Integrand size = 12, antiderivative size = 537 \[ \int \frac {\text {sech}^{-1}(a+b x)^2}{x^3} \, dx=\frac {b^2 \sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x) \text {sech}^{-1}(a+b x)}{a \left (1-a^2\right ) (a+b x) \left (1-\frac {a}{a+b x}\right )}+\frac {b^2 \text {sech}^{-1}(a+b x)^2}{2 a^2}-\frac {\text {sech}^{-1}(a+b x)^2}{2 x^2}+\frac {b^2 \text {sech}^{-1}(a+b x) \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a^2 \left (1-a^2\right )^{3/2}}-\frac {2 b^2 \text {sech}^{-1}(a+b x) \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a^2 \sqrt {1-a^2}}-\frac {b^2 \text {sech}^{-1}(a+b x) \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )}{a^2 \left (1-a^2\right )^{3/2}}+\frac {2 b^2 \text {sech}^{-1}(a+b x) \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )}{a^2 \sqrt {1-a^2}}+\frac {b^2 \log \left (\frac {x}{a+b x}\right )}{a^2 \left (1-a^2\right )}+\frac {b^2 \operatorname {PolyLog}\left (2,\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a^2 \left (1-a^2\right )^{3/2}}-\frac {2 b^2 \operatorname {PolyLog}\left (2,\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a^2 \sqrt {1-a^2}}-\frac {b^2 \operatorname {PolyLog}\left (2,\frac {a e^{\text {sech}^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )}{a^2 \left (1-a^2\right )^{3/2}}+\frac {2 b^2 \operatorname {PolyLog}\left (2,\frac {a e^{\text {sech}^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )}{a^2 \sqrt {1-a^2}} \] Output:
b^2*((-b*x-a+1)/(b*x+a+1))^(1/2)*(b*x+a+1)*arcsech(b*x+a)/a/(-a^2+1)/(b*x+ a)/(1-a/(b*x+a))+1/2*b^2*arcsech(b*x+a)^2/a^2-1/2*arcsech(b*x+a)^2/x^2+b^2 *arcsech(b*x+a)*ln(1-a*(1/(b*x+a)+(1/(b*x+a)-1)^(1/2)*(1/(b*x+a)+1)^(1/2)) /(1-(-a^2+1)^(1/2)))/a^2/(-a^2+1)^(3/2)-2*b^2*arcsech(b*x+a)*ln(1-a*(1/(b* x+a)+(1/(b*x+a)-1)^(1/2)*(1/(b*x+a)+1)^(1/2))/(1-(-a^2+1)^(1/2)))/a^2/(-a^ 2+1)^(1/2)-b^2*arcsech(b*x+a)*ln(1-a*(1/(b*x+a)+(1/(b*x+a)-1)^(1/2)*(1/(b* x+a)+1)^(1/2))/(1+(-a^2+1)^(1/2)))/a^2/(-a^2+1)^(3/2)+2*b^2*arcsech(b*x+a) *ln(1-a*(1/(b*x+a)+(1/(b*x+a)-1)^(1/2)*(1/(b*x+a)+1)^(1/2))/(1+(-a^2+1)^(1 /2)))/a^2/(-a^2+1)^(1/2)+b^2*ln(x/(b*x+a))/a^2/(-a^2+1)+b^2*polylog(2,a*(1 /(b*x+a)+(1/(b*x+a)-1)^(1/2)*(1/(b*x+a)+1)^(1/2))/(1-(-a^2+1)^(1/2)))/a^2/ (-a^2+1)^(3/2)-2*b^2*polylog(2,a*(1/(b*x+a)+(1/(b*x+a)-1)^(1/2)*(1/(b*x+a) +1)^(1/2))/(1-(-a^2+1)^(1/2)))/a^2/(-a^2+1)^(1/2)-b^2*polylog(2,a*(1/(b*x+ a)+(1/(b*x+a)-1)^(1/2)*(1/(b*x+a)+1)^(1/2))/(1+(-a^2+1)^(1/2)))/a^2/(-a^2+ 1)^(3/2)+2*b^2*polylog(2,a*(1/(b*x+a)+(1/(b*x+a)-1)^(1/2)*(1/(b*x+a)+1)^(1 /2))/(1+(-a^2+1)^(1/2)))/a^2/(-a^2+1)^(1/2)
Result contains complex when optimal does not.
Time = 4.84 (sec) , antiderivative size = 1439, normalized size of antiderivative = 2.68 \[ \int \frac {\text {sech}^{-1}(a+b x)^2}{x^3} \, dx =\text {Too large to display} \] Input:
Integrate[ArcSech[a + b*x]^2/x^3,x]
Output:
-1/2*((a + b*x)^2*ArcSech[a + b*x]^2)/(a^2*x^2) + (b*ArcSech[a + b*x]*(-(a *Sqrt[-((-1 + a + b*x)/(1 + a + b*x))]*(1 + a + b*x)) + (-1 + a^2)*(a + b* x)*ArcSech[a + b*x]))/((-1 + a)*a^2*(1 + a)*x) + (b^2*Log[(b*x)/(a + b*x)] )/(a^2 - a^4) - (2*b^2*(2*ArcSech[a + b*x]*ArcTan[((-1 + a)*Coth[ArcSech[a + b*x]/2])/Sqrt[-1 + a^2]] - (2*I)*ArcCos[a^(-1)]*ArcTan[((1 + a)*Tanh[Ar cSech[a + b*x]/2])/Sqrt[-1 + a^2]] + (ArcCos[a^(-1)] + 2*(ArcTan[((-1 + a) *Coth[ArcSech[a + b*x]/2])/Sqrt[-1 + a^2]] + ArcTan[((1 + a)*Tanh[ArcSech[ a + b*x]/2])/Sqrt[-1 + a^2]]))*Log[Sqrt[-1 + a^2]/(Sqrt[2]*Sqrt[a]*E^(ArcS ech[a + b*x]/2)*Sqrt[-((b*x)/(a + b*x))])] + (ArcCos[a^(-1)] - 2*(ArcTan[( (-1 + a)*Coth[ArcSech[a + b*x]/2])/Sqrt[-1 + a^2]] + ArcTan[((1 + a)*Tanh[ ArcSech[a + b*x]/2])/Sqrt[-1 + a^2]]))*Log[(Sqrt[-1 + a^2]*E^(ArcSech[a + b*x]/2))/(Sqrt[2]*Sqrt[a]*Sqrt[-((b*x)/(a + b*x))])] - (ArcCos[a^(-1)] + 2 *ArcTan[((1 + a)*Tanh[ArcSech[a + b*x]/2])/Sqrt[-1 + a^2]])*Log[-(((-1 + a )*(1 + a - I*Sqrt[-1 + a^2])*(-1 + Tanh[ArcSech[a + b*x]/2]))/(a*(-1 + a + I*Sqrt[-1 + a^2]*Tanh[ArcSech[a + b*x]/2])))] - (ArcCos[a^(-1)] - 2*ArcTa n[((1 + a)*Tanh[ArcSech[a + b*x]/2])/Sqrt[-1 + a^2]])*Log[((-1 + a)*(1 + a + I*Sqrt[-1 + a^2])*(1 + Tanh[ArcSech[a + b*x]/2]))/(a*(-1 + a + I*Sqrt[- 1 + a^2]*Tanh[ArcSech[a + b*x]/2]))] + I*(PolyLog[2, ((-1 - I*Sqrt[-1 + a^ 2])*(-1 + a - I*Sqrt[-1 + a^2]*Tanh[ArcSech[a + b*x]/2]))/(a*(-1 + a + I*S qrt[-1 + a^2]*Tanh[ArcSech[a + b*x]/2]))] - PolyLog[2, ((I + Sqrt[-1 + ...
Time = 1.18 (sec) , antiderivative size = 517, normalized size of antiderivative = 0.96, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6875, 25, 5991, 3042, 4679, 2009}
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.
| \(\displaystyle \int \frac {\text {sech}^{-1}(a+b x)^2}{x^3} \, dx\) |
| \(\Big \downarrow \) 6875 |
| \(\displaystyle -b^2 \int \frac {(a+b x) \sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1) \text {sech}^{-1}(a+b x)^2}{b^3 x^3}d\text {sech}^{-1}(a+b x)\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle b^2 \int -\frac {(a+b x) \sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1) \text {sech}^{-1}(a+b x)^2}{b^3 x^3}d\text {sech}^{-1}(a+b x)\) |
| \(\Big \downarrow \) 5991 |
| \(\displaystyle -b^2 \left (\frac {\text {sech}^{-1}(a+b x)^2}{2 b^2 x^2}-\int \frac {\text {sech}^{-1}(a+b x)}{b^2 x^2}d\text {sech}^{-1}(a+b x)\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -b^2 \left (\frac {\text {sech}^{-1}(a+b x)^2}{2 b^2 x^2}-\int \frac {\text {sech}^{-1}(a+b x)}{\left (a-\csc \left (i \text {sech}^{-1}(a+b x)+\frac {\pi }{2}\right )\right )^2}d\text {sech}^{-1}(a+b x)\right )\) |
| \(\Big \downarrow \) 4679 |
| \(\displaystyle -b^2 \left (\frac {\text {sech}^{-1}(a+b x)^2}{2 b^2 x^2}-\int \left (\frac {2 \text {sech}^{-1}(a+b x)}{a^2 \left (\frac {a}{a+b x}-1\right )}+\frac {\text {sech}^{-1}(a+b x)}{a^2}+\frac {\text {sech}^{-1}(a+b x)}{a^2 \left (\frac {a}{a+b x}-1\right )^2}\right )d\text {sech}^{-1}(a+b x)\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -b^2 \left (\frac {2 \operatorname {PolyLog}\left (2,\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a^2 \sqrt {1-a^2}}-\frac {\operatorname {PolyLog}\left (2,\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a^2 \left (1-a^2\right )^{3/2}}-\frac {2 \operatorname {PolyLog}\left (2,\frac {a e^{\text {sech}^{-1}(a+b x)}}{\sqrt {1-a^2}+1}\right )}{a^2 \sqrt {1-a^2}}+\frac {\operatorname {PolyLog}\left (2,\frac {a e^{\text {sech}^{-1}(a+b x)}}{\sqrt {1-a^2}+1}\right )}{a^2 \left (1-a^2\right )^{3/2}}-\frac {\log \left (1-\frac {a}{a+b x}\right )}{a^2 \left (1-a^2\right )}-\frac {\text {sech}^{-1}(a+b x)^2}{2 a^2}-\frac {\sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1) \text {sech}^{-1}(a+b x)}{a \left (1-a^2\right ) (a+b x) \left (1-\frac {a}{a+b x}\right )}+\frac {2 \text {sech}^{-1}(a+b x) \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a^2 \sqrt {1-a^2}}-\frac {\text {sech}^{-1}(a+b x) \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a^2 \left (1-a^2\right )^{3/2}}-\frac {2 \text {sech}^{-1}(a+b x) \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{\sqrt {1-a^2}+1}\right )}{a^2 \sqrt {1-a^2}}+\frac {\text {sech}^{-1}(a+b x) \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{\sqrt {1-a^2}+1}\right )}{a^2 \left (1-a^2\right )^{3/2}}+\frac {\text {sech}^{-1}(a+b x)^2}{2 b^2 x^2}\right )\) |
Input:
Int[ArcSech[a + b*x]^2/x^3,x]
Output:
-(b^2*(-((Sqrt[(1 - a - b*x)/(1 + a + b*x)]*(1 + a + b*x)*ArcSech[a + b*x] )/(a*(1 - a^2)*(a + b*x)*(1 - a/(a + b*x)))) - ArcSech[a + b*x]^2/(2*a^2) + ArcSech[a + b*x]^2/(2*b^2*x^2) - (ArcSech[a + b*x]*Log[1 - (a*E^ArcSech[ a + b*x])/(1 - Sqrt[1 - a^2])])/(a^2*(1 - a^2)^(3/2)) + (2*ArcSech[a + b*x ]*Log[1 - (a*E^ArcSech[a + b*x])/(1 - Sqrt[1 - a^2])])/(a^2*Sqrt[1 - a^2]) + (ArcSech[a + b*x]*Log[1 - (a*E^ArcSech[a + b*x])/(1 + Sqrt[1 - a^2])])/ (a^2*(1 - a^2)^(3/2)) - (2*ArcSech[a + b*x]*Log[1 - (a*E^ArcSech[a + b*x]) /(1 + Sqrt[1 - a^2])])/(a^2*Sqrt[1 - a^2]) - Log[1 - a/(a + b*x)]/(a^2*(1 - a^2)) - PolyLog[2, (a*E^ArcSech[a + b*x])/(1 - Sqrt[1 - a^2])]/(a^2*(1 - a^2)^(3/2)) + (2*PolyLog[2, (a*E^ArcSech[a + b*x])/(1 - Sqrt[1 - a^2])])/ (a^2*Sqrt[1 - a^2]) + PolyLog[2, (a*E^ArcSech[a + b*x])/(1 + Sqrt[1 - a^2] )]/(a^2*(1 - a^2)^(3/2)) - (2*PolyLog[2, (a*E^ArcSech[a + b*x])/(1 + Sqrt[ 1 - a^2])])/(a^2*Sqrt[1 - a^2])))
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(n_.)*((c_.) + (d_.)*(x_))^(m_.) , x_Symbol] :> Int[ExpandIntegrand[(c + d*x)^m, 1/(Sin[e + f*x]^n/(b + a*Si n[e + f*x])^n), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && ILtQ[n, 0] && IGt Q[m, 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]
Time = 0.68 (sec) , antiderivative size = 982, normalized size of antiderivative = 1.83
| method | result | size |
| derivativedivides | \(b^{2} \left (-\frac {\operatorname {arcsech}\left (b x +a \right ) \left (-2 \sqrt {\frac {b x +a +1}{b x +a}}\, \sqrt {-\frac {b x +a -1}{b x +a}}\, a^{2} \left (b x +a \right )+2 \sqrt {\frac {b x +a +1}{b x +a}}\, \sqrt {-\frac {b x +a -1}{b x +a}}\, a \left (b x +a \right )^{2}+2 \,\operatorname {arcsech}\left (b x +a \right ) a^{3} \left (b x +a \right )-\operatorname {arcsech}\left (b x +a \right ) a^{2} \left (b x +a \right )^{2}-2 \,\operatorname {arcsech}\left (b x +a \right ) a \left (b x +a \right )+\operatorname {arcsech}\left (b x +a \right ) \left (b x +a \right )^{2}+2 a^{2}-4 \left (b x +a \right ) a +2 \left (b x +a \right )^{2}\right )}{2 a^{2} \left (a^{2}-1\right ) b^{2} x^{2}}-\frac {\ln \left (a \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )^{2}+a -\frac {2}{b x +a}-2 \sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )}{a^{2} \left (a^{2}-1\right )}+\frac {2 \ln \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )}{a^{2} \left (a^{2}-1\right )}+\frac {\sqrt {-a^{2}+1}\, \operatorname {arcsech}\left (b x +a \right ) \ln \left (\frac {-a \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )+\sqrt {-a^{2}+1}+1}{1+\sqrt {-a^{2}+1}}\right )}{a^{2} \left (a^{2}-1\right )^{2}}-\frac {\sqrt {-a^{2}+1}\, \operatorname {arcsech}\left (b x +a \right ) \ln \left (\frac {a \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )+\sqrt {-a^{2}+1}-1}{-1+\sqrt {-a^{2}+1}}\right )}{a^{2} \left (a^{2}-1\right )^{2}}+\frac {\sqrt {-a^{2}+1}\, \operatorname {dilog}\left (\frac {-a \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )+\sqrt {-a^{2}+1}+1}{1+\sqrt {-a^{2}+1}}\right )}{a^{2} \left (a^{2}-1\right )^{2}}-\frac {\sqrt {-a^{2}+1}\, \operatorname {dilog}\left (\frac {a \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )+\sqrt {-a^{2}+1}-1}{-1+\sqrt {-a^{2}+1}}\right )}{a^{2} \left (a^{2}-1\right )^{2}}-\frac {2 \sqrt {-a^{2}+1}\, \operatorname {arcsech}\left (b x +a \right ) \ln \left (\frac {-a \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )+\sqrt {-a^{2}+1}+1}{1+\sqrt {-a^{2}+1}}\right )}{\left (a^{2}-1\right )^{2}}+\frac {2 \sqrt {-a^{2}+1}\, \operatorname {arcsech}\left (b x +a \right ) \ln \left (\frac {a \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )+\sqrt {-a^{2}+1}-1}{-1+\sqrt {-a^{2}+1}}\right )}{\left (a^{2}-1\right )^{2}}-\frac {2 \sqrt {-a^{2}+1}\, \operatorname {dilog}\left (\frac {-a \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )+\sqrt {-a^{2}+1}+1}{1+\sqrt {-a^{2}+1}}\right )}{\left (a^{2}-1\right )^{2}}+\frac {2 \sqrt {-a^{2}+1}\, \operatorname {dilog}\left (\frac {a \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )+\sqrt {-a^{2}+1}-1}{-1+\sqrt {-a^{2}+1}}\right )}{\left (a^{2}-1\right )^{2}}\right )\) | \(982\) |
| default | \(b^{2} \left (-\frac {\operatorname {arcsech}\left (b x +a \right ) \left (-2 \sqrt {\frac {b x +a +1}{b x +a}}\, \sqrt {-\frac {b x +a -1}{b x +a}}\, a^{2} \left (b x +a \right )+2 \sqrt {\frac {b x +a +1}{b x +a}}\, \sqrt {-\frac {b x +a -1}{b x +a}}\, a \left (b x +a \right )^{2}+2 \,\operatorname {arcsech}\left (b x +a \right ) a^{3} \left (b x +a \right )-\operatorname {arcsech}\left (b x +a \right ) a^{2} \left (b x +a \right )^{2}-2 \,\operatorname {arcsech}\left (b x +a \right ) a \left (b x +a \right )+\operatorname {arcsech}\left (b x +a \right ) \left (b x +a \right )^{2}+2 a^{2}-4 \left (b x +a \right ) a +2 \left (b x +a \right )^{2}\right )}{2 a^{2} \left (a^{2}-1\right ) b^{2} x^{2}}-\frac {\ln \left (a \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )^{2}+a -\frac {2}{b x +a}-2 \sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )}{a^{2} \left (a^{2}-1\right )}+\frac {2 \ln \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )}{a^{2} \left (a^{2}-1\right )}+\frac {\sqrt {-a^{2}+1}\, \operatorname {arcsech}\left (b x +a \right ) \ln \left (\frac {-a \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )+\sqrt {-a^{2}+1}+1}{1+\sqrt {-a^{2}+1}}\right )}{a^{2} \left (a^{2}-1\right )^{2}}-\frac {\sqrt {-a^{2}+1}\, \operatorname {arcsech}\left (b x +a \right ) \ln \left (\frac {a \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )+\sqrt {-a^{2}+1}-1}{-1+\sqrt {-a^{2}+1}}\right )}{a^{2} \left (a^{2}-1\right )^{2}}+\frac {\sqrt {-a^{2}+1}\, \operatorname {dilog}\left (\frac {-a \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )+\sqrt {-a^{2}+1}+1}{1+\sqrt {-a^{2}+1}}\right )}{a^{2} \left (a^{2}-1\right )^{2}}-\frac {\sqrt {-a^{2}+1}\, \operatorname {dilog}\left (\frac {a \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )+\sqrt {-a^{2}+1}-1}{-1+\sqrt {-a^{2}+1}}\right )}{a^{2} \left (a^{2}-1\right )^{2}}-\frac {2 \sqrt {-a^{2}+1}\, \operatorname {arcsech}\left (b x +a \right ) \ln \left (\frac {-a \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )+\sqrt {-a^{2}+1}+1}{1+\sqrt {-a^{2}+1}}\right )}{\left (a^{2}-1\right )^{2}}+\frac {2 \sqrt {-a^{2}+1}\, \operatorname {arcsech}\left (b x +a \right ) \ln \left (\frac {a \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )+\sqrt {-a^{2}+1}-1}{-1+\sqrt {-a^{2}+1}}\right )}{\left (a^{2}-1\right )^{2}}-\frac {2 \sqrt {-a^{2}+1}\, \operatorname {dilog}\left (\frac {-a \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )+\sqrt {-a^{2}+1}+1}{1+\sqrt {-a^{2}+1}}\right )}{\left (a^{2}-1\right )^{2}}+\frac {2 \sqrt {-a^{2}+1}\, \operatorname {dilog}\left (\frac {a \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )+\sqrt {-a^{2}+1}-1}{-1+\sqrt {-a^{2}+1}}\right )}{\left (a^{2}-1\right )^{2}}\right )\) | \(982\) |
Input:
int(arcsech(b*x+a)^2/x^3,x,method=_RETURNVERBOSE)
Output:
b^2*(-1/2*arcsech(b*x+a)*(-2*((b*x+a+1)/(b*x+a))^(1/2)*(-(b*x+a-1)/(b*x+a) )^(1/2)*a^2*(b*x+a)+2*((b*x+a+1)/(b*x+a))^(1/2)*(-(b*x+a-1)/(b*x+a))^(1/2) *a*(b*x+a)^2+2*arcsech(b*x+a)*a^3*(b*x+a)-arcsech(b*x+a)*a^2*(b*x+a)^2-2*a rcsech(b*x+a)*a*(b*x+a)+arcsech(b*x+a)*(b*x+a)^2+2*a^2-4*(b*x+a)*a+2*(b*x+ a)^2)/a^2/(a^2-1)/b^2/x^2-1/a^2/(a^2-1)*ln(a*(1/(b*x+a)+(1/(b*x+a)-1)^(1/2 )*(1/(b*x+a)+1)^(1/2))^2+a-2/(b*x+a)-2*(1/(b*x+a)-1)^(1/2)*(1/(b*x+a)+1)^( 1/2))+2/a^2/(a^2-1)*ln(1/(b*x+a)+(1/(b*x+a)-1)^(1/2)*(1/(b*x+a)+1)^(1/2))+ (-a^2+1)^(1/2)/a^2/(a^2-1)^2*arcsech(b*x+a)*ln((-a*(1/(b*x+a)+(1/(b*x+a)-1 )^(1/2)*(1/(b*x+a)+1)^(1/2))+(-a^2+1)^(1/2)+1)/(1+(-a^2+1)^(1/2)))-(-a^2+1 )^(1/2)/a^2/(a^2-1)^2*arcsech(b*x+a)*ln((a*(1/(b*x+a)+(1/(b*x+a)-1)^(1/2)* (1/(b*x+a)+1)^(1/2))+(-a^2+1)^(1/2)-1)/(-1+(-a^2+1)^(1/2)))+(-a^2+1)^(1/2) /a^2/(a^2-1)^2*dilog((-a*(1/(b*x+a)+(1/(b*x+a)-1)^(1/2)*(1/(b*x+a)+1)^(1/2 ))+(-a^2+1)^(1/2)+1)/(1+(-a^2+1)^(1/2)))-(-a^2+1)^(1/2)/a^2/(a^2-1)^2*dilo g((a*(1/(b*x+a)+(1/(b*x+a)-1)^(1/2)*(1/(b*x+a)+1)^(1/2))+(-a^2+1)^(1/2)-1) /(-1+(-a^2+1)^(1/2)))-2*(-a^2+1)^(1/2)/(a^2-1)^2*arcsech(b*x+a)*ln((-a*(1/ (b*x+a)+(1/(b*x+a)-1)^(1/2)*(1/(b*x+a)+1)^(1/2))+(-a^2+1)^(1/2)+1)/(1+(-a^ 2+1)^(1/2)))+2*(-a^2+1)^(1/2)/(a^2-1)^2*arcsech(b*x+a)*ln((a*(1/(b*x+a)+(1 /(b*x+a)-1)^(1/2)*(1/(b*x+a)+1)^(1/2))+(-a^2+1)^(1/2)-1)/(-1+(-a^2+1)^(1/2 )))-2*(-a^2+1)^(1/2)/(a^2-1)^2*dilog((-a*(1/(b*x+a)+(1/(b*x+a)-1)^(1/2)*(1 /(b*x+a)+1)^(1/2))+(-a^2+1)^(1/2)+1)/(1+(-a^2+1)^(1/2)))+2*(-a^2+1)^(1/...
\[ \int \frac {\text {sech}^{-1}(a+b x)^2}{x^3} \, dx=\int { \frac {\operatorname {arsech}\left (b x + a\right )^{2}}{x^{3}} \,d x } \] Input:
integrate(arcsech(b*x+a)^2/x^3,x, algorithm="fricas")
Output:
integral(arcsech(b*x + a)^2/x^3, x)
\[ \int \frac {\text {sech}^{-1}(a+b x)^2}{x^3} \, dx=\int \frac {\operatorname {asech}^{2}{\left (a + b x \right )}}{x^{3}}\, dx \] Input:
integrate(asech(b*x+a)**2/x**3,x)
Output:
Integral(asech(a + b*x)**2/x**3, x)
\[ \int \frac {\text {sech}^{-1}(a+b x)^2}{x^3} \, dx=\int { \frac {\operatorname {arsech}\left (b x + a\right )^{2}}{x^{3}} \,d x } \] Input:
integrate(arcsech(b*x+a)^2/x^3,x, algorithm="maxima")
Output:
-1/2*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/x^2 - integrate(-(4*(b^3*x^3 + 3*a*b^2*x^2 + a^3 + (3*a^2*b - b)*x - a)*sqrt(b*x + a + 1)*sqrt(-b*x - a + 1)*log(b*x + a)^2 + 4*(b^3*x^3 + 3*a*b^2*x^2 + a^3 + (3*a^2*b - b)*x - a)*log(b*x + a) ^2 + (b^3*x^3 + 2*a*b^2*x^2 + (a^2*b - b)*x - 4*(b^3*x^3 + 3*a*b^2*x^2 + a ^3 + (3*a^2*b - b)*x - a)*log(b*x + a) - (2*(b^3*x^3 + 3*a*b^2*x^2 + a^3 + (3*a^2*b - b)*x - a)*sqrt(b*x + a + 1)*log(b*x + a) - (2*b^3*x^3 + 4*a*b^ 2*x^2 + (2*a^2*b - b)*x - 2*(b^3*x^3 + 3*a*b^2*x^2 + a^3 + (3*a^2*b - b)*x - a)*log(b*x + a))*sqrt(b*x + a + 1))*sqrt(-b*x - a + 1))*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))/(b^3*x^6 + 3*a*b^2*x^5 + (3*a^2*b - b)*x^4 + (a^3 - a)*x^3 + (b^3 *x^6 + 3*a*b^2*x^5 + (3*a^2*b - b)*x^4 + (a^3 - a)*x^3)*sqrt(b*x + a + 1)* sqrt(-b*x - a + 1)), x)
\[ \int \frac {\text {sech}^{-1}(a+b x)^2}{x^3} \, dx=\int { \frac {\operatorname {arsech}\left (b x + a\right )^{2}}{x^{3}} \,d x } \] Input:
integrate(arcsech(b*x+a)^2/x^3,x, algorithm="giac")
Output:
integrate(arcsech(b*x + a)^2/x^3, x)
Timed out. \[ \int \frac {\text {sech}^{-1}(a+b x)^2}{x^3} \, dx=\int \frac {{\mathrm {acosh}\left (\frac {1}{a+b\,x}\right )}^2}{x^3} \,d x \] Input:
int(acosh(1/(a + b*x))^2/x^3,x)
Output:
int(acosh(1/(a + b*x))^2/x^3, x)
\[ \int \frac {\text {sech}^{-1}(a+b x)^2}{x^3} \, dx=\int \frac {\mathit {asech} \left (b x +a \right )^{2}}{x^{3}}d x \] Input:
int(asech(b*x+a)^2/x^3,x)
Output:
int(asech(a + b*x)**2/x**3,x)