3.14.73 \(\int \frac {1}{x^4 (1+x^6)} \, dx\) [1373]

Optimal. Leaf size=16 \[ -\frac {1}{3 x^3}-\frac {1}{3} \tan ^{-1}\left (x^3\right ) \]

[Out]

-1/3/x^3-1/3*arctan(x^3)

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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, 331, 209} \begin {gather*} -\frac {1}{3} \text {ArcTan}\left (x^3\right )-\frac {1}{3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^4*(1 + x^6)),x]

[Out]

-1/3*1/x^3 - ArcTan[x^3]/3

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 331

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

Rubi steps

\begin {align*} \int \frac {1}{x^4 \left (1+x^6\right )} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {1}{x^2 \left (1+x^2\right )} \, dx,x,x^3\right )\\ &=-\frac {1}{3 x^3}-\frac {1}{3} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,x^3\right )\\ &=-\frac {1}{3 x^3}-\frac {1}{3} \tan ^{-1}\left (x^3\right )\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 16, normalized size = 1.00 \begin {gather*} -\frac {1}{3 x^3}+\frac {1}{3} \tan ^{-1}\left (\frac {1}{x^3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^4*(1 + x^6)),x]

[Out]

-1/3*1/x^3 + ArcTan[x^(-3)]/3

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(32\) vs. \(2(12)=24\).
time = 0.18, size = 33, normalized size = 2.06

method	result	size
meijerg	\(-\frac {1}{3 x^{3}}-\frac {\arctan \left (x^{3}\right )}{3}\)	\(13\)
risch	\(-\frac {1}{3 x^{3}}-\frac {\arctan \left (x^{3}\right )}{3}\)	\(13\)
default	\(-\frac {\arctan \left (2 x +\sqrt {3}\right )}{3}-\frac {\arctan \left (2 x -\sqrt {3}\right )}{3}-\frac {1}{3 x^{3}}+\frac {\arctan \left (x \right )}{3}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^4/(x^6+1),x,method=_RETURNVERBOSE)

[Out]

-1/3*arctan(2*x+3^(1/2))-1/3*arctan(2*x-3^(1/2))-1/3/x^3+1/3*arctan(x)

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Maxima [A]
time = 0.52, size = 12, normalized size = 0.75 \begin {gather*} -\frac {1}{3 \, x^{3}} - \frac {1}{3} \, \arctan \left (x^{3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(x^6+1),x, algorithm="maxima")

[Out]

-1/3/x^3 - 1/3*arctan(x^3)

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Fricas [A]
time = 0.38, size = 15, normalized size = 0.94 \begin {gather*} -\frac {x^{3} \arctan \left (x^{3}\right ) + 1}{3 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(x^6+1),x, algorithm="fricas")

[Out]

-1/3*(x^3*arctan(x^3) + 1)/x^3

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Sympy [A]
time = 0.04, size = 14, normalized size = 0.88 \begin {gather*} - \frac {\operatorname {atan}{\left (x^{3} \right )}}{3} - \frac {1}{3 x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**4/(x**6+1),x)

[Out]

-atan(x**3)/3 - 1/(3*x**3)

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Giac [A]
time = 1.10, size = 12, normalized size = 0.75 \begin {gather*} -\frac {1}{3 \, x^{3}} - \frac {1}{3} \, \arctan \left (x^{3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(x^6+1),x, algorithm="giac")

[Out]

-1/3/x^3 - 1/3*arctan(x^3)

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Mupad [B]
time = 1.02, size = 12, normalized size = 0.75 \begin {gather*} -\frac {\mathrm {atan}\left (x^3\right )}{3}-\frac {1}{3\,x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^4*(x^6 + 1)),x)

[Out]

- atan(x^3)/3 - 1/(3*x^3)

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