3.1544 \(\int \frac{x^5}{9-x^{12}} \, dx\)

Optimal. Leaf size=12 \[ \frac{1}{18} \tanh ^{-1}\left (\frac{x^6}{3}\right ) \]

[Out]

ArcTanh[x^6/3]/18

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Rubi [A]  time = 0.0054283, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {275, 206} \[ \frac{1}{18} \tanh ^{-1}\left (\frac{x^6}{3}\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^5/(9 - x^12),x]

[Out]

ArcTanh[x^6/3]/18

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 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^5}{9-x^{12}} \, dx &=\frac{1}{6} \operatorname{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]  time = 0.0035037, size = 23, normalized size = 1.92 \[ \frac{1}{36} \log \left (x^6+3\right )-\frac{1}{36} \log \left (3-x^6\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^5/(9 - x^12),x]

[Out]

-Log[3 - x^6]/36 + Log[3 + x^6]/36

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Maple [B]  time = 0.004, size = 18, normalized size = 1.5 \begin{align*}{\frac{\ln \left ({x}^{6}+3 \right ) }{36}}-{\frac{\ln \left ({x}^{6}-3 \right ) }{36}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 0.948087, size = 23, normalized size = 1.92 \begin{align*} \frac{1}{36} \, \log \left (x^{6} + 3\right ) - \frac{1}{36} \, \log \left (x^{6} - 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/36*log(x^6 + 3) - 1/36*log(x^6 - 3)

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Fricas [B]  time = 1.41374, size = 53, normalized size = 4.42 \begin{align*} \frac{1}{36} \, \log \left (x^{6} + 3\right ) - \frac{1}{36} \, \log \left (x^{6} - 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/36*log(x^6 + 3) - 1/36*log(x^6 - 3)

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Sympy [B]  time = 0.125131, size = 15, normalized size = 1.25 \begin{align*} - \frac{\log{\left (x^{6} - 3 \right )}}{36} + \frac{\log{\left (x^{6} + 3 \right )}}{36} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(x**6 - 3)/36 + log(x**6 + 3)/36

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Giac [B]  time = 1.14702, size = 24, normalized size = 2. \begin{align*} \frac{1}{36} \, \log \left (x^{6} + 3\right ) - \frac{1}{36} \, \log \left ({\left | x^{6} - 3 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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