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

Optimal. Leaf size=2 \[ \tanh ^{-1}(x) \]

[Out]

arctanh(x)

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Rubi [A]  time = 0.00, antiderivative size = 2, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {28, 21, 207} \[ \tanh ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Int[(1 - x^2)/(1 - 2*x^2 + x^4),x]

[Out]

ArcTanh[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])

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 207

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

Rubi steps

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

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Mathematica [B]  time = 0.00, size = 19, normalized size = 9.50 \[ \frac {1}{2} \log (x+1)-\frac {1}{2} \log (1-x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*Log[1 - x] + Log[1 + x]/2

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fricas [B]  time = 0.40, size = 13, normalized size = 6.50 \[ \frac {1}{2} \, \log \left (x + 1\right ) - \frac {1}{2} \, \log \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*log(x + 1) - 1/2*log(x - 1)

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giac [B]  time = 0.15, size = 15, normalized size = 7.50 \[ \frac {1}{2} \, \log \left ({\left | x + 1 \right |}\right ) - \frac {1}{2} \, \log \left ({\left | x - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*log(abs(x + 1)) - 1/2*log(abs(x - 1))

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maple [A]  time = 0.00, size = 3, normalized size = 1.50 \[ \arctanh \relax (x ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctanh(x)

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maxima [B]  time = 1.07, size = 13, normalized size = 6.50 \[ \frac {1}{2} \, \log \left (x + 1\right ) - \frac {1}{2} \, \log \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*log(x + 1) - 1/2*log(x - 1)

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mupad [B]  time = 4.30, size = 2, normalized size = 1.00 \[ \mathrm {atanh}\relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x^2 - 1)/(x^4 - 2*x^2 + 1),x)

[Out]

atanh(x)

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sympy [B]  time = 0.11, size = 12, normalized size = 6.00 \[ - \frac {\log {\left (x - 1 \right )}}{2} + \frac {\log {\left (x + 1 \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(x - 1)/2 + log(x + 1)/2

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