Integrand size = 10, antiderivative size = 133 \[ \int \frac {\text {sech}^{-1}(a+b x)}{x^3} \, dx=\frac {b \sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x)}{2 a \left (1-a^2\right ) x}+\frac {b^2 \text {sech}^{-1}(a+b x)}{2 a^2}-\frac {\text {sech}^{-1}(a+b x)}{2 x^2}-\frac {\left (1-2 a^2\right ) b^2 \text {arctanh}\left (\frac {\sqrt {1+a} \tanh \left (\frac {1}{2} \text {sech}^{-1}(a+b x)\right )}{\sqrt {1-a}}\right )}{a^2 \left (1-a^2\right )^{3/2}} \]
1/2*b^2*arcsech(b*x+a)/a^2-1/2*arcsech(b*x+a)/x^2-(-2*a^2+1)*b^2*arctanh(( 1+a)^(1/2)*tanh(1/2*arcsech(b*x+a))/(1-a)^(1/2))/a^2/(-a^2+1)^(3/2)+1/2*b* (b*x+a+1)*((-b*x-a+1)/(b*x+a+1))^(1/2)/a/(-a^2+1)/x
Leaf count is larger than twice the leaf count of optimal. \(315\) vs. \(2(133)=266\).
Time = 0.90 (sec) , antiderivative size = 315, normalized size of antiderivative = 2.37 \[ \int \frac {\text {sech}^{-1}(a+b x)}{x^3} \, dx=\frac {1}{2} \left (-\frac {b \sqrt {-\frac {-1+a+b x}{1+a+b x}} (1+a+b x)}{(-1+a) a (1+a) x}-\frac {\text {sech}^{-1}(a+b x)}{x^2}-\frac {\left (-1+2 a^2\right ) b^2 \log (x)}{a^2 \left (1-a^2\right )^{3/2}}-\frac {b^2 \log (a+b x)}{a^2}+\frac {b^2 \log \left (1+\sqrt {-\frac {-1+a+b x}{1+a+b x}}+a \sqrt {-\frac {-1+a+b x}{1+a+b x}}+b x \sqrt {-\frac {-1+a+b x}{1+a+b x}}\right )}{a^2}+\frac {\left (-1+2 a^2\right ) b^2 \log \left (1-a^2-a b x+\sqrt {1-a^2} \sqrt {-\frac {-1+a+b x}{1+a+b x}}+a \sqrt {1-a^2} \sqrt {-\frac {-1+a+b x}{1+a+b x}}+\sqrt {1-a^2} b x \sqrt {-\frac {-1+a+b x}{1+a+b x}}\right )}{a^2 \left (1-a^2\right )^{3/2}}\right ) \]
(-((b*Sqrt[-((-1 + a + b*x)/(1 + a + b*x))]*(1 + a + b*x))/((-1 + a)*a*(1 + a)*x)) - ArcSech[a + b*x]/x^2 - ((-1 + 2*a^2)*b^2*Log[x])/(a^2*(1 - a^2) ^(3/2)) - (b^2*Log[a + b*x])/a^2 + (b^2*Log[1 + Sqrt[-((-1 + a + b*x)/(1 + a + b*x))] + a*Sqrt[-((-1 + a + b*x)/(1 + a + b*x))] + b*x*Sqrt[-((-1 + a + b*x)/(1 + a + b*x))]])/a^2 + ((-1 + 2*a^2)*b^2*Log[1 - a^2 - a*b*x + Sq rt[1 - a^2]*Sqrt[-((-1 + a + b*x)/(1 + a + b*x))] + a*Sqrt[1 - a^2]*Sqrt[- ((-1 + a + b*x)/(1 + a + b*x))] + Sqrt[1 - a^2]*b*x*Sqrt[-((-1 + a + b*x)/ (1 + a + b*x))]])/(a^2*(1 - a^2)^(3/2)))/2
Time = 0.76 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.20, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.200, Rules used = {6875, 25, 5991, 3042, 4272, 3042, 4407, 3042, 4318, 3042, 3138, 221}
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.
| \(\displaystyle \int \frac {\text {sech}^{-1}(a+b x)}{x^3} \, dx\) |
| \(\Big \downarrow \) 6875 |
| \(\displaystyle -b^2 \int \frac {(a+b x) \sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1) \text {sech}^{-1}(a+b x)}{b^3 x^3}d\text {sech}^{-1}(a+b x)\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle b^2 \int -\frac {(a+b x) \sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1) \text {sech}^{-1}(a+b x)}{b^3 x^3}d\text {sech}^{-1}(a+b x)\) |
| \(\Big \downarrow \) 5991 |
| \(\displaystyle -b^2 \left (\frac {\text {sech}^{-1}(a+b x)}{2 b^2 x^2}-\frac {1}{2} \int \frac {1}{b^2 x^2}d\text {sech}^{-1}(a+b x)\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -b^2 \left (\frac {\text {sech}^{-1}(a+b x)}{2 b^2 x^2}-\frac {1}{2} \int \frac {1}{\left (a-\csc \left (i \text {sech}^{-1}(a+b x)+\frac {\pi }{2}\right )\right )^2}d\text {sech}^{-1}(a+b x)\right )\) |
| \(\Big \downarrow \) 4272 |
| \(\displaystyle -b^2 \left (\frac {1}{2} \left (-\frac {\int -\frac {-a^2-(a+b x) a+1}{b x}d\text {sech}^{-1}(a+b x)}{a \left (1-a^2\right )}-\frac {\sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1)}{a \left (1-a^2\right ) b x}\right )+\frac {\text {sech}^{-1}(a+b x)}{2 b^2 x^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -b^2 \left (\frac {\text {sech}^{-1}(a+b x)}{2 b^2 x^2}+\frac {1}{2} \left (-\frac {\sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1)}{a \left (1-a^2\right ) b x}-\frac {\int \frac {-a^2-\csc \left (i \text {sech}^{-1}(a+b x)+\frac {\pi }{2}\right ) a+1}{a-\csc \left (i \text {sech}^{-1}(a+b x)+\frac {\pi }{2}\right )}d\text {sech}^{-1}(a+b x)}{a \left (1-a^2\right )}\right )\right )\) |
| \(\Big \downarrow \) 4407 |
| \(\displaystyle -b^2 \left (\frac {1}{2} \left (-\frac {\frac {\left (1-2 a^2\right ) \int -\frac {a+b x}{b x}d\text {sech}^{-1}(a+b x)}{a}+\frac {\left (1-a^2\right ) \text {sech}^{-1}(a+b x)}{a}}{a \left (1-a^2\right )}-\frac {\sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1)}{a \left (1-a^2\right ) b x}\right )+\frac {\text {sech}^{-1}(a+b x)}{2 b^2 x^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -b^2 \left (\frac {\text {sech}^{-1}(a+b x)}{2 b^2 x^2}+\frac {1}{2} \left (-\frac {\sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1)}{a \left (1-a^2\right ) b x}-\frac {\frac {\left (1-a^2\right ) \text {sech}^{-1}(a+b x)}{a}+\frac {\left (1-2 a^2\right ) \int \frac {\csc \left (i \text {sech}^{-1}(a+b x)+\frac {\pi }{2}\right )}{a-\csc \left (i \text {sech}^{-1}(a+b x)+\frac {\pi }{2}\right )}d\text {sech}^{-1}(a+b x)}{a}}{a \left (1-a^2\right )}\right )\right )\) |
| \(\Big \downarrow \) 4318 |
| \(\displaystyle -b^2 \left (\frac {1}{2} \left (-\frac {\frac {\left (1-a^2\right ) \text {sech}^{-1}(a+b x)}{a}-\frac {\left (1-2 a^2\right ) \int \frac {1}{1-\frac {a}{a+b x}}d\text {sech}^{-1}(a+b x)}{a}}{a \left (1-a^2\right )}-\frac {\sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1)}{a \left (1-a^2\right ) b x}\right )+\frac {\text {sech}^{-1}(a+b x)}{2 b^2 x^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -b^2 \left (\frac {\text {sech}^{-1}(a+b x)}{2 b^2 x^2}+\frac {1}{2} \left (-\frac {\sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1)}{a \left (1-a^2\right ) b x}-\frac {\frac {\left (1-a^2\right ) \text {sech}^{-1}(a+b x)}{a}-\frac {\left (1-2 a^2\right ) \int \frac {1}{1-a \sin \left (i \text {sech}^{-1}(a+b x)+\frac {\pi }{2}\right )}d\text {sech}^{-1}(a+b x)}{a}}{a \left (1-a^2\right )}\right )\right )\) |
| \(\Big \downarrow \) 3138 |
| \(\displaystyle -b^2 \left (\frac {1}{2} \left (-\frac {\frac {\left (1-a^2\right ) \text {sech}^{-1}(a+b x)}{a}-\frac {2 \left (1-2 a^2\right ) \int \frac {1}{-\left ((a+1) \tanh ^2\left (\frac {1}{2} \text {sech}^{-1}(a+b x)\right )\right )-a+1}d\tanh \left (\frac {1}{2} \text {sech}^{-1}(a+b x)\right )}{a}}{a \left (1-a^2\right )}-\frac {\sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1)}{a \left (1-a^2\right ) b x}\right )+\frac {\text {sech}^{-1}(a+b x)}{2 b^2 x^2}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -b^2 \left (\frac {1}{2} \left (-\frac {\frac {\left (1-a^2\right ) \text {sech}^{-1}(a+b x)}{a}-\frac {2 \left (1-2 a^2\right ) \text {arctanh}\left (\frac {\sqrt {a+1} \tanh \left (\frac {1}{2} \text {sech}^{-1}(a+b x)\right )}{\sqrt {1-a}}\right )}{a \sqrt {1-a^2}}}{a \left (1-a^2\right )}-\frac {\sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1)}{a \left (1-a^2\right ) b x}\right )+\frac {\text {sech}^{-1}(a+b x)}{2 b^2 x^2}\right )\) |
-(b^2*(ArcSech[a + b*x]/(2*b^2*x^2) + (-((Sqrt[(1 - a - b*x)/(1 + a + b*x) ]*(1 + a + b*x))/(a*(1 - a^2)*b*x)) - (((1 - a^2)*ArcSech[a + b*x])/a - (2 *(1 - 2*a^2)*ArcTanh[(Sqrt[1 + a]*Tanh[ArcSech[a + b*x]/2])/Sqrt[1 - a]])/ (a*Sqrt[1 - a^2]))/(a*(1 - a^2)))/2))
3.1.7.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{ e = FreeFactors[Tan[(c + d*x)/2], x]}, Simp[2*(e/d) Subst[Int[1/(a + b + (a - b)*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> Simp[b^2*Cot[ c + d*x]*((a + b*Csc[c + d*x])^(n + 1)/(a*d*(n + 1)*(a^2 - b^2))), x] + Sim p[1/(a*(n + 1)*(a^2 - b^2)) Int[(a + b*Csc[c + d*x])^(n + 1)*Simp[(a^2 - b^2)*(n + 1) - a*b*(n + 1)*Csc[c + d*x] + b^2*(n + 2)*Csc[c + d*x]^2, x], x ], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && LtQ[n, -1] && Integ erQ[2*n]
Int[csc[(e_.) + (f_.)*(x_)]/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbo l] :> Simp[1/b Int[1/(1 + (a/b)*Sin[e + f*x]), x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[c*(x/a), x] - Simp[(b*c - a*d)/a Int[Csc[e + f* x]/(a + b*Csc[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
Int[((e_.) + (f_.)*(x_))^(m_.)*Sech[(c_.) + (d_.)*(x_)]*((a_) + (b_.)*Sech[ (c_.) + (d_.)*(x_)])^(n_.)*Tanh[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[(-(e + f*x)^m)*((a + b*Sech[c + d*x])^(n + 1)/(b*d*(n + 1))), x] + Simp[f*(m/(b *d*(n + 1))) Int[(e + f*x)^(m - 1)*(a + b*Sech[c + d*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && IGtQ[m, 0] && NeQ[n, -1]
Int[((a_.) + ArcSech[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^( m_.), x_Symbol] :> Simp[-(d^(m + 1))^(-1) Subst[Int[(a + b*x)^p*Sech[x]*T anh[x]*(d*e - c*f + f*Sech[x])^m, x], x, ArcSech[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && IntegerQ[m]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.96 (sec) , antiderivative size = 370, normalized size of antiderivative = 2.78
| method | result | size |
| parts | \(-\frac {\operatorname {arcsech}\left (b x +a \right )}{2 x^{2}}-\frac {b \sqrt {-\frac {b x +a -1}{b x +a}}\, \left (b x +a \right ) \sqrt {\frac {b x +a +1}{b x +a}}\, \operatorname {csgn}\left (b \right )^{2} \left (-\operatorname {arctanh}\left (\frac {1}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right ) a^{4} b x -2 \sqrt {-a^{2}+1}\, \ln \left (\frac {-2 a^{2}+2-2 a b x +2 \sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{x}\right ) a^{2} b x +2 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right ) a^{2} b x +a^{3} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}+\sqrt {-a^{2}+1}\, \ln \left (\frac {-2 a^{2}+2-2 a b x +2 \sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{x}\right ) b x -\operatorname {arctanh}\left (\frac {1}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right ) b x -\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a \right )}{2 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, \left (1+a \right ) \left (-1+a \right ) a^{2} \left (a^{2}-1\right ) x}\) | \(370\) |
| derivativedivides | \(b^{2} \left (-\frac {\operatorname {arcsech}\left (b x +a \right )}{2 b^{2} x^{2}}-\frac {\sqrt {-\frac {b x +a -1}{b x +a}}\, \left (b x +a \right ) \sqrt {\frac {b x +a +1}{b x +a}}\, \left (\operatorname {arctanh}\left (\frac {1}{\sqrt {1-\left (b x +a \right )^{2}}}\right ) a^{5}-\operatorname {arctanh}\left (\frac {1}{\sqrt {1-\left (b x +a \right )^{2}}}\right ) a^{4} \left (b x +a \right )+2 \ln \left (\frac {2 \sqrt {-a^{2}+1}\, \sqrt {1-\left (b x +a \right )^{2}}-2 \left (b x +a \right ) a +2}{b x}\right ) \sqrt {-a^{2}+1}\, a^{3}-2 \ln \left (\frac {2 \sqrt {-a^{2}+1}\, \sqrt {1-\left (b x +a \right )^{2}}-2 \left (b x +a \right ) a +2}{b x}\right ) \sqrt {-a^{2}+1}\, a^{2} \left (b x +a \right )-2 a^{3} \operatorname {arctanh}\left (\frac {1}{\sqrt {1-\left (b x +a \right )^{2}}}\right )+2 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {1-\left (b x +a \right )^{2}}}\right ) a^{2} \left (b x +a \right )+\sqrt {1-\left (b x +a \right )^{2}}\, a^{3}-\ln \left (\frac {2 \sqrt {-a^{2}+1}\, \sqrt {1-\left (b x +a \right )^{2}}-2 \left (b x +a \right ) a +2}{b x}\right ) \sqrt {-a^{2}+1}\, a +\ln \left (\frac {2 \sqrt {-a^{2}+1}\, \sqrt {1-\left (b x +a \right )^{2}}-2 \left (b x +a \right ) a +2}{b x}\right ) \sqrt {-a^{2}+1}\, \left (b x +a \right )+\operatorname {arctanh}\left (\frac {1}{\sqrt {1-\left (b x +a \right )^{2}}}\right ) a -\operatorname {arctanh}\left (\frac {1}{\sqrt {1-\left (b x +a \right )^{2}}}\right ) \left (b x +a \right )-a \sqrt {1-\left (b x +a \right )^{2}}\right )}{2 b x \left (a^{2}-1\right ) \left (1+a \right ) \left (-1+a \right ) \sqrt {1-\left (b x +a \right )^{2}}\, a^{2}}\right )\) | \(483\) |
| default | \(b^{2} \left (-\frac {\operatorname {arcsech}\left (b x +a \right )}{2 b^{2} x^{2}}-\frac {\sqrt {-\frac {b x +a -1}{b x +a}}\, \left (b x +a \right ) \sqrt {\frac {b x +a +1}{b x +a}}\, \left (\operatorname {arctanh}\left (\frac {1}{\sqrt {1-\left (b x +a \right )^{2}}}\right ) a^{5}-\operatorname {arctanh}\left (\frac {1}{\sqrt {1-\left (b x +a \right )^{2}}}\right ) a^{4} \left (b x +a \right )+2 \ln \left (\frac {2 \sqrt {-a^{2}+1}\, \sqrt {1-\left (b x +a \right )^{2}}-2 \left (b x +a \right ) a +2}{b x}\right ) \sqrt {-a^{2}+1}\, a^{3}-2 \ln \left (\frac {2 \sqrt {-a^{2}+1}\, \sqrt {1-\left (b x +a \right )^{2}}-2 \left (b x +a \right ) a +2}{b x}\right ) \sqrt {-a^{2}+1}\, a^{2} \left (b x +a \right )-2 a^{3} \operatorname {arctanh}\left (\frac {1}{\sqrt {1-\left (b x +a \right )^{2}}}\right )+2 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {1-\left (b x +a \right )^{2}}}\right ) a^{2} \left (b x +a \right )+\sqrt {1-\left (b x +a \right )^{2}}\, a^{3}-\ln \left (\frac {2 \sqrt {-a^{2}+1}\, \sqrt {1-\left (b x +a \right )^{2}}-2 \left (b x +a \right ) a +2}{b x}\right ) \sqrt {-a^{2}+1}\, a +\ln \left (\frac {2 \sqrt {-a^{2}+1}\, \sqrt {1-\left (b x +a \right )^{2}}-2 \left (b x +a \right ) a +2}{b x}\right ) \sqrt {-a^{2}+1}\, \left (b x +a \right )+\operatorname {arctanh}\left (\frac {1}{\sqrt {1-\left (b x +a \right )^{2}}}\right ) a -\operatorname {arctanh}\left (\frac {1}{\sqrt {1-\left (b x +a \right )^{2}}}\right ) \left (b x +a \right )-a \sqrt {1-\left (b x +a \right )^{2}}\right )}{2 b x \left (a^{2}-1\right ) \left (1+a \right ) \left (-1+a \right ) \sqrt {1-\left (b x +a \right )^{2}}\, a^{2}}\right )\) | \(483\) |
-1/2*arcsech(b*x+a)/x^2-1/2*b*(-(b*x+a-1)/(b*x+a))^(1/2)*(b*x+a)*((b*x+a+1 )/(b*x+a))^(1/2)*csgn(b)^2*(-arctanh(1/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2))*a^4 *b*x-2*(-a^2+1)^(1/2)*ln(2*(-a*b*x+(-a^2+1)^(1/2)*(-b^2*x^2-2*a*b*x-a^2+1) ^(1/2)-a^2+1)/x)*a^2*b*x+2*arctanh(1/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2))*a^2*b *x+a^3*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)+(-a^2+1)^(1/2)*ln(2*(-a*b*x+(-a^2+1) ^(1/2)*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)-a^2+1)/x)*b*x-arctanh(1/(-b^2*x^2-2* a*b*x-a^2+1)^(1/2))*b*x-(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*a)/(-b^2*x^2-2*a*b* x-a^2+1)^(1/2)/(1+a)/(-1+a)/a^2/(a^2-1)/x
Leaf count of result is larger than twice the leaf count of optimal. 427 vs. \(2 (112) = 224\).
Time = 0.32 (sec) , antiderivative size = 865, normalized size of antiderivative = 6.50 \[ \int \frac {\text {sech}^{-1}(a+b x)}{x^3} \, dx=\left [-\frac {{\left (2 \, a^{2} - 1\right )} \sqrt {-a^{2} + 1} b^{2} x^{2} \log \left (\frac {{\left (2 \, a^{2} - 1\right )} b^{2} x^{2} + 2 \, a^{4} + 4 \, {\left (a^{3} - a\right )} b x - 4 \, a^{2} + 2 \, {\left (a b^{2} x^{2} + a^{3} + {\left (2 \, a^{2} - 1\right )} b x - a\right )} \sqrt {-a^{2} + 1} \sqrt {-\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} + 2}{x^{2}}\right ) - {\left (a^{4} - 2 \, a^{2} + 1\right )} b^{2} x^{2} \log \left (\frac {{\left (b x + a\right )} \sqrt {-\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} + 1}{x}\right ) + {\left (a^{4} - 2 \, a^{2} + 1\right )} b^{2} x^{2} \log \left (\frac {{\left (b x + a\right )} \sqrt {-\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} - 1}{x}\right ) + 2 \, {\left (a^{6} - 2 \, a^{4} + a^{2}\right )} \log \left (\frac {{\left (b x + a\right )} \sqrt {-\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} + 1}{b x + a}\right ) + 2 \, {\left ({\left (a^{3} - a\right )} b^{2} x^{2} + {\left (a^{4} - a^{2}\right )} b x\right )} \sqrt {-\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}}}{4 \, {\left (a^{6} - 2 \, a^{4} + a^{2}\right )} x^{2}}, \frac {2 \, {\left (2 \, a^{2} - 1\right )} \sqrt {a^{2} - 1} b^{2} x^{2} \arctan \left (\frac {{\left (a b^{2} x^{2} + a^{3} + {\left (2 \, a^{2} - 1\right )} b x - a\right )} \sqrt {a^{2} - 1} \sqrt {-\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}}}{{\left (a^{2} - 1\right )} b^{2} x^{2} + a^{4} + 2 \, {\left (a^{3} - a\right )} b x - 2 \, a^{2} + 1}\right ) + {\left (a^{4} - 2 \, a^{2} + 1\right )} b^{2} x^{2} \log \left (\frac {{\left (b x + a\right )} \sqrt {-\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} + 1}{x}\right ) - {\left (a^{4} - 2 \, a^{2} + 1\right )} b^{2} x^{2} \log \left (\frac {{\left (b x + a\right )} \sqrt {-\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} - 1}{x}\right ) - 2 \, {\left (a^{6} - 2 \, a^{4} + a^{2}\right )} \log \left (\frac {{\left (b x + a\right )} \sqrt {-\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} + 1}{b x + a}\right ) - 2 \, {\left ({\left (a^{3} - a\right )} b^{2} x^{2} + {\left (a^{4} - a^{2}\right )} b x\right )} \sqrt {-\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}}}{4 \, {\left (a^{6} - 2 \, a^{4} + a^{2}\right )} x^{2}}\right ] \]
[-1/4*((2*a^2 - 1)*sqrt(-a^2 + 1)*b^2*x^2*log(((2*a^2 - 1)*b^2*x^2 + 2*a^4 + 4*(a^3 - a)*b*x - 4*a^2 + 2*(a*b^2*x^2 + a^3 + (2*a^2 - 1)*b*x - a)*sqr t(-a^2 + 1)*sqrt(-(b^2*x^2 + 2*a*b*x + a^2 - 1)/(b^2*x^2 + 2*a*b*x + a^2)) + 2)/x^2) - (a^4 - 2*a^2 + 1)*b^2*x^2*log(((b*x + a)*sqrt(-(b^2*x^2 + 2*a *b*x + a^2 - 1)/(b^2*x^2 + 2*a*b*x + a^2)) + 1)/x) + (a^4 - 2*a^2 + 1)*b^2 *x^2*log(((b*x + a)*sqrt(-(b^2*x^2 + 2*a*b*x + a^2 - 1)/(b^2*x^2 + 2*a*b*x + a^2)) - 1)/x) + 2*(a^6 - 2*a^4 + a^2)*log(((b*x + a)*sqrt(-(b^2*x^2 + 2 *a*b*x + a^2 - 1)/(b^2*x^2 + 2*a*b*x + a^2)) + 1)/(b*x + a)) + 2*((a^3 - a )*b^2*x^2 + (a^4 - a^2)*b*x)*sqrt(-(b^2*x^2 + 2*a*b*x + a^2 - 1)/(b^2*x^2 + 2*a*b*x + a^2)))/((a^6 - 2*a^4 + a^2)*x^2), 1/4*(2*(2*a^2 - 1)*sqrt(a^2 - 1)*b^2*x^2*arctan((a*b^2*x^2 + a^3 + (2*a^2 - 1)*b*x - a)*sqrt(a^2 - 1)* sqrt(-(b^2*x^2 + 2*a*b*x + a^2 - 1)/(b^2*x^2 + 2*a*b*x + a^2))/((a^2 - 1)* b^2*x^2 + a^4 + 2*(a^3 - a)*b*x - 2*a^2 + 1)) + (a^4 - 2*a^2 + 1)*b^2*x^2* log(((b*x + a)*sqrt(-(b^2*x^2 + 2*a*b*x + a^2 - 1)/(b^2*x^2 + 2*a*b*x + a^ 2)) + 1)/x) - (a^4 - 2*a^2 + 1)*b^2*x^2*log(((b*x + a)*sqrt(-(b^2*x^2 + 2* a*b*x + a^2 - 1)/(b^2*x^2 + 2*a*b*x + a^2)) - 1)/x) - 2*(a^6 - 2*a^4 + a^2 )*log(((b*x + a)*sqrt(-(b^2*x^2 + 2*a*b*x + a^2 - 1)/(b^2*x^2 + 2*a*b*x + a^2)) + 1)/(b*x + a)) - 2*((a^3 - a)*b^2*x^2 + (a^4 - a^2)*b*x)*sqrt(-(b^2 *x^2 + 2*a*b*x + a^2 - 1)/(b^2*x^2 + 2*a*b*x + a^2)))/((a^6 - 2*a^4 + a^2) *x^2)]
\[ \int \frac {\text {sech}^{-1}(a+b x)}{x^3} \, dx=\int \frac {\operatorname {asech}{\left (a + b x \right )}}{x^{3}}\, dx \]
\[ \int \frac {\text {sech}^{-1}(a+b x)}{x^3} \, dx=\int { \frac {\operatorname {arsech}\left (b x + a\right )}{x^{3}} \,d x } \]
-1/2*(3*a^2*b^2 - b^2)*log(x)/(a^6 - 2*a^4 + a^2) + 1/4*((a^4*b^2 - 2*a^3* b^2 + a^2*b^2)*x^2*log(b*x + a + 1) + (a^4*b^2 + 2*a^3*b^2 + a^2*b^2)*x^2* log(-b*x - a + 1) - 2*(a^3*b - a*b)*x - 2*(a^6 - 2*a^4 + a^2)*log(sqrt(b*x + a + 1)*sqrt(-b*x - a + 1)*b*x + sqrt(b*x + a + 1)*sqrt(-b*x - a + 1)*a + b*x + a) + 2*(a^6 - 2*a^4 - (a^4*b^2 - 2*a^2*b^2 + b^2)*x^2 + a^2)*log(b *x + a) + 2*(a^6 - 2*a^4 + a^2)*log(b*x + a))/((a^6 - 2*a^4 + a^2)*x^2) - integrate(1/2*(b^2*x + a*b)/(b^2*x^4 + 2*a*b*x^3 + (a^2 - 1)*x^2 + (b^2*x^ 4 + 2*a*b*x^3 + (a^2 - 1)*x^2)*e^(1/2*log(b*x + a + 1) + 1/2*log(-b*x - a + 1))), x)
\[ \int \frac {\text {sech}^{-1}(a+b x)}{x^3} \, dx=\int { \frac {\operatorname {arsech}\left (b x + a\right )}{x^{3}} \,d x } \]
Timed out. \[ \int \frac {\text {sech}^{-1}(a+b x)}{x^3} \, dx=\int \frac {\mathrm {acosh}\left (\frac {1}{a+b\,x}\right )}{x^3} \,d x \]