3.1359 \(\int \frac {1}{x^7 (1-x^6)} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {266, 44} \[ -\frac {1}{6 x^6}-\frac {1}{6} \log \left (1-x^6\right )+\log (x) \]

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 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])

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]]

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 22, normalized size = 1.00 \[ -\frac {1}{6 x^6}-\frac {1}{6} \log \left (1-x^6\right )+\log (x) \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^7*(1 - x^6)),x]

[Out]

-1/6*1/x^6 + Log[x] - Log[1 - x^6]/6

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fricas [A]  time = 0.76, size = 24, normalized size = 1.09 \[ -\frac {x^{6} \log \left (x^{6} - 1\right ) - 6 \, x^{6} \log \relax (x) + 1}{6 \, x^{6}} \]

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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giac [A]  time = 0.17, size = 26, normalized size = 1.18 \[ -\frac {x^{6} + 1}{6 \, x^{6}} + \frac {1}{6} \, \log \left (x^{6}\right ) - \frac {1}{6} \, \log \left ({\left | x^{6} - 1 \right |}\right ) \]

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) - 1/6*log(abs(x^6 - 1))

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maple [B]  time = 0.01, size = 41, normalized size = 1.86 \[ \ln \relax (x )-\frac {\ln \left (x -1\right )}{6}-\frac {\ln \left (x +1\right )}{6}-\frac {\ln \left (x^{2}-x +1\right )}{6}-\frac {\ln \left (x^{2}+x +1\right )}{6}-\frac {1}{6 x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.08, size = 20, normalized size = 0.91 \[ -\frac {1}{6 \, x^{6}} - \frac {1}{6} \, \log \left (x^{6} - 1\right ) + \frac {1}{6} \, \log \left (x^{6}\right ) \]

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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mupad [B]  time = 1.02, size = 16, normalized size = 0.73 \[ \ln \relax (x)-\frac {\ln \left (x^6-1\right )}{6}-\frac {1}{6\,x^6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-1/(x^7*(x^6 - 1)),x)

[Out]

log(x) - log(x^6 - 1)/6 - 1/(6*x^6)

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sympy [A]  time = 0.24, size = 17, normalized size = 0.77 \[ \log {\relax (x )} - \frac {\log {\left (x^{6} - 1 \right )}}{6} - \frac {1}{6 x^{6}} \]

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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