3.6.77 \(\int \frac {1}{\sqrt {9-b x^2}} \, dx\) [577]

Optimal. Leaf size=17 \[ \frac {\sin ^{-1}\left (\frac {\sqrt {b} x}{3}\right )}{\sqrt {b}} \]

[Out]

arcsin(1/3*x*b^(1/2))/b^(1/2)

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Rubi [A]
time = 0.00, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {222} \begin {gather*} \frac {\text {ArcSin}\left (\frac {\sqrt {b} x}{3}\right )}{\sqrt {b}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[9 - b*x^2],x]

[Out]

ArcSin[(Sqrt[b]*x)/3]/Sqrt[b]

Rule 222

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

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {9-b x^2}} \, dx &=\frac {\sin ^{-1}\left (\frac {\sqrt {b} x}{3}\right )}{\sqrt {b}}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 33, normalized size = 1.94 \begin {gather*} \frac {b \log \left (-\sqrt {-b} x+\sqrt {9-b x^2}\right )}{(-b)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[9 - b*x^2],x]

[Out]

(b*Log[-(Sqrt[-b]*x) + Sqrt[9 - b*x^2]])/(-b)^(3/2)

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Maple [A]
time = 0.04, size = 21, normalized size = 1.24

method	result	size
meijerg	\(\frac {\arcsin \left (\frac {x \sqrt {b}}{3}\right )}{\sqrt {b}}\)	\(12\)
default	\(\frac {\arctan \left (\frac {\sqrt {b}\, x}{\sqrt {-b \,x^{2}+9}}\right )}{\sqrt {b}}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-b*x^2+9)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/b^(1/2)*arctan(b^(1/2)*x/(-b*x^2+9)^(1/2))

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Maxima [A]
time = 0.49, size = 11, normalized size = 0.65 \begin {gather*} \frac {\arcsin \left (\frac {1}{3} \, \sqrt {b} x\right )}{\sqrt {b}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-b*x^2+9)^(1/2),x, algorithm="maxima")

[Out]

arcsin(1/3*sqrt(b)*x)/sqrt(b)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 25 vs. \(2 (11) = 22\).
time = 1.54, size = 58, normalized size = 3.41 \begin {gather*} \left [-\frac {\sqrt {-b} \log \left (-\sqrt {-b} x - \sqrt {-b x^{2} + 9}\right )}{b}, -\frac {2 \, \arctan \left (\frac {\sqrt {-b x^{2} + 9} - 3}{\sqrt {b} x}\right )}{\sqrt {b}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-b*x^2+9)^(1/2),x, algorithm="fricas")

[Out]

[-sqrt(-b)*log(-sqrt(-b)*x - sqrt(-b*x^2 + 9))/b, -2*arctan((sqrt(-b*x^2 + 9) - 3)/(sqrt(b)*x))/sqrt(b)]

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Sympy [C] Result contains complex when optimal does not.
time = 0.44, size = 37, normalized size = 2.18 \begin {gather*} \begin {cases} - \frac {i \operatorname {acosh}{\left (\frac {\sqrt {b} x}{3} \right )}}{\sqrt {b}} & \text {for}\: \left |{b x^{2}}\right | > 9 \\\frac {\operatorname {asin}{\left (\frac {\sqrt {b} x}{3} \right )}}{\sqrt {b}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-b*x**2+9)**(1/2),x)

[Out]

Piecewise((-I*acosh(sqrt(b)*x/3)/sqrt(b), Abs(b*x**2) > 9), (asin(sqrt(b)*x/3)/sqrt(b), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 41 vs. \(2 (11) = 22\).
time = 0.70, size = 41, normalized size = 2.41 \begin {gather*} \frac {1}{2} \, \sqrt {-b x^{2} + 9} x - \frac {9 \, \log \left (-\sqrt {-b} x + \sqrt {-b x^{2} + 9}\right )}{2 \, \sqrt {-b}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-b*x^2+9)^(1/2),x, algorithm="giac")

[Out]

1/2*sqrt(-b*x^2 + 9)*x - 9/2*log(-sqrt(-b)*x + sqrt(-b*x^2 + 9))/sqrt(-b)

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Mupad [B]
time = 0.04, size = 15, normalized size = 0.88 \begin {gather*} \frac {\mathrm {asinh}\left (\frac {\sqrt {-b}\,x}{3}\right )}{\sqrt {-b}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(9 - b*x^2)^(1/2),x)

[Out]

asinh(((-b)^(1/2)*x)/3)/(-b)^(1/2)

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