3.80 \(\int \frac{1}{2+4 x-3 x^2} \, dx\)

Optimal. Leaf size=19 \[ -\frac{\tanh ^{-1}\left (\frac{2-3 x}{\sqrt{10}}\right )}{\sqrt{10}} \]

[Out]

-(ArcTanh[(2 - 3*x)/Sqrt[10]]/Sqrt[10])

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Rubi [A]  time = 0.0192329, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {618, 206} \[ -\frac{\tanh ^{-1}\left (\frac{2-3 x}{\sqrt{10}}\right )}{\sqrt{10}} \]

Antiderivative was successfully verified.

[In]

Int[(2 + 4*x - 3*x^2)^(-1),x]

[Out]

-(ArcTanh[(2 - 3*x)/Sqrt[10]]/Sqrt[10])

Rule 618

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

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])

Rubi steps

\begin{align*} \int \frac{1}{2+4 x-3 x^2} \, dx &=-\left (2 \operatorname{Subst}\left (\int \frac{1}{40-x^2} \, dx,x,4-6 x\right )\right )\\ &=-\frac{\tanh ^{-1}\left (\frac{2-3 x}{\sqrt{10}}\right )}{\sqrt{10}}\\ \end{align*}

Mathematica [A]  time = 0.020412, size = 34, normalized size = 1.79 \[ \frac{\log \left (3 x+\sqrt{10}-2\right )-\log \left (-3 x+\sqrt{10}+2\right )}{2 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

Integrate[(2 + 4*x - 3*x^2)^(-1),x]

[Out]

(-Log[2 + Sqrt[10] - 3*x] + Log[-2 + Sqrt[10] + 3*x])/(2*Sqrt[10])

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Maple [A]  time = 0.046, size = 17, normalized size = 0.9 \begin{align*}{\frac{\sqrt{10}}{10}{\it Artanh} \left ({\frac{ \left ( 6\,x-4 \right ) \sqrt{10}}{20}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*10^(1/2)*arctanh(1/20*(6*x-4)*10^(1/2))

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Maxima [A]  time = 1.49764, size = 36, normalized size = 1.89 \begin{align*} -\frac{1}{20} \, \sqrt{10} \log \left (\frac{3 \, x - \sqrt{10} - 2}{3 \, x + \sqrt{10} - 2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/20*sqrt(10)*log((3*x - sqrt(10) - 2)/(3*x + sqrt(10) - 2))

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Fricas [B]  time = 2.19858, size = 109, normalized size = 5.74 \begin{align*} \frac{1}{20} \, \sqrt{10} \log \left (\frac{9 \, x^{2} + 2 \, \sqrt{10}{\left (3 \, x - 2\right )} - 12 \, x + 14}{3 \, x^{2} - 4 \, x - 2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/20*sqrt(10)*log((9*x^2 + 2*sqrt(10)*(3*x - 2) - 12*x + 14)/(3*x^2 - 4*x - 2))

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Sympy [A]  time = 0.145693, size = 39, normalized size = 2.05 \begin{align*} \frac{\sqrt{10} \log{\left (x - \frac{2}{3} + \frac{\sqrt{10}}{3} \right )}}{20} - \frac{\sqrt{10} \log{\left (x - \frac{\sqrt{10}}{3} - \frac{2}{3} \right )}}{20} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(10)*log(x - 2/3 + sqrt(10)/3)/20 - sqrt(10)*log(x - sqrt(10)/3 - 2/3)/20

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Giac [A]  time = 1.23253, size = 42, normalized size = 2.21 \begin{align*} -\frac{1}{20} \, \sqrt{10} \log \left (\frac{{\left | 6 \, x - 2 \, \sqrt{10} - 4 \right |}}{{\left | 6 \, x + 2 \, \sqrt{10} - 4 \right |}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/20*sqrt(10)*log(abs(6*x - 2*sqrt(10) - 4)/abs(6*x + 2*sqrt(10) - 4))