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

Optimal. Leaf size=27 \[ \frac{x}{2 \left (1-x^4\right )}+\frac{1}{4} \tan ^{-1}(x)+\frac{1}{4} \tanh ^{-1}(x) \]

[Out]

x/(2*(1 - x^4)) + ArcTan[x]/4 + ArcTanh[x]/4

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Rubi [A]  time = 0.0077244, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278, Rules used = {28, 385, 212, 206, 203} \[ \frac{x}{2 \left (1-x^4\right )}+\frac{1}{4} \tan ^{-1}(x)+\frac{1}{4} \tanh ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Int[(1 + x^4)/(1 - 2*x^4 + x^8),x]

[Out]

x/(2*(1 - x^4)) + ArcTan[x]/4 + ArcTanh[x]/4

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 385

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

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

Mathematica [A]  time = 0.0135165, size = 31, normalized size = 1.15 \[ \frac{1}{8} \left (-\frac{4 x}{x^4-1}-\log (1-x)+\log (x+1)+2 \tan ^{-1}(x)\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-4*x)/(-1 + x^4) + 2*ArcTan[x] - Log[1 - x] + Log[1 + x])/8

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Maple [A]  time = 0.011, size = 42, normalized size = 1.6 \begin{align*}{\frac{x}{4\,{x}^{2}+4}}+{\frac{\arctan \left ( x \right ) }{4}}-{\frac{1}{8+8\,x}}+{\frac{\ln \left ( 1+x \right ) }{8}}-{\frac{1}{8\,x-8}}-{\frac{\ln \left ( x-1 \right ) }{8}} \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]

1/4*x/(x^2+1)+1/4*arctan(x)-1/8/(1+x)+1/8*ln(1+x)-1/8/(x-1)-1/8*ln(x-1)

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Maxima [A]  time = 1.49966, size = 36, normalized size = 1.33 \begin{align*} -\frac{x}{2 \,{\left (x^{4} - 1\right )}} + \frac{1}{4} \, \arctan \left (x\right ) + \frac{1}{8} \, \log \left (x + 1\right ) - \frac{1}{8} \, \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*x/(x^4 - 1) + 1/4*arctan(x) + 1/8*log(x + 1) - 1/8*log(x - 1)

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Fricas [B]  time = 1.23094, size = 123, normalized size = 4.56 \begin{align*} \frac{2 \,{\left (x^{4} - 1\right )} \arctan \left (x\right ) +{\left (x^{4} - 1\right )} \log \left (x + 1\right ) -{\left (x^{4} - 1\right )} \log \left (x - 1\right ) - 4 \, x}{8 \,{\left (x^{4} - 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]

1/8*(2*(x^4 - 1)*arctan(x) + (x^4 - 1)*log(x + 1) - (x^4 - 1)*log(x - 1) - 4*x)/(x^4 - 1)

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Sympy [A]  time = 0.142127, size = 26, normalized size = 0.96 \begin{align*} - \frac{x}{2 x^{4} - 2} - \frac{\log{\left (x - 1 \right )}}{8} + \frac{\log{\left (x + 1 \right )}}{8} + \frac{\operatorname{atan}{\left (x \right )}}{4} \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]

-x/(2*x**4 - 2) - log(x - 1)/8 + log(x + 1)/8 + atan(x)/4

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Giac [A]  time = 1.09585, size = 39, normalized size = 1.44 \begin{align*} -\frac{x}{2 \,{\left (x^{4} - 1\right )}} + \frac{1}{4} \, \arctan \left (x\right ) + \frac{1}{8} \, \log \left ({\left | x + 1 \right |}\right ) - \frac{1}{8} \, \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*x/(x^4 - 1) + 1/4*arctan(x) + 1/8*log(abs(x + 1)) - 1/8*log(abs(x - 1))