3.471 \(\int \frac {-10+x^2}{4+9 x^2+2 x^4} \, dx\)

Optimal. Leaf size=22 \[ \tan ^{-1}\left (\frac {x}{2}\right )-\frac {3 \tan ^{-1}\left (\sqrt {2} x\right )}{\sqrt {2}} \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1166, 203} \[ \tan ^{-1}\left (\frac {x}{2}\right )-\frac {3 \tan ^{-1}\left (\sqrt {2} x\right )}{\sqrt {2}} \]

Antiderivative was successfully verified.

[In]

Int[(-10 + x^2)/(4 + 9*x^2 + 2*x^4),x]

[Out]

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

Rule 203

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

Rule 1166

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rubi steps

\begin {align*} \int \frac {-10+x^2}{4+9 x^2+2 x^4} \, dx &=-\left (3 \int \frac {1}{1+2 x^2} \, dx\right )+4 \int \frac {1}{8+2 x^2} \, dx\\ &=\tan ^{-1}\left (\frac {x}{2}\right )-\frac {3 \tan ^{-1}\left (\sqrt {2} x\right )}{\sqrt {2}}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 22, normalized size = 1.00 \[ \tan ^{-1}\left (\frac {x}{2}\right )-\frac {3 \tan ^{-1}\left (\sqrt {2} x\right )}{\sqrt {2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(-10 + x^2)/(4 + 9*x^2 + 2*x^4),x]

[Out]

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

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fricas [A]  time = 0.48, size = 16, normalized size = 0.73 \[ -\frac {3}{2} \, \sqrt {2} \arctan \left (\sqrt {2} x\right ) + \arctan \left (\frac {1}{2} \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2-10)/(2*x^4+9*x^2+4),x, algorithm="fricas")

[Out]

-3/2*sqrt(2)*arctan(sqrt(2)*x) + arctan(1/2*x)

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giac [A]  time = 0.29, size = 16, normalized size = 0.73 \[ -\frac {3}{2} \, \sqrt {2} \arctan \left (\sqrt {2} x\right ) + \arctan \left (\frac {1}{2} \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2-10)/(2*x^4+9*x^2+4),x, algorithm="giac")

[Out]

-3/2*sqrt(2)*arctan(sqrt(2)*x) + arctan(1/2*x)

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maple [A]  time = 0.01, size = 17, normalized size = 0.77 \[ -\frac {3 \sqrt {2}\, \arctan \left (\sqrt {2}\, x \right )}{2}+\arctan \left (\frac {x}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2-10)/(2*x^4+9*x^2+4),x)

[Out]

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

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maxima [A]  time = 1.58, size = 16, normalized size = 0.73 \[ -\frac {3}{2} \, \sqrt {2} \arctan \left (\sqrt {2} x\right ) + \arctan \left (\frac {1}{2} \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2-10)/(2*x^4+9*x^2+4),x, algorithm="maxima")

[Out]

-3/2*sqrt(2)*arctan(sqrt(2)*x) + arctan(1/2*x)

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mupad [B]  time = 0.05, size = 16, normalized size = 0.73 \[ \mathrm {atan}\left (\frac {x}{2}\right )-\frac {3\,\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,x\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2 - 10)/(9*x^2 + 2*x^4 + 4),x)

[Out]

atan(x/2) - (3*2^(1/2)*atan(2^(1/2)*x))/2

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sympy [A]  time = 0.15, size = 20, normalized size = 0.91 \[ \operatorname {atan}{\left (\frac {x}{2} \right )} - \frac {3 \sqrt {2} \operatorname {atan}{\left (\sqrt {2} x \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**2-10)/(2*x**4+9*x**2+4),x)

[Out]

atan(x/2) - 3*sqrt(2)*atan(sqrt(2)*x)/2

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