Integrand size = 14, antiderivative size = 163 \[ \int \frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{x^3} \, dx=\frac {3 b^3 \sqrt {\frac {1-c x}{1+c x}} (1+c x)}{8 x^2}-\frac {3}{8} b^3 c^2 \text {sech}^{-1}(c x)-\frac {3 b^2 (1-c x) (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )}{4 x^2}+\frac {3 b \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )^2}{4 x^2}-\frac {1}{4} c^2 \left (a+b \text {sech}^{-1}(c x)\right )^3-\frac {(1-c x) (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )^3}{2 x^2} \]
-3/8*b^3*c^2*arcsech(c*x)-3/4*b^2*(-c*x+1)*(c*x+1)*(a+b*arcsech(c*x))/x^2- 1/4*c^2*(a+b*arcsech(c*x))^3-1/2*(-c*x+1)*(c*x+1)*(a+b*arcsech(c*x))^3/x^2 +3/8*b^3*(c*x+1)*((-c*x+1)/(c*x+1))^(1/2)/x^2+3/4*b*(c*x+1)*(a+b*arcsech(c *x))^2*((-c*x+1)/(c*x+1))^(1/2)/x^2
Time = 0.53 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.50 \[ \int \frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{x^3} \, dx=\frac {-4 a^3-6 a b^2+3 b \left (2 a^2+b^2\right ) \sqrt {\frac {1-c x}{1+c x}} (1+c x)-6 b \left (2 a^2+b^2-2 a b \sqrt {\frac {1-c x}{1+c x}} (1+c x)\right ) \text {sech}^{-1}(c x)+6 b^2 \left (b \sqrt {\frac {1-c x}{1+c x}} (1+c x)+a \left (-2+c^2 x^2\right )\right ) \text {sech}^{-1}(c x)^2+2 b^3 \left (-2+c^2 x^2\right ) \text {sech}^{-1}(c x)^3-3 b \left (2 a^2+b^2\right ) c^2 x^2 \log (x)+3 b \left (2 a^2+b^2\right ) c^2 x^2 \log \left (1+\sqrt {\frac {1-c x}{1+c x}}+c x \sqrt {\frac {1-c x}{1+c x}}\right )}{8 x^2} \]
(-4*a^3 - 6*a*b^2 + 3*b*(2*a^2 + b^2)*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x) - 6*b*(2*a^2 + b^2 - 2*a*b*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x))*ArcSech[c* x] + 6*b^2*(b*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x) + a*(-2 + c^2*x^2))*ArcS ech[c*x]^2 + 2*b^3*(-2 + c^2*x^2)*ArcSech[c*x]^3 - 3*b*(2*a^2 + b^2)*c^2*x ^2*Log[x] + 3*b*(2*a^2 + b^2)*c^2*x^2*Log[1 + Sqrt[(1 - c*x)/(1 + c*x)] + c*x*Sqrt[(1 - c*x)/(1 + c*x)]])/(8*x^2)
Time = 0.46 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.12, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.786, Rules used = {6839, 5969, 3042, 25, 3792, 17, 25, 3042, 25, 3115, 24}
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.
| \(\displaystyle \int \frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{x^3} \, dx\) |
| \(\Big \downarrow \) 6839 |
| \(\displaystyle -c^2 \int \frac {\sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )^3}{c^2 x^2}d\text {sech}^{-1}(c x)\) |
| \(\Big \downarrow \) 5969 |
| \(\displaystyle -c^2 \left (\frac {(1-c x) (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )^3}{2 c^2 x^2}-\frac {3}{2} b \int \frac {(1-c x) (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )^2}{c^2 x^2}d\text {sech}^{-1}(c x)\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -c^2 \left (\frac {(1-c x) (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )^3}{2 c^2 x^2}-\frac {3}{2} b \int -\left (a+b \text {sech}^{-1}(c x)\right )^2 \sin \left (i \text {sech}^{-1}(c x)\right )^2d\text {sech}^{-1}(c x)\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -c^2 \left (\frac {(1-c x) (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )^3}{2 c^2 x^2}+\frac {3}{2} b \int \left (a+b \text {sech}^{-1}(c x)\right )^2 \sin \left (i \text {sech}^{-1}(c x)\right )^2d\text {sech}^{-1}(c x)\right )\) |
| \(\Big \downarrow \) 3792 |
| \(\displaystyle -c^2 \left (\frac {3}{2} b \left (\frac {1}{2} \int \left (a+b \text {sech}^{-1}(c x)\right )^2d\text {sech}^{-1}(c x)+\frac {1}{2} b^2 \int -\frac {(1-c x) (c x+1)}{c^2 x^2}d\text {sech}^{-1}(c x)+\frac {b (1-c x) (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )}{2 c^2 x^2}-\frac {\sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )^2}{2 c^2 x^2}\right )+\frac {(1-c x) (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )^3}{2 c^2 x^2}\right )\) |
| \(\Big \downarrow \) 17 |
| \(\displaystyle -c^2 \left (\frac {3}{2} b \left (\frac {1}{2} b^2 \int -\frac {(1-c x) (c x+1)}{c^2 x^2}d\text {sech}^{-1}(c x)-\frac {\sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )^2}{2 c^2 x^2}+\frac {b (1-c x) (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )}{2 c^2 x^2}+\frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{6 b}\right )+\frac {(1-c x) (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )^3}{2 c^2 x^2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -c^2 \left (\frac {3}{2} b \left (-\frac {1}{2} b^2 \int \frac {(1-c x) (c x+1)}{c^2 x^2}d\text {sech}^{-1}(c x)-\frac {\sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )^2}{2 c^2 x^2}+\frac {b (1-c x) (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )}{2 c^2 x^2}+\frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{6 b}\right )+\frac {(1-c x) (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )^3}{2 c^2 x^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -c^2 \left (\frac {(1-c x) (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )^3}{2 c^2 x^2}+\frac {3}{2} b \left (-\frac {1}{2} b^2 \int -\sin \left (i \text {sech}^{-1}(c x)\right )^2d\text {sech}^{-1}(c x)-\frac {\sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )^2}{2 c^2 x^2}+\frac {b (1-c x) (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )}{2 c^2 x^2}+\frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{6 b}\right )\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -c^2 \left (\frac {(1-c x) (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )^3}{2 c^2 x^2}+\frac {3}{2} b \left (\frac {1}{2} b^2 \int \sin \left (i \text {sech}^{-1}(c x)\right )^2d\text {sech}^{-1}(c x)-\frac {\sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )^2}{2 c^2 x^2}+\frac {b (1-c x) (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )}{2 c^2 x^2}+\frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{6 b}\right )\right )\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle -c^2 \left (\frac {3}{2} b \left (\frac {1}{2} b^2 \left (\frac {1}{2} \int 1d\text {sech}^{-1}(c x)-\frac {\sqrt {\frac {1-c x}{c x+1}} (c x+1)}{2 c^2 x^2}\right )-\frac {\sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )^2}{2 c^2 x^2}+\frac {b (1-c x) (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )}{2 c^2 x^2}+\frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{6 b}\right )+\frac {(1-c x) (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )^3}{2 c^2 x^2}\right )\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle -c^2 \left (\frac {3}{2} b \left (-\frac {\sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )^2}{2 c^2 x^2}+\frac {b (1-c x) (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )}{2 c^2 x^2}+\frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{6 b}+\frac {1}{2} b^2 \left (\frac {1}{2} \text {sech}^{-1}(c x)-\frac {\sqrt {\frac {1-c x}{c x+1}} (c x+1)}{2 c^2 x^2}\right )\right )+\frac {(1-c x) (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )^3}{2 c^2 x^2}\right )\) |
-(c^2*(((1 - c*x)*(1 + c*x)*(a + b*ArcSech[c*x])^3)/(2*c^2*x^2) + (3*b*((b ^2*(-1/2*(Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x))/(c^2*x^2) + ArcSech[c*x]/2) )/2 + (b*(1 - c*x)*(1 + c*x)*(a + b*ArcSech[c*x]))/(2*c^2*x^2) - (Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*(a + b*ArcSech[c*x])^2)/(2*c^2*x^2) + (a + b*A rcSech[c*x])^3/(6*b)))/2))
3.1.48.3.1 Defintions of rubi rules used
Int[(c_.)*((a_.) + (b_.)*(x_))^(m_.), x_Symbol] :> Simp[c*((a + b*x)^(m + 1 )/(b*(m + 1))), x] /; FreeQ[{a, b, c, m}, x] && NeQ[m, -1]
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbo l] :> Simp[d*m*(c + d*x)^(m - 1)*((b*Sin[e + f*x])^n/(f^2*n^2)), x] + (-Sim p[b*(c + d*x)^m*Cos[e + f*x]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x] + Simp[b^ 2*((n - 1)/n) Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[d^2 *m*((m - 1)/(f^2*n^2)) Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 1]
Int[Cosh[(a_.) + (b_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)* (x_)]^(n_.), x_Symbol] :> Simp[(c + d*x)^m*(Sinh[a + b*x]^(n + 1)/(b*(n + 1 ))), x] - Simp[d*(m/(b*(n + 1))) Int[(c + d*x)^(m - 1)*Sinh[a + b*x]^(n + 1), x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && NeQ[n, -1]
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. \(320\) vs. \(2(147)=294\).
Time = 0.50 (sec) , antiderivative size = 321, normalized size of antiderivative = 1.97
| method | result | size |
| derivativedivides | \(c^{2} \left (-\frac {a^{3}}{2 c^{2} x^{2}}+b^{3} \left (-\frac {\operatorname {arcsech}\left (c x \right )^{3}}{2 c^{2} x^{2}}+\frac {3 \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \operatorname {arcsech}\left (c x \right )^{2}}{4 c x}+\frac {\operatorname {arcsech}\left (c x \right )^{3}}{4}-\frac {3 \,\operatorname {arcsech}\left (c x \right )}{4 c^{2} x^{2}}+\frac {3 \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}}{8 c x}+\frac {3 \,\operatorname {arcsech}\left (c x \right )}{8}\right )+3 a \,b^{2} \left (-\frac {\operatorname {arcsech}\left (c x \right )^{2}}{2 c^{2} x^{2}}+\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \operatorname {arcsech}\left (c x \right )}{2 c x}+\frac {\operatorname {arcsech}\left (c x \right )^{2}}{4}-\frac {1}{4 c^{2} x^{2}}\right )+3 b \,a^{2} \left (-\frac {\operatorname {arcsech}\left (c x \right )}{2 c^{2} x^{2}}+\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \left (\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right ) c^{2} x^{2}+\sqrt {-c^{2} x^{2}+1}\right )}{4 c x \sqrt {-c^{2} x^{2}+1}}\right )\right )\) | \(321\) |
| default | \(c^{2} \left (-\frac {a^{3}}{2 c^{2} x^{2}}+b^{3} \left (-\frac {\operatorname {arcsech}\left (c x \right )^{3}}{2 c^{2} x^{2}}+\frac {3 \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \operatorname {arcsech}\left (c x \right )^{2}}{4 c x}+\frac {\operatorname {arcsech}\left (c x \right )^{3}}{4}-\frac {3 \,\operatorname {arcsech}\left (c x \right )}{4 c^{2} x^{2}}+\frac {3 \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}}{8 c x}+\frac {3 \,\operatorname {arcsech}\left (c x \right )}{8}\right )+3 a \,b^{2} \left (-\frac {\operatorname {arcsech}\left (c x \right )^{2}}{2 c^{2} x^{2}}+\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \operatorname {arcsech}\left (c x \right )}{2 c x}+\frac {\operatorname {arcsech}\left (c x \right )^{2}}{4}-\frac {1}{4 c^{2} x^{2}}\right )+3 b \,a^{2} \left (-\frac {\operatorname {arcsech}\left (c x \right )}{2 c^{2} x^{2}}+\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \left (\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right ) c^{2} x^{2}+\sqrt {-c^{2} x^{2}+1}\right )}{4 c x \sqrt {-c^{2} x^{2}+1}}\right )\right )\) | \(321\) |
| parts | \(-\frac {a^{3}}{2 x^{2}}+b^{3} c^{2} \left (-\frac {\operatorname {arcsech}\left (c x \right )^{3}}{2 c^{2} x^{2}}+\frac {3 \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \operatorname {arcsech}\left (c x \right )^{2}}{4 c x}+\frac {\operatorname {arcsech}\left (c x \right )^{3}}{4}-\frac {3 \,\operatorname {arcsech}\left (c x \right )}{4 c^{2} x^{2}}+\frac {3 \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}}{8 c x}+\frac {3 \,\operatorname {arcsech}\left (c x \right )}{8}\right )+3 a \,b^{2} c^{2} \left (-\frac {\operatorname {arcsech}\left (c x \right )^{2}}{2 c^{2} x^{2}}+\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \operatorname {arcsech}\left (c x \right )}{2 c x}+\frac {\operatorname {arcsech}\left (c x \right )^{2}}{4}-\frac {1}{4 c^{2} x^{2}}\right )+3 b \,a^{2} c^{2} \left (-\frac {\operatorname {arcsech}\left (c x \right )}{2 c^{2} x^{2}}+\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \left (\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right ) c^{2} x^{2}+\sqrt {-c^{2} x^{2}+1}\right )}{4 c x \sqrt {-c^{2} x^{2}+1}}\right )\) | \(323\) |
c^2*(-1/2*a^3/c^2/x^2+b^3*(-1/2/c^2/x^2*arcsech(c*x)^3+3/4*(-(c*x-1)/c/x)^ (1/2)*((c*x+1)/c/x)^(1/2)/c/x*arcsech(c*x)^2+1/4*arcsech(c*x)^3-3/4/c^2/x^ 2*arcsech(c*x)+3/8*(-(c*x-1)/c/x)^(1/2)*((c*x+1)/c/x)^(1/2)/c/x+3/8*arcsec h(c*x))+3*a*b^2*(-1/2/c^2/x^2*arcsech(c*x)^2+1/2*(-(c*x-1)/c/x)^(1/2)*((c* x+1)/c/x)^(1/2)/c/x*arcsech(c*x)+1/4*arcsech(c*x)^2-1/4/c^2/x^2)+3*b*a^2*( -1/2/c^2/x^2*arcsech(c*x)+1/4*(-(c*x-1)/c/x)^(1/2)/c/x*((c*x+1)/c/x)^(1/2) *(arctanh(1/(-c^2*x^2+1)^(1/2))*c^2*x^2+(-c^2*x^2+1)^(1/2))/(-c^2*x^2+1)^( 1/2)))
Time = 0.27 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.66 \[ \int \frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{x^3} \, dx=\frac {2 \, {\left (b^{3} c^{2} x^{2} - 2 \, b^{3}\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right )^{3} + 3 \, {\left (2 \, a^{2} b + b^{3}\right )} c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 4 \, a^{3} - 6 \, a b^{2} + 6 \, {\left (a b^{2} c^{2} x^{2} + b^{3} c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 2 \, a b^{2}\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right )^{2} + 3 \, {\left (4 \, a b^{2} c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + {\left (2 \, a^{2} b + b^{3}\right )} c^{2} x^{2} - 4 \, a^{2} b - 2 \, b^{3}\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right )}{8 \, x^{2}} \]
1/8*(2*(b^3*c^2*x^2 - 2*b^3)*log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) + 1)/ (c*x))^3 + 3*(2*a^2*b + b^3)*c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) - 4*a^3 - 6*a*b^2 + 6*(a*b^2*c^2*x^2 + b^3*c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) - 2*a* b^2)*log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) + 1)/(c*x))^2 + 3*(4*a*b^2*c* x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) + (2*a^2*b + b^3)*c^2*x^2 - 4*a^2*b - 2*b ^3)*log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) + 1)/(c*x)))/x^2
\[ \int \frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{x^3} \, dx=\int \frac {\left (a + b \operatorname {asech}{\left (c x \right )}\right )^{3}}{x^{3}}\, dx \]
\[ \int \frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{x^3} \, dx=\int { \frac {{\left (b \operatorname {arsech}\left (c x\right ) + a\right )}^{3}}{x^{3}} \,d x } \]
-3/8*a^2*b*((2*c^4*x*sqrt(1/(c^2*x^2) - 1)/(c^2*x^2*(1/(c^2*x^2) - 1) - 1) - c^3*log(c*x*sqrt(1/(c^2*x^2) - 1) + 1) + c^3*log(c*x*sqrt(1/(c^2*x^2) - 1) - 1))/c + 4*arcsech(c*x)/x^2) - 1/2*a^3/x^2 + integrate(b^3*log(sqrt(1 /(c*x) + 1)*sqrt(1/(c*x) - 1) + 1/(c*x))^3/x^3 + 3*a*b^2*log(sqrt(1/(c*x) + 1)*sqrt(1/(c*x) - 1) + 1/(c*x))^2/x^3, x)
\[ \int \frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{x^3} \, dx=\int { \frac {{\left (b \operatorname {arsech}\left (c x\right ) + a\right )}^{3}}{x^{3}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{x^3} \, dx=\int \frac {{\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}^3}{x^3} \,d x \]