3.114 \(\int \frac{1}{\sqrt{5-6 x+9 x^2}} \, dx\)

Optimal. Leaf size=14 \[ \frac{1}{3} \sinh ^{-1}\left (\frac{1}{2} (3 x-1)\right ) \]

[Out]

ArcSinh[(-1 + 3*x)/2]/3

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Rubi [A]  time = 0.0070962, antiderivative size = 14, normalized size of antiderivative = 1., 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 = {619, 215} \[ \frac{1}{3} \sinh ^{-1}\left (\frac{1}{2} (3 x-1)\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[5 - 6*x + 9*x^2],x]

[Out]

ArcSinh[(-1 + 3*x)/2]/3

Rule 619

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[1/(2*c*((-4*c)/(b^2 - 4*a*c))^p), Subst[Int[Si
mp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]

Rule 215

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

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{5-6 x+9 x^2}} \, dx &=\frac{1}{36} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{144}}} \, dx,x,-6+18 x\right )\\ &=\frac{1}{3} \sinh ^{-1}\left (\frac{1}{2} (-1+3 x)\right )\\ \end{align*}

Mathematica [A]  time = 0.0051132, size = 14, normalized size = 1. \[ \frac{1}{3} \sinh ^{-1}\left (\frac{1}{2} (3 x-1)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[5 - 6*x + 9*x^2],x]

[Out]

ArcSinh[(-1 + 3*x)/2]/3

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Maple [A]  time = 0.045, size = 9, normalized size = 0.6 \begin{align*}{\frac{1}{3}{\it Arcsinh} \left ( -{\frac{1}{2}}+{\frac{3\,x}{2}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(9*x^2-6*x+5)^(1/2),x)

[Out]

1/3*arcsinh(-1/2+3/2*x)

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Maxima [A]  time = 1.69956, size = 11, normalized size = 0.79 \begin{align*} \frac{1}{3} \, \operatorname{arsinh}\left (\frac{3}{2} \, x - \frac{1}{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(9*x^2-6*x+5)^(1/2),x, algorithm="maxima")

[Out]

1/3*arcsinh(3/2*x - 1/2)

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Fricas [B]  time = 2.11903, size = 59, normalized size = 4.21 \begin{align*} -\frac{1}{3} \, \log \left (-3 \, x + \sqrt{9 \, x^{2} - 6 \, x + 5} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(9*x^2-6*x+5)^(1/2),x, algorithm="fricas")

[Out]

-1/3*log(-3*x + sqrt(9*x^2 - 6*x + 5) + 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{9 x^{2} - 6 x + 5}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(9*x**2-6*x+5)**(1/2),x)

[Out]

Integral(1/sqrt(9*x**2 - 6*x + 5), x)

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Giac [B]  time = 1.27011, size = 27, normalized size = 1.93 \begin{align*} -\frac{1}{3} \, \log \left (-3 \, x + \sqrt{9 \, x^{2} - 6 \, x + 5} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(9*x^2-6*x+5)^(1/2),x, algorithm="giac")

[Out]

-1/3*log(-3*x + sqrt(9*x^2 - 6*x + 5) + 1)