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

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 = {6169, 641, 216} \[ a \sqrt {1-\frac {1}{a^2 x^2}}-a \csc ^{-1}(a x) \]

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 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]

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 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]

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*}

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Mathematica [A]  time = 0.02, size = 27, normalized size = 1.12 \[ a \left (\sqrt {1-\frac {1}{a^2 x^2}}-\sin ^{-1}\left (\frac {1}{a x}\right )\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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fricas [B]  time = 0.56, size = 46, normalized size = 1.92 \[ \frac {2 \, a x \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) + {\left (a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{x} \]

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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giac [B]  time = 0.14, size = 53, normalized size = 2.21 \[ 2 \, a {\left (\frac {\sqrt {\frac {a x - 1}{a x + 1}}}{\frac {a x - 1}{a x + 1} + 1} + \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right )\right )} \]

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*(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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maple [B]  time = 0.05, size = 220, normalized size = 9.17 \[ \frac {\left (a x -1\right ) \left (\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, x^{2} a^{2}+\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, x a -\left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}-\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, x a -a x \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )-\ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x \,a^{2}+\ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x \,a^{2}\right )}{\sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, x \sqrt {a^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a*x-1)*((a^2*x^2-1)^(1/2)*(a^2)^(1/2)*x^2*a^2+((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2)*x*a-(a^2*x^2-1)^(3/2)*(a^2)
^(1/2)-(a^2*x^2-1)^(1/2)*(a^2)^(1/2)*x*a-a*x*(a^2)^(1/2)*arctan(1/(a^2*x^2-1)^(1/2))-ln((a^2*x+(a^2*x^2-1)^(1/
2)*(a^2)^(1/2))/(a^2)^(1/2))*x*a^2+ln((a^2*x+((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2))/(a^2)^(1/2))*x*a^2)/((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]  time = 0.41, size = 53, normalized size = 2.21 \[ 2 \, a {\left (\frac {\sqrt {\frac {a x - 1}{a x + 1}}}{\frac {a x - 1}{a x + 1} + 1} + \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right )\right )} \]

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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mupad [B]  time = 0.05, size = 55, normalized size = 2.29 \[ 2\,a\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )+\frac {2\,a\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{\frac {a\,x-1}{a\,x+1}+1} \]

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

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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