\(\int \frac {(a+b \text {sech}^{-1}(c x))^3}{x} \, dx\) [46]

Optimal result

Integrand size = 14, antiderivative size = 114 \[ \int \frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{x} \, dx=\frac {\left (a+b \text {sech}^{-1}(c x)\right )^4}{4 b}-\left (a+b \text {sech}^{-1}(c x)\right )^3 \log \left (1+e^{2 \text {sech}^{-1}(c x)}\right )-\frac {3}{2} b \left (a+b \text {sech}^{-1}(c x)\right )^2 \operatorname {PolyLog}\left (2,-e^{2 \text {sech}^{-1}(c x)}\right )+\frac {3}{2} b^2 \left (a+b \text {sech}^{-1}(c x)\right ) \operatorname {PolyLog}\left (3,-e^{2 \text {sech}^{-1}(c x)}\right )-\frac {3}{4} b^3 \operatorname {PolyLog}\left (4,-e^{2 \text {sech}^{-1}(c x)}\right ) \] Output:

1/4*(a+b*arcsech(c*x))^4/b-(a+b*arcsech(c*x))^3*ln(1+(1/c/x+(-1+1/c/x)^(1/ 
2)*(1+1/c/x)^(1/2))^2)-3/2*b*(a+b*arcsech(c*x))^2*polylog(2,-(1/c/x+(-1+1/ 
c/x)^(1/2)*(1+1/c/x)^(1/2))^2)+3/2*b^2*(a+b*arcsech(c*x))*polylog(3,-(1/c/ 
x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))^2)-3/4*b^3*polylog(4,-(1/c/x+(-1+1/c/x 
)^(1/2)*(1+1/c/x)^(1/2))^2)
 

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.60 \[ \int \frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{x} \, dx=\frac {1}{4} \left (-6 a^2 b \text {sech}^{-1}(c x)^2-4 a b^2 \text {sech}^{-1}(c x)^3-b^3 \text {sech}^{-1}(c x)^4-12 a^2 b \text {sech}^{-1}(c x) \log \left (1+e^{-2 \text {sech}^{-1}(c x)}\right )-12 a b^2 \text {sech}^{-1}(c x)^2 \log \left (1+e^{-2 \text {sech}^{-1}(c x)}\right )-4 b^3 \text {sech}^{-1}(c x)^3 \log \left (1+e^{-2 \text {sech}^{-1}(c x)}\right )+4 a^3 \log (c x)+6 b \left (a+b \text {sech}^{-1}(c x)\right )^2 \operatorname {PolyLog}\left (2,-e^{-2 \text {sech}^{-1}(c x)}\right )+6 b^2 \left (a+b \text {sech}^{-1}(c x)\right ) \operatorname {PolyLog}\left (3,-e^{-2 \text {sech}^{-1}(c x)}\right )+3 b^3 \operatorname {PolyLog}\left (4,-e^{-2 \text {sech}^{-1}(c x)}\right )\right ) \] Input:

Integrate[(a + b*ArcSech[c*x])^3/x,x]
 

Output:

(-6*a^2*b*ArcSech[c*x]^2 - 4*a*b^2*ArcSech[c*x]^3 - b^3*ArcSech[c*x]^4 - 1 
2*a^2*b*ArcSech[c*x]*Log[1 + E^(-2*ArcSech[c*x])] - 12*a*b^2*ArcSech[c*x]^ 
2*Log[1 + E^(-2*ArcSech[c*x])] - 4*b^3*ArcSech[c*x]^3*Log[1 + E^(-2*ArcSec 
h[c*x])] + 4*a^3*Log[c*x] + 6*b*(a + b*ArcSech[c*x])^2*PolyLog[2, -E^(-2*A 
rcSech[c*x])] + 6*b^2*(a + b*ArcSech[c*x])*PolyLog[3, -E^(-2*ArcSech[c*x]) 
] + 3*b^3*PolyLog[4, -E^(-2*ArcSech[c*x])])/4
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.69 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.14, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {6839, 3042, 26, 4201, 2620, 3011, 7163, 2720, 7143}

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.

\(\displaystyle \int \frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{x} \, dx\)

\(\Big \downarrow \) 6839

\(\displaystyle -\int \sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )^3d\text {sech}^{-1}(c x)\)

\(\Big \downarrow \) 3042

\(\displaystyle -\int -i \left (a+b \text {sech}^{-1}(c x)\right )^3 \tan \left (i \text {sech}^{-1}(c x)\right )d\text {sech}^{-1}(c x)\)

\(\Big \downarrow \) 26

\(\displaystyle i \int \left (a+b \text {sech}^{-1}(c x)\right )^3 \tan \left (i \text {sech}^{-1}(c x)\right )d\text {sech}^{-1}(c x)\)

\(\Big \downarrow \) 4201

\(\displaystyle i \left (2 i \int \frac {e^{2 \text {sech}^{-1}(c x)} \left (a+b \text {sech}^{-1}(c x)\right )^3}{1+e^{2 \text {sech}^{-1}(c x)}}d\text {sech}^{-1}(c x)-\frac {i \left (a+b \text {sech}^{-1}(c x)\right )^4}{4 b}\right )\)

\(\Big \downarrow \) 2620

\(\displaystyle i \left (2 i \left (\frac {1}{2} \log \left (e^{2 \text {sech}^{-1}(c x)}+1\right ) \left (a+b \text {sech}^{-1}(c x)\right )^3-\frac {3}{2} b \int \left (a+b \text {sech}^{-1}(c x)\right )^2 \log \left (1+e^{2 \text {sech}^{-1}(c x)}\right )d\text {sech}^{-1}(c x)\right )-\frac {i \left (a+b \text {sech}^{-1}(c x)\right )^4}{4 b}\right )\)

\(\Big \downarrow \) 3011

\(\displaystyle i \left (2 i \left (\frac {1}{2} \log \left (e^{2 \text {sech}^{-1}(c x)}+1\right ) \left (a+b \text {sech}^{-1}(c x)\right )^3-\frac {3}{2} b \left (b \int \left (a+b \text {sech}^{-1}(c x)\right ) \operatorname {PolyLog}\left (2,-e^{2 \text {sech}^{-1}(c x)}\right )d\text {sech}^{-1}(c x)-\frac {1}{2} \operatorname {PolyLog}\left (2,-e^{2 \text {sech}^{-1}(c x)}\right ) \left (a+b \text {sech}^{-1}(c x)\right )^2\right )\right )-\frac {i \left (a+b \text {sech}^{-1}(c x)\right )^4}{4 b}\right )\)

\(\Big \downarrow \) 7163

\(\displaystyle i \left (2 i \left (\frac {1}{2} \log \left (e^{2 \text {sech}^{-1}(c x)}+1\right ) \left (a+b \text {sech}^{-1}(c x)\right )^3-\frac {3}{2} b \left (b \left (\frac {1}{2} \operatorname {PolyLog}\left (3,-e^{2 \text {sech}^{-1}(c x)}\right ) \left (a+b \text {sech}^{-1}(c x)\right )-\frac {1}{2} b \int \operatorname {PolyLog}\left (3,-e^{2 \text {sech}^{-1}(c x)}\right )d\text {sech}^{-1}(c x)\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,-e^{2 \text {sech}^{-1}(c x)}\right ) \left (a+b \text {sech}^{-1}(c x)\right )^2\right )\right )-\frac {i \left (a+b \text {sech}^{-1}(c x)\right )^4}{4 b}\right )\)

\(\Big \downarrow \) 2720

\(\displaystyle i \left (2 i \left (\frac {1}{2} \log \left (e^{2 \text {sech}^{-1}(c x)}+1\right ) \left (a+b \text {sech}^{-1}(c x)\right )^3-\frac {3}{2} b \left (b \left (\frac {1}{2} \operatorname {PolyLog}\left (3,-e^{2 \text {sech}^{-1}(c x)}\right ) \left (a+b \text {sech}^{-1}(c x)\right )-\frac {1}{4} b \int e^{-2 \text {sech}^{-1}(c x)} \operatorname {PolyLog}\left (3,-e^{2 \text {sech}^{-1}(c x)}\right )de^{2 \text {sech}^{-1}(c x)}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,-e^{2 \text {sech}^{-1}(c x)}\right ) \left (a+b \text {sech}^{-1}(c x)\right )^2\right )\right )-\frac {i \left (a+b \text {sech}^{-1}(c x)\right )^4}{4 b}\right )\)

\(\Big \downarrow \) 7143

\(\displaystyle i \left (2 i \left (\frac {1}{2} \log \left (e^{2 \text {sech}^{-1}(c x)}+1\right ) \left (a+b \text {sech}^{-1}(c x)\right )^3-\frac {3}{2} b \left (b \left (\frac {1}{2} \operatorname {PolyLog}\left (3,-e^{2 \text {sech}^{-1}(c x)}\right ) \left (a+b \text {sech}^{-1}(c x)\right )-\frac {1}{4} b \operatorname {PolyLog}\left (4,-e^{2 \text {sech}^{-1}(c x)}\right )\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,-e^{2 \text {sech}^{-1}(c x)}\right ) \left (a+b \text {sech}^{-1}(c x)\right )^2\right )\right )-\frac {i \left (a+b \text {sech}^{-1}(c x)\right )^4}{4 b}\right )\)

Input:

Int[(a + b*ArcSech[c*x])^3/x,x]
 

Output:

I*(((-1/4*I)*(a + b*ArcSech[c*x])^4)/b + (2*I)*(((a + b*ArcSech[c*x])^3*Lo 
g[1 + E^(2*ArcSech[c*x])])/2 - (3*b*(-1/2*((a + b*ArcSech[c*x])^2*PolyLog[ 
2, -E^(2*ArcSech[c*x])]) + b*(((a + b*ArcSech[c*x])*PolyLog[3, -E^(2*ArcSe 
ch[c*x])])/2 - (b*PolyLog[4, -E^(2*ArcSech[c*x])])/4)))/2))
 

Defintions of rubi rules used

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 2620

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] - Si 
mp[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 2720

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] 
   Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct 
ionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ 
[{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 3011

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] + Simp[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 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4201

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (Complex[0, fz_])*(f_.)*(x_)], x 
_Symbol] :> Simp[(-I)*((c + d*x)^(m + 1)/(d*(m + 1))), x] + Simp[2*I   Int[ 
(c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*fz*x)))), x], x] 
 /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
 

rule 6839

Int[((a_.) + ArcSech[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ 
-(c^(m + 1))^(-1)   Subst[Int[(a + b*x)^n*Sech[x]^(m + 1)*Tanh[x], x], x, A 
rcSech[c*x]], x] /; FreeQ[{a, b, c}, x] && IntegerQ[n] && IntegerQ[m] && (G 
tQ[n, 0] || LtQ[m, -1])
 

rule 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> 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 7163

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(428\) vs. \(2(206)=412\).

Time = 0.49 (sec) , antiderivative size = 429, normalized size of antiderivative = 3.76

method	result	size
parts	\(a^{3} \ln \left (x \right )+b^{3} \left (\frac {\operatorname {arcsech}\left (c x \right )^{4}}{4}-\operatorname {arcsech}\left (c x \right )^{3} \ln \left (1+\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )-\frac {3 \operatorname {arcsech}\left (c x \right )^{2} \operatorname {polylog}\left (2, -\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )}{2}+\frac {3 \,\operatorname {arcsech}\left (c x \right ) \operatorname {polylog}\left (3, -\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )}{2}-\frac {3 \operatorname {polylog}\left (4, -\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )}{4}\right )+3 a \,b^{2} \left (\frac {\operatorname {arcsech}\left (c x \right )^{3}}{3}-\operatorname {arcsech}\left (c x \right )^{2} \ln \left (1+\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )-\operatorname {arcsech}\left (c x \right ) \operatorname {polylog}\left (2, -\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )+\frac {\operatorname {polylog}\left (3, -\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )}{2}\right )+3 a^{2} b \left (\frac {\operatorname {arcsech}\left (c x \right )^{2}}{2}-\operatorname {arcsech}\left (c x \right ) \ln \left (1+\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )-\frac {\operatorname {polylog}\left (2, -\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )}{2}\right )\)	\(429\)
derivativedivides	\(a^{3} \ln \left (c x \right )+b^{3} \left (\frac {\operatorname {arcsech}\left (c x \right )^{4}}{4}-\operatorname {arcsech}\left (c x \right )^{3} \ln \left (1+\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )-\frac {3 \operatorname {arcsech}\left (c x \right )^{2} \operatorname {polylog}\left (2, -\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )}{2}+\frac {3 \,\operatorname {arcsech}\left (c x \right ) \operatorname {polylog}\left (3, -\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )}{2}-\frac {3 \operatorname {polylog}\left (4, -\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )}{4}\right )+3 a \,b^{2} \left (\frac {\operatorname {arcsech}\left (c x \right )^{3}}{3}-\operatorname {arcsech}\left (c x \right )^{2} \ln \left (1+\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )-\operatorname {arcsech}\left (c x \right ) \operatorname {polylog}\left (2, -\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )+\frac {\operatorname {polylog}\left (3, -\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )}{2}\right )+3 a^{2} b \left (\frac {\operatorname {arcsech}\left (c x \right )^{2}}{2}-\operatorname {arcsech}\left (c x \right ) \ln \left (1+\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )-\frac {\operatorname {polylog}\left (2, -\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )}{2}\right )\)	\(431\)
default	\(a^{3} \ln \left (c x \right )+b^{3} \left (\frac {\operatorname {arcsech}\left (c x \right )^{4}}{4}-\operatorname {arcsech}\left (c x \right )^{3} \ln \left (1+\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )-\frac {3 \operatorname {arcsech}\left (c x \right )^{2} \operatorname {polylog}\left (2, -\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )}{2}+\frac {3 \,\operatorname {arcsech}\left (c x \right ) \operatorname {polylog}\left (3, -\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )}{2}-\frac {3 \operatorname {polylog}\left (4, -\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )}{4}\right )+3 a \,b^{2} \left (\frac {\operatorname {arcsech}\left (c x \right )^{3}}{3}-\operatorname {arcsech}\left (c x \right )^{2} \ln \left (1+\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )-\operatorname {arcsech}\left (c x \right ) \operatorname {polylog}\left (2, -\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )+\frac {\operatorname {polylog}\left (3, -\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )}{2}\right )+3 a^{2} b \left (\frac {\operatorname {arcsech}\left (c x \right )^{2}}{2}-\operatorname {arcsech}\left (c x \right ) \ln \left (1+\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )-\frac {\operatorname {polylog}\left (2, -\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )}{2}\right )\)	\(431\)

Input:

int((a+b*arcsech(c*x))^3/x,x,method=_RETURNVERBOSE)
 

Output:

a^3*ln(x)+b^3*(1/4*arcsech(c*x)^4-arcsech(c*x)^3*ln(1+(1/c/x+(-1+1/c/x)^(1 
/2)*(1+1/c/x)^(1/2))^2)-3/2*arcsech(c*x)^2*polylog(2,-(1/c/x+(-1+1/c/x)^(1 
/2)*(1+1/c/x)^(1/2))^2)+3/2*arcsech(c*x)*polylog(3,-(1/c/x+(-1+1/c/x)^(1/2 
)*(1+1/c/x)^(1/2))^2)-3/4*polylog(4,-(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/ 
2))^2))+3*a*b^2*(1/3*arcsech(c*x)^3-arcsech(c*x)^2*ln(1+(1/c/x+(-1+1/c/x)^ 
(1/2)*(1+1/c/x)^(1/2))^2)-arcsech(c*x)*polylog(2,-(1/c/x+(-1+1/c/x)^(1/2)* 
(1+1/c/x)^(1/2))^2)+1/2*polylog(3,-(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2) 
)^2))+3*a^2*b*(1/2*arcsech(c*x)^2-arcsech(c*x)*ln(1+(1/c/x+(-1+1/c/x)^(1/2 
)*(1+1/c/x)^(1/2))^2)-1/2*polylog(2,-(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/ 
2))^2))
 

Fricas [F]

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

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

Output:

integral((b^3*arcsech(c*x)^3 + 3*a*b^2*arcsech(c*x)^2 + 3*a^2*b*arcsech(c* 
x) + a^3)/x, x)
 

Sympy [F]

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

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

Output:

Integral((a + b*asech(c*x))**3/x, x)
 

Maxima [F]

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

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

Output:

a^3*log(x) + integrate(b^3*log(sqrt(1/(c*x) + 1)*sqrt(1/(c*x) - 1) + 1/(c* 
x))^3/x + 3*a*b^2*log(sqrt(1/(c*x) + 1)*sqrt(1/(c*x) - 1) + 1/(c*x))^2/x + 
 3*a^2*b*log(sqrt(1/(c*x) + 1)*sqrt(1/(c*x) - 1) + 1/(c*x))/x, x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

int((a + b*acosh(1/(c*x)))^3/x,x)
 

Output:

int((a + b*acosh(1/(c*x)))^3/x, x)
 

Reduce [F]

\[ \int \frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{x} \, dx=3 \left (\int \frac {\mathit {asech} \left (c x \right )}{x}d x \right ) a^{2} b +\left (\int \frac {\mathit {asech} \left (c x \right )^{3}}{x}d x \right ) b^{3}+3 \left (\int \frac {\mathit {asech} \left (c x \right )^{2}}{x}d x \right ) a \,b^{2}+\mathrm {log}\left (x \right ) a^{3} \] Input:

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

Output:

3*int(asech(c*x)/x,x)*a**2*b + int(asech(c*x)**3/x,x)*b**3 + 3*int(asech(c 
*x)**2/x,x)*a*b**2 + log(x)*a**3