3.1373 \(\int \frac{1}{x^4 (1+x^6)} \, dx\)

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

[Out]

-1/(3*x^3) - ArcTan[x^3]/3

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Rubi [A]  time = 0.0064709, antiderivative size = 16, normalized size of antiderivative = 1., 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 = {275, 325, 203} \[ -\frac{1}{3 x^3}-\frac{1}{3} \tan ^{-1}\left (x^3\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/(x^4*(1 + x^6)),x]

[Out]

-1/(3*x^3) - ArcTan[x^3]/3

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 325

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]

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{1}{x^4 \left (1+x^6\right )} \, dx &=\frac{1}{3} \operatorname{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} \operatorname{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*}

Mathematica [A]  time = 0.0043492, size = 16, normalized size = 1. \[ \frac{1}{3} \tan ^{-1}\left (\frac{1}{x^3}\right )-\frac{1}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^4*(1 + x^6)),x]

[Out]

-1/(3*x^3) + ArcTan[x^(-3)]/3

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Maple [B]  time = 0.013, size = 33, normalized size = 2.1 \begin{align*}{\frac{\arctan \left ( x \right ) }{3}}-{\frac{1}{3\,{x}^{3}}}-{\frac{\arctan \left ( 2\,x-\sqrt{3} \right ) }{3}}-{\frac{\arctan \left ( 2\,x+\sqrt{3} \right ) }{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^4/(x^6+1),x)

[Out]

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

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Maxima [A]  time = 2.41633, size = 16, normalized size = 1. \begin{align*} -\frac{1}{3 \, x^{3}} - \frac{1}{3} \, \arctan \left (x^{3}\right ) \end{align*}

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 = 1.44824, size = 43, normalized size = 2.69 \begin{align*} -\frac{x^{3} \arctan \left (x^{3}\right ) + 1}{3 \, x^{3}} \end{align*}

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 = 0.115805, size = 14, normalized size = 0.88 \begin{align*} - \frac{\operatorname{atan}{\left (x^{3} \right )}}{3} - \frac{1}{3 x^{3}} \end{align*}

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 = 1.12215, size = 16, normalized size = 1. \begin{align*} -\frac{1}{3 \, x^{3}} - \frac{1}{3} \, \arctan \left (x^{3}\right ) \end{align*}

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)