\(\int \frac {x^8}{1-x^6} \, dx\) [1344]

Optimal result

Integrand size = 13, antiderivative size = 16 \[ \int \frac {x^8}{1-x^6} \, dx=-\frac {x^3}{3}+\frac {\text {arctanh}\left (x^3\right )}{3} \]

[Out]

-1/3*x^3+1/3*arctanh(x^3)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {281, 327, 212} \[ \int \frac {x^8}{1-x^6} \, dx=\frac {\text {arctanh}\left (x^3\right )}{3}-\frac {x^3}{3} \]

[In]

Int[x^8/(1 - x^6),x]

[Out]

-1/3*x^3 + ArcTanh[x^3]/3

Rule 212

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

Rule 281

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 327

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]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int \frac {x^2}{1-x^2} \, dx,x,x^3\right ) \\ & = -\frac {x^3}{3}+\frac {1}{3} \text {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] (verified)

Time = 0.01 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.88 \[ \int \frac {x^8}{1-x^6} \, dx=-\frac {x^3}{3}-\frac {1}{6} \log \left (1-x^3\right )+\frac {1}{6} \log \left (1+x^3\right ) \]

[In]

Integrate[x^8/(1 - x^6),x]

[Out]

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

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 4.39 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12

method	result	size
meijerg	\(\frac {i \left (2 i x^{3}-2 i \operatorname {arctanh}\left (x^{3}\right )\right )}{6}\)	\(18\)
default	\(-\frac {x^{3}}{3}-\frac {\ln \left (x^{3}-1\right )}{6}+\frac {\ln \left (x^{3}+1\right )}{6}\)	\(23\)
risch	\(-\frac {x^{3}}{3}-\frac {\ln \left (x^{3}-1\right )}{6}+\frac {\ln \left (x^{3}+1\right )}{6}\)	\(23\)
norman	\(-\frac {x^{3}}{3}-\frac {\ln \left (-1+x \right )}{6}+\frac {\ln \left (1+x \right )}{6}+\frac {\ln \left (x^{2}-x +1\right )}{6}-\frac {\ln \left (x^{2}+x +1\right )}{6}\)	\(39\)
parallelrisch	\(-\frac {x^{3}}{3}-\frac {\ln \left (-1+x \right )}{6}+\frac {\ln \left (1+x \right )}{6}+\frac {\ln \left (x^{2}-x +1\right )}{6}-\frac {\ln \left (x^{2}+x +1\right )}{6}\)	\(39\)

[In]

int(x^8/(-x^6+1),x,method=_RETURNVERBOSE)

[Out]

1/6*I*(2*I*x^3-2*I*arctanh(x^3))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.38 \[ \int \frac {x^8}{1-x^6} \, dx=-\frac {1}{3} \, x^{3} + \frac {1}{6} \, \log \left (x^{3} + 1\right ) - \frac {1}{6} \, \log \left (x^{3} - 1\right ) \]

[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)

Sympy [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.25 \[ \int \frac {x^8}{1-x^6} \, dx=- \frac {x^{3}}{3} - \frac {\log {\left (x^{3} - 1 \right )}}{6} + \frac {\log {\left (x^{3} + 1 \right )}}{6} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.38 \[ \int \frac {x^8}{1-x^6} \, dx=-\frac {1}{3} \, x^{3} + \frac {1}{6} \, \log \left (x^{3} + 1\right ) - \frac {1}{6} \, \log \left (x^{3} - 1\right ) \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.50 \[ \int \frac {x^8}{1-x^6} \, dx=-\frac {1}{3} \, x^{3} + \frac {1}{6} \, \log \left ({\left | x^{3} + 1 \right |}\right ) - \frac {1}{6} \, \log \left ({\left | x^{3} - 1 \right |}\right ) \]

[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))

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {x^8}{1-x^6} \, dx=\frac {\mathrm {atanh}\left (x^3\right )}{3}-\frac {x^3}{3} \]

[In]

int(-x^8/(x^6 - 1),x)

[Out]

atanh(x^3)/3 - x^3/3