3.1499 \(\int \frac{1}{x^9 (1+x^8)} \, dx\)

Optimal. Leaf size=22 \[ -\frac{1}{8 x^8}+\frac{1}{8} \log \left (x^8+1\right )-\log (x) \]

[Out]

-1/(8*x^8) - Log[x] + Log[1 + x^8]/8

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Rubi [A]  time = 0.0097526, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {266, 44} \[ -\frac{1}{8 x^8}+\frac{1}{8} \log \left (x^8+1\right )-\log (x) \]

Antiderivative was successfully verified.

[In]

Int[1/(x^9*(1 + x^8)),x]

[Out]

-1/(8*x^8) - Log[x] + Log[1 + x^8]/8

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^9 \left (1+x^8\right )} \, dx &=\frac{1}{8} \operatorname{Subst}\left (\int \frac{1}{x^2 (1+x)} \, dx,x,x^8\right )\\ &=\frac{1}{8} \operatorname{Subst}\left (\int \left (\frac{1}{x^2}-\frac{1}{x}+\frac{1}{1+x}\right ) \, dx,x,x^8\right )\\ &=-\frac{1}{8 x^8}-\log (x)+\frac{1}{8} \log \left (1+x^8\right )\\ \end{align*}

Mathematica [A]  time = 0.0033918, size = 22, normalized size = 1. \[ -\frac{1}{8 x^8}+\frac{1}{8} \log \left (x^8+1\right )-\log (x) \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^9*(1 + x^8)),x]

[Out]

-1/(8*x^8) - Log[x] + Log[1 + x^8]/8

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Maple [A]  time = 0.005, size = 19, normalized size = 0.9 \begin{align*} -{\frac{1}{8\,{x}^{8}}}-\ln \left ( x \right ) +{\frac{\ln \left ({x}^{8}+1 \right ) }{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^9/(x^8+1),x)

[Out]

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

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Maxima [A]  time = 0.980192, size = 27, normalized size = 1.23 \begin{align*} -\frac{1}{8 \, x^{8}} + \frac{1}{8} \, \log \left (x^{8} + 1\right ) - \frac{1}{8} \, \log \left (x^{8}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^9/(x^8+1),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.21145, size = 63, normalized size = 2.86 \begin{align*} \frac{x^{8} \log \left (x^{8} + 1\right ) - 8 \, x^{8} \log \left (x\right ) - 1}{8 \, x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^9/(x^8+1),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.153234, size = 17, normalized size = 0.77 \begin{align*} - \log{\left (x \right )} + \frac{\log{\left (x^{8} + 1 \right )}}{8} - \frac{1}{8 x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**9/(x**8+1),x)

[Out]

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

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Giac [A]  time = 1.18975, size = 34, normalized size = 1.55 \begin{align*} \frac{x^{8} - 1}{8 \, x^{8}} + \frac{1}{8} \, \log \left (x^{8} + 1\right ) - \frac{1}{8} \, \log \left (x^{8}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^9/(x^8+1),x, algorithm="giac")

[Out]

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