3.1.9 \(\int \frac {\text {sech}^{-1}(a x)^2}{x^4} \, dx\) [9]

3.1.9.1 Optimal result

Integrand size = 10, antiderivative size = 102 \[ \int \frac {\text {sech}^{-1}(a x)^2}{x^4} \, dx=-\frac {2}{27 x^3}-\frac {4 a^2}{9 x}+\frac {2 \sqrt {\frac {1-a x}{1+a x}} (1+a x) \text {sech}^{-1}(a x)}{9 x^3}+\frac {4 a^2 \sqrt {\frac {1-a x}{1+a x}} (1+a x) \text {sech}^{-1}(a x)}{9 x}-\frac {\text {sech}^{-1}(a x)^2}{3 x^3} \]

output

-2/27/x^3-4/9*a^2/x-1/3*arcsech(a*x)^2/x^3+2/9*(a*x+1)*arcsech(a*x)*((-a*x 
+1)/(a*x+1))^(1/2)/x^3+4/9*a^2*(a*x+1)*arcsech(a*x)*((-a*x+1)/(a*x+1))^(1/ 
2)/x
 

3.1.9.2 Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.72 \[ \int \frac {\text {sech}^{-1}(a x)^2}{x^4} \, dx=\frac {-2 \left (1+6 a^2 x^2\right )+6 \sqrt {\frac {1-a x}{1+a x}} \left (1+a x+2 a^2 x^2+2 a^3 x^3\right ) \text {sech}^{-1}(a x)-9 \text {sech}^{-1}(a x)^2}{27 x^3} \]

input

Integrate[ArcSech[a*x]^2/x^4,x]
 

output

(-2*(1 + 6*a^2*x^2) + 6*Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x + 2*a^2*x^2 + 2 
*a^3*x^3)*ArcSech[a*x] - 9*ArcSech[a*x]^2)/(27*x^3)
 

3.1.9.3 Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.19, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {6839, 5896, 3042, 3791, 3042, 3777, 26, 3042, 26, 3118}

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.

\(\displaystyle \int \frac {\text {sech}^{-1}(a x)^2}{x^4} \, dx\)

\(\Big \downarrow \) 6839

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

\(\Big \downarrow \) 5896

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3791

\(\displaystyle -a^3 \left (\frac {\text {sech}^{-1}(a x)^2}{3 a^3 x^3}-\frac {2}{3} \left (\frac {2}{3} \int \frac {\text {sech}^{-1}(a x)}{a x}d\text {sech}^{-1}(a x)-\frac {1}{9 a^3 x^3}+\frac {\sqrt {\frac {1-a x}{a x+1}} (a x+1) \text {sech}^{-1}(a x)}{3 a^3 x^3}\right )\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle -a^3 \left (\frac {\text {sech}^{-1}(a x)^2}{3 a^3 x^3}-\frac {2}{3} \left (\frac {2}{3} \int \text {sech}^{-1}(a x) \sin \left (i \text {sech}^{-1}(a x)+\frac {\pi }{2}\right )d\text {sech}^{-1}(a x)-\frac {1}{9 a^3 x^3}+\frac {\sqrt {\frac {1-a x}{a x+1}} (a x+1) \text {sech}^{-1}(a x)}{3 a^3 x^3}\right )\right )\)

\(\Big \downarrow \) 3777

\(\displaystyle -a^3 \left (\frac {\text {sech}^{-1}(a x)^2}{3 a^3 x^3}-\frac {2}{3} \left (\frac {2}{3} \left (\frac {\sqrt {\frac {1-a x}{a x+1}} (a x+1) \text {sech}^{-1}(a x)}{a x}-i \int -\frac {i \sqrt {\frac {1-a x}{a x+1}} (a x+1)}{a x}d\text {sech}^{-1}(a x)\right )-\frac {1}{9 a^3 x^3}+\frac {\sqrt {\frac {1-a x}{a x+1}} (a x+1) \text {sech}^{-1}(a x)}{3 a^3 x^3}\right )\right )\)

\(\Big \downarrow \) 26

\(\displaystyle -a^3 \left (\frac {\text {sech}^{-1}(a x)^2}{3 a^3 x^3}-\frac {2}{3} \left (\frac {2}{3} \left (\frac {\sqrt {\frac {1-a x}{a x+1}} (a x+1) \text {sech}^{-1}(a x)}{a x}-\int \frac {\sqrt {\frac {1-a x}{a x+1}} (a x+1)}{a x}d\text {sech}^{-1}(a x)\right )-\frac {1}{9 a^3 x^3}+\frac {\sqrt {\frac {1-a x}{a x+1}} (a x+1) \text {sech}^{-1}(a x)}{3 a^3 x^3}\right )\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle -a^3 \left (\frac {\text {sech}^{-1}(a x)^2}{3 a^3 x^3}-\frac {2}{3} \left (\frac {2}{3} \left (\frac {\sqrt {\frac {1-a x}{a x+1}} (a x+1) \text {sech}^{-1}(a x)}{a x}-\int -i \sin \left (i \text {sech}^{-1}(a x)\right )d\text {sech}^{-1}(a x)\right )-\frac {1}{9 a^3 x^3}+\frac {\sqrt {\frac {1-a x}{a x+1}} (a x+1) \text {sech}^{-1}(a x)}{3 a^3 x^3}\right )\right )\)

\(\Big \downarrow \) 26

\(\displaystyle -a^3 \left (\frac {\text {sech}^{-1}(a x)^2}{3 a^3 x^3}-\frac {2}{3} \left (\frac {2}{3} \left (\frac {\sqrt {\frac {1-a x}{a x+1}} (a x+1) \text {sech}^{-1}(a x)}{a x}+i \int \sin \left (i \text {sech}^{-1}(a x)\right )d\text {sech}^{-1}(a x)\right )-\frac {1}{9 a^3 x^3}+\frac {\sqrt {\frac {1-a x}{a x+1}} (a x+1) \text {sech}^{-1}(a x)}{3 a^3 x^3}\right )\right )\)

\(\Big \downarrow \) 3118

\(\displaystyle -a^3 \left (\frac {\text {sech}^{-1}(a x)^2}{3 a^3 x^3}-\frac {2}{3} \left (-\frac {1}{9 a^3 x^3}+\frac {\sqrt {\frac {1-a x}{a x+1}} (a x+1) \text {sech}^{-1}(a x)}{3 a^3 x^3}+\frac {2}{3} \left (\frac {\sqrt {\frac {1-a x}{a x+1}} (a x+1) \text {sech}^{-1}(a x)}{a x}-\frac {1}{a x}\right )\right )\right )\)

input

Int[ArcSech[a*x]^2/x^4,x]
 

output

-(a^3*(ArcSech[a*x]^2/(3*a^3*x^3) - (2*(-1/9*1/(a^3*x^3) + (Sqrt[(1 - a*x) 
/(1 + a*x)]*(1 + a*x)*ArcSech[a*x])/(3*a^3*x^3) + (2*(-(1/(a*x)) + (Sqrt[( 
1 - a*x)/(1 + a*x)]*(1 + a*x)*ArcSech[a*x])/(a*x)))/3))/3))
 

3.1.9.3.1 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 3042

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

rule 3118

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Cos[c + d*x]/d, x] /; FreeQ 
[{c, d}, x]
 

rule 3777

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[( 
-(c + d*x)^m)*(Cos[e + f*x]/f), x] + Simp[d*(m/f)   Int[(c + d*x)^(m - 1)*C 
os[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
 

rule 3791

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

rule 5896

Int[Cosh[(a_.) + (b_.)*(x_)^(n_.)]^(p_.)*(x_)^(m_.)*Sinh[(a_.) + (b_.)*(x_) 
^(n_.)], x_Symbol] :> Simp[x^(m - n + 1)*(Cosh[a + b*x^n]^(p + 1)/(b*n*(p + 
 1))), x] - Simp[(m - n + 1)/(b*n*(p + 1))   Int[x^(m - n)*Cosh[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])
 

3.1.9.4 Maple [A] (verified)

Time = 0.83 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.10

method	result	size
derivativedivides	\(a^{3} \left (-\frac {\operatorname {arcsech}\left (a x \right )^{2}}{3 a^{3} x^{3}}+\frac {4 \sqrt {-\frac {a x -1}{a x}}\, \sqrt {\frac {a x +1}{a x}}\, \operatorname {arcsech}\left (a x \right )}{9}+\frac {2 \sqrt {-\frac {a x -1}{a x}}\, \sqrt {\frac {a x +1}{a x}}\, \operatorname {arcsech}\left (a x \right )}{9 a^{2} x^{2}}-\frac {4}{9 a x}-\frac {2}{27 x^{3} a^{3}}\right )\)	\(112\)
default	\(a^{3} \left (-\frac {\operatorname {arcsech}\left (a x \right )^{2}}{3 a^{3} x^{3}}+\frac {4 \sqrt {-\frac {a x -1}{a x}}\, \sqrt {\frac {a x +1}{a x}}\, \operatorname {arcsech}\left (a x \right )}{9}+\frac {2 \sqrt {-\frac {a x -1}{a x}}\, \sqrt {\frac {a x +1}{a x}}\, \operatorname {arcsech}\left (a x \right )}{9 a^{2} x^{2}}-\frac {4}{9 a x}-\frac {2}{27 x^{3} a^{3}}\right )\)	\(112\)

input

int(arcsech(a*x)^2/x^4,x,method=_RETURNVERBOSE)
 

output

a^3*(-1/3/a^3/x^3*arcsech(a*x)^2+4/9*(-(a*x-1)/a/x)^(1/2)*((a*x+1)/a/x)^(1 
/2)*arcsech(a*x)+2/9/a^2/x^2*(-(a*x-1)/a/x)^(1/2)*((a*x+1)/a/x)^(1/2)*arcs 
ech(a*x)-4/9/a/x-2/27/x^3/a^3)
 

3.1.9.5 Fricas [A] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.14 \[ \int \frac {\text {sech}^{-1}(a x)^2}{x^4} \, dx=-\frac {12 \, a^{2} x^{2} - 6 \, {\left (2 \, a^{3} x^{3} + a x\right )} \sqrt {-\frac {a^{2} x^{2} - 1}{a^{2} x^{2}}} \log \left (\frac {a x \sqrt {-\frac {a^{2} x^{2} - 1}{a^{2} x^{2}}} + 1}{a x}\right ) + 9 \, \log \left (\frac {a x \sqrt {-\frac {a^{2} x^{2} - 1}{a^{2} x^{2}}} + 1}{a x}\right )^{2} + 2}{27 \, x^{3}} \]

input

integrate(arcsech(a*x)^2/x^4,x, algorithm="fricas")
 

output

-1/27*(12*a^2*x^2 - 6*(2*a^3*x^3 + 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)) + 9*log((a*x*sqrt(-(a^2*x 
^2 - 1)/(a^2*x^2)) + 1)/(a*x))^2 + 2)/x^3
 

3.1.9.6 Sympy [F]

\[ \int \frac {\text {sech}^{-1}(a x)^2}{x^4} \, dx=\int \frac {\operatorname {asech}^{2}{\left (a x \right )}}{x^{4}}\, dx \]

input

integrate(asech(a*x)**2/x**4,x)
 

output

Integral(asech(a*x)**2/x**4, x)
 

3.1.9.7 Maxima [F]

\[ \int \frac {\text {sech}^{-1}(a x)^2}{x^4} \, dx=\int { \frac {\operatorname {arsech}\left (a x\right )^{2}}{x^{4}} \,d x } \]

input

integrate(arcsech(a*x)^2/x^4,x, algorithm="maxima")
 

output

integrate(arcsech(a*x)^2/x^4, x)
 

3.1.9.8 Giac [F]

\[ \int \frac {\text {sech}^{-1}(a x)^2}{x^4} \, dx=\int { \frac {\operatorname {arsech}\left (a x\right )^{2}}{x^{4}} \,d x } \]

input

integrate(arcsech(a*x)^2/x^4,x, algorithm="giac")
 

output

integrate(arcsech(a*x)^2/x^4, x)
 

3.1.9.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\text {sech}^{-1}(a x)^2}{x^4} \, dx=\int \frac {{\mathrm {acosh}\left (\frac {1}{a\,x}\right )}^2}{x^4} \,d x \]

input

int(acosh(1/(a*x))^2/x^4,x)
 

output

int(acosh(1/(a*x))^2/x^4, x)