3.14.47 \(\int \frac {x^5}{1-x^6} \, dx\) [1347]

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 12, normalized size of antiderivative = 1.00, 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 = {266} \begin {gather*} -\frac {1}{6} \log \left (1-x^6\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^5/(1 - x^6),x]

[Out]

-1/6*Log[1 - x^6]

Rule 266

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*}

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Mathematica [A]
time = 0.00, size = 12, normalized size = 1.00 \begin {gather*} -\frac {1}{6} \log \left (1-x^6\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^5/(1 - x^6),x]

[Out]

-1/6*Log[1 - x^6]

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Maple [A]
time = 0.18, size = 18, normalized size = 1.50

method	result	size
risch	\(-\frac {\ln \left (x^{6}-1\right )}{6}\)	\(9\)
derivativedivides	\(-\frac {\ln \left (-x^{6}+1\right )}{6}\)	\(11\)
meijerg	\(-\frac {\ln \left (-x^{6}+1\right )}{6}\)	\(11\)
default	\(-\frac {\ln \left (x^{3}-1\right )}{6}-\frac {\ln \left (x^{3}+1\right )}{6}\)	\(18\)
norman	\(-\frac {\ln \left (x -1\right )}{6}-\frac {\ln \left (x +1\right )}{6}-\frac {\ln \left (x^{2}-x +1\right )}{6}-\frac {\ln \left (x^{2}+x +1\right )}{6}\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.30, size = 8, normalized size = 0.67 \begin {gather*} -\frac {1}{6} \, \log \left (x^{6} - 1\right ) \end {gather*}

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 = 0.35, size = 8, normalized size = 0.67 \begin {gather*} -\frac {1}{6} \, \log \left (x^{6} - 1\right ) \end {gather*}

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 = 0.03, size = 8, normalized size = 0.67 \begin {gather*} - \frac {\log {\left (x^{6} - 1 \right )}}{6} \end {gather*}

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 = 1.61, size = 9, normalized size = 0.75 \begin {gather*} -\frac {1}{6} \, \log \left ({\left | x^{6} - 1 \right |}\right ) \end {gather*}

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))

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Mupad [B]
time = 0.04, size = 8, normalized size = 0.67 \begin {gather*} -\frac {\ln \left (x^6-1\right )}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-x^5/(x^6 - 1),x)

[Out]

-log(x^6 - 1)/6

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