3.279 \(\int \frac{1}{x^5 (1+2 x^4+x^8)} \, dx\)

Optimal. Leaf size=33 \[ -\frac{1}{4 \left (x^4+1\right )}-\frac{1}{4 x^4}+\frac{1}{2} \log \left (x^4+1\right )-2 \log (x) \]

[Out]

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

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Rubi [A]  time = 0.0162449, antiderivative size = 33, 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} \[ -\frac{1}{4 \left (x^4+1\right )}-\frac{1}{4 x^4}+\frac{1}{2} \log \left (x^4+1\right )-2 \log (x) \]

Antiderivative was successfully verified.

[In]

Int[1/(x^5*(1 + 2*x^4 + x^8)),x]

[Out]

-1/(4*x^4) - 1/(4*(1 + x^4)) - 2*Log[x] + Log[1 + x^4]/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 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^5 \left (1+2 x^4+x^8\right )} \, dx &=\int \frac{1}{x^5 \left (1+x^4\right )^2} \, dx\\ &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{x^2 (1+x)^2} \, dx,x,x^4\right )\\ &=\frac{1}{4} \operatorname{Subst}\left (\int \left (\frac{1}{x^2}-\frac{2}{x}+\frac{1}{(1+x)^2}+\frac{2}{1+x}\right ) \, dx,x,x^4\right )\\ &=-\frac{1}{4 x^4}-\frac{1}{4 \left (1+x^4\right )}-2 \log (x)+\frac{1}{2} \log \left (1+x^4\right )\\ \end{align*}

Mathematica [A]  time = 0.0118251, size = 33, normalized size = 1. \[ -\frac{1}{4 \left (x^4+1\right )}-\frac{1}{4 x^4}+\frac{1}{2} \log \left (x^4+1\right )-2 \log (x) \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^5*(1 + 2*x^4 + x^8)),x]

[Out]

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

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Maple [A]  time = 0.013, size = 28, normalized size = 0.9 \begin{align*} -{\frac{1}{4\,{x}^{4}}}-{\frac{1}{4\,{x}^{4}+4}}-2\,\ln \left ( x \right ) +{\frac{\ln \left ({x}^{4}+1 \right ) }{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^5/(x^8+2*x^4+1),x)

[Out]

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

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Maxima [A]  time = 1.00189, size = 45, normalized size = 1.36 \begin{align*} -\frac{2 \, x^{4} + 1}{4 \,{\left (x^{8} + x^{4}\right )}} + \frac{1}{2} \, \log \left (x^{4} + 1\right ) - \frac{1}{2} \, \log \left (x^{4}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(x^8+2*x^4+1),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.43895, size = 111, normalized size = 3.36 \begin{align*} -\frac{2 \, x^{4} - 2 \,{\left (x^{8} + x^{4}\right )} \log \left (x^{4} + 1\right ) + 8 \,{\left (x^{8} + x^{4}\right )} \log \left (x\right ) + 1}{4 \,{\left (x^{8} + x^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(x^8+2*x^4+1),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.158586, size = 29, normalized size = 0.88 \begin{align*} - \frac{2 x^{4} + 1}{4 x^{8} + 4 x^{4}} - 2 \log{\left (x \right )} + \frac{\log{\left (x^{4} + 1 \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**5/(x**8+2*x**4+1),x)

[Out]

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

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Giac [A]  time = 1.07711, size = 45, normalized size = 1.36 \begin{align*} -\frac{2 \, x^{4} + 1}{4 \,{\left (x^{8} + x^{4}\right )}} + \frac{1}{2} \, \log \left (x^{4} + 1\right ) - \frac{1}{2} \, \log \left (x^{4}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(x^8+2*x^4+1),x, algorithm="giac")

[Out]

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