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

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 = {217, 203} \[ \tan ^{-1}\left (\frac {x}{\sqrt {a^2-x^2}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[a^2 - x^2],x]

[Out]

ArcTan[x/Sqrt[a^2 - x^2]]

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])

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 )\\ &=\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 \[ \tan ^{-1}\left (\frac {x}{\sqrt {a^2-x^2}}\right ) \]

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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fricas [A]  time = 0.40, size = 23, normalized size = 1.44 \[ -2 \, \arctan \left (-\frac {a - \sqrt {a^{2} - x^{2}}}{x}\right ) \]

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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giac [A]  time = 0.04, size = 9, normalized size = 0.56 \[ \arcsin \left (\frac {x}{a}\right ) \mathrm {sgn}\relax (a) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arcsin(x/a)*sgn(a)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.22, size = 6, normalized size = 0.38 \[ \arcsin \left (\frac {x}{a}\right ) \]

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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mupad [B]  time = 0.16, size = 14, normalized size = 0.88 \[ \mathrm {atan}\left (\frac {x}{\sqrt {a^2-x^2}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a^2 - x^2)^(1/2),x)

[Out]

atan(x/(a^2 - x^2)^(1/2))

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sympy [A]  time = 1.15, size = 19, normalized size = 1.19 \[ \begin {cases} - i \operatorname {acosh}{\left (\frac {x}{a} \right )} & \text {for}\: \left |{\frac {x^{2}}{a^{2}}}\right | > 1 \\\operatorname {asin}{\left (\frac {x}{a} \right )} & \text {otherwise} \end {cases} \]

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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