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

Optimal. Leaf size=20 \[ \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )+\csc ^{-1}(a x) \]

[Out]

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

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Rubi [A]  time = 0.05, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6169, 844, 216, 266, 63, 208} \[ \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )+\csc ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

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

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 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 844

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[g/e, Int[(d
+ e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a,
c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !IGtQ[m, 0]

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

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Mathematica [A]  time = 0.02, size = 34, normalized size = 1.70 \[ \log \left (x \left (\sqrt {\frac {a^2 x^2-1}{a^2 x^2}}+1\right )\right )+\sin ^{-1}\left (\frac {1}{a x}\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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fricas [B]  time = 0.57, size = 57, normalized size = 2.85 \[ -2 \, \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) + \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 0.16, size = 59, normalized size = 2.95 \[ -2 \, \arctan \left (-x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1}\right ) \mathrm {sgn}\left (a x + 1\right ) - \frac {a \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right ) \mathrm {sgn}\left (a x + 1\right )}{{\left | a \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.05, size = 133, normalized size = 6.65 \[ -\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) \left (\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}-\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}-\arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right ) \sqrt {a^{2}}-a \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right )\right )}{\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.40, size = 70, normalized size = 3.50 \[ -a {\left (\frac {2 \, \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right )}{a} - \frac {\log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a} + \frac {\log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.03, size = 37, normalized size = 1.85 \[ 2\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )-2\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a*x - 1)/(a*x + 1))^(1/2)/x,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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