3.4.26 \(\int \frac {x^4}{16+x^{10}} \, dx\) [326]

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

[Out]

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

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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 = {281, 209} \begin {gather*} \frac {1}{20} \text {ArcTan}\left (\frac {x^5}{4}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^4/(16 + x^10),x]

[Out]

ArcTan[x^5/4]/20

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[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*} \int \frac {x^4}{16+x^{10}} \, dx &=\frac {1}{5} \text {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*}

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Mathematica [A]
time = 0.00, size = 12, normalized size = 1.00 \begin {gather*} \frac {1}{20} \tan ^{-1}\left (\frac {x^5}{4}\right ) \end {gather*}

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 = 0.06, size = 9, normalized size = 0.75

method	result	size
default	\(\frac {\arctan \left (\frac {x^{5}}{4}\right )}{20}\)	\(9\)
meijerg	\(\frac {\arctan \left (\frac {x^{5}}{4}\right )}{20}\)	\(9\)
risch	\(\frac {\arctan \left (\frac {x^{5}}{4}\right )}{20}\)	\(9\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(x^10+16),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 1.91, size = 8, normalized size = 0.67 \begin {gather*} \frac {1}{20} \, \arctan \left (\frac {1}{4} \, x^{5}\right ) \end {gather*}

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 = 1.03, size = 8, normalized size = 0.67 \begin {gather*} \frac {1}{20} \, \arctan \left (\frac {1}{4} \, x^{5}\right ) \end {gather*}

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 = 0.04, size = 7, normalized size = 0.58 \begin {gather*} \frac {\operatorname {atan}{\left (\frac {x^{5}}{4} \right )}}{20} \end {gather*}

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 = 0.87, size = 8, normalized size = 0.67 \begin {gather*} \frac {1}{20} \, \arctan \left (\frac {1}{4} \, x^{5}\right ) \end {gather*}

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)

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Mupad [B]
time = 0.18, size = 8, normalized size = 0.67 \begin {gather*} \frac {\mathrm {atan}\left (\frac {x^5}{4}\right )}{20} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(x^10 + 16),x)

[Out]

atan(x^5/4)/20

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