3.1.30 \(\int \frac {1+2 x^2}{1+5 x^2+4 x^4} \, dx\)

Optimal. Leaf size=15 \[ \frac {1}{3} \tan ^{-1}(x)+\frac {1}{3} \tan ^{-1}(2 x) \]

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Rubi [A]  time = 0.01, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1163, 203} \begin {gather*} \frac {1}{3} \tan ^{-1}(x)+\frac {1}{3} \tan ^{-1}(2 x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + 2*x^2)/(1 + 5*x^2 + 4*x^4),x]

[Out]

ArcTan[x]/3 + ArcTan[2*x]/3

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 1163

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] && EqQ[c*d^2 - a*e^2, 0] && GtQ[b^2
- 4*a*c, 0]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 17, normalized size = 1.13 \begin {gather*} -\frac {1}{3} \tan ^{-1}\left (\frac {3 x}{2 x^2-1}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + 2*x^2)/(1 + 5*x^2 + 4*x^4),x]

[Out]

-1/3*ArcTan[(3*x)/(-1 + 2*x^2)]

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1+2 x^2}{1+5 x^2+4 x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(1 + 2*x^2)/(1 + 5*x^2 + 4*x^4),x]

[Out]

IntegrateAlgebraic[(1 + 2*x^2)/(1 + 5*x^2 + 4*x^4), x]

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fricas [A]  time = 0.71, size = 19, normalized size = 1.27 \begin {gather*} \frac {1}{3} \, \arctan \left (\frac {4}{3} \, x^{3} + \frac {7}{3} \, x\right ) + \frac {1}{3} \, \arctan \left (\frac {2}{3} \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^2+1)/(4*x^4+5*x^2+1),x, algorithm="fricas")

[Out]

1/3*arctan(4/3*x^3 + 7/3*x) + 1/3*arctan(2/3*x)

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giac [A]  time = 0.15, size = 11, normalized size = 0.73 \begin {gather*} \frac {1}{3} \, \arctan \left (2 \, x\right ) + \frac {1}{3} \, \arctan \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^2+1)/(4*x^4+5*x^2+1),x, algorithm="giac")

[Out]

1/3*arctan(2*x) + 1/3*arctan(x)

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maple [A]  time = 0.01, size = 12, normalized size = 0.80 \begin {gather*} \frac {\arctan \relax (x )}{3}+\frac {\arctan \left (2 x \right )}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^2+1)/(4*x^4+5*x^2+1),x)

[Out]

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

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maxima [A]  time = 2.49, size = 11, normalized size = 0.73 \begin {gather*} \frac {1}{3} \, \arctan \left (2 \, x\right ) + \frac {1}{3} \, \arctan \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^2+1)/(4*x^4+5*x^2+1),x, algorithm="maxima")

[Out]

1/3*arctan(2*x) + 1/3*arctan(x)

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mupad [B]  time = 0.07, size = 19, normalized size = 1.27 \begin {gather*} \frac {\mathrm {atan}\left (\frac {2\,x}{3}\right )}{3}+\frac {\mathrm {atan}\left (\frac {4\,x^3}{3}+\frac {7\,x}{3}\right )}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^2 + 1)/(5*x^2 + 4*x^4 + 1),x)

[Out]

atan((2*x)/3)/3 + atan((7*x)/3 + (4*x^3)/3)/3

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sympy [B]  time = 0.12, size = 22, normalized size = 1.47 \begin {gather*} \frac {\operatorname {atan}{\left (\frac {2 x}{3} \right )}}{3} + \frac {\operatorname {atan}{\left (\frac {4 x^{3}}{3} + \frac {7 x}{3} \right )}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x**2+1)/(4*x**4+5*x**2+1),x)

[Out]

atan(2*x/3)/3 + atan(4*x**3/3 + 7*x/3)/3

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