3.6.87 \(\int \frac {1}{\sqrt {-a-b x^2}} \, dx\) [587]

Optimal. Leaf size=28 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {-a-b x^2}}\right )}{\sqrt {b}} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {223, 209} \begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {-a-b x^2}}\right )}{\sqrt {b}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[-a - b*x^2],x]

[Out]

ArcTan[(Sqrt[b]*x)/Sqrt[-a - b*x^2]]/Sqrt[b]

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

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Mathematica [A]
time = 0.00, size = 28, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {-a-b x^2}}\right )}{\sqrt {b}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[-a - b*x^2],x]

[Out]

ArcTan[(Sqrt[b]*x)/Sqrt[-a - b*x^2]]/Sqrt[b]

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Maple [A]
time = 0.03, size = 23, normalized size = 0.82

method	result	size
default	\(\frac {\arctan \left (\frac {x \sqrt {b}}{\sqrt {-b \,x^{2}-a}}\right )}{\sqrt {b}}\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [C] Result contains complex when optimal does not.
time = 0.31, size = 14, normalized size = 0.50 \begin {gather*} -\frac {i \, \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {b}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-I*arcsinh(b*x/sqrt(a*b))/sqrt(b)

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Fricas [A]
time = 1.50, size = 74, normalized size = 2.64 \begin {gather*} \left [-\frac {\sqrt {-b} \log \left (-2 \, b x^{2} + 2 \, \sqrt {-b x^{2} - a} \sqrt {-b} x - a\right )}{2 \, b}, -\frac {\arctan \left (\frac {\sqrt {-b x^{2} - a} \sqrt {b} x}{b x^{2} + a}\right )}{\sqrt {b}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/2*sqrt(-b)*log(-2*b*x^2 + 2*sqrt(-b*x^2 - a)*sqrt(-b)*x - a)/b, -arctan(sqrt(-b*x^2 - a)*sqrt(b)*x/(b*x^2
+ a))/sqrt(b)]

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Sympy [C] Result contains complex when optimal does not.
time = 0.45, size = 20, normalized size = 0.71 \begin {gather*} - \frac {i \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{\sqrt {b}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-I*asinh(sqrt(b)*x/sqrt(a))/sqrt(b)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 47 vs. \(2 (22) = 44\).
time = 0.76, size = 47, normalized size = 1.68 \begin {gather*} \frac {1}{2} \, \sqrt {-b x^{2} - a} x + \frac {a \log \left ({\left | -\sqrt {-b} x + \sqrt {-b x^{2} - a} \right |}\right )}{2 \, \sqrt {-b}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 5.13, size = 27, normalized size = 0.96 \begin {gather*} \frac {\ln \left (\sqrt {-b\,x^2-a}+\sqrt {-b}\,x\right )}{\sqrt {-b}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(- a - b*x^2)^(1/2),x)

[Out]

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

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