3.52 \(\int \frac {1}{x \sqrt {a^2-x^2}} \, dx\)

Optimal. Leaf size=23 \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {a^2-x^2}}{a}\right )}{a} \]

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Rubi [A]  time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {266, 63, 206} \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {a^2-x^2}}{a}\right )}{a} \]

Antiderivative was successfully verified.

[In]

Int[1/(x*Sqrt[a^2 - x^2]),x]

[Out]

-(ArcTanh[Sqrt[a^2 - x^2]/a]/a)

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 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

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]]

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 23, normalized size = 1.00 \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {a^2-x^2}}{a}\right )}{a} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*Sqrt[a^2 - x^2]),x]

[Out]

-(ArcTanh[Sqrt[a^2 - x^2]/a]/a)

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IntegrateAlgebraic [A]  time = 0.02, size = 23, normalized size = 1.00 \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {a^2-x^2}}{a}\right )}{a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(ArcTanh[Sqrt[a^2 - x^2]/a]/a)

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fricas [A]  time = 0.88, size = 25, normalized size = 1.09 \[ \frac {\log \left (-\frac {a - \sqrt {a^{2} - x^{2}}}{x}\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(-(a - sqrt(a^2 - x^2))/x)/a

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giac [B]  time = 0.85, size = 43, normalized size = 1.87 \[ -\frac {\log \left ({\left | a + \sqrt {a^{2} - x^{2}} \right |}\right )}{2 \, a} + \frac {\log \left ({\left | -a + \sqrt {a^{2} - x^{2}} \right |}\right )}{2 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.32, size = 37, normalized size = 1.61

method	result	size
default	\(-\frac {\ln \left (\frac {2 a^{2}+2 \sqrt {a^{2}}\, \sqrt {a^{2}-x^{2}}}{x}\right )}{\sqrt {a^{2}}}\)	\(37\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.43, size = 34, normalized size = 1.48 \[ -\frac {\log \left (\frac {2 \, a^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {a^{2} - x^{2}} a}{{\left | x \right |}}\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.44, size = 21, normalized size = 0.91 \[ -\frac {\mathrm {atanh}\left (\frac {\sqrt {a^2-x^2}}{a}\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(a^2 - x^2)^(1/2)),x)

[Out]

-atanh((a^2 - x^2)^(1/2)/a)/a

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sympy [A]  time = 1.09, size = 22, normalized size = 0.96 \[ \begin {cases} - \frac {\operatorname {acosh}{\left (\frac {a}{x} \right )}}{a} & \text {for}\: \left |{\frac {a^{2}}{x^{2}}}\right | > 1 \\\frac {i \operatorname {asin}{\left (\frac {a}{x} \right )}}{a} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-acosh(a/x)/a, Abs(a**2/x**2) > 1), (I*asin(a/x)/a, True))

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