3.154 \(\int \frac{x^2}{3+4 x^3+x^6} \, dx\)

Optimal. Leaf size=10 \[ -\frac{1}{3} \tanh ^{-1}\left (x^3+2\right ) \]

[Out]

-ArcTanh[2 + x^3]/3

________________________________________________________________________________________

Rubi [B]  time = 0.0139787, antiderivative size = 21, normalized size of antiderivative = 2.1, number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {1352, 616, 31} \[ \frac{1}{6} \log \left (x^3+1\right )-\frac{1}{6} \log \left (x^3+3\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^2/(3 + 4*x^3 + x^6),x]

[Out]

Log[1 + x^3]/6 - Log[3 + x^3]/6

Rule 1352

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[(a + b*x +
c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[Simplify[m - n + 1], 0]

Rule 616

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[c/q, Int[1/Simp
[b/2 - q/2 + c*x, x], x], x] - Dist[c/q, Int[1/Simp[b/2 + q/2 + c*x, x], x], x]] /; FreeQ[{a, b, c}, x] && NeQ
[b^2 - 4*a*c, 0] && PosQ[b^2 - 4*a*c] && PerfectSquareQ[b^2 - 4*a*c]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

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

Mathematica [B]  time = 0.0035071, size = 21, normalized size = 2.1 \[ \frac{1}{6} \log \left (x^3+1\right )-\frac{1}{6} \log \left (x^3+3\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/(3 + 4*x^3 + x^6),x]

[Out]

Log[1 + x^3]/6 - Log[3 + x^3]/6

________________________________________________________________________________________

Maple [B]  time = 0.004, size = 18, normalized size = 1.8 \begin{align*}{\frac{\ln \left ({x}^{3}+1 \right ) }{6}}-{\frac{\ln \left ({x}^{3}+3 \right ) }{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(x^6+4*x^3+3),x)

[Out]

1/6*ln(x^3+1)-1/6*ln(x^3+3)

________________________________________________________________________________________

Maxima [B]  time = 1.38805, size = 23, normalized size = 2.3 \begin{align*} -\frac{1}{6} \, \log \left (x^{3} + 3\right ) + \frac{1}{6} \, \log \left (x^{3} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(x^6+4*x^3+3),x, algorithm="maxima")

[Out]

-1/6*log(x^3 + 3) + 1/6*log(x^3 + 1)

________________________________________________________________________________________

Fricas [B]  time = 1.4608, size = 51, normalized size = 5.1 \begin{align*} -\frac{1}{6} \, \log \left (x^{3} + 3\right ) + \frac{1}{6} \, \log \left (x^{3} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(x^6+4*x^3+3),x, algorithm="fricas")

[Out]

-1/6*log(x^3 + 3) + 1/6*log(x^3 + 1)

________________________________________________________________________________________

Sympy [A]  time = 0.111578, size = 15, normalized size = 1.5 \begin{align*} \frac{\log{\left (x^{3} + 1 \right )}}{6} - \frac{\log{\left (x^{3} + 3 \right )}}{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/(x**6+4*x**3+3),x)

[Out]

log(x**3 + 1)/6 - log(x**3 + 3)/6

________________________________________________________________________________________

Giac [B]  time = 1.09131, size = 26, normalized size = 2.6 \begin{align*} -\frac{1}{6} \, \log \left ({\left | x^{3} + 3 \right |}\right ) + \frac{1}{6} \, \log \left ({\left | x^{3} + 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(x^6+4*x^3+3),x, algorithm="giac")

[Out]

-1/6*log(abs(x^3 + 3)) + 1/6*log(abs(x^3 + 1))