3.6.86 \(\int \frac {1}{\sqrt {-a+b x^2}} \, dx\) [586]

Optimal. Leaf size=27 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {-a+b x^2}}\right )}{\sqrt {b}} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 27, 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*} \frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {b x^2-a}}\right )}{\sqrt {b}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[-a + b*x^2],x]

[Out]

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

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+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 {\tanh ^{-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 = 27, normalized size = 1.00 \begin {gather*} \frac {\tanh ^{-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]

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

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Maple [A]
time = 0.03, size = 23, normalized size = 0.85

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.30, size = 26, normalized size = 0.96 \begin {gather*} \frac {\log \left (2 \, b x + 2 \, \sqrt {b x^{2} - a} \sqrt {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]

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

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Fricas [A]
time = 1.21, size = 63, normalized size = 2.33 \begin {gather*} \left [\frac {\log \left (2 \, b x^{2} + 2 \, \sqrt {b x^{2} - a} \sqrt {b} x - a\right )}{2 \, \sqrt {b}}, -\frac {\sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} - a}}\right )}{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*log(2*b*x^2 + 2*sqrt(b*x^2 - a)*sqrt(b)*x - a)/sqrt(b), -sqrt(-b)*arctan(sqrt(-b)*x/sqrt(b*x^2 - a))/b]

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Sympy [C] Result contains complex when optimal does not.
time = 0.48, size = 46, normalized size = 1.70 \begin {gather*} \begin {cases} \frac {\operatorname {acosh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{\sqrt {b}} & \text {for}\: \left |{\frac {b x^{2}}{a}}\right | > 1 \\- \frac {i \operatorname {asin}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \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-a)**(1/2),x)

[Out]

Piecewise((acosh(sqrt(b)*x/sqrt(a))/sqrt(b), Abs(b*x**2/a) > 1), (-I*asin(sqrt(b)*x/sqrt(a))/sqrt(b), True))

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Giac [A]
time = 0.70, size = 41, normalized size = 1.52 \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 = 0.12, size = 22, normalized size = 0.81 \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/(b*x^2 - a)^(1/2),x)

[Out]

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

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