3.1.38 \(\int \frac {e^{-\coth ^{-1}(a x)}}{x^2} \, dx\) [38]

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

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6304, 655, 222} \begin {gather*} a \left (-\csc ^{-1}(a x)\right )-a \sqrt {1-\frac {1}{a^2 x^2}} \end {gather*}

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 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*} \int \frac {e^{-\coth ^{-1}(a x)}}{x^2} \, dx &=-\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*}

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Mathematica [A]
time = 0.01, size = 26, normalized size = 1.04 \begin {gather*} -a \left (\sqrt {1-\frac {1}{a^2 x^2}}+\text {ArcSin}\left (\frac {1}{a x}\right )\right ) \end {gather*}

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)]))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(219\) vs. \(2(23)=46\).
time = 0.08, size = 220, normalized size = 8.80

method	result	size
risch	\(-\frac {\left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}{x}-\frac {a \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right ) \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{a x -1}\)	\(77\)
default	\(\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) \left (-\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{2} x^{2}+\sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \sqrt {a^{2}}\, a x -\ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right ) a^{2} x +\left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}-\sqrt {a^{2}}\, \sqrt {a^{2} x^{2}-1}\, a x -\sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right ) a x +\ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{2} x \right )}{\sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \sqrt {a^{2}}\, x}\)	\(220\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 55 vs. \(2 (23) = 46\).
time = 0.47, size = 55, normalized size = 2.20 \begin {gather*} -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 {gather*}

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))))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 47 vs. \(2 (23) = 46\).
time = 0.34, size = 47, normalized size = 1.88 \begin {gather*} \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 {gather*}

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {\frac {a x - 1}{a x + 1}}}{x^{2}}\, dx \end {gather*}

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)

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Warning, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Ch
eck [abs(sa

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Mupad [B]
time = 1.20, size = 55, normalized size = 2.20 \begin {gather*} 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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a*x - 1)/(a*x + 1))^(1/2)/x^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)

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