3.434 \(\int \frac{1+x^2}{3 x+x^3} \, dx\)

Optimal. Leaf size=12 \[ \frac{1}{3} \log \left (x^3+3 x\right ) \]

[Out]

Log[3*x + x^3]/3

________________________________________________________________________________________

Rubi [A]  time = 0.006125, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {1587} \[ \frac{1}{3} \log \left (x^3+3 x\right ) \]

Antiderivative was successfully verified.

[In]

Int[(1 + x^2)/(3*x + x^3),x]

[Out]

Log[3*x + x^3]/3

Rule 1587

Int[(Pp_)/(Qq_), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[(Coeff[Pp, x, p]*Log[RemoveConte
nt[Qq, x]])/(q*Coeff[Qq, x, q]), x] /; EqQ[p, q - 1] && EqQ[Pp, Simplify[(Coeff[Pp, x, p]*D[Qq, x])/(q*Coeff[Q
q, x, q])]]] /; PolyQ[Pp, x] && PolyQ[Qq, x]

Rubi steps

\begin{align*} \int \frac{1+x^2}{3 x+x^3} \, dx &=\frac{1}{3} \log \left (3 x+x^3\right )\\ \end{align*}

Mathematica [A]  time = 0.0034966, size = 17, normalized size = 1.42 \[ \frac{1}{3} \log \left (x^2+3\right )+\frac{\log (x)}{3} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + x^2)/(3*x + x^3),x]

[Out]

Log[x]/3 + Log[3 + x^2]/3

________________________________________________________________________________________

Maple [A]  time = 0.002, size = 11, normalized size = 0.9 \begin{align*}{\frac{\ln \left ( x \left ({x}^{2}+3 \right ) \right ) }{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2+1)/(x^3+3*x),x)

[Out]

1/3*ln(x*(x^2+3))

________________________________________________________________________________________

Maxima [A]  time = 0.993105, size = 14, normalized size = 1.17 \begin{align*} \frac{1}{3} \, \log \left (x^{3} + 3 \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+1)/(x^3+3*x),x, algorithm="maxima")

[Out]

1/3*log(x^3 + 3*x)

________________________________________________________________________________________

Fricas [A]  time = 0.948328, size = 27, normalized size = 2.25 \begin{align*} \frac{1}{3} \, \log \left (x^{3} + 3 \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+1)/(x^3+3*x),x, algorithm="fricas")

[Out]

1/3*log(x^3 + 3*x)

________________________________________________________________________________________

Sympy [A]  time = 0.083129, size = 8, normalized size = 0.67 \begin{align*} \frac{\log{\left (x^{3} + 3 x \right )}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**2+1)/(x**3+3*x),x)

[Out]

log(x**3 + 3*x)/3

________________________________________________________________________________________

Giac [A]  time = 1.1117, size = 20, normalized size = 1.67 \begin{align*} \frac{1}{3} \, \log \left (x^{2} + 3\right ) + \frac{1}{6} \, \log \left (x^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+1)/(x^3+3*x),x, algorithm="giac")

[Out]

1/3*log(x^2 + 3) + 1/6*log(x^2)