3.1344 \(\int \frac{x^8}{1-x^6} \, dx\)

Optimal. Leaf size=16 \[ \frac{1}{3} \tanh ^{-1}\left (x^3\right )-\frac{x^3}{3} \]

[Out]

-x^3/3 + ArcTanh[x^3]/3

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Rubi [A]  time = 0.007649, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {275, 321, 206} \[ \frac{1}{3} \tanh ^{-1}\left (x^3\right )-\frac{x^3}{3} \]

Antiderivative was successfully verified.

[In]

Int[x^8/(1 - x^6),x]

[Out]

-x^3/3 + ArcTanh[x^3]/3

Rule 275

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{x^8}{1-x^6} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x^2}{1-x^2} \, dx,x,x^3\right )\\ &=-\frac{x^3}{3}+\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,x^3\right )\\ &=-\frac{x^3}{3}+\frac{1}{3} \tanh ^{-1}\left (x^3\right )\\ \end{align*}

Mathematica [A]  time = 0.0046284, size = 30, normalized size = 1.88 \[ -\frac{x^3}{3}-\frac{1}{6} \log \left (1-x^3\right )+\frac{1}{6} \log \left (x^3+1\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^8/(1 - x^6),x]

[Out]

-x^3/3 - Log[1 - x^3]/6 + Log[1 + x^3]/6

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Maple [A]  time = 0.003, size = 23, normalized size = 1.4 \begin{align*} -{\frac{{x}^{3}}{3}}-{\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^8/(-x^6+1),x)

[Out]

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

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Maxima [A]  time = 0.953767, size = 30, normalized size = 1.88 \begin{align*} -\frac{1}{3} \, x^{3} + \frac{1}{6} \, \log \left (x^{3} + 1\right ) - \frac{1}{6} \, \log \left (x^{3} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8/(-x^6+1),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.39435, size = 65, normalized size = 4.06 \begin{align*} -\frac{1}{3} \, x^{3} + \frac{1}{6} \, \log \left (x^{3} + 1\right ) - \frac{1}{6} \, \log \left (x^{3} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8/(-x^6+1),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.098674, size = 20, normalized size = 1.25 \begin{align*} - \frac{x^{3}}{3} - \frac{\log{\left (x^{3} - 1 \right )}}{6} + \frac{\log{\left (x^{3} + 1 \right )}}{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**8/(-x**6+1),x)

[Out]

-x**3/3 - log(x**3 - 1)/6 + log(x**3 + 1)/6

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Giac [A]  time = 1.19222, size = 32, normalized size = 2. \begin{align*} -\frac{1}{3} \, x^{3} + \frac{1}{6} \, \log \left ({\left | x^{3} + 1 \right |}\right ) - \frac{1}{6} \, \log \left ({\left | x^{3} - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8/(-x^6+1),x, algorithm="giac")

[Out]

-1/3*x^3 + 1/6*log(abs(x^3 + 1)) - 1/6*log(abs(x^3 - 1))