3.326 \(\int \frac{x^4}{16+x^{10}} \, dx\)

Optimal. Leaf size=12 \[ \frac{1}{20} \tan ^{-1}\left (\frac{x^5}{4}\right ) \]

[Out]

ArcTan[x^5/4]/20

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Rubi [A]  time = 0.0050937, antiderivative size = 12, normalized size of antiderivative = 1., 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} \[ \frac{1}{20} \tan ^{-1}\left (\frac{x^5}{4}\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^4/(16 + x^10),x]

[Out]

ArcTan[x^5/4]/20

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 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])

Rubi steps

\begin{align*} \int \frac{x^4}{16+x^{10}} \, dx &=\frac{1}{5} \operatorname{Subst}\left (\int \frac{1}{16+x^2} \, dx,x,x^5\right )\\ &=\frac{1}{20} \tan ^{-1}\left (\frac{x^5}{4}\right )\\ \end{align*}

Mathematica [A]  time = 0.0036869, size = 12, normalized size = 1. \[ \frac{1}{20} \tan ^{-1}\left (\frac{x^5}{4}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^4/(16 + x^10),x]

[Out]

ArcTan[x^5/4]/20

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Maple [A]  time = 0.001, size = 9, normalized size = 0.8 \begin{align*}{\frac{1}{20}\arctan \left ({\frac{{x}^{5}}{4}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(x^10+16),x)

[Out]

1/20*arctan(1/4*x^5)

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Maxima [A]  time = 1.40738, size = 11, normalized size = 0.92 \begin{align*} \frac{1}{20} \, \arctan \left (\frac{1}{4} \, x^{5}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(x^10+16),x, algorithm="maxima")

[Out]

1/20*arctan(1/4*x^5)

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Fricas [A]  time = 2.14365, size = 30, normalized size = 2.5 \begin{align*} \frac{1}{20} \, \arctan \left (\frac{1}{4} \, x^{5}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(x^10+16),x, algorithm="fricas")

[Out]

1/20*arctan(1/4*x^5)

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Sympy [A]  time = 0.111918, size = 7, normalized size = 0.58 \begin{align*} \frac{\operatorname{atan}{\left (\frac{x^{5}}{4} \right )}}{20} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4/(x**10+16),x)

[Out]

atan(x**5/4)/20

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Giac [A]  time = 1.06897, size = 11, normalized size = 0.92 \begin{align*} \frac{1}{20} \, \arctan \left (\frac{1}{4} \, x^{5}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(x^10+16),x, algorithm="giac")

[Out]

1/20*arctan(1/4*x^5)