\(\int \frac {\text {sech}^{-1}(a+b x)^3}{x^3} \, dx\) [19]

Optimal result

Integrand size = 12, antiderivative size = 965 \[ \int \frac {\text {sech}^{-1}(a+b x)^3}{x^3} \, dx=-\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 \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 )}+\frac {3 b^2 \text {sech}^{-1}(a+b x) \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 {6 b^2 \text {sech}^{-1}(a+b x) \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 {3 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 )}-\frac {3 b^2 \text {sech}^{-1}(a+b x) \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 {6 b^2 \text {sech}^{-1}(a+b x) \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 {3 b^2 \operatorname {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 \operatorname {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 \operatorname {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 \operatorname {PolyLog}\left (3,\frac {a e^{\text {sech}^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )}{a^2 \sqrt {1-a^2}} \]

[Out]

-3/2*b^2*arcsech(b*x+a)^2/a^2/(-a^2+1)+1/2*b^2*arcsech(b*x+a)^3/a^2-1/2*arcsech(b*x+a)^3/x^2+3*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*b^2*arcsec
h(b*x+a)^2*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)+3
*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*b^2*arcsech(b*x+a)^2*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)+3*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*b^2*arcsech(b*x+a)*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)+3*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*b^2*arcsech(b*x+a)*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)-3*b^2*polylog(3,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)+3*b^2*polylog(3,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)-3*b^2*arcsech(b*x+a)^2*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)+3*b^2*arcsech(b*x+a)^2*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)-6*b^2*arcsech(b*x+a)*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)+6*b^2*arcsech(b*x+a)*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)+6*b^2*polylog(3,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)-6*b^2*polylog(3,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)+3/2*b^2*(b*x+a+1)*arcsech(b*x+a)^2*
((-b*x-a+1)/(b*x+a+1))^(1/2)/a/(-a^2+1)/(b*x+a)/(1-a/(b*x+a))

Rubi [A] (verified)

Time = 1.05 (sec) , antiderivative size = 965, normalized size of antiderivative = 1.00, number of steps used = 32, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.083, Rules used = {6456, 5576, 4276, 3405, 3401, 2296, 2221, 2611, 2320, 6724, 5681, 2317, 2438} \[ \int \frac {\text {sech}^{-1}(a+b x)^3}{x^3} \, dx=\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 \operatorname {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 \operatorname {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 \operatorname {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 \operatorname {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 \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 )}+\frac {3 b^2 \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 )}+\frac {6 b^2 \operatorname {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 \operatorname {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 \operatorname {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 \operatorname {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}} \]

[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 2221

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/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], 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 2296

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 2317

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 2320

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 2438

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]

Rule 2611

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 3401

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)/(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))))/E^(I*Pi*(k - 1/2)), x], x] /; FreeQ[{a, b, c, d, e, f, fz}, x] && IntegerQ[
2*k] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 3405

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 4276

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 5576

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 5681

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 6456

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 6724

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]

Rubi steps \begin{align*} \text {integral}& = -\left (b^2 \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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) \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 {6 b^2 \text {sech}^{-1}(a+b x) \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 {\left (3 b^2\right ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-\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 ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-\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 \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 )}+\frac {3 b^2 \text {sech}^{-1}(a+b x) \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 {6 b^2 \text {sech}^{-1}(a+b x) \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 {3 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 )}-\frac {3 b^2 \text {sech}^{-1}(a+b x) \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 {6 b^2 \text {sech}^{-1}(a+b x) \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 {\left (3 b^2\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-\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 ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-\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 ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,\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 ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,\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 \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 )}+\frac {3 b^2 \text {sech}^{-1}(a+b x) \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 {6 b^2 \text {sech}^{-1}(a+b x) \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 {3 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 )}-\frac {3 b^2 \text {sech}^{-1}(a+b x) \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 {6 b^2 \text {sech}^{-1}(a+b x) \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 {6 b^2 \operatorname {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 {6 b^2 \operatorname {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 {\left (3 b^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,\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 ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,\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 \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 )}+\frac {3 b^2 \text {sech}^{-1}(a+b x) \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 {6 b^2 \text {sech}^{-1}(a+b x) \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 {3 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 )}-\frac {3 b^2 \text {sech}^{-1}(a+b x) \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 {6 b^2 \text {sech}^{-1}(a+b x) \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 {3 b^2 \operatorname {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 \operatorname {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 \operatorname {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 \operatorname {PolyLog}\left (3,\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]

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

[In]

Integrate[ArcSech[a + b*x]^3/x^3,x]

[Out]

Integrate[ArcSech[a + b*x]^3/x^3, x]

Maple [F]

\[\int \frac {\operatorname {arcsech}\left (b x +a \right )^{3}}{x^{3}}d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int \frac {\text {sech}^{-1}(a+b x)^3}{x^3} \, dx=\int { \frac {\operatorname {arsech}\left (b x + a\right )^{3}}{x^{3}} \,d x } \]

[In]

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

[Out]

integral(arcsech(b*x + a)^3/x^3, x)

Sympy [F]

\[ \int \frac {\text {sech}^{-1}(a+b x)^3}{x^3} \, dx=\int \frac {\operatorname {asech}^{3}{\left (a + b x \right )}}{x^{3}}\, dx \]

[In]

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

[Out]

Integral(asech(a + b*x)**3/x**3, x)

Maxima [F]

\[ \int \frac {\text {sech}^{-1}(a+b x)^3}{x^3} \, dx=\int { \frac {\operatorname {arsech}\left (b x + a\right )^{3}}{x^{3}} \,d x } \]

[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)

Giac [F]

\[ \int \frac {\text {sech}^{-1}(a+b x)^3}{x^3} \, dx=\int { \frac {\operatorname {arsech}\left (b x + a\right )^{3}}{x^{3}} \,d x } \]

[In]

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

[Out]

integrate(arcsech(b*x + a)^3/x^3, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\text {sech}^{-1}(a+b x)^3}{x^3} \, dx=\int \frac {{\mathrm {acosh}\left (\frac {1}{a+b\,x}\right )}^3}{x^3} \,d x \]

[In]

int(acosh(1/(a + b*x))^3/x^3,x)

[Out]

int(acosh(1/(a + b*x))^3/x^3, x)