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

Optimal. Leaf size=24 \[ \frac{1}{4 \left (x^4+1\right )}-\frac{1}{4} \log \left (x^4+1\right )+\log (x) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0118203, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {28, 266, 44} \[ \frac{1}{4 \left (x^4+1\right )}-\frac{1}{4} \log \left (x^4+1\right )+\log (x) \]

Antiderivative was successfully verified.

[In]

Int[1/(x*(1 + 2*x^4 + x^8)),x]

[Out]

1/(4*(1 + x^4)) + Log[x] - Log[1 + x^4]/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 266

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

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.0094163, size = 24, normalized size = 1. \[ \frac{1}{4 \left (x^4+1\right )}-\frac{1}{4} \log \left (x^4+1\right )+\log (x) \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*(1 + 2*x^4 + x^8)),x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.013, size = 21, normalized size = 0.9 \begin{align*}{\frac{1}{4\,{x}^{4}+4}}+\ln \left ( x \right ) -{\frac{\ln \left ({x}^{4}+1 \right ) }{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4/(x^4+1)+ln(x)-1/4*ln(x^4+1)

________________________________________________________________________________________

Maxima [A]  time = 1.03468, size = 32, normalized size = 1.33 \begin{align*} \frac{1}{4 \,{\left (x^{4} + 1\right )}} - \frac{1}{4} \, \log \left (x^{4} + 1\right ) + \frac{1}{4} \, \log \left (x^{4}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.41221, size = 89, normalized size = 3.71 \begin{align*} -\frac{{\left (x^{4} + 1\right )} \log \left (x^{4} + 1\right ) - 4 \,{\left (x^{4} + 1\right )} \log \left (x\right ) - 1}{4 \,{\left (x^{4} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0.131085, size = 19, normalized size = 0.79 \begin{align*} \log{\left (x \right )} - \frac{\log{\left (x^{4} + 1 \right )}}{4} + \frac{1}{4 x^{4} + 4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.10797, size = 39, normalized size = 1.62 \begin{align*} \frac{x^{4} + 2}{4 \,{\left (x^{4} + 1\right )}} - \frac{1}{4} \, \log \left (x^{4} + 1\right ) + \frac{1}{4} \, \log \left (x^{4}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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