3.2.21 \(\int \frac {1}{\sqrt {-a^2+x^2}} \, dx\) [121]

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

[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 = {223, 212} \begin {gather*} \tanh ^{-1}\left (\frac {x}{\sqrt {x^2-a^2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[-a^2 + x^2],x]

[Out]

ArcTanh[x/Sqrt[-a^2 + x^2]]

Rule 212

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

Rule 223

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 &=\text {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] Leaf count is larger than twice the leaf count of optimal. \(46\) vs. \(2(16)=32\).
time = 0.00, size = 46, normalized size = 2.88 \begin {gather*} -\frac {1}{2} \log \left (1-\frac {x}{\sqrt {-a^2+x^2}}\right )+\frac {1}{2} \log \left (1+\frac {x}{\sqrt {-a^2+x^2}}\right ) \end {gather*}

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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Maple [A]
time = 0.04, size = 15, normalized size = 0.94

method	result	size
default	\(\ln \left (x +\sqrt {-a^{2}+x^{2}}\right )\)	\(15\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 2.00, size = 18, normalized size = 1.12 \begin {gather*} \log \left (2 \, x + 2 \, \sqrt {-a^{2} + x^{2}}\right ) \end {gather*}

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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Fricas [A]
time = 0.55, size = 18, normalized size = 1.12 \begin {gather*} -\log \left (-x + \sqrt {-a^{2} + x^{2}}\right ) \end {gather*}

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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Sympy [C] Result contains complex when optimal does not.
time = 0.45, size = 19, normalized size = 1.19 \begin {gather*} \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} \end {gather*}

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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 37 vs. \(2 (14) = 28\).
time = 0.86, size = 37, normalized size = 2.31 \begin {gather*} \frac {1}{2} \, a^{2} \log \left ({\left | -x + \sqrt {-a^{2} + x^{2}} \right |}\right ) + \frac {1}{2} \, \sqrt {-a^{2} + x^{2}} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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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