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

Optimal. Leaf size=1 \[ x \]

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Rubi [A]  time = 0.00, antiderivative size = 1, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {21, 8} \begin {gather*} x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - x^2)^2/(-1 + x^2)^2,x]

[Out]

x

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 1, normalized size = 1.00 \begin {gather*} x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 - x^2)^2/(-1 + x^2)^2,x]

[Out]

x

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IntegrateAlgebraic [F]  time = 0.24, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (1-x^2\right )^2}{\left (-1+x^2\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.70, size = 1, normalized size = 1.00 \begin {gather*} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x

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giac [A]  time = 0.30, size = 1, normalized size = 1.00 \begin {gather*} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x

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maple [A]  time = 0.00, size = 2, normalized size = 2.00 \begin {gather*} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x

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maxima [A]  time = 1.05, size = 1, normalized size = 1.00 \begin {gather*} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x

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mupad [B]  time = 0.00, size = 1, normalized size = 1.00 \begin {gather*} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1,x)

[Out]

x

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sympy [A]  time = 0.06, size = 0, normalized size = 0.00 \begin {gather*} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x

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