3.26 \(\int \frac{1-x^4}{1-2 x^4+x^8} \, dx\)

Optimal. Leaf size=13 \[ \frac{1}{2} \tan ^{-1}(x)+\frac{1}{2} \tanh ^{-1}(x) \]

[Out]

ArcTan[x]/2 + ArcTanh[x]/2

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Rubi [A]  time = 0.0046864, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {28, 21, 212, 206, 203} \[ \frac{1}{2} \tan ^{-1}(x)+\frac{1}{2} \tanh ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Int[(1 - x^4)/(1 - 2*x^4 + x^8),x]

[Out]

ArcTan[x]/2 + ArcTanh[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 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 212

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b), 2]
]}, Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&
!GtQ[a/b, 0]

Rule 206

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

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])

Rubi steps

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

Mathematica [A]  time = 0.0046195, size = 25, normalized size = 1.92 \[ -\frac{1}{4} \log (1-x)+\frac{1}{4} \log (x+1)+\frac{1}{2} \tan ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - x^4)/(1 - 2*x^4 + x^8),x]

[Out]

ArcTan[x]/2 - Log[1 - x]/4 + Log[1 + x]/4

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Maple [A]  time = 0.003, size = 10, normalized size = 0.8 \begin{align*}{\frac{\arctan \left ( x \right ) }{2}}+{\frac{{\it Artanh} \left ( x \right ) }{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-x^4+1)/(x^8-2*x^4+1),x)

[Out]

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

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Maxima [A]  time = 1.47873, size = 23, normalized size = 1.77 \begin{align*} \frac{1}{2} \, \arctan \left (x\right ) + \frac{1}{4} \, \log \left (x + 1\right ) - \frac{1}{4} \, \log \left (x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^4+1)/(x^8-2*x^4+1),x, algorithm="maxima")

[Out]

1/2*arctan(x) + 1/4*log(x + 1) - 1/4*log(x - 1)

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Fricas [A]  time = 1.30202, size = 66, normalized size = 5.08 \begin{align*} \frac{1}{2} \, \arctan \left (x\right ) + \frac{1}{4} \, \log \left (x + 1\right ) - \frac{1}{4} \, \log \left (x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^4+1)/(x^8-2*x^4+1),x, algorithm="fricas")

[Out]

1/2*arctan(x) + 1/4*log(x + 1) - 1/4*log(x - 1)

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Sympy [B]  time = 0.123423, size = 17, normalized size = 1.31 \begin{align*} - \frac{\log{\left (x - 1 \right )}}{4} + \frac{\log{\left (x + 1 \right )}}{4} + \frac{\operatorname{atan}{\left (x \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x**4+1)/(x**8-2*x**4+1),x)

[Out]

-log(x - 1)/4 + log(x + 1)/4 + atan(x)/2

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Giac [B]  time = 1.12839, size = 26, normalized size = 2. \begin{align*} \frac{1}{2} \, \arctan \left (x\right ) + \frac{1}{4} \, \log \left ({\left | x + 1 \right |}\right ) - \frac{1}{4} \, \log \left ({\left | x - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^4+1)/(x^8-2*x^4+1),x, algorithm="giac")

[Out]

1/2*arctan(x) + 1/4*log(abs(x + 1)) - 1/4*log(abs(x - 1))