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

Optimal. Leaf size=16 \[ \tanh ^{-1}\left (\frac {x}{\sqrt {x^2-a^2}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.00, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {217, 206} \[ \tanh ^{-1}\left (\frac {x}{\sqrt {x^2-a^2}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[-a^2 + x^2],x]

[Out]

ArcTanh[x/Sqrt[-a^2 + x^2]]

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 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rubi steps

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

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Mathematica [B]  time = 0.00, size = 46, normalized size = 2.88 \[ \frac {1}{2} \log \left (\frac {x}{\sqrt {x^2-a^2}}+1\right )-\frac {1}{2} \log \left (1-\frac {x}{\sqrt {x^2-a^2}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[-a^2 + x^2],x]

[Out]

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

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fricas [A]  time = 0.42, size = 18, normalized size = 1.12 \[ -\log \left (-x + \sqrt {-a^{2} + x^{2}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(-x + sqrt(-a^2 + x^2))

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giac [A]  time = 0.99, size = 19, normalized size = 1.19 \[ -\log \left ({\left | -x + \sqrt {-a^{2} + x^{2}} \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(abs(-x + sqrt(-a^2 + x^2)))

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maple [A]  time = 0.00, size = 15, normalized size = 0.94 \[ \ln \left (x +\sqrt {-a^{2}+x^{2}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.48, size = 18, normalized size = 1.12 \[ \log \left (2 \, x + 2 \, \sqrt {-a^{2} + x^{2}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(2*x + 2*sqrt(-a^2 + x^2))

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mupad [B]  time = 0.08, size = 14, normalized size = 0.88 \[ \ln \left (x+\sqrt {x^2-a^2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2 - a^2)^(1/2),x)

[Out]

log(x + (x^2 - a^2)^(1/2))

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sympy [A]  time = 1.08, size = 19, normalized size = 1.19 \[ \begin {cases} \operatorname {acosh}{\left (\frac {x}{a} \right )} & \text {for}\: \left |{\frac {x^{2}}{a^{2}}}\right | > 1 \\- i \operatorname {asin}{\left (\frac {x}{a} \right )} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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