3.1347 \(\int \frac{x^5}{1-x^6} \, dx\)

Optimal. Leaf size=12 \[ -\frac{1}{6} \log \left (1-x^6\right ) \]

[Out]

-Log[1 - x^6]/6

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Rubi [A]  time = 0.0025075, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {260} \[ -\frac{1}{6} \log \left (1-x^6\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^5/(1 - x^6),x]

[Out]

-Log[1 - x^6]/6

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rubi steps

\begin{align*} \int \frac{x^5}{1-x^6} \, dx &=-\frac{1}{6} \log \left (1-x^6\right )\\ \end{align*}

Mathematica [A]  time = 0.002893, size = 12, normalized size = 1. \[ -\frac{1}{6} \log \left (1-x^6\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^5/(1 - x^6),x]

[Out]

-Log[1 - x^6]/6

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Maple [A]  time = 0.003, size = 18, normalized size = 1.5 \begin{align*} -{\frac{\ln \left ({x}^{3}-1 \right ) }{6}}-{\frac{\ln \left ({x}^{3}+1 \right ) }{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(-x^6+1),x)

[Out]

-1/6*ln(x^3-1)-1/6*ln(x^3+1)

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Maxima [A]  time = 0.979428, size = 11, normalized size = 0.92 \begin{align*} -\frac{1}{6} \, \log \left (x^{6} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(-x^6+1),x, algorithm="maxima")

[Out]

-1/6*log(x^6 - 1)

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Fricas [A]  time = 1.40847, size = 26, normalized size = 2.17 \begin{align*} -\frac{1}{6} \, \log \left (x^{6} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(-x^6+1),x, algorithm="fricas")

[Out]

-1/6*log(x^6 - 1)

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Sympy [A]  time = 0.086711, size = 8, normalized size = 0.67 \begin{align*} - \frac{\log{\left (x^{6} - 1 \right )}}{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5/(-x**6+1),x)

[Out]

-log(x**6 - 1)/6

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Giac [A]  time = 1.18427, size = 12, normalized size = 1. \begin{align*} -\frac{1}{6} \, \log \left ({\left | x^{6} - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(-x^6+1),x, algorithm="giac")

[Out]

-1/6*log(abs(x^6 - 1))