3.1.80 \(\int \frac {1}{2+4 x-3 x^2} \, dx\) [80]

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

[Out]

-1/10*arctanh(1/10*(2-3*x)*10^(1/2))*10^(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 = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {632, 212} \begin {gather*} -\frac {\tanh ^{-1}\left (\frac {2-3 x}{\sqrt {10}}\right )}{\sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2 + 4*x - 3*x^2)^(-1),x]

[Out]

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

Rule 212

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

Rule 632

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]

Rubi steps

\begin {align*} \int \frac {1}{2+4 x-3 x^2} \, dx &=-\left (2 \text {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*}

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Mathematica [A]
time = 0.01, size = 34, normalized size = 1.79 \begin {gather*} \frac {-\log \left (2+\sqrt {10}-3 x\right )+\log \left (-2+\sqrt {10}+3 x\right )}{2 \sqrt {10}} \end {gather*}

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 = 0.55, size = 17, normalized size = 0.89

method	result	size
default	\(\frac {\sqrt {10}\, \arctanh \left (\frac {\left (6 x -4\right ) \sqrt {10}}{20}\right )}{10}\)	\(17\)
risch	\(\frac {\sqrt {10}\, \ln \left (3 x -2+\sqrt {10}\right )}{20}-\frac {\sqrt {10}\, \ln \left (3 x -2-\sqrt {10}\right )}{20}\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.51, size = 27, normalized size = 1.42 \begin {gather*} -\frac {1}{20} \, \sqrt {10} \log \left (\frac {3 \, x - \sqrt {10} - 2}{3 \, x + \sqrt {10} - 2}\right ) \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 39 vs. \(2 (16) = 32\).
time = 1.98, size = 39, normalized size = 2.05 \begin {gather*} \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 {gather*}

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 = 0.04, size = 39, normalized size = 2.05 \begin {gather*} \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 {gather*}

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 = 0.84, size = 31, normalized size = 1.63 \begin {gather*} -\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 {gather*}

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))

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Mupad [B]
time = 0.21, size = 15, normalized size = 0.79 \begin {gather*} \frac {\sqrt {10}\,\mathrm {atanh}\left (\sqrt {10}\,\left (\frac {3\,x}{10}-\frac {1}{5}\right )\right )}{10} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(4*x - 3*x^2 + 2),x)

[Out]

(10^(1/2)*atanh(10^(1/2)*((3*x)/10 - 1/5)))/10

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