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

Optimal. Leaf size=27 \[ \frac{x}{4 \left (1-x^4\right )}+\frac{3}{8} \tan ^{-1}(x)+\frac{3}{8} \tanh ^{-1}(x) \]

[Out]

x/(4*(1 - x^4)) + (3*ArcTan[x])/8 + (3*ArcTanh[x])/8

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Rubi [A]  time = 0.0053873, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {28, 199, 212, 206, 203} \[ \frac{x}{4 \left (1-x^4\right )}+\frac{3}{8} \tan ^{-1}(x)+\frac{3}{8} \tanh ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Int[(1 - 2*x^4 + x^8)^(-1),x]

[Out]

x/(4*(1 - x^4)) + (3*ArcTan[x])/8 + (3*ArcTanh[x])/8

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 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

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}{1-2 x^4+x^8} \, dx &=\int \frac{1}{\left (-1+x^4\right )^2} \, dx\\ &=\frac{x}{4 \left (1-x^4\right )}-\frac{3}{4} \int \frac{1}{-1+x^4} \, dx\\ &=\frac{x}{4 \left (1-x^4\right )}+\frac{3}{8} \int \frac{1}{1-x^2} \, dx+\frac{3}{8} \int \frac{1}{1+x^2} \, dx\\ &=\frac{x}{4 \left (1-x^4\right )}+\frac{3}{8} \tan ^{-1}(x)+\frac{3}{8} \tanh ^{-1}(x)\\ \end{align*}

Mathematica [A]  time = 0.00972, size = 33, normalized size = 1.22 \[ \frac{1}{16} \left (-\frac{4 x}{x^4-1}-3 \log (1-x)+3 \log (x+1)+6 \tan ^{-1}(x)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - 2*x^4 + x^8)^(-1),x]

[Out]

((-4*x)/(-1 + x^4) + 6*ArcTan[x] - 3*Log[1 - x] + 3*Log[1 + x])/16

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Maple [A]  time = 0.012, size = 42, normalized size = 1.6 \begin{align*}{\frac{x}{8\,{x}^{2}+8}}+{\frac{3\,\arctan \left ( x \right ) }{8}}-{\frac{1}{16+16\,x}}+{\frac{3\,\ln \left ( 1+x \right ) }{16}}-{\frac{1}{16\,x-16}}-{\frac{3\,\ln \left ( x-1 \right ) }{16}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*x/(x^2+1)+3/8*arctan(x)-1/16/(1+x)+3/16*ln(1+x)-1/16/(x-1)-3/16*ln(x-1)

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Maxima [A]  time = 1.51261, size = 36, normalized size = 1.33 \begin{align*} -\frac{x}{4 \,{\left (x^{4} - 1\right )}} + \frac{3}{8} \, \arctan \left (x\right ) + \frac{3}{16} \, \log \left (x + 1\right ) - \frac{3}{16} \, \log \left (x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*x/(x^4 - 1) + 3/8*arctan(x) + 3/16*log(x + 1) - 3/16*log(x - 1)

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Fricas [B]  time = 1.48817, size = 130, normalized size = 4.81 \begin{align*} \frac{6 \,{\left (x^{4} - 1\right )} \arctan \left (x\right ) + 3 \,{\left (x^{4} - 1\right )} \log \left (x + 1\right ) - 3 \,{\left (x^{4} - 1\right )} \log \left (x - 1\right ) - 4 \, x}{16 \,{\left (x^{4} - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*(6*(x^4 - 1)*arctan(x) + 3*(x^4 - 1)*log(x + 1) - 3*(x^4 - 1)*log(x - 1) - 4*x)/(x^4 - 1)

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Sympy [A]  time = 0.15957, size = 31, normalized size = 1.15 \begin{align*} - \frac{x}{4 x^{4} - 4} - \frac{3 \log{\left (x - 1 \right )}}{16} + \frac{3 \log{\left (x + 1 \right )}}{16} + \frac{3 \operatorname{atan}{\left (x \right )}}{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x/(4*x**4 - 4) - 3*log(x - 1)/16 + 3*log(x + 1)/16 + 3*atan(x)/8

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Giac [A]  time = 1.11741, size = 39, normalized size = 1.44 \begin{align*} -\frac{x}{4 \,{\left (x^{4} - 1\right )}} + \frac{3}{8} \, \arctan \left (x\right ) + \frac{3}{16} \, \log \left ({\left | x + 1 \right |}\right ) - \frac{3}{16} \, \log \left ({\left | x - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*x/(x^4 - 1) + 3/8*arctan(x) + 3/16*log(abs(x + 1)) - 3/16*log(abs(x - 1))