\(\int \frac {x^5}{9-x^{12}} \, dx\) [1544]

Optimal result

Integrand size = 13, antiderivative size = 12 \[ \int \frac {x^5}{9-x^{12}} \, dx=\frac {1}{18} \text {arctanh}\left (\frac {x^6}{3}\right ) \]

[Out]

1/18*arctanh(1/3*x^6)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {281, 212} \[ \int \frac {x^5}{9-x^{12}} \, dx=\frac {1}{18} \text {arctanh}\left (\frac {x^6}{3}\right ) \]

[In]

Int[x^5/(9 - x^12),x]

[Out]

ArcTanh[x^6/3]/18

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]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{6} \text {Subst}\left (\int \frac {1}{9-x^2} \, dx,x,x^6\right ) \\ & = \frac {1}{18} \tanh ^{-1}\left (\frac {x^6}{3}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.92 \[ \int \frac {x^5}{9-x^{12}} \, dx=-\frac {1}{36} \log \left (3-x^6\right )+\frac {1}{36} \log \left (3+x^6\right ) \]

[In]

Integrate[x^5/(9 - x^12),x]

[Out]

-1/36*Log[3 - x^6] + Log[3 + x^6]/36

Maple [A] (verified)

Time = 3.50 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.75

method	result	size
meijerg	\(\frac {\operatorname {arctanh}\left (\frac {x^{6}}{3}\right )}{18}\)	\(9\)
default	\(\frac {\ln \left (x^{6}+3\right )}{36}-\frac {\ln \left (x^{6}-3\right )}{36}\)	\(18\)
norman	\(\frac {\ln \left (x^{6}+3\right )}{36}-\frac {\ln \left (x^{6}-3\right )}{36}\)	\(18\)
risch	\(\frac {\ln \left (x^{6}+3\right )}{36}-\frac {\ln \left (x^{6}-3\right )}{36}\)	\(18\)
parallelrisch	\(\frac {\ln \left (x^{6}+3\right )}{36}-\frac {\ln \left (x^{6}-3\right )}{36}\)	\(18\)

[In]

int(x^5/(-x^12+9),x,method=_RETURNVERBOSE)

[Out]

1/18*arctanh(1/3*x^6)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 17 vs. \(2 (8) = 16\).

Time = 0.27 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.42 \[ \int \frac {x^5}{9-x^{12}} \, dx=\frac {1}{36} \, \log \left (x^{6} + 3\right ) - \frac {1}{36} \, \log \left (x^{6} - 3\right ) \]

[In]

integrate(x^5/(-x^12+9),x, algorithm="fricas")

[Out]

1/36*log(x^6 + 3) - 1/36*log(x^6 - 3)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 15 vs. \(2 (7) = 14\).

Time = 0.06 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.25 \[ \int \frac {x^5}{9-x^{12}} \, dx=- \frac {\log {\left (x^{6} - 3 \right )}}{36} + \frac {\log {\left (x^{6} + 3 \right )}}{36} \]

[In]

integrate(x**5/(-x**12+9),x)

[Out]

-log(x**6 - 3)/36 + log(x**6 + 3)/36

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 17 vs. \(2 (8) = 16\).

Time = 0.27 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.42 \[ \int \frac {x^5}{9-x^{12}} \, dx=\frac {1}{36} \, \log \left (x^{6} + 3\right ) - \frac {1}{36} \, \log \left (x^{6} - 3\right ) \]

[In]

integrate(x^5/(-x^12+9),x, algorithm="maxima")

[Out]

1/36*log(x^6 + 3) - 1/36*log(x^6 - 3)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 18 vs. \(2 (8) = 16\).

Time = 0.29 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.50 \[ \int \frac {x^5}{9-x^{12}} \, dx=\frac {1}{36} \, \log \left (x^{6} + 3\right ) - \frac {1}{36} \, \log \left ({\left | x^{6} - 3 \right |}\right ) \]

[In]

integrate(x^5/(-x^12+9),x, algorithm="giac")

[Out]

1/36*log(x^6 + 3) - 1/36*log(abs(x^6 - 3))

Mupad [B] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.67 \[ \int \frac {x^5}{9-x^{12}} \, dx=\frac {\mathrm {atanh}\left (\frac {x^6}{3}\right )}{18} \]

[In]

int(-x^5/(x^12 - 9),x)

[Out]

atanh(x^6/3)/18