\(\int \frac {e^{\coth ^{-1}(a x)}}{x^2} \, dx\) [6]

Optimal result

Integrand size = 10, antiderivative size = 24 \[ \int \frac {e^{\coth ^{-1}(a x)}}{x^2} \, dx=a \sqrt {1-\frac {1}{a^2 x^2}}-a \csc ^{-1}(a x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {6304, 655, 222} \[ \int \frac {e^{\coth ^{-1}(a x)}}{x^2} \, dx=a \sqrt {1-\frac {1}{a^2 x^2}}-a \csc ^{-1}(a x) \]

[In]

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

[Out]

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

Rule 222

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]

Rule 655

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 6304

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]

Rubi steps \begin{align*} \text {integral}& = -\text {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}}-\text {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] (verified)

Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.12 \[ \int \frac {e^{\coth ^{-1}(a x)}}{x^2} \, dx=a \left (\sqrt {1-\frac {1}{a^2 x^2}}-\arcsin \left (\frac {1}{a x}\right )\right ) \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(75\) vs. \(2(22)=44\).

Time = 0.13 (sec) , antiderivative size = 76, normalized size of antiderivative = 3.17

method	result	size
risch	\(\frac {a x -1}{x \sqrt {\frac {a x -1}{a x +1}}}-\frac {a \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right ) \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}\)	\(76\)
default	\(-\frac {\left (a x -1\right ) \left (-\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{2} x^{2}+\left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}+\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a x +\ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{2} x +a x \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )-\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a x -\ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{2} x \right )}{\sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, x \sqrt {a^{2}}}\)	\(220\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (22) = 44\).

Time = 0.25 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.92 \[ \int \frac {e^{\coth ^{-1}(a x)}}{x^2} \, dx=\frac {2 \, a x \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) + {\left (a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{x} \]

[In]

integrate(1/((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]

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

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (22) = 44\).

Time = 0.28 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.21 \[ \int \frac {e^{\coth ^{-1}(a x)}}{x^2} \, dx=2 \, a {\left (\frac {\sqrt {\frac {a x - 1}{a x + 1}}}{\frac {a x - 1}{a x + 1} + 1} + \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right )\right )} \]

[In]

integrate(1/((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))))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (22) = 44\).

Time = 0.26 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.75 \[ \int \frac {e^{\coth ^{-1}(a x)}}{x^2} \, dx=\frac {2 \, a \arctan \left (-x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1}\right )}{\mathrm {sgn}\left (a x + 1\right )} + \frac {2 \, {\left | a \right |}}{{\left ({\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{2} + 1\right )} \mathrm {sgn}\left (a x + 1\right )} \]

[In]

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

Mupad [B] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 55, normalized size of antiderivative = 2.29 \[ \int \frac {e^{\coth ^{-1}(a x)}}{x^2} \, dx=2\,a\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )+\frac {2\,a\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{\frac {a\,x-1}{a\,x+1}+1} \]

[In]

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

[Out]

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