\(\int \frac {e^{-\text {sech}^{-1}(a x)}}{x^2} \, dx\) [82]

Optimal result

Integrand size = 12, antiderivative size = 72 \[ \int \frac {e^{-\text {sech}^{-1}(a x)}}{x^2} \, dx=-\frac {a}{\left (1+\sqrt {\frac {1-a x}{1+a x}}\right )^2}+\frac {a}{1+\sqrt {\frac {1-a x}{1+a x}}}-a \text {arctanh}\left (\sqrt {\frac {1-a x}{1+a x}}\right ) \]

[Out]

-a*arctanh(((-a*x+1)/(a*x+1))^(1/2))-a/(1+((-a*x+1)/(a*x+1))^(1/2))^2+a/(1+((-a*x+1)/(a*x+1))^(1/2))

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6472, 78, 213} \[ \int \frac {e^{-\text {sech}^{-1}(a x)}}{x^2} \, dx=a \left (-\text {arctanh}\left (\sqrt {\frac {1-a x}{a x+1}}\right )\right )+\frac {a}{\sqrt {\frac {1-a x}{a x+1}}+1}-\frac {a}{\left (\sqrt {\frac {1-a x}{a x+1}}+1\right )^2} \]

[In]

Int[1/(E^ArcSech[a*x]*x^2),x]

[Out]

-(a/(1 + Sqrt[(1 - a*x)/(1 + a*x)])^2) + a/(1 + Sqrt[(1 - a*x)/(1 + a*x)]) - a*ArcTanh[Sqrt[(1 - a*x)/(1 + a*x
)]]

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rule 213

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(-1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])]
, x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rule 6472

Int[E^(ArcSech[u_]*(n_.))*(x_)^(m_.), x_Symbol] :> Int[x^m*(1/u + Sqrt[(1 - u)/(1 + u)] + (1/u)*Sqrt[(1 - u)/(
1 + u)])^n, x] /; FreeQ[m, x] && IntegerQ[n]

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{x^2 \left (\frac {1}{a x}+\sqrt {\frac {1-a x}{1+a x}}+\frac {\sqrt {\frac {1-a x}{1+a x}}}{a x}\right )} \, dx \\ & = (4 a) \text {Subst}\left (\int \frac {x}{(-1+x) (1+x)^3} \, dx,x,\sqrt {\frac {1-a x}{1+a x}}\right ) \\ & = (4 a) \text {Subst}\left (\int \left (\frac {1}{2 (1+x)^3}-\frac {1}{4 (1+x)^2}+\frac {1}{4 \left (-1+x^2\right )}\right ) \, dx,x,\sqrt {\frac {1-a x}{1+a x}}\right ) \\ & = -\frac {a}{\left (1+\sqrt {\frac {1-a x}{1+a x}}\right )^2}+\frac {a}{1+\sqrt {\frac {1-a x}{1+a x}}}+a \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {\frac {1-a x}{1+a x}}\right ) \\ & = -\frac {a}{\left (1+\sqrt {\frac {1-a x}{1+a x}}\right )^2}+\frac {a}{1+\sqrt {\frac {1-a x}{1+a x}}}-a \text {arctanh}\left (\sqrt {\frac {1-a x}{1+a x}}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.28 \[ \int \frac {e^{-\text {sech}^{-1}(a x)}}{x^2} \, dx=\frac {1}{2} \left (-\frac {1}{a x^2}+\frac {\sqrt {\frac {1-a x}{1+a x}} (1+a x)}{a x^2}+a \log (x)-a \log \left (1+\sqrt {\frac {1-a x}{1+a x}}+a x \sqrt {\frac {1-a x}{1+a x}}\right )\right ) \]

[In]

Integrate[1/(E^ArcSech[a*x]*x^2),x]

[Out]

(-(1/(a*x^2)) + (Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x))/(a*x^2) + a*Log[x] - a*Log[1 + Sqrt[(1 - a*x)/(1 + a*x)]
+ a*x*Sqrt[(1 - a*x)/(1 + a*x)]])/2

Maple [A] (verified)

Time = 0.89 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.33

method	result	size
default	\(a \left (-\frac {1}{2 a^{2} x^{2}}-\frac {\sqrt {-\frac {a x -1}{a x}}\, \sqrt {\frac {a x +1}{a x}}\, \left (a^{2} x^{2} \operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )-\sqrt {-a^{2} x^{2}+1}\right )}{2 a x \sqrt {-a^{2} x^{2}+1}}\right )\)	\(96\)

[In]

int(1/(1/a/x+(1/a/x-1)^(1/2)*(1+1/a/x)^(1/2))/x^2,x,method=_RETURNVERBOSE)

[Out]

a*(-1/2/a^2/x^2-1/2/a*(-(a*x-1)/a/x)^(1/2)/x*((a*x+1)/a/x)^(1/2)*(a^2*x^2*arctanh(1/(-a^2*x^2+1)^(1/2))-(-a^2*
x^2+1)^(1/2))/(-a^2*x^2+1)^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.78 \[ \int \frac {e^{-\text {sech}^{-1}(a x)}}{x^2} \, dx=-\frac {a^{2} x^{2} \log \left (a x \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} + 1\right ) - a^{2} x^{2} \log \left (a x \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} - 1\right ) - 2 \, a x \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} + 2}{4 \, a x^{2}} \]

[In]

integrate(1/(1/a/x+(1/a/x-1)^(1/2)*(1+1/a/x)^(1/2))/x^2,x, algorithm="fricas")

[Out]

-1/4*(a^2*x^2*log(a*x*sqrt((a*x + 1)/(a*x))*sqrt(-(a*x - 1)/(a*x)) + 1) - a^2*x^2*log(a*x*sqrt((a*x + 1)/(a*x)
)*sqrt(-(a*x - 1)/(a*x)) - 1) - 2*a*x*sqrt((a*x + 1)/(a*x))*sqrt(-(a*x - 1)/(a*x)) + 2)/(a*x^2)

Sympy [F]

\[ \int \frac {e^{-\text {sech}^{-1}(a x)}}{x^2} \, dx=a \int \frac {1}{a x^{2} \sqrt {-1 + \frac {1}{a x}} \sqrt {1 + \frac {1}{a x}} + x}\, dx \]

[In]

integrate(1/(1/a/x+(1/a/x-1)**(1/2)*(1+1/a/x)**(1/2))/x**2,x)

[Out]

a*Integral(1/(a*x**2*sqrt(-1 + 1/(a*x))*sqrt(1 + 1/(a*x)) + x), x)

Maxima [F]

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

[In]

integrate(1/(1/a/x+(1/a/x-1)^(1/2)*(1+1/a/x)^(1/2))/x^2,x, algorithm="maxima")

[Out]

integrate(1/(x^2*(sqrt(1/(a*x) + 1)*sqrt(1/(a*x) - 1) + 1/(a*x))), x)

Giac [F]

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

[In]

integrate(1/(1/a/x+(1/a/x-1)^(1/2)*(1+1/a/x)^(1/2))/x^2,x, algorithm="giac")

[Out]

integrate(1/(x^2*(sqrt(1/(a*x) + 1)*sqrt(1/(a*x) - 1) + 1/(a*x))), x)

Mupad [B] (verification not implemented)

Time = 16.76 (sec) , antiderivative size = 323, normalized size of antiderivative = 4.49 \[ \int \frac {e^{-\text {sech}^{-1}(a x)}}{x^2} \, dx=2\,a\,\mathrm {atanh}\left (\frac {\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}}{\sqrt {\frac {1}{a\,x}+1}-1}\right )-a\,\mathrm {acosh}\left (\frac {1}{a\,x}\right )-\frac {1}{2\,a\,x^2}-\frac {a\,\left (\frac {14\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^3}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^3}+\frac {14\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^5}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^5}+\frac {2\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^7}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^7}+\frac {2\,\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}{\sqrt {\frac {1}{a\,x}+1}-1}\right )}{1+\frac {6\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^4}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^4}-\frac {4\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^6}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^6}+\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^8}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^8}-\frac {4\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^2}} \]

[In]

int(1/(x^2*((1/(a*x) - 1)^(1/2)*(1/(a*x) + 1)^(1/2) + 1/(a*x))),x)

[Out]

2*a*atanh(((1/(a*x) - 1)^(1/2) - 1i)/((1/(a*x) + 1)^(1/2) - 1)) - a*acosh(1/(a*x)) - 1/(2*a*x^2) - (a*((14*((1
/(a*x) - 1)^(1/2) - 1i)^3)/((1/(a*x) + 1)^(1/2) - 1)^3 + (14*((1/(a*x) - 1)^(1/2) - 1i)^5)/((1/(a*x) + 1)^(1/2
) - 1)^5 + (2*((1/(a*x) - 1)^(1/2) - 1i)^7)/((1/(a*x) + 1)^(1/2) - 1)^7 + (2*((1/(a*x) - 1)^(1/2) - 1i))/((1/(
a*x) + 1)^(1/2) - 1)))/((6*((1/(a*x) - 1)^(1/2) - 1i)^4)/((1/(a*x) + 1)^(1/2) - 1)^4 - (4*((1/(a*x) - 1)^(1/2)
- 1i)^2)/((1/(a*x) + 1)^(1/2) - 1)^2 - (4*((1/(a*x) - 1)^(1/2) - 1i)^6)/((1/(a*x) + 1)^(1/2) - 1)^6 + ((1/(a*
x) - 1)^(1/2) - 1i)^8/((1/(a*x) + 1)^(1/2) - 1)^8 + 1)