3.2.22 \(\int \frac {1}{\sqrt {-2+4 x-3 x^2}} \, dx\) [122]

Optimal. Leaf size=33 \[ -\frac {\tan ^{-1}\left (\frac {2-3 x}{\sqrt {3} \sqrt {-2+4 x-3 x^2}}\right )}{\sqrt {3}} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 33, 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 = {635, 210} \begin {gather*} -\frac {\text {ArcTan}\left (\frac {2-3 x}{\sqrt {3} \sqrt {-3 x^2+4 x-2}}\right )}{\sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[-2 + 4*x - 3*x^2],x]

[Out]

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

Rule 210

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

Rule 635

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

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {-2+4 x-3 x^2}} \, dx &=2 \text {Subst}\left (\int \frac {1}{-12-x^2} \, dx,x,\frac {4-6 x}{\sqrt {-2+4 x-3 x^2}}\right )\\ &=-\frac {\tan ^{-1}\left (\frac {2-3 x}{\sqrt {3} \sqrt {-2+4 x-3 x^2}}\right )}{\sqrt {3}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.07, size = 33, normalized size = 1.00 \begin {gather*} \frac {i \log \left (2-3 x-i \sqrt {-6+12 x-9 x^2}\right )}{\sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[-2 + 4*x - 3*x^2],x]

[Out]

(I*Log[2 - 3*x - I*Sqrt[-6 + 12*x - 9*x^2]])/Sqrt[3]

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Maple [A]
time = 0.62, size = 26, normalized size = 0.79

method	result	size
default	\(\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x -\frac {2}{3}\right )}{\sqrt {-3 x^{2}+4 x -2}}\right )}{3}\)	\(26\)
trager	\(\frac {\RootOf \left (\textit {\_Z}^{2}+3\right ) \ln \left (-3 x \RootOf \left (\textit {\_Z}^{2}+3\right )+3 \sqrt {-3 x^{2}+4 x -2}+2 \RootOf \left (\textit {\_Z}^{2}+3\right )\right )}{3}\)	\(42\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [C] Result contains complex when optimal does not.
time = 0.55, size = 16, normalized size = 0.48 \begin {gather*} -\frac {1}{3} i \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {2} {\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-2)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [C] Result contains complex when optimal does not.
time = 1.54, size = 67, normalized size = 2.03 \begin {gather*} \frac {1}{6} i \, \sqrt {3} \log \left (-\frac {2 \, {\left (i \, \sqrt {3} \sqrt {-3 \, x^{2} + 4 \, x - 2} + 3 \, x - 2\right )}}{x}\right ) - \frac {1}{6} i \, \sqrt {3} \log \left (-\frac {2 \, {\left (-i \, \sqrt {3} \sqrt {-3 \, x^{2} + 4 \, x - 2} + 3 \, x - 2\right )}}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*I*sqrt(3)*log(-2*(I*sqrt(3)*sqrt(-3*x^2 + 4*x - 2) + 3*x - 2)/x) - 1/6*I*sqrt(3)*log(-2*(-I*sqrt(3)*sqrt(-
3*x^2 + 4*x - 2) + 3*x - 2)/x)

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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 - 2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(-3*x**2 + 4*x - 2), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.14, size = 17, normalized size = 0.52 \begin {gather*} -\frac {\sqrt {3}\,\mathrm {asin}\left (\sqrt {2}\,\left (\frac {3\,x}{2}-1\right )\,1{}\mathrm {i}\right )}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(4*x - 3*x^2 - 2)^(1/2),x)

[Out]

-(3^(1/2)*asin(2^(1/2)*((3*x)/2 - 1)*1i))/3

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