right )}{\left (a+b \text {sech}^{-1}(c x)\right )^3}d\text {sech}^{-1}(c x)\) |
| \(\Big \downarrow \) 3778 |
| \(\displaystyle \frac {1}{2} i c^2 \left (\frac {i \int \frac {\cosh \left (2 \text {sech}^{-1}(c x)\right )}{\left (a+b \text {sech}^{-1}(c x)\right )^2}d\text {sech}^{-1}(c x)}{b}-\frac {i \sinh \left (2 \text {sech}^{-1}(c x)\right )}{2 b \left (a+b \text {sech}^{-1}(c x)\right )^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{2} i c^2 \left (\frac {i \int \frac {\sin \left (2 i \text {sech}^{-1}(c x)+\frac {\pi }{2}\right )}{\left (a+b \text {sech}^{-1}(c x)\right )^2}d\text {sech}^{-1}(c x)}{b}-\frac {i \sinh \left (2 \text {sech}^{-1}(c x)\right )}{2 b \left (a+b \text {sech}^{-1}(c x)\right )^2}\right )\) |
| \(\Big \downarrow \) 3778 |
| \(\displaystyle \frac {1}{2} i c^2 \left (\frac {i \left (-\frac {\cosh \left (2 \text {sech}^{-1}(c x)\right )}{b \left (a+b \text {sech}^{-1}(c x)\right )}+\frac {2 i \int -\frac {i \sinh \left (2 \text {sech}^{-1}(c x)\right )}{a+b \text {sech}^{-1}(c x)}d\text {sech}^{-1}(c x)}{b}\right )}{b}-\frac {i \sinh \left (2 \text {sech}^{-1}(c x)\right )}{2 b \left (a+b \text {sech}^{-1}(c x)\right )^2}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {1}{2} i c^2 \left (\frac {i \left (\frac {2 \int \frac {\sinh \left (2 \text {sech}^{-1}(c x)\right )}{a+b \text {sech}^{-1}(c x)}d\text {sech}^{-1}(c x)}{b}-\frac {\cosh \left (2 \text {sech}^{-1}(c x)\right )}{b \left (a+b \text {sech}^{-1}(c x)\right )}\right )}{b}-\frac {i \sinh \left (2 \text {sech}^{-1}(c x)\right )}{2 b \left (a+b \text {sech}^{-1}(c x)\right )^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{2} i c^2 \left (\frac {i \left (-\frac {\cosh \left (2 \text {sech}^{-1}(c x)\right )}{b \left (a+b \text {sech}^{-1}(c x)\right )}+\frac {2 \int -\frac {i \sin \left (2 i \text {sech}^{-1}(c x)\right )}{a+b \text {sech}^{-1}(c x)}d\text {sech}^{-1}(c x)}{b}\right )}{b}-\frac {i \sinh \left (2 \text {sech}^{-1}(c x)\right )}{2 b \left (a+b \text {sech}^{-1}(c x)\right )^2}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {1}{2} i c^2 \left (\frac {i \left (-\frac {\cosh \left (2 \text {sech}^{-1}(c x)\right )}{b \left (a+b \text {sech}^{-1}(c x)\right )}-\frac {2 i \int \frac {\sin \left (2 i \text {sech}^{-1}(c x)\right )}{a+b \text {sech}^{-1}(c x)}d\text {sech}^{-1}(c x)}{b}\right )}{b}-\frac {i \sinh \left (2 \text {sech}^{-1}(c x)\right )}{2 b \left (a+b \text {sech}^{-1}(c x)\right )^2}\right )\) |
| \(\Big \downarrow \) 3784 |
| \(\displaystyle \frac {1}{2} i c^2 \left (\frac {i \left (-\frac {\cosh \left (2 \text {sech}^{-1}(c x)\right )}{b \left (a+b \text {sech}^{-1}(c x)\right )}-\frac {2 i \left (\cosh \left (\frac {2 a}{b}\right ) \int \frac {i \sinh \left (\frac {2 a}{b}+2 \text {sech}^{-1}(c x)\right )}{a+b \text {sech}^{-1}(c x)}d\text {sech}^{-1}(c x)-i \sinh \left (\frac {2 a}{b}\right ) \int \frac {\cosh \left (\frac {2 a}{b}+2 \text {sech}^{-1}(c x)\right )}{a+b \text {sech}^{-1}(c x)}d\text {sech}^{-1}(c x)\right )}{b}\right )}{b}-\frac {i \sinh \left (2 \text {sech}^{-1}(c x)\right )}{2 b \left (a+b \text {sech}^{-1}(c x)\right )^2}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {1}{2} i c^2 \left (\frac {i \left (-\frac {\cosh \left (2 \text {sech}^{-1}(c x)\right )}{b \left (a+b \text {sech}^{-1}(c x)\right )}-\frac {2 i \left (i \cosh \left (\frac {2 a}{b}\right ) \int \frac {\sinh \left (\frac {2 a}{b}+2 \text {sech}^{-1}(c x)\right )}{a+b \text {sech}^{-1}(c x)}d\text {sech}^{-1}(c x)-i \sinh \left (\frac {2 a}{b}\right ) \int \frac {\cosh \left (\frac {2 a}{b}+2 \text {sech}^{-1}(c x)\right )}{a+b \text {sech}^{-1}(c x)}d\text {sech}^{-1}(c x)\right )}{b}\right )}{b}-\frac {i \sinh \left (2 \text {sech}^{-1}(c x)\right )}{2 b \left (a+b \text {sech}^{-1}(c x)\right )^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{2} i c^2 \left (\frac {i \left (-\frac {\cosh \left (2 \text {sech}^{-1}(c x)\right )}{b \left (a+b \text {sech}^{-1}(c x)\right )}-\frac {2 i \left (i \cosh \left (\frac {2 a}{b}\right ) \int -\frac {i \sin \left (\frac {2 i a}{b}+2 i \text {sech}^{-1}(c x)\right )}{a+b \text {sech}^{-1}(c x)}d\text {sech}^{-1}(c x)-i \sinh \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 i a}{b}+2 i \text {sech}^{-1}(c x)+\frac {\pi }{2}\right )}{a+b \text {sech}^{-1}(c x)}d\text {sech}^{-1}(c x)\right )}{b}\right )}{b}-\frac {i \sinh \left (2 \text {sech}^{-1}(c x)\right )}{2 b \left (a+b \text {sech}^{-1}(c x)\right )^2}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {1}{2} i c^2 \left (\frac {i \left (-\frac {\cosh \left (2 \text {sech}^{-1}(c x)\right )}{b \left (a+b \text {sech}^{-1}(c x)\right )}-\frac {2 i \left (\cosh \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 i a}{b}+2 i \text {sech}^{-1}(c x)\right )}{a+b \text {sech}^{-1}(c x)}d\text {sech}^{-1}(c x)-i \sinh \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 i a}{b}+2 i \text {sech}^{-1}(c x)+\frac {\pi }{2}\right )}{a+b \text {sech}^{-1}(c x)}d\text {sech}^{-1}(c x)\right )}{b}\right )}{b}-\frac {i \sinh \left (2 \text {sech}^{-1}(c x)\right )}{2 b \left (a+b \text {sech}^{-1}(c x)\right )^2}\right )\) |
| \(\Big \downarrow \) 3779 |
| \(\displaystyle \frac {1}{2} i c^2 \left (\frac {i \left (-\frac {\cosh \left (2 \text {sech}^{-1}(c x)\right )}{b \left (a+b \text {sech}^{-1}(c x)\right )}-\frac {2 i \left (\frac {i \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \text {sech}^{-1}(c x)\right )}{b}-i \sinh \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 i a}{b}+2 i \text {sech}^{-1}(c x)+\frac {\pi }{2}\right )}{a+b \text {sech}^{-1}(c x)}d\text {sech}^{-1}(c x)\right )}{b}\right )}{b}-\frac {i \sinh \left (2 \text {sech}^{-1}(c x)\right )}{2 b \left (a+b \text {sech}^{-1}(c x)\right )^2}\right )\) |
| \(\Big \downarrow \) 3782 |
| \(\displaystyle \frac {1}{2} i c^2 \left (\frac {i \left (-\frac {\cosh \left (2 \text {sech}^{-1}(c x)\right )}{b \left (a+b \text {sech}^{-1}(c x)\right )}-\frac {2 i \left (\frac {i \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \text {sech}^{-1}(c x)\right )}{b}-\frac {i \sinh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 a}{b}+2 \text {sech}^{-1}(c x)\right )}{b}\right )}{b}\right )}{b}-\frac {i \sinh \left (2 \text {sech}^{-1}(c x)\right )}{2 b \left (a+b \text {sech}^{-1}(c x)\right )^2}\right )\) |
(I/2)*c^2*(((-1/2*I)*Sinh[2*ArcSech[c*x]])/(b*(a + b*ArcSech[c*x])^2) + (I *(-(Cosh[2*ArcSech[c*x]]/(b*(a + b*ArcSech[c*x]))) - ((2*I)*(((-I)*CoshInt egral[(2*a)/b + 2*ArcSech[c*x]]*Sinh[(2*a)/b])/b + (I*Cosh[(2*a)/b]*SinhIn tegral[(2*a)/b + 2*ArcSech[c*x]])/b))/b))/b)
3.1.67.3.1 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[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(c + d*x)^(m + 1)*(Sin[e + f*x]/(d*(m + 1))), x] - Simp[f/(d*(m + 1)) Int[( c + d*x)^(m + 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && LtQ[m, - 1]
Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbo l] :> Simp[I*(SinhIntegral[c*f*(fz/d) + f*fz*x]/d), x] /; FreeQ[{c, d, e, f , fz}, x] && EqQ[d*e - c*f*fz*I, 0]
Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbo l] :> Simp[CoshIntegral[c*f*(fz/d) + f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz }, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]
Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[Cos[(d* e - c*f)/d] Int[Sin[c*(f/d) + f*x]/(c + d*x), x], x] + Simp[Sin[(d*e - c* f)/d] Int[Cos[c*(f/d) + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f}, x] && NeQ[d*e - c*f, 0]
Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sinh[a + b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] & & IGtQ[p, 0]
Int[((a_.) + ArcSech[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ -(c^(m + 1))^(-1) Subst[Int[(a + b*x)^n*Sech[x]^(m + 1)*Tanh[x], x], x, A rcSech[c*x]], x] /; FreeQ[{a, b, c}, x] && IntegerQ[n] && IntegerQ[m] && (G tQ[n, 0] || LtQ[m, -1])
Leaf count of result is larger than twice the leaf count of optimal. \(276\) vs. \(2(108)=216\).
Time = 0.79 (sec) , antiderivative size = 277, normalized size of antiderivative = 2.47
| method | result | size |
| derivativedivides | \(c^{2} \left (-\frac {\left (2 \sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}+c^{2} x^{2}-2\right ) \left (2 b \,\operatorname {arcsech}\left (c x \right )+2 a -b \right )}{8 c^{2} x^{2} b^{2} \left (b^{2} \operatorname {arcsech}\left (c x \right )^{2}+2 a b \,\operatorname {arcsech}\left (c x \right )+a^{2}\right )}-\frac {{\mathrm e}^{\frac {2 a}{b}} \operatorname {Ei}_{1}\left (\frac {2 a}{b}+2 \,\operatorname {arcsech}\left (c x \right )\right )}{2 b^{3}}-\frac {c^{2} x^{2}-2-2 \sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}}{8 b \,c^{2} x^{2} \left (a +b \,\operatorname {arcsech}\left (c x \right )\right )^{2}}-\frac {c^{2} x^{2}-2-2 \sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}}{4 b^{2} c^{2} x^{2} \left (a +b \,\operatorname {arcsech}\left (c x \right )\right )}+\frac {{\mathrm e}^{-\frac {2 a}{b}} \operatorname {Ei}_{1}\left (-2 \,\operatorname {arcsech}\left (c x \right )-\frac {2 a}{b}\right )}{2 b^{3}}\right )\) | \(277\) |
| default | \(c^{2} \left (-\frac {\left (2 \sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}+c^{2} x^{2}-2\right ) \left (2 b \,\operatorname {arcsech}\left (c x \right )+2 a -b \right )}{8 c^{2} x^{2} b^{2} \left (b^{2} \operatorname {arcsech}\left (c x \right )^{2}+2 a b \,\operatorname {arcsech}\left (c x \right )+a^{2}\right )}-\frac {{\mathrm e}^{\frac {2 a}{b}} \operatorname {Ei}_{1}\left (\frac {2 a}{b}+2 \,\operatorname {arcsech}\left (c x \right )\right )}{2 b^{3}}-\frac {c^{2} x^{2}-2-2 \sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}}{8 b \,c^{2} x^{2} \left (a +b \,\operatorname {arcsech}\left (c x \right )\right )^{2}}-\frac {c^{2} x^{2}-2-2 \sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}}{4 b^{2} c^{2} x^{2} \left (a +b \,\operatorname {arcsech}\left (c x \right )\right )}+\frac {{\mathrm e}^{-\frac {2 a}{b}} \operatorname {Ei}_{1}\left (-2 \,\operatorname {arcsech}\left (c x \right )-\frac {2 a}{b}\right )}{2 b^{3}}\right )\) | \(277\) |
c^2*(-1/8*(2*(-(c*x-1)/c/x)^(1/2)*c*x*((c*x+1)/c/x)^(1/2)+c^2*x^2-2)*(2*b* arcsech(c*x)+2*a-b)/c^2/x^2/b^2/(b^2*arcsech(c*x)^2+2*a*b*arcsech(c*x)+a^2 )-1/2/b^3*exp(2*a/b)*Ei(1,2*a/b+2*arcsech(c*x))-1/8/b*(c^2*x^2-2-2*(-(c*x- 1)/c/x)^(1/2)*c*x*((c*x+1)/c/x)^(1/2))/c^2/x^2/(a+b*arcsech(c*x))^2-1/4/b^ 2*(c^2*x^2-2-2*(-(c*x-1)/c/x)^(1/2)*c*x*((c*x+1)/c/x)^(1/2))/c^2/x^2/(a+b* arcsech(c*x))+1/2/b^3*exp(-2*a/b)*Ei(1,-2*arcsech(c*x)-2*a/b))
\[ \int \frac {1}{x^3 \left (a+b \text {sech}^{-1}(c x)\right )^3} \, dx=\int { \frac {1}{{\left (b \operatorname {arsech}\left (c x\right ) + a\right )}^{3} x^{3}} \,d x } \]
integral(1/(b^3*x^3*arcsech(c*x)^3 + 3*a*b^2*x^3*arcsech(c*x)^2 + 3*a^2*b* x^3*arcsech(c*x) + a^3*x^3), x)
\[ \int \frac {1}{x^3 \left (a+b \text {sech}^{-1}(c x)\right )^3} \, dx=\int \frac {1}{x^{3} \left (a + b \operatorname {asech}{\left (c x \right )}\right )^{3}}\, dx \]
\[ \int \frac {1}{x^3 \left (a+b \text {sech}^{-1}(c x)\right )^3} \, dx=\int { \frac {1}{{\left (b \operatorname {arsech}\left (c x\right ) + a\right )}^{3} x^{3}} \,d x } \]
-1/2*((b*c^6*(2*log(c) - 1) - 2*a*c^6)*x^7 - 3*(b*c^4*(2*log(c) - 1) - 2*a *c^4)*x^5 + ((b*c^2*(2*log(c) - 1) - 2*a*c^2)*x^3 - (b*(2*log(c) - 1) - 2* a)*x + 2*(b*c^2*x^3 - b*x)*log(x))*(c*x + 1)^(3/2)*(-c*x + 1)^(3/2) + 3*(b *c^2*(2*log(c) - 1) - 2*a*c^2)*x^3 - ((b*c^6*log(c) - a*c^6)*x^7 - (b*c^4* (5*log(c) - 2) - 5*a*c^4)*x^5 + 5*(b*c^2*(2*log(c) - 1) - 2*a*c^2)*x^3 - 3 *(b*(2*log(c) - 1) - 2*a)*x + (b*c^6*x^7 - 5*b*c^4*x^5 + 10*b*c^2*x^3 - 6* b*x)*log(x))*(c*x + 1)*(c*x - 1) + ((b*c^6*(3*log(c) - 1) - 3*a*c^6)*x^7 - (b*c^4*(11*log(c) - 5) - 11*a*c^4)*x^5 + 7*(b*c^2*(2*log(c) - 1) - 2*a*c^ 2)*x^3 - 3*(b*(2*log(c) - 1) - 2*a)*x + (3*b*c^6*x^7 - 11*b*c^4*x^5 + 14*b *c^2*x^3 - 6*b*x)*log(x))*sqrt(c*x + 1)*sqrt(-c*x + 1) - (b*(2*log(c) - 1) - 2*a)*x - (2*b*c^6*x^7 - 6*b*c^4*x^5 + 6*b*c^2*x^3 + 2*(b*c^2*x^3 - b*x) *(c*x + 1)^(3/2)*(-c*x + 1)^(3/2) - (b*c^6*x^7 - 5*b*c^4*x^5 + 10*b*c^2*x^ 3 - 6*b*x)*(c*x + 1)*(c*x - 1) + (3*b*c^6*x^7 - 11*b*c^4*x^5 + 14*b*c^2*x^ 3 - 6*b*x)*sqrt(c*x + 1)*sqrt(-c*x + 1) - 2*b*x)*log(sqrt(c*x + 1)*sqrt(-c *x + 1) + 1) + 2*(b*c^6*x^7 - 3*b*c^4*x^5 + 3*b*c^2*x^3 - b*x)*log(x))/((b ^4*c^6*x^6 - 3*b^4*c^4*x^4 + 3*b^4*c^2*x^2 - b^4)*x^3*log(x)^2 + 2*((b^4*c ^6*log(c) - a*b^3*c^6)*x^6 - 3*(b^4*c^4*log(c) - a*b^3*c^4)*x^4 - b^4*log( c) + a*b^3 + 3*(b^4*c^2*log(c) - a*b^3*c^2)*x^2)*x^3*log(x) - (b^4*x^3*log (x)^2 + 2*(b^4*log(c) - a*b^3)*x^3*log(x) + (b^4*log(c)^2 - 2*a*b^3*log(c) + a^2*b^2)*x^3)*(c*x + 1)^(3/2)*(-c*x + 1)^(3/2) + ((b^4*c^6*log(c)^2 ...
\[ \int \frac {1}{x^3 \left (a+b \text {sech}^{-1}(c x)\right )^3} \, dx=\int { \frac {1}{{\left (b \operatorname {arsech}\left (c x\right ) + a\right )}^{3} x^{3}} \,d x } \]
Timed out. \[ \int \frac {1}{x^3 \left (a+b \text {sech}^{-1}(c x)\right )^3} \, dx=\int \frac {1}{x^3\,{\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}^3} \,d x \]