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]

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

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Rubi [A]  time = 0.01, antiderivative size = 12, normalized size of antiderivative = 1.00, 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 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])

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]

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*}

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Mathematica [A]  time = 0.00, 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]

-1/36*Log[3 - x^6] + Log[3 + x^6]/36

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fricas [B]  time = 0.89, size = 17, normalized size = 1.42 \[ \frac {1}{36} \, \log \left (x^{6} + 3\right ) - \frac {1}{36} \, \log \left (x^{6} - 3\right ) \]

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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giac [B]  time = 0.20, size = 18, normalized size = 1.50 \[ \frac {1}{36} \, \log \left (x^{6} + 3\right ) - \frac {1}{36} \, \log \left ({\left | x^{6} - 3 \right |}\right ) \]

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))

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maple [B]  time = 0.01, size = 18, normalized size = 1.50 \[ -\frac {\ln \left (x^{6}-3\right )}{36}+\frac {\ln \left (x^{6}+3\right )}{36} \]

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 = 1.01, size = 17, normalized size = 1.42 \[ \frac {1}{36} \, \log \left (x^{6} + 3\right ) - \frac {1}{36} \, \log \left (x^{6} - 3\right ) \]

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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mupad [B]  time = 1.08, size = 8, normalized size = 0.67 \[ \frac {\mathrm {atanh}\left (\frac {x^6}{3}\right )}{18} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-x^5/(x^12 - 9),x)

[Out]

atanh(x^6/3)/18

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sympy [B]  time = 0.14, size = 15, normalized size = 1.25 \[ - \frac {\log {\left (x^{6} - 3 \right )}}{36} + \frac {\log {\left (x^{6} + 3 \right )}}{36} \]

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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