3.2.14 \(\int \frac {1}{\sqrt {5-6 x+9 x^2}} \, dx\) [114]

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

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.00, antiderivative size = 14, 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 = {633, 221} \begin {gather*} \frac {1}{3} \sinh ^{-1}\left (\frac {1}{2} (3 x-1)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[5 - 6*x + 9*x^2],x]

[Out]

ArcSinh[(-1 + 3*x)/2]/3

Rule 221

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]

Rule 633

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]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {5-6 x+9 x^2}} \, dx &=\frac {1}{36} \text {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 = 0.05, size = 24, normalized size = 1.71 \begin {gather*} -\frac {1}{3} \log \left (1-3 x+\sqrt {5-6 x+9 x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[5 - 6*x + 9*x^2],x]

[Out]

-1/3*Log[1 - 3*x + Sqrt[5 - 6*x + 9*x^2]]

________________________________________________________________________________________

Maple [A]
time = 0.51, size = 9, normalized size = 0.64

method	result	size
default	\(\frac {\arcsinh \left (\frac {3 x}{2}-\frac {1}{2}\right )}{3}\)	\(9\)
trager	\(-\frac {\ln \left (\sqrt {9 x^{2}-6 x +5}+1-3 x \right )}{3}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.48, size = 8, normalized size = 0.57 \begin {gather*} \frac {1}{3} \, \operatorname {arsinh}\left (\frac {3}{2} \, x - \frac {1}{2}\right ) \end {gather*}

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)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 20 vs. \(2 (8) = 16\).
time = 2.70, size = 20, normalized size = 1.43 \begin {gather*} -\frac {1}{3} \, \log \left (-3 \, x + \sqrt {9 \, x^{2} - 6 \, x + 5} + 1\right ) \end {gather*}

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)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {9 x^{2} - 6 x + 5}}\, dx \end {gather*}

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)

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 40 vs. \(2 (8) = 16\).
time = 1.51, size = 40, normalized size = 2.86 \begin {gather*} \frac {1}{6} \, \sqrt {9 \, x^{2} - 6 \, x + 5} {\left (3 \, x - 1\right )} - \frac {2}{3} \, \log \left (-3 \, x + \sqrt {9 \, x^{2} - 6 \, x + 5} + 1\right ) \end {gather*}

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]

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

________________________________________________________________________________________

Mupad [B]
time = 0.20, size = 20, normalized size = 1.43 \begin {gather*} \frac {\ln \left (3\,x+\sqrt {9\,x^2-6\,x+5}-1\right )}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(9*x^2 - 6*x + 5)^(1/2),x)

[Out]

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

________________________________________________________________________________________