3.13 \(\int \frac{\text{sech}^{-1}(a+b x)^2}{x^2} \, dx\)

Optimal. Leaf size=224 \[ \frac{2 b \text{PolyLog}\left (2,\frac{a e^{\text{sech}^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )}{a \sqrt{1-a^2}}-\frac{2 b \text{PolyLog}\left (2,\frac{a e^{\text{sech}^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right )}{a \sqrt{1-a^2}}+\frac{2 b \text{sech}^{-1}(a+b x) \log \left (1-\frac{a e^{\text{sech}^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )}{a \sqrt{1-a^2}}-\frac{2 b \text{sech}^{-1}(a+b x) \log \left (1-\frac{a e^{\text{sech}^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right )}{a \sqrt{1-a^2}}-\frac{b \text{sech}^{-1}(a+b x)^2}{a}-\frac{\text{sech}^{-1}(a+b x)^2}{x} \]

[Out]

-((b*ArcSech[a + b*x]^2)/a) - ArcSech[a + b*x]^2/x + (2*b*ArcSech[a + b*x]*Log[1 - (a*E^ArcSech[a + b*x])/(1 -
Sqrt[1 - a^2])])/(a*Sqrt[1 - a^2]) - (2*b*ArcSech[a + b*x]*Log[1 - (a*E^ArcSech[a + b*x])/(1 + Sqrt[1 - a^2])
])/(a*Sqrt[1 - a^2]) + (2*b*PolyLog[2, (a*E^ArcSech[a + b*x])/(1 - Sqrt[1 - a^2])])/(a*Sqrt[1 - a^2]) - (2*b*P
olyLog[2, (a*E^ArcSech[a + b*x])/(1 + Sqrt[1 - a^2])])/(a*Sqrt[1 - a^2])

________________________________________________________________________________________

Rubi [A]  time = 0.388906, antiderivative size = 224, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667, Rules used = {6321, 5468, 4191, 3320, 2264, 2190, 2279, 2391} \[ \frac{2 b \text{PolyLog}\left (2,\frac{a e^{\text{sech}^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )}{a \sqrt{1-a^2}}-\frac{2 b \text{PolyLog}\left (2,\frac{a e^{\text{sech}^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right )}{a \sqrt{1-a^2}}+\frac{2 b \text{sech}^{-1}(a+b x) \log \left (1-\frac{a e^{\text{sech}^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )}{a \sqrt{1-a^2}}-\frac{2 b \text{sech}^{-1}(a+b x) \log \left (1-\frac{a e^{\text{sech}^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right )}{a \sqrt{1-a^2}}-\frac{b \text{sech}^{-1}(a+b x)^2}{a}-\frac{\text{sech}^{-1}(a+b x)^2}{x} \]

Antiderivative was successfully verified.

[In]

Int[ArcSech[a + b*x]^2/x^2,x]

[Out]

-((b*ArcSech[a + b*x]^2)/a) - ArcSech[a + b*x]^2/x + (2*b*ArcSech[a + b*x]*Log[1 - (a*E^ArcSech[a + b*x])/(1 -
Sqrt[1 - a^2])])/(a*Sqrt[1 - a^2]) - (2*b*ArcSech[a + b*x]*Log[1 - (a*E^ArcSech[a + b*x])/(1 + Sqrt[1 - a^2])
])/(a*Sqrt[1 - a^2]) + (2*b*PolyLog[2, (a*E^ArcSech[a + b*x])/(1 - Sqrt[1 - a^2])])/(a*Sqrt[1 - a^2]) - (2*b*P
olyLog[2, (a*E^ArcSech[a + b*x])/(1 + Sqrt[1 - a^2])])/(a*Sqrt[1 - a^2])

Rule 6321

Int[((a_.) + ArcSech[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> -Dist[(d^(m + 1)
)^(-1), Subst[Int[(a + b*x)^p*Sech[x]*Tanh[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]

Rule 5468

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] + Dist[(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]

Rule 4191

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*Sin[e + f*x])^n), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && ILtQ[n, 0] &
& IGtQ[m, 0]

Rule 3320

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]), x_Symbol]
:> Dist[2, Int[((c + d*x)^m*E^(-(I*e) + f*fz*x))/(E^(I*Pi*(k - 1/2))*(b + (2*a*E^(-(I*e) + f*fz*x))/E^(I*Pi*(k
- 1/2)) - (b*E^(2*(-(I*e) + f*fz*x)))/E^(2*I*k*Pi))), x], x] /; FreeQ[{a, b, c, d, e, f, fz}, x] && IntegerQ[
2*k] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 2264

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.)*(F_)^(v_)), x_Symbol] :> With[{q =
Rt[b^2 - 4*a*c, 2]}, Dist[(2*c)/q, Int[((f + g*x)^m*F^u)/(b - q + 2*c*F^u), x], x] - Dist[(2*c)/q, Int[((f +
g*x)^m*F^u)/(b + q + 2*c*F^u), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[v, 2*u] && LinearQ[u, x] && NeQ[
b^2 - 4*a*c, 0] && IGtQ[m, 0]

Rule 2190

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*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), x]
- Dist[(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]

Rule 2279

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rubi steps

\begin{align*} \int \frac{\text{sech}^{-1}(a+b x)^2}{x^2} \, dx &=-\left (b \operatorname{Subst}\left (\int \frac{x^2 \text{sech}(x) \tanh (x)}{(-a+\text{sech}(x))^2} \, dx,x,\text{sech}^{-1}(a+b x)\right )\right )\\ &=-\frac{\text{sech}^{-1}(a+b x)^2}{x}+(2 b) \operatorname{Subst}\left (\int \frac{x}{-a+\text{sech}(x)} \, dx,x,\text{sech}^{-1}(a+b x)\right )\\ &=-\frac{\text{sech}^{-1}(a+b x)^2}{x}+(2 b) \operatorname{Subst}\left (\int \left (-\frac{x}{a}+\frac{x}{a (1-a \cosh (x))}\right ) \, dx,x,\text{sech}^{-1}(a+b x)\right )\\ &=-\frac{b \text{sech}^{-1}(a+b x)^2}{a}-\frac{\text{sech}^{-1}(a+b x)^2}{x}+\frac{(2 b) \operatorname{Subst}\left (\int \frac{x}{1-a \cosh (x)} \, dx,x,\text{sech}^{-1}(a+b x)\right )}{a}\\ &=-\frac{b \text{sech}^{-1}(a+b x)^2}{a}-\frac{\text{sech}^{-1}(a+b x)^2}{x}+\frac{(4 b) \operatorname{Subst}\left (\int \frac{e^x x}{-a+2 e^x-a e^{2 x}} \, dx,x,\text{sech}^{-1}(a+b x)\right )}{a}\\ &=-\frac{b \text{sech}^{-1}(a+b x)^2}{a}-\frac{\text{sech}^{-1}(a+b x)^2}{x}-\frac{(4 b) \operatorname{Subst}\left (\int \frac{e^x x}{2-2 \sqrt{1-a^2}-2 a e^x} \, dx,x,\text{sech}^{-1}(a+b x)\right )}{\sqrt{1-a^2}}+\frac{(4 b) \operatorname{Subst}\left (\int \frac{e^x x}{2+2 \sqrt{1-a^2}-2 a e^x} \, dx,x,\text{sech}^{-1}(a+b x)\right )}{\sqrt{1-a^2}}\\ &=-\frac{b \text{sech}^{-1}(a+b x)^2}{a}-\frac{\text{sech}^{-1}(a+b x)^2}{x}+\frac{2 b \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 \sqrt{1-a^2}}-\frac{2 b \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 \sqrt{1-a^2}}-\frac{(2 b) \operatorname{Subst}\left (\int \log \left (1-\frac{2 a e^x}{2-2 \sqrt{1-a^2}}\right ) \, dx,x,\text{sech}^{-1}(a+b x)\right )}{a \sqrt{1-a^2}}+\frac{(2 b) \operatorname{Subst}\left (\int \log \left (1-\frac{2 a e^x}{2+2 \sqrt{1-a^2}}\right ) \, dx,x,\text{sech}^{-1}(a+b x)\right )}{a \sqrt{1-a^2}}\\ &=-\frac{b \text{sech}^{-1}(a+b x)^2}{a}-\frac{\text{sech}^{-1}(a+b x)^2}{x}+\frac{2 b \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 \sqrt{1-a^2}}-\frac{2 b \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 \sqrt{1-a^2}}-\frac{(2 b) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{2 a x}{2-2 \sqrt{1-a^2}}\right )}{x} \, dx,x,e^{\text{sech}^{-1}(a+b x)}\right )}{a \sqrt{1-a^2}}+\frac{(2 b) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{2 a x}{2+2 \sqrt{1-a^2}}\right )}{x} \, dx,x,e^{\text{sech}^{-1}(a+b x)}\right )}{a \sqrt{1-a^2}}\\ &=-\frac{b \text{sech}^{-1}(a+b x)^2}{a}-\frac{\text{sech}^{-1}(a+b x)^2}{x}+\frac{2 b \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 \sqrt{1-a^2}}-\frac{2 b \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 \sqrt{1-a^2}}+\frac{2 b \text{Li}_2\left (\frac{a e^{\text{sech}^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )}{a \sqrt{1-a^2}}-\frac{2 b \text{Li}_2\left (\frac{a e^{\text{sech}^{-1}(a+b x)}}{1+\sqrt{1-a^2}}\right )}{a \sqrt{1-a^2}}\\ \end{align*}

Mathematica [C]  time = 2.62484, size = 678, normalized size = 3.03 \[ \frac{-\frac{(a+b x) \text{sech}^{-1}(a+b x)^2}{x}+\frac{2 b \left (i \left (\text{PolyLog}\left (2,\frac{\left (-1-i \sqrt{a^2-1}\right ) \left (-i \sqrt{a^2-1} \tanh \left (\frac{1}{2} \text{sech}^{-1}(a+b x)\right )+a-1\right )}{a \left (i \sqrt{a^2-1} \tanh \left (\frac{1}{2} \text{sech}^{-1}(a+b x)\right )+a-1\right )}\right )-\text{PolyLog}\left (2,\frac{\left (\sqrt{a^2-1}+i\right ) \left (-i \sqrt{a^2-1} \tanh \left (\frac{1}{2} \text{sech}^{-1}(a+b x)\right )+a-1\right )}{a \left (\sqrt{a^2-1} \tanh \left (\frac{1}{2} \text{sech}^{-1}(a+b x)\right )-i (a-1)\right )}\right )\right )+2 \text{sech}^{-1}(a+b x) \tan ^{-1}\left (\frac{(a-1) \coth \left (\frac{1}{2} \text{sech}^{-1}(a+b x)\right )}{\sqrt{a^2-1}}\right )-2 i \cos ^{-1}\left (\frac{1}{a}\right ) \tan ^{-1}\left (\frac{(a+1) \tanh \left (\frac{1}{2} \text{sech}^{-1}(a+b x)\right )}{\sqrt{a^2-1}}\right )-\log \left (-\frac{(a-1) \left (-i \sqrt{a^2-1}+a+1\right ) \left (\tanh \left (\frac{1}{2} \text{sech}^{-1}(a+b x)\right )-1\right )}{a \left (i \sqrt{a^2-1} \tanh \left (\frac{1}{2} \text{sech}^{-1}(a+b x)\right )+a-1\right )}\right ) \left (2 \tan ^{-1}\left (\frac{(a+1) \tanh \left (\frac{1}{2} \text{sech}^{-1}(a+b x)\right )}{\sqrt{a^2-1}}\right )+\cos ^{-1}\left (\frac{1}{a}\right )\right )-\log \left (\frac{(a-1) \left (i \sqrt{a^2-1}+a+1\right ) \left (\tanh \left (\frac{1}{2} \text{sech}^{-1}(a+b x)\right )+1\right )}{a \left (i \sqrt{a^2-1} \tanh \left (\frac{1}{2} \text{sech}^{-1}(a+b x)\right )+a-1\right )}\right ) \left (\cos ^{-1}\left (\frac{1}{a}\right )-2 \tan ^{-1}\left (\frac{(a+1) \tanh \left (\frac{1}{2} \text{sech}^{-1}(a+b x)\right )}{\sqrt{a^2-1}}\right )\right )+\log \left (\frac{\sqrt{a^2-1} e^{-\frac{1}{2} \text{sech}^{-1}(a+b x)}}{\sqrt{2} \sqrt{a} \sqrt{-\frac{b x}{a+b x}}}\right ) \left (2 \left (\tan ^{-1}\left (\frac{(a-1) \coth \left (\frac{1}{2} \text{sech}^{-1}(a+b x)\right )}{\sqrt{a^2-1}}\right )+\tan ^{-1}\left (\frac{(a+1) \tanh \left (\frac{1}{2} \text{sech}^{-1}(a+b x)\right )}{\sqrt{a^2-1}}\right )\right )+\cos ^{-1}\left (\frac{1}{a}\right )\right )+\log \left (\frac{\sqrt{a^2-1} e^{\frac{1}{2} \text{sech}^{-1}(a+b x)}}{\sqrt{2} \sqrt{a} \sqrt{-\frac{b x}{a+b x}}}\right ) \left (\cos ^{-1}\left (\frac{1}{a}\right )-2 \left (\tan ^{-1}\left (\frac{(a-1) \coth \left (\frac{1}{2} \text{sech}^{-1}(a+b x)\right )}{\sqrt{a^2-1}}\right )+\tan ^{-1}\left (\frac{(a+1) \tanh \left (\frac{1}{2} \text{sech}^{-1}(a+b x)\right )}{\sqrt{a^2-1}}\right )\right )\right )\right )}{\sqrt{a^2-1}}}{a} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcSech[a + b*x]^2/x^2,x]

[Out]

(-(((a + b*x)*ArcSech[a + b*x]^2)/x) + (2*b*(2*ArcSech[a + b*x]*ArcTan[((-1 + a)*Coth[ArcSech[a + b*x]/2])/Sqr
t[-1 + a^2]] - (2*I)*ArcCos[a^(-1)]*ArcTan[((1 + a)*Tanh[ArcSech[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^(ArcSech[a + b*x]/2)*Sqrt[-((b*x)/(a + b*x))])] + (Arc
Cos[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*ArcTan[((1 + a)*Tanh[ArcSech[a + b*x]/2])/Sqrt[-1 + a^2]])*Log[((-1 + a)*(1 + a + I*Sq
rt[-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*(P
olyLog[2, ((-1 - I*Sqrt[-1 + a^2])*(-1 + a - I*Sqrt[-1 + a^2]*Tanh[ArcSech[a + b*x]/2]))/(a*(-1 + a + I*Sqrt[-
1 + a^2]*Tanh[ArcSech[a + b*x]/2]))] - PolyLog[2, ((I + Sqrt[-1 + a^2])*(-1 + a - I*Sqrt[-1 + a^2]*Tanh[ArcSec
h[a + b*x]/2]))/(a*((-I)*(-1 + a) + Sqrt[-1 + a^2]*Tanh[ArcSech[a + b*x]/2]))])))/Sqrt[-1 + a^2])/a

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Maple [A]  time = 0.357, size = 367, normalized size = 1.6 \begin{align*} -{\frac{b \left ({\rm arcsech} \left (bx+a\right ) \right ) ^{2}}{a}}-{\frac{ \left ({\rm arcsech} \left (bx+a\right ) \right ) ^{2}}{x}}+2\,{\frac{b\sqrt{-{a}^{2}+1}{\rm arcsech} \left (bx+a\right )}{a \left ({a}^{2}-1 \right ) }\ln \left ({\frac{-a \left ( \left ( bx+a \right ) ^{-1}+\sqrt{ \left ( bx+a \right ) ^{-1}-1}\sqrt{ \left ( bx+a \right ) ^{-1}+1} \right ) +\sqrt{-{a}^{2}+1}+1}{1+\sqrt{-{a}^{2}+1}}} \right ) }-2\,{\frac{b\sqrt{-{a}^{2}+1}{\rm arcsech} \left (bx+a\right )}{a \left ({a}^{2}-1 \right ) }\ln \left ({\frac{a \left ( \left ( bx+a \right ) ^{-1}+\sqrt{ \left ( bx+a \right ) ^{-1}-1}\sqrt{ \left ( bx+a \right ) ^{-1}+1} \right ) +\sqrt{-{a}^{2}+1}-1}{-1+\sqrt{-{a}^{2}+1}}} \right ) }+2\,{\frac{b\sqrt{-{a}^{2}+1}}{a \left ({a}^{2}-1 \right ) }{\it dilog} \left ({\frac{-a \left ( \left ( bx+a \right ) ^{-1}+\sqrt{ \left ( bx+a \right ) ^{-1}-1}\sqrt{ \left ( bx+a \right ) ^{-1}+1} \right ) +\sqrt{-{a}^{2}+1}+1}{1+\sqrt{-{a}^{2}+1}}} \right ) }-2\,{\frac{b\sqrt{-{a}^{2}+1}}{a \left ({a}^{2}-1 \right ) }{\it dilog} \left ({\frac{a \left ( \left ( bx+a \right ) ^{-1}+\sqrt{ \left ( bx+a \right ) ^{-1}-1}\sqrt{ \left ( bx+a \right ) ^{-1}+1} \right ) +\sqrt{-{a}^{2}+1}-1}{-1+\sqrt{-{a}^{2}+1}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arcsech(b*x+a)^2/x^2,x)

[Out]

-b*arcsech(b*x+a)^2/a-arcsech(b*x+a)^2/x+2*b*(-a^2+1)^(1/2)/a/(a^2-1)*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*b*(-a^2+1)^(1/2)/a/(a^2-1)*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*b*(-a
^2+1)^(1/2)/a/(a^2-1)*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*b*(-a^2+1)^(1/2)/a/(a^2-1)*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)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{\log \left (\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\right )^{2}}{x} - \int -\frac{2 \,{\left (2 \,{\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + a^{3} +{\left (3 \, a^{2} b - b\right )} x - a\right )} \sqrt{b x + a + 1} \sqrt{-b x - a + 1} \log \left (b x + a\right )^{2} + 2 \,{\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + a^{3} +{\left (3 \, a^{2} b - b\right )} x - a\right )} \log \left (b x + a\right )^{2} +{\left (b^{3} x^{3} + 2 \, a b^{2} x^{2} +{\left (a^{2} b - b\right )} x - 2 \,{\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + a^{3} +{\left (3 \, a^{2} b - b\right )} x - a\right )} \log \left (b x + a\right ) -{\left ({\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + a^{3} +{\left (3 \, a^{2} b - b\right )} x - a\right )} \sqrt{b x + a + 1} \log \left (b x + a\right ) -{\left (2 \, b^{3} x^{3} + 4 \, a b^{2} x^{2} +{\left (2 \, a^{2} b - b\right )} x -{\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + a^{3} +{\left (3 \, a^{2} b - b\right )} x - a\right )} \log \left (b x + a\right )\right )} \sqrt{b x + a + 1}\right )} \sqrt{-b x - a + 1}\right )} \log \left (\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\right )\right )}}{b^{3} x^{5} + 3 \, a b^{2} x^{4} +{\left (3 \, a^{2} b - b\right )} x^{3} +{\left (a^{3} - a\right )} x^{2} +{\left (b^{3} x^{5} + 3 \, a b^{2} x^{4} +{\left (3 \, a^{2} b - b\right )} x^{3} +{\left (a^{3} - a\right )} x^{2}\right )} \sqrt{b x + a + 1} \sqrt{-b x - a + 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsech(b*x+a)^2/x^2,x, algorithm="maxima")

[Out]

-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 - integr
ate(-2*(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)*sqrt(-b*x - a + 1)*log(b*x + a
)^2 + 2*(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 - 2*(b^3*x^3 + 3*a*b^2*x^2 + a^3 + (3*a^2*b - b)*x - a)*log(b*x + a) - ((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 - (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^5 + 3*a*b^2*x^4 + (3*a
^2*b - b)*x^3 + (a^3 - a)*x^2 + (b^3*x^5 + 3*a*b^2*x^4 + (3*a^2*b - b)*x^3 + (a^3 - a)*x^2)*sqrt(b*x + a + 1)*
sqrt(-b*x - a + 1)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\operatorname{arsech}\left (b x + a\right )^{2}}{x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsech(b*x+a)^2/x^2,x, algorithm="fricas")

[Out]

integral(arcsech(b*x + a)^2/x^2, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{asech}^{2}{\left (a + b x \right )}}{x^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(asech(b*x+a)**2/x**2,x)

[Out]

Integral(asech(a + b*x)**2/x**2, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arsech}\left (b x + a\right )^{2}}{x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsech(b*x+a)^2/x^2,x, algorithm="giac")

[Out]

integrate(arcsech(b*x + a)^2/x^2, x)