3.1543 \(\int \frac {x^5}{9+x^{12}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.00, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {275, 203} \[ \frac {1}{18} \tan ^{-1}\left (\frac {x^6}{3}\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^5/(9 + x^12),x]

[Out]

ArcTan[x^6/3]/18

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[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} \tan ^{-1}\left (\frac {x^6}{3}\right )\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 12, normalized size = 1.00 \[ \frac {1}{18} \tan ^{-1}\left (\frac {x^6}{3}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^5/(9 + x^12),x]

[Out]

ArcTan[x^6/3]/18

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fricas [A]  time = 1.08, size = 8, normalized size = 0.67 \[ \frac {1}{18} \, \arctan \left (\frac {1}{3} \, x^{6}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.17, size = 8, normalized size = 0.67 \[ \frac {1}{18} \, \arctan \left (\frac {1}{3} \, x^{6}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.00, size = 9, normalized size = 0.75 \[ \frac {\arctan \left (\frac {x^{6}}{3}\right )}{18} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 2.30, size = 8, normalized size = 0.67 \[ \frac {1}{18} \, \arctan \left (\frac {1}{3} \, x^{6}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.06, size = 8, normalized size = 0.67 \[ \frac {\mathrm {atan}\left (\frac {x^6}{3}\right )}{18} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(x^12 + 9),x)

[Out]

atan(x^6/3)/18

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sympy [A]  time = 0.14, size = 7, normalized size = 0.58 \[ \frac {\operatorname {atan}{\left (\frac {x^{6}}{3} \right )}}{18} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

atan(x**6/3)/18

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