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

Optimal. Leaf size=22 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{x^2-a^2}}{a}\right )}{a} \]

[Out]

ArcTan[Sqrt[-a^2 + x^2]/a]/a

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Rubi [A]  time = 0.0125473, antiderivative size = 22, normalized size of antiderivative = 1., 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, 203} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{x^2-a^2}}{a}\right )}{a} \]

Antiderivative was successfully verified.

[In]

Int[1/(x*Sqrt[-a^2 + x^2]),x]

[Out]

ArcTan[Sqrt[-a^2 + x^2]/a]/a

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 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 203

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

Rubi steps

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

Mathematica [A]  time = 0.003801, size = 22, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{\sqrt{x^2-a^2}}{a}\right )}{a} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*Sqrt[-a^2 + x^2]),x]

[Out]

ArcTan[Sqrt[-a^2 + x^2]/a]/a

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Maple [A]  time = 0.003, size = 41, normalized size = 1.9 \begin{align*} -{\ln \left ({\frac{1}{x} \left ( -2\,{a}^{2}+2\,\sqrt{-{a}^{2}}\sqrt{-{a}^{2}+{x}^{2}} \right ) } \right ){\frac{1}{\sqrt{-{a}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.96431, size = 53, normalized size = 2.41 \begin{align*} \frac{2 \, \arctan \left (-\frac{x - \sqrt{-a^{2} + x^{2}}}{a}\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*arctan(-(x - sqrt(-a^2 + x^2))/a)/a

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Sympy [A]  time = 1.08734, size = 24, normalized size = 1.09 \begin{align*} \begin{cases} \frac{i \operatorname{acosh}{\left (\frac{a}{x} \right )}}{a} & \text{for}\: \frac{\left |{a^{2}}\right |}{\left |{x^{2}}\right |} > 1 \\- \frac{\operatorname{asin}{\left (\frac{a}{x} \right )}}{a} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.04928, size = 27, normalized size = 1.23 \begin{align*} \frac{\arctan \left (\frac{\sqrt{-a^{2} + x^{2}}}{a}\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctan(sqrt(-a^2 + x^2)/a)/a