3.14 \(\int \frac{6-3 x^2+x^4}{4+5 x^2-5 x^4+x^6} \, dx\)

Optimal. Leaf size=39 \[ \tan ^{-1}\left (\frac{1}{2} x \left (x^4-3 x^2+1\right )\right )-\tan ^{-1}\left (\sqrt{3}-2 x\right )+\tan ^{-1}\left (2 x+\sqrt{3}\right ) \]

[Out]

-ArcTan[Sqrt[3] - 2*x] + ArcTan[Sqrt[3] + 2*x] + ArcTan[(x*(1 - 3*x^2 + x^4))/2]

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Rubi [A]  time = 0.025965, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.036, Rules used = {2095} \[ \tan ^{-1}\left (\frac{1}{2} x \left (x^4-3 x^2+1\right )\right )-\tan ^{-1}\left (\sqrt{3}-2 x\right )+\tan ^{-1}\left (2 x+\sqrt{3}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(6 - 3*x^2 + x^4)/(4 + 5*x^2 - 5*x^4 + x^6),x]

[Out]

-ArcTan[Sqrt[3] - 2*x] + ArcTan[Sqrt[3] + 2*x] + ArcTan[(x*(1 - 3*x^2 + x^4))/2]

Rule 2095

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)/((d_) + (e_.)*(x_)^2 + (f_.)*(x_)^4 + (g_.)*(x_)^6), x_Symbol] :> Wit
h[{q = Rt[(-(a*c*f^2) + 12*a^2*g^2 + f*(3*c^2*d - 2*a*b*g))/(c*g*(3*c*d - a*f)), 2], r = Rt[(a*c*f^2 + 4*g*(b*
c*d + a^2*g) - f*(3*c^2*d + 2*a*b*g))/(c*g*(3*c*d - a*f)), 2]}, Simp[(c*ArcTan[(r + 2*x)/q])/(g*q), x] + (-Sim
p[(c*ArcTan[(r - 2*x)/q])/(g*q), x] - Simp[(c*ArcTan[((3*c*d - a*f)*x*(b*c^2*d*f - a*b^2*f*g - 2*a^2*c*f*g + 6
*a^2*b*g^2 + c*(3*c^2*d*f - a*c*f^2 - b*c*d*g + 2*a^2*g^2)*x^2 + c^2*g*(3*c*d - a*f)*x^4))/(g*q*(b*c*d - 2*a^2
*g)*(b*c*d - a*b*f + 4*a^2*g))])/(g*q), x])] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[9*c^3*d^2 - c*(b^2 + 6*
a*c)*d*f + a^2*c*f^2 + 2*a*b*(3*c*d + a*f)*g - 12*a^3*g^2, 0] && EqQ[3*c^4*d^2*e - 3*a^2*c^2*d*f*g + a^3*c*f^2
*g + 2*a^3*g^2*(b*f - 6*a*g) - c^3*d*(2*b*d*f + a*e*f - 12*a*d*g), 0] && NeQ[3*c*d - a*f, 0] && NeQ[b*c*d - 2*
a^2*g, 0] && NeQ[b*c*d - a*b*f + 4*a^2*g, 0] && PosQ[(-(a*c*f^2) + 12*a^2*g^2 + f*(3*c^2*d - 2*a*b*g))/(c*g*(3
*c*d - a*f))]

Rubi steps

\begin{align*} \int \frac{6-3 x^2+x^4}{4+5 x^2-5 x^4+x^6} \, dx &=-\tan ^{-1}\left (\sqrt{3}-2 x\right )+\tan ^{-1}\left (\sqrt{3}+2 x\right )+\tan ^{-1}\left (\frac{1}{2} x \left (1-3 x^2+x^4\right )\right )\\ \end{align*}

Mathematica [A]  time = 0.0107197, size = 41, normalized size = 1.05 \[ \frac{1}{2} \tan ^{-1}\left (\frac{x \left (x^2-3\right )}{x^2-2}\right )-\frac{1}{2} \tan ^{-1}\left (\frac{x \left (x^2-3\right )}{2-x^2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(6 - 3*x^2 + x^4)/(4 + 5*x^2 - 5*x^4 + x^6),x]

[Out]

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

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Maple [A]  time = 0.015, size = 23, normalized size = 0.6 \begin{align*} \arctan \left ({\frac{{x}^{5}}{2}}-{\frac{3\,{x}^{3}}{2}}+{\frac{x}{2}} \right ) +\arctan \left ({x}^{3} \right ) +\arctan \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4-3*x^2+6)/(x^6-5*x^4+5*x^2+4),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4} - 3 \, x^{2} + 6}{x^{6} - 5 \, x^{4} + 5 \, x^{2} + 4}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-3*x^2+6)/(x^6-5*x^4+5*x^2+4),x, algorithm="maxima")

[Out]

integrate((x^4 - 3*x^2 + 6)/(x^6 - 5*x^4 + 5*x^2 + 4), x)

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Fricas [A]  time = 2.06022, size = 82, normalized size = 2.1 \begin{align*} \arctan \left (\frac{1}{2} \, x^{5} - \frac{3}{2} \, x^{3} + \frac{1}{2} \, x\right ) + \arctan \left (x^{3}\right ) + \arctan \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-3*x^2+6)/(x^6-5*x^4+5*x^2+4),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.119388, size = 24, normalized size = 0.62 \begin{align*} \operatorname{atan}{\left (x \right )} + \operatorname{atan}{\left (x^{3} \right )} + \operatorname{atan}{\left (\frac{x^{5}}{2} - \frac{3 x^{3}}{2} + \frac{x}{2} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**4-3*x**2+6)/(x**6-5*x**4+5*x**2+4),x)

[Out]

atan(x) + atan(x**3) + atan(x**5/2 - 3*x**3/2 + x/2)

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Giac [A]  time = 1.11341, size = 30, normalized size = 0.77 \begin{align*} \arctan \left (\frac{1}{2} \, x^{5} - \frac{3}{2} \, x^{3} + \frac{1}{2} \, x\right ) + \arctan \left (x^{3}\right ) + \arctan \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-3*x^2+6)/(x^6-5*x^4+5*x^2+4),x, algorithm="giac")

[Out]

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