3.102 \(\int \log (2+3 x^2) \, dx\)

Optimal. Leaf size=33 \[ x \log \left (3 x^2+2\right )-2 x+2 \sqrt{\frac{2}{3}} \tan ^{-1}\left (\sqrt{\frac{3}{2}} x\right ) \]

[Out]

-2*x + 2*Sqrt[2/3]*ArcTan[Sqrt[3/2]*x] + x*Log[2 + 3*x^2]

________________________________________________________________________________________

Rubi [A]  time = 0.0111443, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {2448, 321, 203} \[ x \log \left (3 x^2+2\right )-2 x+2 \sqrt{\frac{2}{3}} \tan ^{-1}\left (\sqrt{\frac{3}{2}} x\right ) \]

Antiderivative was successfully verified.

[In]

Int[Log[2 + 3*x^2],x]

[Out]

-2*x + 2*Sqrt[2/3]*ArcTan[Sqrt[3/2]*x] + x*Log[2 + 3*x^2]

Rule 2448

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)], x_Symbol] :> Simp[x*Log[c*(d + e*x^n)^p], x] - Dist[e*n*p, Int[
x^n/(d + e*x^n), x], x] /; FreeQ[{c, d, e, n, p}, x]

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && 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 \log \left (2+3 x^2\right ) \, dx &=x \log \left (2+3 x^2\right )-6 \int \frac{x^2}{2+3 x^2} \, dx\\ &=-2 x+x \log \left (2+3 x^2\right )+4 \int \frac{1}{2+3 x^2} \, dx\\ &=-2 x+2 \sqrt{\frac{2}{3}} \tan ^{-1}\left (\sqrt{\frac{3}{2}} x\right )+x \log \left (2+3 x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0121088, size = 33, normalized size = 1. \[ x \log \left (3 x^2+2\right )-2 x+2 \sqrt{\frac{2}{3}} \tan ^{-1}\left (\sqrt{\frac{3}{2}} x\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Log[2 + 3*x^2],x]

[Out]

-2*x + 2*Sqrt[2/3]*ArcTan[Sqrt[3/2]*x] + x*Log[2 + 3*x^2]

________________________________________________________________________________________

Maple [A]  time = 0., size = 27, normalized size = 0.8 \begin{align*} -2\,x+x\ln \left ( 3\,{x}^{2}+2 \right ) +{\frac{2\,\sqrt{6}}{3}\arctan \left ({\frac{x\sqrt{6}}{2}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(3*x^2+2),x)

[Out]

-2*x+x*ln(3*x^2+2)+2/3*arctan(1/2*x*6^(1/2))*6^(1/2)

________________________________________________________________________________________

Maxima [A]  time = 1.41188, size = 35, normalized size = 1.06 \begin{align*} x \log \left (3 \, x^{2} + 2\right ) + \frac{2}{3} \, \sqrt{6} \arctan \left (\frac{1}{2} \, \sqrt{6} x\right ) - 2 \, x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*log(3*x^2 + 2) + 2/3*sqrt(6)*arctan(1/2*sqrt(6)*x) - 2*x

________________________________________________________________________________________

Fricas [A]  time = 1.89351, size = 103, normalized size = 3.12 \begin{align*} \frac{2}{3} \, \sqrt{3} \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{3} \sqrt{2} x\right ) + x \log \left (3 \, x^{2} + 2\right ) - 2 \, x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0.125984, size = 31, normalized size = 0.94 \begin{align*} x \log{\left (3 x^{2} + 2 \right )} - 2 x + \frac{2 \sqrt{6} \operatorname{atan}{\left (\frac{\sqrt{6} x}{2} \right )}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(3*x**2+2),x)

[Out]

x*log(3*x**2 + 2) - 2*x + 2*sqrt(6)*atan(sqrt(6)*x/2)/3

________________________________________________________________________________________

Giac [A]  time = 1.04873, size = 35, normalized size = 1.06 \begin{align*} x \log \left (3 \, x^{2} + 2\right ) + \frac{2}{3} \, \sqrt{6} \arctan \left (\frac{1}{2} \, \sqrt{6} x\right ) - 2 \, x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*log(3*x^2 + 2) + 2/3*sqrt(6)*arctan(1/2*sqrt(6)*x) - 2*x