3.468 \(\int \sqrt {-9+4 x^2} \, dx\)

Optimal. Leaf size=36 \[ \frac {1}{2} x \sqrt {4 x^2-9}-\frac {9}{4} \tanh ^{-1}\left (\frac {2 x}{\sqrt {4 x^2-9}}\right ) \]

[Out]

-9/4*arctanh(2*x/(4*x^2-9)^(1/2))+1/2*x*(4*x^2-9)^(1/2)

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Rubi [A]  time = 0.01, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {195, 217, 206} \[ \frac {1}{2} x \sqrt {4 x^2-9}-\frac {9}{4} \tanh ^{-1}\left (\frac {2 x}{\sqrt {4 x^2-9}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[-9 + 4*x^2],x]

[Out]

(x*Sqrt[-9 + 4*x^2])/2 - (9*ArcTanh[(2*x)/Sqrt[-9 + 4*x^2]])/4

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[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 \sqrt {-9+4 x^2} \, dx &=\frac {1}{2} x \sqrt {-9+4 x^2}-\frac {9}{2} \int \frac {1}{\sqrt {-9+4 x^2}} \, dx\\ &=\frac {1}{2} x \sqrt {-9+4 x^2}-\frac {9}{2} \operatorname {Subst}\left (\int \frac {1}{1-4 x^2} \, dx,x,\frac {x}{\sqrt {-9+4 x^2}}\right )\\ &=\frac {1}{2} x \sqrt {-9+4 x^2}-\frac {9}{4} \tanh ^{-1}\left (\frac {2 x}{\sqrt {-9+4 x^2}}\right )\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 37, normalized size = 1.03 \[ \frac {1}{2} x \sqrt {4 x^2-9}-\frac {9}{4} \log \left (\sqrt {4 x^2-9}+2 x\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[-9 + 4*x^2],x]

[Out]

(x*Sqrt[-9 + 4*x^2])/2 - (9*Log[2*x + Sqrt[-9 + 4*x^2]])/4

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fricas [A]  time = 0.74, size = 29, normalized size = 0.81 \[ \frac {1}{2} \, \sqrt {4 \, x^{2} - 9} x + \frac {9}{4} \, \log \left (-2 \, x + \sqrt {4 \, x^{2} - 9}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 1.11, size = 30, normalized size = 0.83 \[ \frac {1}{2} \, \sqrt {4 \, x^{2} - 9} x + \frac {9}{4} \, \log \left ({\left | -2 \, x + \sqrt {4 \, x^{2} - 9} \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(4*x^2 - 9)*x + 9/4*log(abs(-2*x + sqrt(4*x^2 - 9)))

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maple [A]  time = 0.00, size = 35, normalized size = 0.97 \[ \frac {\sqrt {4 x^{2}-9}\, x}{2}-\frac {9 \sqrt {4}\, \ln \left (\sqrt {4}\, x +\sqrt {4 x^{2}-9}\right )}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(4*x^2-9)^(1/2)*x-9/8*4^(1/2)*ln(4^(1/2)*x+(4*x^2-9)^(1/2))

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maxima [A]  time = 2.97, size = 31, normalized size = 0.86 \[ \frac {1}{2} \, \sqrt {4 \, x^{2} - 9} x - \frac {9}{4} \, \log \left (8 \, x + 4 \, \sqrt {4 \, x^{2} - 9}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(4*x^2 - 9)*x - 9/4*log(8*x + 4*sqrt(4*x^2 - 9))

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mupad [B]  time = 5.03, size = 29, normalized size = 0.81 \[ \frac {x\,\sqrt {4\,x^2-9}}{2}-\frac {9\,\ln \left (2\,x+\sqrt {4\,x^2-9}\right )}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x^2 - 9)^(1/2),x)

[Out]

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

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sympy [A]  time = 0.22, size = 22, normalized size = 0.61 \[ \frac {x \sqrt {4 x^{2} - 9}}{2} - \frac {9 \operatorname {acosh}{\left (\frac {2 x}{3} \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*sqrt(4*x**2 - 9)/2 - 9*acosh(2*x/3)/4

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