3.1472 \(\int \frac{x^7}{1-x^8} \, dx\)

Optimal. Leaf size=12 \[ -\frac{1}{8} \log \left (1-x^8\right ) \]

[Out]

-Log[1 - x^8]/8

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Rubi [A]  time = 0.0025672, 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} \[ -\frac{1}{8} \log \left (1-x^8\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^7/(1 - x^8),x]

[Out]

-Log[1 - x^8]/8

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^7}{1-x^8} \, dx &=-\frac{1}{8} \log \left (1-x^8\right )\\ \end{align*}

Mathematica [A]  time = 0.0029171, size = 12, normalized size = 1. \[ -\frac{1}{8} \log \left (1-x^8\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^7/(1 - x^8),x]

[Out]

-Log[1 - x^8]/8

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Maple [B]  time = 0.005, size = 30, normalized size = 2.5 \begin{align*} -{\frac{\ln \left ( -1+x \right ) }{8}}-{\frac{\ln \left ( 1+x \right ) }{8}}-{\frac{\ln \left ({x}^{2}+1 \right ) }{8}}-{\frac{\ln \left ({x}^{4}+1 \right ) }{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7/(-x^8+1),x)

[Out]

-1/8*ln(-1+x)-1/8*ln(1+x)-1/8*ln(x^2+1)-1/8*ln(x^4+1)

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Maxima [A]  time = 0.941738, size = 11, normalized size = 0.92 \begin{align*} -\frac{1}{8} \, \log \left (x^{8} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(-x^8+1),x, algorithm="maxima")

[Out]

-1/8*log(x^8 - 1)

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Fricas [A]  time = 1.18963, size = 26, normalized size = 2.17 \begin{align*} -\frac{1}{8} \, \log \left (x^{8} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(-x^8+1),x, algorithm="fricas")

[Out]

-1/8*log(x^8 - 1)

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Sympy [A]  time = 0.095827, size = 8, normalized size = 0.67 \begin{align*} - \frac{\log{\left (x^{8} - 1 \right )}}{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**7/(-x**8+1),x)

[Out]

-log(x**8 - 1)/8

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Giac [A]  time = 1.18282, size = 12, normalized size = 1. \begin{align*} -\frac{1}{8} \, \log \left ({\left | x^{8} - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(-x^8+1),x, algorithm="giac")

[Out]

-1/8*log(abs(x^8 - 1))