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

Optimal. Leaf size=25 \[ \frac{x^2}{4 \left (1-x^4\right )}+\frac{1}{4} \tanh ^{-1}\left (x^2\right ) \]

[Out]

x^2/(4*(1 - x^4)) + ArcTanh[x^2]/4

________________________________________________________________________________________

Rubi [A]  time = 0.0081537, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {28, 275, 199, 207} \[ \frac{x^2}{4 \left (1-x^4\right )}+\frac{1}{4} \tanh ^{-1}\left (x^2\right ) \]

Antiderivative was successfully verified.

[In]

Int[x/(1 - 2*x^4 + x^8),x]

[Out]

x^2/(4*(1 - x^4)) + ArcTanh[x^2]/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 275

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

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

Mathematica [A]  time = 0.0083112, size = 33, normalized size = 1.32 \[ \frac{1}{8} \left (-\frac{2 x^2}{x^4-1}-\log \left (1-x^2\right )+\log \left (x^2+1\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x/(1 - 2*x^4 + x^8),x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.009, size = 36, normalized size = 1.4 \begin{align*} -{\frac{1}{8\,{x}^{2}+8}}+{\frac{\ln \left ({x}^{2}+1 \right ) }{8}}-{\frac{1}{8\,{x}^{2}-8}}-{\frac{\ln \left ({x}^{2}-1 \right ) }{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.00547, size = 39, normalized size = 1.56 \begin{align*} -\frac{x^{2}}{4 \,{\left (x^{4} - 1\right )}} + \frac{1}{8} \, \log \left (x^{2} + 1\right ) - \frac{1}{8} \, \log \left (x^{2} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [B]  time = 1.52176, size = 100, normalized size = 4. \begin{align*} -\frac{2 \, x^{2} -{\left (x^{4} - 1\right )} \log \left (x^{2} + 1\right ) +{\left (x^{4} - 1\right )} \log \left (x^{2} - 1\right )}{8 \,{\left (x^{4} - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0.126262, size = 26, normalized size = 1.04 \begin{align*} - \frac{x^{2}}{4 x^{4} - 4} - \frac{\log{\left (x^{2} - 1 \right )}}{8} + \frac{\log{\left (x^{2} + 1 \right )}}{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.12905, size = 41, normalized size = 1.64 \begin{align*} -\frac{x^{2}}{4 \,{\left (x^{4} - 1\right )}} + \frac{1}{8} \, \log \left (x^{2} + 1\right ) - \frac{1}{8} \, \log \left ({\left | x^{2} - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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