3.74 \(\int \frac{1}{3-19 x^2+32 x^4-16 x^6} \, dx\)

Optimal. Leaf size=31 \[ \frac{1}{3} \tanh ^{-1}(x)+\frac{1}{3} \tanh ^{-1}(2 x)-\frac{\tanh ^{-1}\left (\frac{2 x}{\sqrt{3}}\right )}{\sqrt{3}} \]

[Out]

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

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Rubi [A]  time = 0.0228096, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2057, 207} \[ \frac{1}{3} \tanh ^{-1}(x)+\frac{1}{3} \tanh ^{-1}(2 x)-\frac{\tanh ^{-1}\left (\frac{2 x}{\sqrt{3}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Int[(3 - 19*x^2 + 32*x^4 - 16*x^6)^(-1),x]

[Out]

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

Rule 2057

Int[(P_)^(p_), x_Symbol] :> With[{u = Factor[P /. x -> Sqrt[x]]}, Int[ExpandIntegrand[(u /. x -> x^2)^p, x], x
] /;  !SumQ[NonfreeFactors[u, x]]] /; PolyQ[P, x^2] && ILtQ[p, 0]

Rule 207

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

Rubi steps

\begin{align*} \int \frac{1}{3-19 x^2+32 x^4-16 x^6} \, dx &=\int \left (-\frac{1}{3 \left (-1+x^2\right )}+\frac{2}{-3+4 x^2}-\frac{2}{3 \left (-1+4 x^2\right )}\right ) \, dx\\ &=-\left (\frac{1}{3} \int \frac{1}{-1+x^2} \, dx\right )-\frac{2}{3} \int \frac{1}{-1+4 x^2} \, dx+2 \int \frac{1}{-3+4 x^2} \, dx\\ &=\frac{1}{3} \tanh ^{-1}(x)+\frac{1}{3} \tanh ^{-1}(2 x)-\frac{\tanh ^{-1}\left (\frac{2 x}{\sqrt{3}}\right )}{\sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.0134657, size = 62, normalized size = 2. \[ \frac{1}{6} \left (-\log \left (2 x^2-3 x+1\right )+\log \left (2 x^2+3 x+1\right )+\sqrt{3} \log \left (\sqrt{3}-2 x\right )-\sqrt{3} \log \left (2 x+\sqrt{3}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(3 - 19*x^2 + 32*x^4 - 16*x^6)^(-1),x]

[Out]

(Sqrt[3]*Log[Sqrt[3] - 2*x] - Sqrt[3]*Log[Sqrt[3] + 2*x] - Log[1 - 3*x + 2*x^2] + Log[1 + 3*x + 2*x^2])/6

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Maple [A]  time = 0.01, size = 42, normalized size = 1.4 \begin{align*} -{\frac{\ln \left ( x-1 \right ) }{6}}-{\frac{\sqrt{3}}{3}{\it Artanh} \left ({\frac{2\,x\sqrt{3}}{3}} \right ) }-{\frac{\ln \left ( 2\,x-1 \right ) }{6}}+{\frac{\ln \left ( 1+2\,x \right ) }{6}}+{\frac{\ln \left ( 1+x \right ) }{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-16*x^6+32*x^4-19*x^2+3),x)

[Out]

-1/6*ln(x-1)-1/3*arctanh(2/3*x*3^(1/2))*3^(1/2)-1/6*ln(2*x-1)+1/6*ln(1+2*x)+1/6*ln(1+x)

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Maxima [B]  time = 1.73327, size = 73, normalized size = 2.35 \begin{align*} \frac{1}{6} \, \sqrt{3} \log \left (\frac{2 \, x - \sqrt{3}}{2 \, x + \sqrt{3}}\right ) + \frac{1}{6} \, \log \left (2 \, x + 1\right ) - \frac{1}{6} \, \log \left (2 \, x - 1\right ) + \frac{1}{6} \, \log \left (x + 1\right ) - \frac{1}{6} \, \log \left (x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-16*x^6+32*x^4-19*x^2+3),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.63208, size = 149, normalized size = 4.81 \begin{align*} \frac{1}{6} \, \sqrt{3} \log \left (\frac{4 \, x^{2} - 4 \, \sqrt{3} x + 3}{4 \, x^{2} - 3}\right ) + \frac{1}{6} \, \log \left (2 \, x^{2} + 3 \, x + 1\right ) - \frac{1}{6} \, \log \left (2 \, x^{2} - 3 \, x + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-16*x^6+32*x^4-19*x^2+3),x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 0.139756, size = 63, normalized size = 2.03 \begin{align*} \frac{\sqrt{3} \log{\left (x - \frac{\sqrt{3}}{2} \right )}}{6} - \frac{\sqrt{3} \log{\left (x + \frac{\sqrt{3}}{2} \right )}}{6} - \frac{\log{\left (x^{2} - \frac{3 x}{2} + \frac{1}{2} \right )}}{6} + \frac{\log{\left (x^{2} + \frac{3 x}{2} + \frac{1}{2} \right )}}{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-16*x**6+32*x**4-19*x**2+3),x)

[Out]

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

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Giac [B]  time = 1.12844, size = 84, normalized size = 2.71 \begin{align*} \frac{1}{6} \, \sqrt{3} \log \left (\frac{{\left | 8 \, x - 4 \, \sqrt{3} \right |}}{{\left | 8 \, x + 4 \, \sqrt{3} \right |}}\right ) + \frac{1}{6} \, \log \left ({\left | 2 \, x + 1 \right |}\right ) - \frac{1}{6} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) + \frac{1}{6} \, \log \left ({\left | x + 1 \right |}\right ) - \frac{1}{6} \, \log \left ({\left | x - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-16*x^6+32*x^4-19*x^2+3),x, algorithm="giac")

[Out]

1/6*sqrt(3)*log(abs(8*x - 4*sqrt(3))/abs(8*x + 4*sqrt(3))) + 1/6*log(abs(2*x + 1)) - 1/6*log(abs(2*x - 1)) + 1
/6*log(abs(x + 1)) - 1/6*log(abs(x - 1))