3.1376 \(\int \frac{1}{x^7 (1+x^6)} \, dx\)

Optimal. Leaf size=22 \[ -\frac{1}{6 x^6}+\frac{1}{6} \log \left (x^6+1\right )-\log (x) \]

[Out]

-1/(6*x^6) - Log[x] + Log[1 + x^6]/6

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Rubi [A]  time = 0.0104281, 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} \[ -\frac{1}{6 x^6}+\frac{1}{6} \log \left (x^6+1\right )-\log (x) \]

Antiderivative was successfully verified.

[In]

Int[1/(x^7*(1 + x^6)),x]

[Out]

-1/(6*x^6) - Log[x] + Log[1 + x^6]/6

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^7 \left (1+x^6\right )} \, dx &=\frac{1}{6} \operatorname{Subst}\left (\int \frac{1}{x^2 (1+x)} \, dx,x,x^6\right )\\ &=\frac{1}{6} \operatorname{Subst}\left (\int \left (\frac{1}{x^2}-\frac{1}{x}+\frac{1}{1+x}\right ) \, dx,x,x^6\right )\\ &=-\frac{1}{6 x^6}-\log (x)+\frac{1}{6} \log \left (1+x^6\right )\\ \end{align*}

Mathematica [A]  time = 0.0034667, size = 22, normalized size = 1. \[ -\frac{1}{6 x^6}+\frac{1}{6} \log \left (x^6+1\right )-\log (x) \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^7*(1 + x^6)),x]

[Out]

-1/(6*x^6) - Log[x] + Log[1 + x^6]/6

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Maple [A]  time = 0.009, size = 32, normalized size = 1.5 \begin{align*}{\frac{\ln \left ({x}^{2}+1 \right ) }{6}}-{\frac{1}{6\,{x}^{6}}}-\ln \left ( x \right ) +{\frac{\ln \left ({x}^{4}-{x}^{2}+1 \right ) }{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^7/(x^6+1),x)

[Out]

1/6*ln(x^2+1)-1/6/x^6-ln(x)+1/6*ln(x^4-x^2+1)

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Maxima [A]  time = 1.19439, size = 27, normalized size = 1.23 \begin{align*} -\frac{1}{6 \, x^{6}} + \frac{1}{6} \, \log \left (x^{6} + 1\right ) - \frac{1}{6} \, \log \left (x^{6}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^7/(x^6+1),x, algorithm="maxima")

[Out]

-1/6/x^6 + 1/6*log(x^6 + 1) - 1/6*log(x^6)

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Fricas [A]  time = 1.46069, size = 63, normalized size = 2.86 \begin{align*} \frac{x^{6} \log \left (x^{6} + 1\right ) - 6 \, x^{6} \log \left (x\right ) - 1}{6 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^7/(x^6+1),x, algorithm="fricas")

[Out]

1/6*(x^6*log(x^6 + 1) - 6*x^6*log(x) - 1)/x^6

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Sympy [A]  time = 0.129262, size = 17, normalized size = 0.77 \begin{align*} - \log{\left (x \right )} + \frac{\log{\left (x^{6} + 1 \right )}}{6} - \frac{1}{6 x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**7/(x**6+1),x)

[Out]

-log(x) + log(x**6 + 1)/6 - 1/(6*x**6)

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Giac [A]  time = 1.20296, size = 34, normalized size = 1.55 \begin{align*} \frac{x^{6} - 1}{6 \, x^{6}} + \frac{1}{6} \, \log \left (x^{6} + 1\right ) - \frac{1}{6} \, \log \left (x^{6}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^7/(x^6+1),x, algorithm="giac")

[Out]

1/6*(x^6 - 1)/x^6 + 1/6*log(x^6 + 1) - 1/6*log(x^6)