3.6.88 \(\int \frac {1}{\sqrt {a^2-x^2}} \, dx\) [588]

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

[Out]

arctan(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, 209} \begin {gather*} \text {ArcTan}\left (\frac {x}{\sqrt {a^2-x^2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[a^2 - x^2],x]

[Out]

ArcTan[x/Sqrt[a^2 - x^2]]

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[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 )\\ &=\tan ^{-1}\left (\frac {x}{\sqrt {a^2-x^2}}\right )\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 16, normalized size = 1.00 \begin {gather*} \tan ^{-1}\left (\frac {x}{\sqrt {a^2-x^2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[a^2 - x^2],x]

[Out]

ArcTan[x/Sqrt[a^2 - x^2]]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctan(x/(a^2-x^2)^(1/2))

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Maxima [A]
time = 0.53, size = 6, normalized size = 0.38 \begin {gather*} \arcsin \left (\frac {x}{a}\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]

arcsin(x/a)

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Fricas [A]
time = 1.21, size = 23, normalized size = 1.44 \begin {gather*} -2 \, \arctan \left (-\frac {a - \sqrt {a^{2} - x^{2}}}{x}\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]

-2*arctan(-(a - sqrt(a^2 - x^2))/x)

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Sympy [C] Result contains complex when optimal does not.
time = 0.44, size = 19, normalized size = 1.19 \begin {gather*} \begin {cases} - i \operatorname {acosh}{\left (\frac {x}{a} \right )} & \text {for}\: \left |{\frac {x^{2}}{a^{2}}}\right | > 1 \\\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((-I*acosh(x/a), Abs(x**2/a**2) > 1), (asin(x/a), True))

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Giac [A]
time = 0.65, size = 28, normalized size = 1.75 \begin {gather*} \frac {1}{2} \, a^{2} \arcsin \left (\frac {x}{a}\right ) \mathrm {sgn}\left (a\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*arcsin(x/a)*sgn(a) + 1/2*sqrt(a^2 - x^2)*x

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Mupad [B]
time = 4.74, size = 14, normalized size = 0.88 \begin {gather*} \mathrm {atan}\left (\frac {x}{\sqrt {a^2-x^2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a^2 - x^2)^(1/2),x)

[Out]

atan(x/(a^2 - x^2)^(1/2))

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