3.1.81 \(\int \frac {x^5}{1+x^4} \, dx\) [81]

Optimal. Leaf size=16 \[ \frac {x^2}{2}-\frac {1}{2} \tan ^{-1}\left (x^2\right ) \]

[Out]

1/2*x^2-1/2*arctan(x^2)

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Rubi [A]
time = 0.00, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {281, 327, 209} \begin {gather*} \frac {x^2}{2}-\frac {1}{2} \tan ^{-1}\left (x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^5/(1 + x^4),x]

[Out]

x^2/2 - ArcTan[x^2]/2

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]

Rule 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rubi steps

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

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Mathematica [A]
time = 0.00, size = 16, normalized size = 1.00 \begin {gather*} \frac {x^2}{2}-\frac {1}{2} \tan ^{-1}\left (x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^5/(1 + x^4),x]

[Out]

x^2/2 - ArcTan[x^2]/2

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Mathics [A]
time = 1.75, size = 12, normalized size = 0.75 \begin {gather*} \frac {x^2}{2}-\frac {\text {ArcTan}\left [x^2\right ]}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[x^5/(1 + x^4),x]')

[Out]

x ^ 2 / 2 - ArcTan[x ^ 2] / 2

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Maple [A]
time = 0.03, size = 13, normalized size = 0.81

method	result	size
default	\(\frac {x^{2}}{2}-\frac {\arctan \left (x^{2}\right )}{2}\)	\(13\)
meijerg	\(\frac {x^{2}}{2}-\frac {\arctan \left (x^{2}\right )}{2}\)	\(13\)
risch	\(\frac {x^{2}}{2}-\frac {\arctan \left (x^{2}\right )}{2}\)	\(13\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(x^4+1),x,method=_RETURNVERBOSE)

[Out]

1/2*x^2-1/2*arctan(x^2)

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Maxima [A]
time = 0.35, size = 12, normalized size = 0.75 \begin {gather*} \frac {1}{2} \, x^{2} - \frac {1}{2} \, \arctan \left (x^{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(x^4+1),x, algorithm="maxima")

[Out]

1/2*x^2 - 1/2*arctan(x^2)

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Fricas [A]
time = 0.31, size = 12, normalized size = 0.75 \begin {gather*} \frac {1}{2} \, x^{2} - \frac {1}{2} \, \arctan \left (x^{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(x^4+1),x, algorithm="fricas")

[Out]

1/2*x^2 - 1/2*arctan(x^2)

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Sympy [A]
time = 0.05, size = 10, normalized size = 0.62 \begin {gather*} \frac {x^{2}}{2} - \frac {\operatorname {atan}{\left (x^{2} \right )}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5/(x**4+1),x)

[Out]

x**2/2 - atan(x**2)/2

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Giac [A]
time = 0.00, size = 15, normalized size = 0.94 \begin {gather*} \frac {x^{2}}{2}-\frac {\arctan \left (x^{2}\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(x^4+1),x)

[Out]

1/2*x^2 - 1/2*arctan(x^2)

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Mupad [B]
time = 0.02, size = 12, normalized size = 0.75 \begin {gather*} \frac {x^2}{2}-\frac {\mathrm {atan}\left (x^2\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(x^4 + 1),x)

[Out]

x^2/2 - atan(x^2)/2

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