3.6.60 \(\int \frac {1}{x \sqrt {-9+4 x^2}} \, dx\) [560]

Optimal. Leaf size=20 \[ \frac {1}{3} \tan ^{-1}\left (\frac {1}{3} \sqrt {-9+4 x^2}\right ) \]

[Out]

1/3*arctan(1/3*(4*x^2-9)^(1/2))

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Rubi [A]
time = 0.01, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {272, 65, 209} \begin {gather*} \frac {1}{3} \text {ArcTan}\left (\frac {1}{3} \sqrt {4 x^2-9}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x*Sqrt[-9 + 4*x^2]),x]

[Out]

ArcTan[Sqrt[-9 + 4*x^2]/3]/3

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

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 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int \frac {1}{x \sqrt {-9+4 x^2}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{x \sqrt {-9+4 x}} \, dx,x,x^2\right )\\ &=\frac {1}{4} \text {Subst}\left (\int \frac {1}{\frac {9}{4}+\frac {x^2}{4}} \, dx,x,\sqrt {-9+4 x^2}\right )\\ &=\frac {1}{3} \tan ^{-1}\left (\frac {1}{3} \sqrt {-9+4 x^2}\right )\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 20, normalized size = 1.00 \begin {gather*} \frac {1}{3} \tan ^{-1}\left (\frac {1}{3} \sqrt {-9+4 x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x*Sqrt[-9 + 4*x^2]),x]

[Out]

ArcTan[Sqrt[-9 + 4*x^2]/3]/3

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Maple [A]
time = 0.12, size = 15, normalized size = 0.75

method	result	size
default	\(-\frac {\arctan \left (\frac {3}{\sqrt {4 x^{2}-9}}\right )}{3}\)	\(15\)
trager	\(\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\sqrt {4 x^{2}-9}+3 \RootOf \left (\textit {\_Z}^{2}+1\right )}{x}\right )}{3}\)	\(32\)
meijerg	\(\frac {\sqrt {-\mathrm {signum}\left (-1+\frac {4 x^{2}}{9}\right )}\, \left (-2 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {1-\frac {4 x^{2}}{9}}}{2}\right )+\left (2 \ln \left (x \right )-2 \ln \left (3\right )+i \pi \right ) \sqrt {\pi }\right )}{6 \sqrt {\pi }\, \sqrt {\mathrm {signum}\left (-1+\frac {4 x^{2}}{9}\right )}}\)	\(65\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(4*x^2-9)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/3*arctan(3/(4*x^2-9)^(1/2))

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Maxima [A]
time = 0.50, size = 9, normalized size = 0.45 \begin {gather*} -\frac {1}{3} \, \arcsin \left (\frac {3}{2 \, {\left | x \right |}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(4*x^2-9)^(1/2),x, algorithm="maxima")

[Out]

-1/3*arcsin(3/2/abs(x))

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Fricas [A]
time = 1.55, size = 18, normalized size = 0.90 \begin {gather*} \frac {2}{3} \, \arctan \left (-\frac {2}{3} \, x + \frac {1}{3} \, \sqrt {4 \, x^{2} - 9}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(4*x^2-9)^(1/2),x, algorithm="fricas")

[Out]

2/3*arctan(-2/3*x + 1/3*sqrt(4*x^2 - 9))

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Sympy [C] Result contains complex when optimal does not.
time = 0.46, size = 26, normalized size = 1.30 \begin {gather*} \begin {cases} \frac {i \operatorname {acosh}{\left (\frac {3}{2 x} \right )}}{3} & \text {for}\: \frac {1}{\left |{x^{2}}\right |} > \frac {4}{9} \\- \frac {\operatorname {asin}{\left (\frac {3}{2 x} \right )}}{3} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(4*x**2-9)**(1/2),x)

[Out]

Piecewise((I*acosh(3/(2*x))/3, 1/Abs(x**2) > 4/9), (-asin(3/(2*x))/3, True))

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Giac [A]
time = 0.78, size = 14, normalized size = 0.70 \begin {gather*} \frac {1}{3} \, \arctan \left (\frac {1}{3} \, \sqrt {4 \, x^{2} - 9}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(4*x^2-9)^(1/2),x, algorithm="giac")

[Out]

1/3*arctan(1/3*sqrt(4*x^2 - 9))

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Mupad [B]
time = 0.12, size = 20, normalized size = 1.00 \begin {gather*} \frac {\ln \left (\frac {\sqrt {4\,x^2-9}+3{}\mathrm {i}}{x}\right )\,1{}\mathrm {i}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(4*x^2 - 9)^(1/2)),x)

[Out]

(log(((4*x^2 - 9)^(1/2) + 3i)/x)*1i)/3

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