3.322 \(\int \frac {9+x^4}{x^2 (9+x^2)} \, dx\)

Optimal. Leaf size=17 \[ x-\frac {1}{x}-\frac {10}{3} \tan ^{-1}\left (\frac {x}{3}\right ) \]

[Out]

-1/x+x-10/3*arctan(1/3*x)

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Rubi [A]  time = 0.01, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1262, 203} \[ x-\frac {1}{x}-\frac {10}{3} \tan ^{-1}\left (\frac {x}{3}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(9 + x^4)/(x^2*(9 + x^2)),x]

[Out]

-x^(-1) + x - (10*ArcTan[x/3])/3

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])

Rule 1262

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[(f*x)^m*(d + e*x^2)^q*(a + c*x^4)^p, x], x] /; FreeQ[{a, c, d, e, f, m, q}, x] && IGtQ[p, 0] && IGtQ[q, -2]

Rubi steps

\begin {align*} \int \frac {9+x^4}{x^2 \left (9+x^2\right )} \, dx &=\int \left (1+\frac {1}{x^2}-\frac {10}{9+x^2}\right ) \, dx\\ &=-\frac {1}{x}+x-10 \int \frac {1}{9+x^2} \, dx\\ &=-\frac {1}{x}+x-\frac {10}{3} \tan ^{-1}\left (\frac {x}{3}\right )\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 17, normalized size = 1.00 \[ x-\frac {1}{x}-\frac {10}{3} \tan ^{-1}\left (\frac {x}{3}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(9 + x^4)/(x^2*(9 + x^2)),x]

[Out]

-x^(-1) + x - (10*ArcTan[x/3])/3

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fricas [A]  time = 0.89, size = 19, normalized size = 1.12 \[ \frac {3 \, x^{2} - 10 \, x \arctan \left (\frac {1}{3} \, x\right ) - 3}{3 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+9)/x^2/(x^2+9),x, algorithm="fricas")

[Out]

1/3*(3*x^2 - 10*x*arctan(1/3*x) - 3)/x

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giac [A]  time = 0.30, size = 13, normalized size = 0.76 \[ x - \frac {1}{x} - \frac {10}{3} \, \arctan \left (\frac {1}{3} \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+9)/x^2/(x^2+9),x, algorithm="giac")

[Out]

x - 1/x - 10/3*arctan(1/3*x)

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maple [A]  time = 0.01, size = 14, normalized size = 0.82 \[ x -\frac {10 \arctan \left (\frac {x}{3}\right )}{3}-\frac {1}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4+9)/x^2/(x^2+9),x)

[Out]

-1/x+x-10/3*arctan(1/3*x)

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maxima [A]  time = 2.04, size = 13, normalized size = 0.76 \[ x - \frac {1}{x} - \frac {10}{3} \, \arctan \left (\frac {1}{3} \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+9)/x^2/(x^2+9),x, algorithm="maxima")

[Out]

x - 1/x - 10/3*arctan(1/3*x)

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mupad [B]  time = 0.03, size = 13, normalized size = 0.76 \[ x-\frac {10\,\mathrm {atan}\left (\frac {x}{3}\right )}{3}-\frac {1}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4 + 9)/(x^2*(x^2 + 9)),x)

[Out]

x - (10*atan(x/3))/3 - 1/x

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sympy [A]  time = 0.11, size = 12, normalized size = 0.71 \[ x - \frac {10 \operatorname {atan}{\left (\frac {x}{3} \right )}}{3} - \frac {1}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**4+9)/x**2/(x**2+9),x)

[Out]

x - 10*atan(x/3)/3 - 1/x

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