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

Optimal. Leaf size=24 \[ a \sqrt {1-\frac {1}{a^2 x^2}}-a \csc ^{-1}(a x) \]

[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 = 24, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {6304, 655, 222} \begin {gather*} a \sqrt {1-\frac {1}{a^2 x^2}}-a \csc ^{-1}(a x) \end {gather*}

Antiderivative was successfully verified.

[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*} \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 = 27, normalized size = 1.12 \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[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. \(2(22)=44\).
time = 0.09, size = 220, normalized size = 9.17

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	\(\frac {\left (a x -1\right ) \left (\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{2} x^{2}+\sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \sqrt {a^{2}}\, a x -\left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}-\sqrt {a^{2}}\, \sqrt {a^{2} x^{2}-1}\, a x -\ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{2} x -a x \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )+\ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right ) a^{2} x \right )}{\sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, x \sqrt {a^{2}}}\)	\(220\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (22) = 44\).
time = 0.46, size = 53, normalized size = 2.21 \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(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))))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (22) = 44\).
time = 0.35, size = 46, normalized size = 1.92 \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(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

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (22) = 44\).
time = 0.40, size = 66, normalized size = 2.75 \begin {gather*} \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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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Mupad [B]
time = 0.05, size = 55, normalized size = 2.29 \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(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)

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