### 3.38 $$\int \frac{e^{-\coth ^{-1}(a x)}}{x^2} \, dx$$

Optimal. Leaf size=25 $a \left (-\csc ^{-1}(a x)\right )-a \sqrt{1-\frac{1}{a^2 x^2}}$

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0267834, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.25, Rules used = {6169, 641, 216} $a \left (-\csc ^{-1}(a x)\right )-a \sqrt{1-\frac{1}{a^2 x^2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Rule 6169

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*(x_)^(m_.), x_Symbol] :> -Subst[Int[(1 + x/a)^((n + 1)/2)/(x^(m + 2)*(1 - x/
a)^((n - 1)/2)*Sqrt[1 - x^2/a^2]), x], x, 1/x] /; FreeQ[a, x] && IntegerQ[(n - 1)/2] && IntegerQ[m]

Rule 641

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*(a + c*x^2)^(p + 1))/(2*c*(p + 1)),
x] + Dist[d, Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, p}, x] && NeQ[p, -1]

Rule 216

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

Rubi steps

\begin{align*} \int \frac{e^{-\coth ^{-1}(a x)}}{x^2} \, dx &=-\operatorname{Subst}\left (\int \frac{1-\frac{x}{a}}{\sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=-a \sqrt{1-\frac{1}{a^2 x^2}}-\operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=-a \sqrt{1-\frac{1}{a^2 x^2}}-a \csc ^{-1}(a x)\\ \end{align*}

Mathematica [A]  time = 0.0201107, size = 26, normalized size = 1.04 $-a \left (\sqrt{1-\frac{1}{a^2 x^2}}+\sin ^{-1}\left (\frac{1}{a x}\right )\right )$

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [B]  time = 0.128, size = 220, normalized size = 8.8 \begin{align*}{\frac{ax+1}{x}\sqrt{{\frac{ax-1}{ax+1}}} \left ( -\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}{x}^{2}{a}^{2}+ \left ({a}^{2}{x}^{2}-1 \right ) ^{{\frac{3}{2}}}\sqrt{{a}^{2}}-\sqrt{{a}^{2}}\sqrt{{a}^{2}{x}^{2}-1}xa+\ln \left ({ \left ({a}^{2}x+\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}} \right ){\frac{1}{\sqrt{{a}^{2}}}}} \right ) x{a}^{2}-ax\sqrt{{a}^{2}}\arctan \left ({\frac{1}{\sqrt{{a}^{2}{x}^{2}-1}}} \right ) +\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }xa-\ln \left ({ \left ({a}^{2}x+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) } \right ){\frac{1}{\sqrt{{a}^{2}}}}} \right ) x{a}^{2} \right ){\frac{1}{\sqrt{{a}^{2}}}}{\frac{1}{\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((a*x-1)/(a*x+1))^(1/2)*(a*x+1)*(-(a^2*x^2-1)^(1/2)*(a^2)^(1/2)*x^2*a^2+(a^2*x^2-1)^(3/2)*(a^2)^(1/2)-(a^2)^(1
/2)*(a^2*x^2-1)^(1/2)*x*a+ln((a^2*x+(a^2*x^2-1)^(1/2)*(a^2)^(1/2))/(a^2)^(1/2))*x*a^2-a*x*(a^2)^(1/2)*arctan(1
/(a^2*x^2-1)^(1/2))+(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2)*x*a-ln((a^2*x+(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2))/(a^
2)^(1/2))*x*a^2)/((a*x-1)*(a*x+1))^(1/2)/x/(a^2)^(1/2)

________________________________________________________________________________________

Maxima [B]  time = 1.5047, size = 74, normalized size = 2.96 \begin{align*} -2 \, a{\left (\frac{\sqrt{\frac{a x - 1}{a x + 1}}}{\frac{a x - 1}{a x + 1} + 1} - \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right )\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*a*(sqrt((a*x - 1)/(a*x + 1))/((a*x - 1)/(a*x + 1) + 1) - arctan(sqrt((a*x - 1)/(a*x + 1))))

________________________________________________________________________________________

Fricas [B]  time = 1.86385, size = 112, normalized size = 4.48 \begin{align*} \frac{2 \, a x \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right ) -{\left (a x + 1\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*a*x*arctan(sqrt((a*x - 1)/(a*x + 1))) - (a*x + 1)*sqrt((a*x - 1)/(a*x + 1)))/x

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\frac{a x - 1}{a x + 1}}}{x^{2}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [B]  time = 1.1525, size = 84, normalized size = 3.36 \begin{align*} 2 \, a \arctan \left (-x{\left | a \right |} + \sqrt{a^{2} x^{2} - 1}\right ) \mathrm{sgn}\left (a x + 1\right ) - \frac{2 \,{\left | a \right |} \mathrm{sgn}\left (a x + 1\right )}{{\left (x{\left | a \right |} - \sqrt{a^{2} x^{2} - 1}\right )}^{2} + 1} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a*arctan(-x*abs(a) + sqrt(a^2*x^2 - 1))*sgn(a*x + 1) - 2*abs(a)*sgn(a*x + 1)/((x*abs(a) - sqrt(a^2*x^2 - 1))
^2 + 1)