3.1.14 \(\int \text {sech}^{-1}(a x)^3 \, dx\) [14]

3.1.14.1 Optimal result

Integrand size = 6, antiderivative size = 111 \[ \int \text {sech}^{-1}(a x)^3 \, dx=x \text {sech}^{-1}(a x)^3-\frac {6 \text {sech}^{-1}(a x)^2 \arctan \left (e^{\text {sech}^{-1}(a x)}\right )}{a}+\frac {6 i \text {sech}^{-1}(a x) \operatorname {PolyLog}\left (2,-i e^{\text {sech}^{-1}(a x)}\right )}{a}-\frac {6 i \text {sech}^{-1}(a x) \operatorname {PolyLog}\left (2,i e^{\text {sech}^{-1}(a x)}\right )}{a}-\frac {6 i \operatorname {PolyLog}\left (3,-i e^{\text {sech}^{-1}(a x)}\right )}{a}+\frac {6 i \operatorname {PolyLog}\left (3,i e^{\text {sech}^{-1}(a x)}\right )}{a} \]

output

x*arcsech(a*x)^3-6*arcsech(a*x)^2*arctan(1/a/x+(1/a/x-1)^(1/2)*(1+1/a/x)^( 
1/2))/a+6*I*arcsech(a*x)*polylog(2,-I*(1/a/x+(1/a/x-1)^(1/2)*(1+1/a/x)^(1/ 
2)))/a-6*I*arcsech(a*x)*polylog(2,I*(1/a/x+(1/a/x-1)^(1/2)*(1+1/a/x)^(1/2) 
))/a-6*I*polylog(3,-I*(1/a/x+(1/a/x-1)^(1/2)*(1+1/a/x)^(1/2)))/a+6*I*polyl 
og(3,I*(1/a/x+(1/a/x-1)^(1/2)*(1+1/a/x)^(1/2)))/a
 

3.1.14.2 Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.15 \[ \int \text {sech}^{-1}(a x)^3 \, dx=x \text {sech}^{-1}(a x)^3-\frac {3 i \left (-\text {sech}^{-1}(a x)^2 \left (\log \left (1-i e^{-\text {sech}^{-1}(a x)}\right )-\log \left (1+i e^{-\text {sech}^{-1}(a x)}\right )\right )-2 \text {sech}^{-1}(a x) \left (\operatorname {PolyLog}\left (2,-i e^{-\text {sech}^{-1}(a x)}\right )-\operatorname {PolyLog}\left (2,i e^{-\text {sech}^{-1}(a x)}\right )\right )-2 \left (\operatorname {PolyLog}\left (3,-i e^{-\text {sech}^{-1}(a x)}\right )-\operatorname {PolyLog}\left (3,i e^{-\text {sech}^{-1}(a x)}\right )\right )\right )}{a} \]

input

Integrate[ArcSech[a*x]^3,x]
 

output

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

3.1.14.3 Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.94, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.167, Rules used = {6833, 5941, 3042, 4668, 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.

\(\displaystyle \int \text {sech}^{-1}(a x)^3 \, dx\)

\(\Big \downarrow \) 6833

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

\(\Big \downarrow \) 5941

\(\displaystyle -\frac {3 \int a x \text {sech}^{-1}(a x)^2d\text {sech}^{-1}(a x)-a x \text {sech}^{-1}(a x)^3}{a}\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4668

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

input

Int[ArcSech[a*x]^3,x]
 

output

-((-(a*x*ArcSech[a*x]^3) + 3*(2*ArcSech[a*x]^2*ArcTan[E^ArcSech[a*x]] + (2 
*I)*(-(ArcSech[a*x]*PolyLog[2, (-I)*E^ArcSech[a*x]]) + PolyLog[3, (-I)*E^A 
rcSech[a*x]]) - (2*I)*(-(ArcSech[a*x]*PolyLog[2, I*E^ArcSech[a*x]]) + Poly 
Log[3, I*E^ArcSech[a*x]])))/a)
 

3.1.14.3.1 Defintions of rubi rules used

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 4668

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

rule 5941

Int[(x_)^(m_.)*Sech[(a_.) + (b_.)*(x_)^(n_.)]^(p_.)*Tanh[(a_.) + (b_.)*(x_) 
^(n_.)]^(q_.), x_Symbol] :> Simp[(-x^(m - n + 1))*(Sech[a + b*x^n]^p/(b*n*p 
)), x] + Simp[(m - n + 1)/(b*n*p)   Int[x^(m - n)*Sech[a + b*x^n]^p, x], x] 
 /; FreeQ[{a, b, p}, x] && RationalQ[m] && IntegerQ[n] && GeQ[m - n, 0] && 
EqQ[q, 1]
 

rule 6833

Int[((a_.) + ArcSech[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[-c^(-1)   S 
ubst[Int[(a + b*x)^n*Sech[x]*Tanh[x], x], x, ArcSech[c*x]], x] /; FreeQ[{a, 
 b, c, n}, x] && IGtQ[n, 0]
 

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]
 

3.1.14.4 Maple [F]

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

input

int(arcsech(a*x)^3,x)
 

output

int(arcsech(a*x)^3,x)
 

3.1.14.5 Fricas [F]

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

input

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

output

integral(arcsech(a*x)^3, x)
 

3.1.14.6 Sympy [F]

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

input

integrate(asech(a*x)**3,x)
 

output

Integral(asech(a*x)**3, x)
 

3.1.14.7 Maxima [F]

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

input

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

output

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

3.1.14.8 Giac [F]

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

input

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

output

integrate(arcsech(a*x)^3, x)
 

3.1.14.9 Mupad [F(-1)]

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

input

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

output

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