3.19 \(\int \frac{\text{sech}^{-1}(a+b x)^3}{x^3} \, dx\)

Optimal. Leaf size=965 \[ \text{result too large to display} \]

[Out]

(-3*b^2*ArcSech[a + b*x]^2)/(2*a^2*(1 - a^2)) + (3*b^2*Sqrt[(1 - a - b*x)/(1 + a + b*x)]*(1 + a + b*x)*ArcSech
[a + b*x]^2)/(2*a*(1 - a^2)*(a + b*x)*(1 - a/(a + b*x))) + (b^2*ArcSech[a + b*x]^3)/(2*a^2) - ArcSech[a + b*x]
^3/(2*x^2) + (3*b^2*ArcSech[a + b*x]*Log[1 - (a*E^ArcSech[a + b*x])/(1 - Sqrt[1 - a^2])])/(a^2*(1 - a^2)) + (3
*b^2*ArcSech[a + b*x]^2*Log[1 - (a*E^ArcSech[a + b*x])/(1 - Sqrt[1 - a^2])])/(2*a^2*(1 - a^2)^(3/2)) - (3*b^2*
ArcSech[a + b*x]^2*Log[1 - (a*E^ArcSech[a + b*x])/(1 - Sqrt[1 - a^2])])/(a^2*Sqrt[1 - a^2]) + (3*b^2*ArcSech[a
+ b*x]*Log[1 - (a*E^ArcSech[a + b*x])/(1 + Sqrt[1 - a^2])])/(a^2*(1 - a^2)) - (3*b^2*ArcSech[a + b*x]^2*Log[1
- (a*E^ArcSech[a + b*x])/(1 + Sqrt[1 - a^2])])/(2*a^2*(1 - a^2)^(3/2)) + (3*b^2*ArcSech[a + b*x]^2*Log[1 - (a
*E^ArcSech[a + b*x])/(1 + Sqrt[1 - a^2])])/(a^2*Sqrt[1 - a^2]) + (3*b^2*PolyLog[2, (a*E^ArcSech[a + b*x])/(1 -
Sqrt[1 - a^2])])/(a^2*(1 - a^2)) + (3*b^2*ArcSech[a + b*x]*PolyLog[2, (a*E^ArcSech[a + b*x])/(1 - Sqrt[1 - a^
2])])/(a^2*(1 - a^2)^(3/2)) - (6*b^2*ArcSech[a + b*x]*PolyLog[2, (a*E^ArcSech[a + b*x])/(1 - Sqrt[1 - a^2])])/
(a^2*Sqrt[1 - a^2]) + (3*b^2*PolyLog[2, (a*E^ArcSech[a + b*x])/(1 + Sqrt[1 - a^2])])/(a^2*(1 - a^2)) - (3*b^2*
ArcSech[a + b*x]*PolyLog[2, (a*E^ArcSech[a + b*x])/(1 + Sqrt[1 - a^2])])/(a^2*(1 - a^2)^(3/2)) + (6*b^2*ArcSec
h[a + b*x]*PolyLog[2, (a*E^ArcSech[a + b*x])/(1 + Sqrt[1 - a^2])])/(a^2*Sqrt[1 - a^2]) - (3*b^2*PolyLog[3, (a*
E^ArcSech[a + b*x])/(1 - Sqrt[1 - a^2])])/(a^2*(1 - a^2)^(3/2)) + (6*b^2*PolyLog[3, (a*E^ArcSech[a + b*x])/(1
- Sqrt[1 - a^2])])/(a^2*Sqrt[1 - a^2]) + (3*b^2*PolyLog[3, (a*E^ArcSech[a + b*x])/(1 + Sqrt[1 - a^2])])/(a^2*(
1 - a^2)^(3/2)) - (6*b^2*PolyLog[3, (a*E^ArcSech[a + b*x])/(1 + Sqrt[1 - a^2])])/(a^2*Sqrt[1 - a^2])

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Rubi [A]  time = 1.33793, antiderivative size = 965, normalized size of antiderivative = 1., number of steps used = 32, number of rules used = 13, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1.083, Rules used = {6321, 5468, 4191, 3324, 3320, 2264, 2190, 2531, 2282, 6589, 5562, 2279, 2391} \[ \frac{b^2 \text{sech}^{-1}(a+b x)^3}{2 a^2}-\frac{\text{sech}^{-1}(a+b x)^3}{2 x^2}-\frac{3 b^2 \text{sech}^{-1}(a+b x)^2}{2 a^2 \left (1-a^2\right )}-\frac{3 b^2 \log \left (1-\frac{a e^{\text{sech}^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right ) \text{sech}^{-1}(a+b x)^2}{a^2 \sqrt{1-a^2}}+\frac{3 b^2 \log \left (1-\frac{a e^{\text{sech}^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right ) \text{sech}^{-1}(a+b x)^2}{2 a^2 \left (1-a^2\right )^{3/2}}+\frac{3 b^2 \log \left (1-\frac{a e^{\text{sech}^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right ) \text{sech}^{-1}(a+b x)^2}{a^2 \sqrt{1-a^2}}-\frac{3 b^2 \log \left (1-\frac{a e^{\text{sech}^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right ) \text{sech}^{-1}(a+b x)^2}{2 a^2 \left (1-a^2\right )^{3/2}}+\frac{3 b^2 \sqrt{\frac{-a-b x+1}{a+b x+1}} (a+b x+1) \text{sech}^{-1}(a+b x)^2}{2 a \left (1-a^2\right ) (a+b x) \left (1-\frac{a}{a+b x}\right )}+\frac{3 b^2 \log \left (1-\frac{a e^{\text{sech}^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right ) \text{sech}^{-1}(a+b x)}{a^2 \left (1-a^2\right )}+\frac{3 b^2 \log \left (1-\frac{a e^{\text{sech}^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right ) \text{sech}^{-1}(a+b x)}{a^2 \left (1-a^2\right )}-\frac{6 b^2 \text{PolyLog}\left (2,\frac{a e^{\text{sech}^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right ) \text{sech}^{-1}(a+b x)}{a^2 \sqrt{1-a^2}}+\frac{3 b^2 \text{PolyLog}\left (2,\frac{a e^{\text{sech}^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right ) \text{sech}^{-1}(a+b x)}{a^2 \left (1-a^2\right )^{3/2}}+\frac{6 b^2 \text{PolyLog}\left (2,\frac{a e^{\text{sech}^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right ) \text{sech}^{-1}(a+b x)}{a^2 \sqrt{1-a^2}}-\frac{3 b^2 \text{PolyLog}\left (2,\frac{a e^{\text{sech}^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right ) \text{sech}^{-1}(a+b x)}{a^2 \left (1-a^2\right )^{3/2}}+\frac{3 b^2 \text{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 )}+\frac{3 b^2 \text{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 )}+\frac{6 b^2 \text{PolyLog}\left (3,\frac{a e^{\text{sech}^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )}{a^2 \sqrt{1-a^2}}-\frac{3 b^2 \text{PolyLog}\left (3,\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{6 b^2 \text{PolyLog}\left (3,\frac{a e^{\text{sech}^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right )}{a^2 \sqrt{1-a^2}}+\frac{3 b^2 \text{PolyLog}\left (3,\frac{a e^{\text{sech}^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right )}{a^2 \left (1-a^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[ArcSech[a + b*x]^3/x^3,x]

[Out]

(-3*b^2*ArcSech[a + b*x]^2)/(2*a^2*(1 - a^2)) + (3*b^2*Sqrt[(1 - a - b*x)/(1 + a + b*x)]*(1 + a + b*x)*ArcSech
[a + b*x]^2)/(2*a*(1 - a^2)*(a + b*x)*(1 - a/(a + b*x))) + (b^2*ArcSech[a + b*x]^3)/(2*a^2) - ArcSech[a + b*x]
^3/(2*x^2) + (3*b^2*ArcSech[a + b*x]*Log[1 - (a*E^ArcSech[a + b*x])/(1 - Sqrt[1 - a^2])])/(a^2*(1 - a^2)) + (3
*b^2*ArcSech[a + b*x]^2*Log[1 - (a*E^ArcSech[a + b*x])/(1 - Sqrt[1 - a^2])])/(2*a^2*(1 - a^2)^(3/2)) - (3*b^2*
ArcSech[a + b*x]^2*Log[1 - (a*E^ArcSech[a + b*x])/(1 - Sqrt[1 - a^2])])/(a^2*Sqrt[1 - a^2]) + (3*b^2*ArcSech[a
+ b*x]*Log[1 - (a*E^ArcSech[a + b*x])/(1 + Sqrt[1 - a^2])])/(a^2*(1 - a^2)) - (3*b^2*ArcSech[a + b*x]^2*Log[1
- (a*E^ArcSech[a + b*x])/(1 + Sqrt[1 - a^2])])/(2*a^2*(1 - a^2)^(3/2)) + (3*b^2*ArcSech[a + b*x]^2*Log[1 - (a
*E^ArcSech[a + b*x])/(1 + Sqrt[1 - a^2])])/(a^2*Sqrt[1 - a^2]) + (3*b^2*PolyLog[2, (a*E^ArcSech[a + b*x])/(1 -
Sqrt[1 - a^2])])/(a^2*(1 - a^2)) + (3*b^2*ArcSech[a + b*x]*PolyLog[2, (a*E^ArcSech[a + b*x])/(1 - Sqrt[1 - a^
2])])/(a^2*(1 - a^2)^(3/2)) - (6*b^2*ArcSech[a + b*x]*PolyLog[2, (a*E^ArcSech[a + b*x])/(1 - Sqrt[1 - a^2])])/
(a^2*Sqrt[1 - a^2]) + (3*b^2*PolyLog[2, (a*E^ArcSech[a + b*x])/(1 + Sqrt[1 - a^2])])/(a^2*(1 - a^2)) - (3*b^2*
ArcSech[a + b*x]*PolyLog[2, (a*E^ArcSech[a + b*x])/(1 + Sqrt[1 - a^2])])/(a^2*(1 - a^2)^(3/2)) + (6*b^2*ArcSec
h[a + b*x]*PolyLog[2, (a*E^ArcSech[a + b*x])/(1 + Sqrt[1 - a^2])])/(a^2*Sqrt[1 - a^2]) - (3*b^2*PolyLog[3, (a*
E^ArcSech[a + b*x])/(1 - Sqrt[1 - a^2])])/(a^2*(1 - a^2)^(3/2)) + (6*b^2*PolyLog[3, (a*E^ArcSech[a + b*x])/(1
- Sqrt[1 - a^2])])/(a^2*Sqrt[1 - a^2]) + (3*b^2*PolyLog[3, (a*E^ArcSech[a + b*x])/(1 + Sqrt[1 - a^2])])/(a^2*(
1 - a^2)^(3/2)) - (6*b^2*PolyLog[3, (a*E^ArcSech[a + b*x])/(1 + Sqrt[1 - a^2])])/(a^2*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 3324

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^2, x_Symbol] :> Simp[(b*(c + d*x)^m*Cos[
e + f*x])/(f*(a^2 - b^2)*(a + b*Sin[e + f*x])), x] + (Dist[a/(a^2 - b^2), Int[(c + d*x)^m/(a + b*Sin[e + f*x])
, x], x] - Dist[(b*d*m)/(f*(a^2 - b^2)), Int[((c + d*x)^(m - 1)*Cos[e + f*x])/(a + b*Sin[e + f*x]), x], x]) /;
FreeQ[{a, b, c, d, e, f}, x] && NeQ[a^2 - b^2, 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 2531

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> -Simp[((
f + g*x)^m*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)])/(b*c*n*Log[F]), x] + Dist[(g*m)/(b*c*n*Log[F]), Int[(f + g*x)
^(m - 1)*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 6589

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rule 5562

Int[(((e_.) + (f_.)*(x_))^(m_.)*Sinh[(c_.) + (d_.)*(x_)])/(Cosh[(c_.) + (d_.)*(x_)]*(b_.) + (a_)), x_Symbol] :
> -Simp[(e + f*x)^(m + 1)/(b*f*(m + 1)), x] + (Int[((e + f*x)^m*E^(c + d*x))/(a - Rt[a^2 - b^2, 2] + b*E^(c +
d*x)), x] + Int[((e + f*x)^m*E^(c + d*x))/(a + Rt[a^2 - b^2, 2] + b*E^(c + d*x)), x]) /; FreeQ[{a, b, c, d, e,
f}, x] && IGtQ[m, 0] && NeQ[a^2 - b^2, 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)^3}{x^3} \, dx &=-\left (b^2 \operatorname{Subst}\left (\int \frac{x^3 \text{sech}(x) \tanh (x)}{(-a+\text{sech}(x))^3} \, dx,x,\text{sech}^{-1}(a+b x)\right )\right )\\ &=-\frac{\text{sech}^{-1}(a+b x)^3}{2 x^2}+\frac{1}{2} \left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{x^2}{(-a+\text{sech}(x))^2} \, dx,x,\text{sech}^{-1}(a+b x)\right )\\ &=-\frac{\text{sech}^{-1}(a+b x)^3}{2 x^2}+\frac{1}{2} \left (3 b^2\right ) \operatorname{Subst}\left (\int \left (\frac{x^2}{a^2}+\frac{x^2}{a^2 (-1+a \cosh (x))^2}+\frac{2 x^2}{a^2 (-1+a \cosh (x))}\right ) \, dx,x,\text{sech}^{-1}(a+b x)\right )\\ &=\frac{b^2 \text{sech}^{-1}(a+b x)^3}{2 a^2}-\frac{\text{sech}^{-1}(a+b x)^3}{2 x^2}+\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{x^2}{(-1+a \cosh (x))^2} \, dx,x,\text{sech}^{-1}(a+b x)\right )}{2 a^2}+\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{x^2}{-1+a \cosh (x)} \, dx,x,\text{sech}^{-1}(a+b x)\right )}{a^2}\\ &=\frac{3 b^2 \sqrt{\frac{1-a-b x}{1+a+b x}} (1+a+b x) \text{sech}^{-1}(a+b x)^2}{2 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)^3}{2 a^2}-\frac{\text{sech}^{-1}(a+b x)^3}{2 x^2}+\frac{\left (6 b^2\right ) \operatorname{Subst}\left (\int \frac{e^x x^2}{a-2 e^x+a e^{2 x}} \, dx,x,\text{sech}^{-1}(a+b x)\right )}{a^2}-\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{x^2}{-1+a \cosh (x)} \, dx,x,\text{sech}^{-1}(a+b x)\right )}{2 a^2 \left (1-a^2\right )}+\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{x \sinh (x)}{-1+a \cosh (x)} \, dx,x,\text{sech}^{-1}(a+b x)\right )}{a \left (1-a^2\right )}\\ &=-\frac{3 b^2 \text{sech}^{-1}(a+b x)^2}{2 a^2 \left (1-a^2\right )}+\frac{3 b^2 \sqrt{\frac{1-a-b x}{1+a+b x}} (1+a+b x) \text{sech}^{-1}(a+b x)^2}{2 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)^3}{2 a^2}-\frac{\text{sech}^{-1}(a+b x)^3}{2 x^2}-\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{e^x x^2}{a-2 e^x+a e^{2 x}} \, dx,x,\text{sech}^{-1}(a+b x)\right )}{a^2 \left (1-a^2\right )}+\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{e^x x}{-1-\sqrt{1-a^2}+a e^x} \, dx,x,\text{sech}^{-1}(a+b x)\right )}{a \left (1-a^2\right )}+\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{e^x x}{-1+\sqrt{1-a^2}+a e^x} \, dx,x,\text{sech}^{-1}(a+b x)\right )}{a \left (1-a^2\right )}+\frac{\left (6 b^2\right ) \operatorname{Subst}\left (\int \frac{e^x x^2}{-2-2 \sqrt{1-a^2}+2 a e^x} \, dx,x,\text{sech}^{-1}(a+b x)\right )}{a \sqrt{1-a^2}}-\frac{\left (6 b^2\right ) \operatorname{Subst}\left (\int \frac{e^x x^2}{-2+2 \sqrt{1-a^2}+2 a e^x} \, dx,x,\text{sech}^{-1}(a+b x)\right )}{a \sqrt{1-a^2}}\\ &=-\frac{3 b^2 \text{sech}^{-1}(a+b x)^2}{2 a^2 \left (1-a^2\right )}+\frac{3 b^2 \sqrt{\frac{1-a-b x}{1+a+b x}} (1+a+b x) \text{sech}^{-1}(a+b x)^2}{2 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)^3}{2 a^2}-\frac{\text{sech}^{-1}(a+b x)^3}{2 x^2}+\frac{3 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 )}-\frac{3 b^2 \text{sech}^{-1}(a+b x)^2 \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{3 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 )}+\frac{3 b^2 \text{sech}^{-1}(a+b x)^2 \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{\left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{e^x x^2}{-2-2 \sqrt{1-a^2}+2 a e^x} \, dx,x,\text{sech}^{-1}(a+b x)\right )}{a \left (1-a^2\right )^{3/2}}+\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{e^x x^2}{-2+2 \sqrt{1-a^2}+2 a e^x} \, dx,x,\text{sech}^{-1}(a+b x)\right )}{a \left (1-a^2\right )^{3/2}}-\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int \log \left (1+\frac{a e^x}{-1-\sqrt{1-a^2}}\right ) \, dx,x,\text{sech}^{-1}(a+b x)\right )}{a^2 \left (1-a^2\right )}-\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int \log \left (1+\frac{a e^x}{-1+\sqrt{1-a^2}}\right ) \, dx,x,\text{sech}^{-1}(a+b x)\right )}{a^2 \left (1-a^2\right )}-\frac{\left (6 b^2\right ) \operatorname{Subst}\left (\int x \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^2 \sqrt{1-a^2}}+\frac{\left (6 b^2\right ) \operatorname{Subst}\left (\int x \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^2 \sqrt{1-a^2}}\\ &=-\frac{3 b^2 \text{sech}^{-1}(a+b x)^2}{2 a^2 \left (1-a^2\right )}+\frac{3 b^2 \sqrt{\frac{1-a-b x}{1+a+b x}} (1+a+b x) \text{sech}^{-1}(a+b x)^2}{2 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)^3}{2 a^2}-\frac{\text{sech}^{-1}(a+b x)^3}{2 x^2}+\frac{3 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 )}+\frac{3 b^2 \text{sech}^{-1}(a+b x)^2 \log \left (1-\frac{a e^{\text{sech}^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )}{2 a^2 \left (1-a^2\right )^{3/2}}-\frac{3 b^2 \text{sech}^{-1}(a+b x)^2 \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{3 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 )}-\frac{3 b^2 \text{sech}^{-1}(a+b x)^2 \log \left (1-\frac{a e^{\text{sech}^{-1}(a+b x)}}{1+\sqrt{1-a^2}}\right )}{2 a^2 \left (1-a^2\right )^{3/2}}+\frac{3 b^2 \text{sech}^{-1}(a+b x)^2 \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{6 b^2 \text{sech}^{-1}(a+b x) \text{Li}_2\left (\frac{a e^{\text{sech}^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )}{a^2 \sqrt{1-a^2}}+\frac{6 b^2 \text{sech}^{-1}(a+b x) \text{Li}_2\left (\frac{a e^{\text{sech}^{-1}(a+b x)}}{1+\sqrt{1-a^2}}\right )}{a^2 \sqrt{1-a^2}}+\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int x \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^2 \left (1-a^2\right )^{3/2}}-\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int x \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^2 \left (1-a^2\right )^{3/2}}-\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{a x}{-1-\sqrt{1-a^2}}\right )}{x} \, dx,x,e^{\text{sech}^{-1}(a+b x)}\right )}{a^2 \left (1-a^2\right )}-\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{a x}{-1+\sqrt{1-a^2}}\right )}{x} \, dx,x,e^{\text{sech}^{-1}(a+b x)}\right )}{a^2 \left (1-a^2\right )}-\frac{\left (6 b^2\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-\frac{2 a e^x}{-2-2 \sqrt{1-a^2}}\right ) \, dx,x,\text{sech}^{-1}(a+b x)\right )}{a^2 \sqrt{1-a^2}}+\frac{\left (6 b^2\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-\frac{2 a e^x}{-2+2 \sqrt{1-a^2}}\right ) \, dx,x,\text{sech}^{-1}(a+b x)\right )}{a^2 \sqrt{1-a^2}}\\ &=-\frac{3 b^2 \text{sech}^{-1}(a+b x)^2}{2 a^2 \left (1-a^2\right )}+\frac{3 b^2 \sqrt{\frac{1-a-b x}{1+a+b x}} (1+a+b x) \text{sech}^{-1}(a+b x)^2}{2 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)^3}{2 a^2}-\frac{\text{sech}^{-1}(a+b x)^3}{2 x^2}+\frac{3 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 )}+\frac{3 b^2 \text{sech}^{-1}(a+b x)^2 \log \left (1-\frac{a e^{\text{sech}^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )}{2 a^2 \left (1-a^2\right )^{3/2}}-\frac{3 b^2 \text{sech}^{-1}(a+b x)^2 \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{3 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 )}-\frac{3 b^2 \text{sech}^{-1}(a+b x)^2 \log \left (1-\frac{a e^{\text{sech}^{-1}(a+b x)}}{1+\sqrt{1-a^2}}\right )}{2 a^2 \left (1-a^2\right )^{3/2}}+\frac{3 b^2 \text{sech}^{-1}(a+b x)^2 \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{3 b^2 \text{Li}_2\left (\frac{a e^{\text{sech}^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )}{a^2 \left (1-a^2\right )}+\frac{3 b^2 \text{sech}^{-1}(a+b x) \text{Li}_2\left (\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{6 b^2 \text{sech}^{-1}(a+b x) \text{Li}_2\left (\frac{a e^{\text{sech}^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )}{a^2 \sqrt{1-a^2}}+\frac{3 b^2 \text{Li}_2\left (\frac{a e^{\text{sech}^{-1}(a+b x)}}{1+\sqrt{1-a^2}}\right )}{a^2 \left (1-a^2\right )}-\frac{3 b^2 \text{sech}^{-1}(a+b x) \text{Li}_2\left (\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{6 b^2 \text{sech}^{-1}(a+b x) \text{Li}_2\left (\frac{a e^{\text{sech}^{-1}(a+b x)}}{1+\sqrt{1-a^2}}\right )}{a^2 \sqrt{1-a^2}}+\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-\frac{2 a e^x}{-2-2 \sqrt{1-a^2}}\right ) \, dx,x,\text{sech}^{-1}(a+b x)\right )}{a^2 \left (1-a^2\right )^{3/2}}-\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-\frac{2 a e^x}{-2+2 \sqrt{1-a^2}}\right ) \, dx,x,\text{sech}^{-1}(a+b x)\right )}{a^2 \left (1-a^2\right )^{3/2}}+\frac{\left (6 b^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{a x}{1-\sqrt{1-a^2}}\right )}{x} \, dx,x,e^{\text{sech}^{-1}(a+b x)}\right )}{a^2 \sqrt{1-a^2}}-\frac{\left (6 b^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{a x}{1+\sqrt{1-a^2}}\right )}{x} \, dx,x,e^{\text{sech}^{-1}(a+b x)}\right )}{a^2 \sqrt{1-a^2}}\\ &=-\frac{3 b^2 \text{sech}^{-1}(a+b x)^2}{2 a^2 \left (1-a^2\right )}+\frac{3 b^2 \sqrt{\frac{1-a-b x}{1+a+b x}} (1+a+b x) \text{sech}^{-1}(a+b x)^2}{2 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)^3}{2 a^2}-\frac{\text{sech}^{-1}(a+b x)^3}{2 x^2}+\frac{3 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 )}+\frac{3 b^2 \text{sech}^{-1}(a+b x)^2 \log \left (1-\frac{a e^{\text{sech}^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )}{2 a^2 \left (1-a^2\right )^{3/2}}-\frac{3 b^2 \text{sech}^{-1}(a+b x)^2 \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{3 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 )}-\frac{3 b^2 \text{sech}^{-1}(a+b x)^2 \log \left (1-\frac{a e^{\text{sech}^{-1}(a+b x)}}{1+\sqrt{1-a^2}}\right )}{2 a^2 \left (1-a^2\right )^{3/2}}+\frac{3 b^2 \text{sech}^{-1}(a+b x)^2 \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{3 b^2 \text{Li}_2\left (\frac{a e^{\text{sech}^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )}{a^2 \left (1-a^2\right )}+\frac{3 b^2 \text{sech}^{-1}(a+b x) \text{Li}_2\left (\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{6 b^2 \text{sech}^{-1}(a+b x) \text{Li}_2\left (\frac{a e^{\text{sech}^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )}{a^2 \sqrt{1-a^2}}+\frac{3 b^2 \text{Li}_2\left (\frac{a e^{\text{sech}^{-1}(a+b x)}}{1+\sqrt{1-a^2}}\right )}{a^2 \left (1-a^2\right )}-\frac{3 b^2 \text{sech}^{-1}(a+b x) \text{Li}_2\left (\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{6 b^2 \text{sech}^{-1}(a+b x) \text{Li}_2\left (\frac{a e^{\text{sech}^{-1}(a+b x)}}{1+\sqrt{1-a^2}}\right )}{a^2 \sqrt{1-a^2}}+\frac{6 b^2 \text{Li}_3\left (\frac{a e^{\text{sech}^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )}{a^2 \sqrt{1-a^2}}-\frac{6 b^2 \text{Li}_3\left (\frac{a e^{\text{sech}^{-1}(a+b x)}}{1+\sqrt{1-a^2}}\right )}{a^2 \sqrt{1-a^2}}-\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{a x}{1-\sqrt{1-a^2}}\right )}{x} \, dx,x,e^{\text{sech}^{-1}(a+b x)}\right )}{a^2 \left (1-a^2\right )^{3/2}}+\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{a x}{1+\sqrt{1-a^2}}\right )}{x} \, dx,x,e^{\text{sech}^{-1}(a+b x)}\right )}{a^2 \left (1-a^2\right )^{3/2}}\\ &=-\frac{3 b^2 \text{sech}^{-1}(a+b x)^2}{2 a^2 \left (1-a^2\right )}+\frac{3 b^2 \sqrt{\frac{1-a-b x}{1+a+b x}} (1+a+b x) \text{sech}^{-1}(a+b x)^2}{2 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)^3}{2 a^2}-\frac{\text{sech}^{-1}(a+b x)^3}{2 x^2}+\frac{3 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 )}+\frac{3 b^2 \text{sech}^{-1}(a+b x)^2 \log \left (1-\frac{a e^{\text{sech}^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )}{2 a^2 \left (1-a^2\right )^{3/2}}-\frac{3 b^2 \text{sech}^{-1}(a+b x)^2 \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{3 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 )}-\frac{3 b^2 \text{sech}^{-1}(a+b x)^2 \log \left (1-\frac{a e^{\text{sech}^{-1}(a+b x)}}{1+\sqrt{1-a^2}}\right )}{2 a^2 \left (1-a^2\right )^{3/2}}+\frac{3 b^2 \text{sech}^{-1}(a+b x)^2 \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{3 b^2 \text{Li}_2\left (\frac{a e^{\text{sech}^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )}{a^2 \left (1-a^2\right )}+\frac{3 b^2 \text{sech}^{-1}(a+b x) \text{Li}_2\left (\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{6 b^2 \text{sech}^{-1}(a+b x) \text{Li}_2\left (\frac{a e^{\text{sech}^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )}{a^2 \sqrt{1-a^2}}+\frac{3 b^2 \text{Li}_2\left (\frac{a e^{\text{sech}^{-1}(a+b x)}}{1+\sqrt{1-a^2}}\right )}{a^2 \left (1-a^2\right )}-\frac{3 b^2 \text{sech}^{-1}(a+b x) \text{Li}_2\left (\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{6 b^2 \text{sech}^{-1}(a+b x) \text{Li}_2\left (\frac{a e^{\text{sech}^{-1}(a+b x)}}{1+\sqrt{1-a^2}}\right )}{a^2 \sqrt{1-a^2}}-\frac{3 b^2 \text{Li}_3\left (\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{6 b^2 \text{Li}_3\left (\frac{a e^{\text{sech}^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )}{a^2 \sqrt{1-a^2}}+\frac{3 b^2 \text{Li}_3\left (\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{6 b^2 \text{Li}_3\left (\frac{a e^{\text{sech}^{-1}(a+b x)}}{1+\sqrt{1-a^2}}\right )}{a^2 \sqrt{1-a^2}}\\ \end{align*}

Mathematica [F]  time = 9.65002, size = 0, normalized size = 0. \[ \int \frac{\text{sech}^{-1}(a+b x)^3}{x^3} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[ArcSech[a + b*x]^3/x^3,x]

[Out]

Integrate[ArcSech[a + b*x]^3/x^3, x]

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Maple [F]  time = 0.694, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ({\rm arcsech} \left (bx+a\right ) \right ) ^{3}}{{x}^{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arcsech(b*x+a)^3/x^3,x)

[Out]

int(arcsech(b*x+a)^3/x^3,x)

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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 )^{3}}{2 \, x^{2}} - \int \frac{16 \,{\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 )^{3} + 16 \,{\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 )^{3} - 3 \,{\left (b^{3} x^{3} + 2 \, a b^{2} x^{2} +{\left (a^{2} b - b\right )} x - 4 \,{\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 (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} \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 - 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 )\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 )^{2} - 24 \,{\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} \sqrt{-b x - a + 1} \log \left (b x + a\right )^{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}\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 )}{2 \,{\left (b^{3} x^{6} + 3 \, a b^{2} x^{5} +{\left (3 \, a^{2} b - b\right )} x^{4} +{\left (a^{3} - a\right )} x^{3} +{\left (b^{3} x^{6} + 3 \, a b^{2} x^{5} +{\left (3 \, a^{2} b - b\right )} x^{4} +{\left (a^{3} - a\right )} x^{3}\right )} \sqrt{b x + a + 1} \sqrt{-b x - a + 1}\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsech(b*x+a)^3/x^3,x, algorithm="maxima")

[Out]

-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)^3/x^2 -
integrate(1/2*(16*(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)^3 + 16*(b^3*x^3 + 3*a*b^2*x^2 + a^3 + (3*a^2*b - b)*x - a)*log(b*x + a)^3 - 3*(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))*lo
g(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 - 24*((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 + (b^3*x^3 +
3*a*b^2*x^2 + a^3 + (3*a^2*b - b)*x - a)*log(b*x + 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))/(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)

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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 )^{3}}{x^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsech(b*x+a)^3/x^3,x, algorithm="fricas")

[Out]

integral(arcsech(b*x + a)^3/x^3, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{asech}^{3}{\left (a + b x \right )}}{x^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(asech(b*x+a)**3/x**3,x)

[Out]

Integral(asech(a + b*x)**3/x**3, 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 )^{3}}{x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsech(b*x+a)^3/x^3,x, algorithm="giac")

[Out]

integrate(arcsech(b*x + a)^3/x^3, x)