3.1.39 \(\int \frac {1+2 x^2}{1-4 x^2+4 x^4} \, dx\)

Optimal. Leaf size=11 \[ \frac {x}{1-2 x^2} \]

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Rubi [A]  time = 0.01, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {28, 383} \begin {gather*} \frac {x}{1-2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x/(1 - 2*x^2)

Rule 28

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/c^p, Int[u*(b/2 + c*x^n)^(2*
p), x], x] /; FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 383

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(c*x*(a + b*x^n)^(p + 1))/a, x]
/; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[a*d - b*c*(n*(p + 1) + 1), 0]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 12, normalized size = 1.09 \begin {gather*} -\frac {x}{2 x^2-1} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(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}{1-4 x^2+4 x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.58, size = 12, normalized size = 1.09 \begin {gather*} -\frac {x}{2 \, x^{2} - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x/(2*x^2 - 1)

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giac [A]  time = 0.16, size = 12, normalized size = 1.09 \begin {gather*} -\frac {x}{2 \, x^{2} - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x/(2*x^2 - 1)

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maple [A]  time = 0.01, size = 11, normalized size = 1.00 \begin {gather*} -\frac {x}{2 \left (x^{2}-\frac {1}{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.93, size = 12, normalized size = 1.09 \begin {gather*} -\frac {x}{2 \, x^{2} - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x/(2*x^2 - 1)

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mupad [B]  time = 4.30, size = 12, normalized size = 1.09 \begin {gather*} -\frac {x}{2\,\left (x^2-\frac {1}{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x/(2*(x^2 - 1/2))

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sympy [A]  time = 0.09, size = 8, normalized size = 0.73 \begin {gather*} - \frac {x}{2 x^{2} - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x/(2*x**2 - 1)

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