3.1.55 \(\int \frac {1}{\sqrt {5-4 x+3 x^2}} \, dx\) [55]

Optimal. Leaf size=19 \[ -\frac {\sinh ^{-1}\left (\frac {2-3 x}{\sqrt {11}}\right )}{\sqrt {3}} \]

[Out]

-1/3*arcsinh(1/11*(2-3*x)*11^(1/2))*3^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 19, 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 {\sinh ^{-1}\left (\frac {2-3 x}{\sqrt {11}}\right )}{\sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[5 - 4*x + 3*x^2],x]

[Out]

-(ArcSinh[(2 - 3*x)/Sqrt[11]]/Sqrt[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 {5-4 x+3 x^2}} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{44}}} \, dx,x,-4+6 x\right )}{2 \sqrt {33}}\\ &=-\frac {\sinh ^{-1}\left (\frac {2-3 x}{\sqrt {11}}\right )}{\sqrt {3}}\\ \end {align*}

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Mathematica [A]
time = 0.06, size = 33, normalized size = 1.74 \begin {gather*} -\frac {\log \left (2-3 x+\sqrt {3} \sqrt {5-4 x+3 x^2}\right )}{\sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[5 - 4*x + 3*x^2],x]

[Out]

-(Log[2 - 3*x + Sqrt[3]*Sqrt[5 - 4*x + 3*x^2]]/Sqrt[3])

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Maple [A]
time = 0.23, size = 15, normalized size = 0.79

method	result	size
default	\(\frac {\sqrt {3}\, \arcsinh \left (\frac {3 \sqrt {11}\, \left (x -\frac {2}{3}\right )}{11}\right )}{3}\)	\(15\)
trager	\(\frac {\RootOf \left (\textit {\_Z}^{2}-3\right ) \ln \left (3 \RootOf \left (\textit {\_Z}^{2}-3\right ) x +3 \sqrt {3 x^{2}-4 x +5}-2 \RootOf \left (\textit {\_Z}^{2}-3\right )\right )}{3}\)	\(42\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(3*x^2-4*x+5)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/3*3^(1/2)*arcsinh(3/11*11^(1/2)*(x-2/3))

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Maxima [A]
time = 3.86, size = 16, normalized size = 0.84 \begin {gather*} \frac {1}{3} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{11} \, \sqrt {11} {\left (3 \, x - 2\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3*x^2-4*x+5)^(1/2),x, algorithm="maxima")

[Out]

1/3*sqrt(3)*arcsinh(1/11*sqrt(11)*(3*x - 2))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 38 vs. \(2 (16) = 32\).
time = 0.69, size = 38, normalized size = 2.00 \begin {gather*} \frac {1}{6} \, \sqrt {3} \log \left (-2 \, \sqrt {3} \sqrt {3 \, x^{2} - 4 \, x + 5} {\left (3 \, x - 2\right )} - 18 \, x^{2} + 24 \, x - 19\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3*x^2-4*x+5)^(1/2),x, algorithm="fricas")

[Out]

1/6*sqrt(3)*log(-2*sqrt(3)*sqrt(3*x^2 - 4*x + 5)*(3*x - 2) - 18*x^2 + 24*x - 19)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {3 x^{2} - 4 x + 5}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3*x**2-4*x+5)**(1/2),x)

[Out]

Integral(1/sqrt(3*x**2 - 4*x + 5), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (16) = 32\).
time = 0.77, size = 53, normalized size = 2.79 \begin {gather*} \frac {1}{6} \, \sqrt {3 \, x^{2} - 4 \, x + 5} {\left (3 \, x - 2\right )} - \frac {11}{18} \, \sqrt {3} \log \left (-\sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} - 4 \, x + 5}\right )} + 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3*x^2-4*x+5)^(1/2),x, algorithm="giac")

[Out]

1/6*sqrt(3*x^2 - 4*x + 5)*(3*x - 2) - 11/18*sqrt(3)*log(-sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 - 4*x + 5)) + 2)

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Mupad [B]
time = 0.29, size = 26, normalized size = 1.37 \begin {gather*} \frac {\sqrt {3}\,\ln \left (\sqrt {3}\,\left (x-\frac {2}{3}\right )+\sqrt {3\,x^2-4\,x+5}\right )}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(3*x^2 - 4*x + 5)^(1/2),x)

[Out]

(3^(1/2)*log(3^(1/2)*(x - 2/3) + (3*x^2 - 4*x + 5)^(1/2)))/3

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