3.1.6 \(\int \frac {\text {sech}^{-1}(a x)^2}{x} \, dx\) [6]

3.1.6.1 Optimal result

Integrand size = 10, antiderivative size = 64 \[ \int \frac {\text {sech}^{-1}(a x)^2}{x} \, dx=\frac {1}{3} \text {sech}^{-1}(a x)^3-\text {sech}^{-1}(a x)^2 \log \left (1+e^{2 \text {sech}^{-1}(a x)}\right )-\text {sech}^{-1}(a x) \operatorname {PolyLog}\left (2,-e^{2 \text {sech}^{-1}(a x)}\right )+\frac {1}{2} \operatorname {PolyLog}\left (3,-e^{2 \text {sech}^{-1}(a x)}\right ) \]

output

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

3.1.6.2 Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.98 \[ \int \frac {\text {sech}^{-1}(a x)^2}{x} \, dx=-\frac {1}{3} \text {sech}^{-1}(a x)^3-\text {sech}^{-1}(a x)^2 \log \left (1+e^{-2 \text {sech}^{-1}(a x)}\right )+\text {sech}^{-1}(a x) \operatorname {PolyLog}\left (2,-e^{-2 \text {sech}^{-1}(a x)}\right )+\frac {1}{2} \operatorname {PolyLog}\left (3,-e^{-2 \text {sech}^{-1}(a x)}\right ) \]

input

Integrate[ArcSech[a*x]^2/x,x]
 

output

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

3.1.6.3 Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.46 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.23, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {6839, 3042, 26, 4201, 2620, 3011, 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.

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

\(\Big \downarrow \) 6839

\(\displaystyle -\int \sqrt {\frac {1-a x}{a x+1}} (a x+1) \text {sech}^{-1}(a x)^2d\text {sech}^{-1}(a x)\)

\(\Big \downarrow \) 3042

\(\displaystyle -\int -i \text {sech}^{-1}(a x)^2 \tan \left (i \text {sech}^{-1}(a x)\right )d\text {sech}^{-1}(a x)\)

\(\Big \downarrow \) 26

\(\displaystyle i \int \text {sech}^{-1}(a x)^2 \tan \left (i \text {sech}^{-1}(a x)\right )d\text {sech}^{-1}(a x)\)

\(\Big \downarrow \) 4201

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

input

Int[ArcSech[a*x]^2/x,x]
 

output

I*((-1/3*I)*ArcSech[a*x]^3 + (2*I)*((ArcSech[a*x]^2*Log[1 + E^(2*ArcSech[a 
*x])])/2 + (ArcSech[a*x]*PolyLog[2, -E^(2*ArcSech[a*x])])/2 - PolyLog[3, - 
E^(2*ArcSech[a*x])]/4))
 

3.1.6.3.1 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]
 

3.1.6.4 Maple [A] (verified)

Time = 0.26 (sec) , antiderivative size = 136, normalized size of antiderivative = 2.12

method	result	size
derivativedivides	\(\frac {\operatorname {arcsech}\left (a x \right )^{3}}{3}-\operatorname {arcsech}\left (a x \right )^{2} \ln \left (1+\left (\frac {1}{a x}+\sqrt {\frac {1}{a x}-1}\, \sqrt {1+\frac {1}{a x}}\right )^{2}\right )-\operatorname {arcsech}\left (a x \right ) \operatorname {polylog}\left (2, -\left (\frac {1}{a x}+\sqrt {\frac {1}{a x}-1}\, \sqrt {1+\frac {1}{a x}}\right )^{2}\right )+\frac {\operatorname {polylog}\left (3, -\left (\frac {1}{a x}+\sqrt {\frac {1}{a x}-1}\, \sqrt {1+\frac {1}{a x}}\right )^{2}\right )}{2}\)	\(136\)
default	\(\frac {\operatorname {arcsech}\left (a x \right )^{3}}{3}-\operatorname {arcsech}\left (a x \right )^{2} \ln \left (1+\left (\frac {1}{a x}+\sqrt {\frac {1}{a x}-1}\, \sqrt {1+\frac {1}{a x}}\right )^{2}\right )-\operatorname {arcsech}\left (a x \right ) \operatorname {polylog}\left (2, -\left (\frac {1}{a x}+\sqrt {\frac {1}{a x}-1}\, \sqrt {1+\frac {1}{a x}}\right )^{2}\right )+\frac {\operatorname {polylog}\left (3, -\left (\frac {1}{a x}+\sqrt {\frac {1}{a x}-1}\, \sqrt {1+\frac {1}{a x}}\right )^{2}\right )}{2}\)	\(136\)

input

int(arcsech(a*x)^2/x,x,method=_RETURNVERBOSE)
 

output

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

3.1.6.5 Fricas [F]

\[ \int \frac {\text {sech}^{-1}(a x)^2}{x} \, dx=\int { \frac {\operatorname {arsech}\left (a x\right )^{2}}{x} \,d x } \]

input

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

output

integral(arcsech(a*x)^2/x, x)
 

3.1.6.6 Sympy [F]

\[ \int \frac {\text {sech}^{-1}(a x)^2}{x} \, dx=\int \frac {\operatorname {asech}^{2}{\left (a x \right )}}{x}\, dx \]

input

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

output

Integral(asech(a*x)**2/x, x)
 

3.1.6.7 Maxima [F]

\[ \int \frac {\text {sech}^{-1}(a x)^2}{x} \, dx=\int { \frac {\operatorname {arsech}\left (a x\right )^{2}}{x} \,d x } \]

input

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

output

integrate(arcsech(a*x)^2/x, x)
 

3.1.6.8 Giac [F]

\[ \int \frac {\text {sech}^{-1}(a x)^2}{x} \, dx=\int { \frac {\operatorname {arsech}\left (a x\right )^{2}}{x} \,d x } \]

input

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

output

integrate(arcsech(a*x)^2/x, x)
 

3.1.6.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\text {sech}^{-1}(a x)^2}{x} \, dx=\int \frac {{\mathrm {acosh}\left (\frac {1}{a\,x}\right )}^2}{x} \,d x \]

input

int(acosh(1/(a*x))^2/x,x)
 

output

int(acosh(1/(a*x))^2/x, x)