3.1.17 \(\int \frac {\text {sech}^{-1}(a x)^3}{x^3} \, dx\) [17]

3.1.17.1 Optimal result

Integrand size = 10, antiderivative size = 137 \[ \int \frac {\text {sech}^{-1}(a x)^3}{x^3} \, dx=\frac {3 \sqrt {\frac {1-a x}{1+a x}} (1+a x)}{8 x^2}-\frac {3}{8} a^2 \text {sech}^{-1}(a x)-\frac {3 (1-a x) (1+a x) \text {sech}^{-1}(a x)}{4 x^2}+\frac {3 \sqrt {\frac {1-a x}{1+a x}} (1+a x) \text {sech}^{-1}(a x)^2}{4 x^2}-\frac {1}{4} a^2 \text {sech}^{-1}(a x)^3-\frac {(1-a x) (1+a x) \text {sech}^{-1}(a x)^3}{2 x^2} \]

output

-3/8*a^2*arcsech(a*x)-3/4*(-a*x+1)*(a*x+1)*arcsech(a*x)/x^2-1/4*a^2*arcsec 
h(a*x)^3-1/2*(-a*x+1)*(a*x+1)*arcsech(a*x)^3/x^2+3/8*(a*x+1)*((-a*x+1)/(a* 
x+1))^(1/2)/x^2+3/4*(a*x+1)*arcsech(a*x)^2*((-a*x+1)/(a*x+1))^(1/2)/x^2
 

3.1.17.2 Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.07 \[ \int \frac {\text {sech}^{-1}(a x)^3}{x^3} \, dx=\frac {3 \sqrt {\frac {1-a x}{1+a x}} (1+a x)-6 \text {sech}^{-1}(a x)+6 \sqrt {\frac {1-a x}{1+a x}} (1+a x) \text {sech}^{-1}(a x)^2+2 \left (-2+a^2 x^2\right ) \text {sech}^{-1}(a x)^3-3 a^2 x^2 \log (x)+3 a^2 x^2 \log \left (1+\sqrt {\frac {1-a x}{1+a x}}+a x \sqrt {\frac {1-a x}{1+a x}}\right )}{8 x^2} \]

input

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

output

(3*Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x) - 6*ArcSech[a*x] + 6*Sqrt[(1 - a*x) 
/(1 + a*x)]*(1 + a*x)*ArcSech[a*x]^2 + 2*(-2 + a^2*x^2)*ArcSech[a*x]^3 - 3 
*a^2*x^2*Log[x] + 3*a^2*x^2*Log[1 + Sqrt[(1 - a*x)/(1 + a*x)] + a*x*Sqrt[( 
1 - a*x)/(1 + a*x)]])/(8*x^2)
 

3.1.17.3 Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.15, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.100, Rules used = {6839, 5895, 3042, 25, 3792, 15, 25, 3042, 25, 3115, 24}

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.

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

\(\Big \downarrow \) 6839

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

\(\Big \downarrow \) 5895

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3792

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

\(\Big \downarrow \) 15

\(\displaystyle -a^2 \left (\frac {3}{2} \left (\frac {1}{2} \int -\frac {(1-a x) (a x+1)}{a^2 x^2}d\text {sech}^{-1}(a x)-\frac {\sqrt {\frac {1-a x}{a x+1}} (a x+1) \text {sech}^{-1}(a x)^2}{2 a^2 x^2}+\frac {(1-a x) (a x+1) \text {sech}^{-1}(a x)}{2 a^2 x^2}+\frac {1}{6} \text {sech}^{-1}(a x)^3\right )+\frac {(1-a x) (a x+1) \text {sech}^{-1}(a x)^3}{2 a^2 x^2}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle -a^2 \left (\frac {3}{2} \left (-\frac {1}{2} \int \frac {(1-a x) (a x+1)}{a^2 x^2}d\text {sech}^{-1}(a x)-\frac {\sqrt {\frac {1-a x}{a x+1}} (a x+1) \text {sech}^{-1}(a x)^2}{2 a^2 x^2}+\frac {(1-a x) (a x+1) \text {sech}^{-1}(a x)}{2 a^2 x^2}+\frac {1}{6} \text {sech}^{-1}(a x)^3\right )+\frac {(1-a x) (a x+1) \text {sech}^{-1}(a x)^3}{2 a^2 x^2}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle -a^2 \left (\frac {(1-a x) (a x+1) \text {sech}^{-1}(a x)^3}{2 a^2 x^2}+\frac {3}{2} \left (-\frac {1}{2} \int -\sin \left (i \text {sech}^{-1}(a x)\right )^2d\text {sech}^{-1}(a x)-\frac {\sqrt {\frac {1-a x}{a x+1}} (a x+1) \text {sech}^{-1}(a x)^2}{2 a^2 x^2}+\frac {(1-a x) (a x+1) \text {sech}^{-1}(a x)}{2 a^2 x^2}+\frac {1}{6} \text {sech}^{-1}(a x)^3\right )\right )\)

\(\Big \downarrow \) 25

\(\displaystyle -a^2 \left (\frac {(1-a x) (a x+1) \text {sech}^{-1}(a x)^3}{2 a^2 x^2}+\frac {3}{2} \left (\frac {1}{2} \int \sin \left (i \text {sech}^{-1}(a x)\right )^2d\text {sech}^{-1}(a x)-\frac {\sqrt {\frac {1-a x}{a x+1}} (a x+1) \text {sech}^{-1}(a x)^2}{2 a^2 x^2}+\frac {(1-a x) (a x+1) \text {sech}^{-1}(a x)}{2 a^2 x^2}+\frac {1}{6} \text {sech}^{-1}(a x)^3\right )\right )\)

\(\Big \downarrow \) 3115

\(\displaystyle -a^2 \left (\frac {3}{2} \left (\frac {1}{2} \left (\frac {1}{2} \int 1d\text {sech}^{-1}(a x)-\frac {\sqrt {\frac {1-a x}{a x+1}} (a x+1)}{2 a^2 x^2}\right )-\frac {\sqrt {\frac {1-a x}{a x+1}} (a x+1) \text {sech}^{-1}(a x)^2}{2 a^2 x^2}+\frac {(1-a x) (a x+1) \text {sech}^{-1}(a x)}{2 a^2 x^2}+\frac {1}{6} \text {sech}^{-1}(a x)^3\right )+\frac {(1-a x) (a x+1) \text {sech}^{-1}(a x)^3}{2 a^2 x^2}\right )\)

\(\Big \downarrow \) 24

\(\displaystyle -a^2 \left (\frac {(1-a x) (a x+1) \text {sech}^{-1}(a x)^3}{2 a^2 x^2}+\frac {3}{2} \left (-\frac {\sqrt {\frac {1-a x}{a x+1}} (a x+1) \text {sech}^{-1}(a x)^2}{2 a^2 x^2}+\frac {(1-a x) (a x+1) \text {sech}^{-1}(a x)}{2 a^2 x^2}+\frac {1}{2} \left (\frac {1}{2} \text {sech}^{-1}(a x)-\frac {\sqrt {\frac {1-a x}{a x+1}} (a x+1)}{2 a^2 x^2}\right )+\frac {1}{6} \text {sech}^{-1}(a x)^3\right )\right )\)

input

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

output

-(a^2*(((1 - a*x)*(1 + a*x)*ArcSech[a*x]^3)/(2*a^2*x^2) + (3*((-1/2*(Sqrt[ 
(1 - a*x)/(1 + a*x)]*(1 + a*x))/(a^2*x^2) + ArcSech[a*x]/2)/2 + ((1 - a*x) 
*(1 + a*x)*ArcSech[a*x])/(2*a^2*x^2) - (Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x 
)*ArcSech[a*x]^2)/(2*a^2*x^2) + ArcSech[a*x]^3/6))/2))
 

3.1.17.3.1 Defintions of rubi rules used

rule 15

Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ 
{a, m}, x] && NeQ[m, -1]
 

rule 24

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
 

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 3042

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

rule 3115

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* 
x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n)   Int[(b*Sin 
[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 
2*n]
 

rule 3792

Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbo 
l] :> Simp[d*m*(c + d*x)^(m - 1)*((b*Sin[e + f*x])^n/(f^2*n^2)), x] + (-Sim 
p[b*(c + d*x)^m*Cos[e + f*x]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x] + Simp[b^ 
2*((n - 1)/n)   Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[d^2 
*m*((m - 1)/(f^2*n^2))   Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x], x]) 
/; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 1]
 

rule 5895

Int[Cosh[(a_.) + (b_.)*(x_)^(n_.)]*(x_)^(m_.)*Sinh[(a_.) + (b_.)*(x_)^(n_.) 
]^(p_.), x_Symbol] :> Simp[x^(m - n + 1)*(Sinh[a + b*x^n]^(p + 1)/(b*n*(p + 
 1))), x] - Simp[(m - n + 1)/(b*n*(p + 1))   Int[x^(m - n)*Sinh[a + b*x^n]^ 
(p + 1), x], x] /; FreeQ[{a, b, p}, x] && LtQ[0, n, m + 1] && NeQ[p, -1]
 

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])
 

3.1.17.4 Maple [A] (verified)

Time = 0.25 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.92

method	result	size
derivativedivides	\(a^{2} \left (-\frac {\operatorname {arcsech}\left (a x \right )^{3}}{2 a^{2} x^{2}}+\frac {3 \sqrt {-\frac {a x -1}{a x}}\, \sqrt {\frac {a x +1}{a x}}\, \operatorname {arcsech}\left (a x \right )^{2}}{4 a x}+\frac {\operatorname {arcsech}\left (a x \right )^{3}}{4}-\frac {3 \,\operatorname {arcsech}\left (a x \right )}{4 a^{2} x^{2}}+\frac {3 \sqrt {-\frac {a x -1}{a x}}\, \sqrt {\frac {a x +1}{a x}}}{8 a x}+\frac {3 \,\operatorname {arcsech}\left (a x \right )}{8}\right )\)	\(126\)
default	\(a^{2} \left (-\frac {\operatorname {arcsech}\left (a x \right )^{3}}{2 a^{2} x^{2}}+\frac {3 \sqrt {-\frac {a x -1}{a x}}\, \sqrt {\frac {a x +1}{a x}}\, \operatorname {arcsech}\left (a x \right )^{2}}{4 a x}+\frac {\operatorname {arcsech}\left (a x \right )^{3}}{4}-\frac {3 \,\operatorname {arcsech}\left (a x \right )}{4 a^{2} x^{2}}+\frac {3 \sqrt {-\frac {a x -1}{a x}}\, \sqrt {\frac {a x +1}{a x}}}{8 a x}+\frac {3 \,\operatorname {arcsech}\left (a x \right )}{8}\right )\)	\(126\)

input

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

output

a^2*(-1/2/a^2/x^2*arcsech(a*x)^3+3/4*(-(a*x-1)/a/x)^(1/2)*((a*x+1)/a/x)^(1 
/2)/a/x*arcsech(a*x)^2+1/4*arcsech(a*x)^3-3/4/a^2/x^2*arcsech(a*x)+3/8*(-( 
a*x-1)/a/x)^(1/2)*((a*x+1)/a/x)^(1/2)/a/x+3/8*arcsech(a*x))
 

3.1.17.5 Fricas [A] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.27 \[ \int \frac {\text {sech}^{-1}(a x)^3}{x^3} \, dx=\frac {6 \, a x \sqrt {-\frac {a^{2} x^{2} - 1}{a^{2} x^{2}}} \log \left (\frac {a x \sqrt {-\frac {a^{2} x^{2} - 1}{a^{2} x^{2}}} + 1}{a x}\right )^{2} + 2 \, {\left (a^{2} x^{2} - 2\right )} \log \left (\frac {a x \sqrt {-\frac {a^{2} x^{2} - 1}{a^{2} x^{2}}} + 1}{a x}\right )^{3} + 3 \, a x \sqrt {-\frac {a^{2} x^{2} - 1}{a^{2} x^{2}}} + 3 \, {\left (a^{2} x^{2} - 2\right )} \log \left (\frac {a x \sqrt {-\frac {a^{2} x^{2} - 1}{a^{2} x^{2}}} + 1}{a x}\right )}{8 \, x^{2}} \]

input

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

output

1/8*(6*a*x*sqrt(-(a^2*x^2 - 1)/(a^2*x^2))*log((a*x*sqrt(-(a^2*x^2 - 1)/(a^ 
2*x^2)) + 1)/(a*x))^2 + 2*(a^2*x^2 - 2)*log((a*x*sqrt(-(a^2*x^2 - 1)/(a^2* 
x^2)) + 1)/(a*x))^3 + 3*a*x*sqrt(-(a^2*x^2 - 1)/(a^2*x^2)) + 3*(a^2*x^2 - 
2)*log((a*x*sqrt(-(a^2*x^2 - 1)/(a^2*x^2)) + 1)/(a*x)))/x^2
 

3.1.17.6 Sympy [F]

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

input

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

output

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

3.1.17.7 Maxima [F]

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

input

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

output

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

3.1.17.8 Giac [F]

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

input

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

output

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

3.1.17.9 Mupad [F(-1)]

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

input

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

output

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