3.3.43 \(\int \log (2+3 x^2) \, dx\) [243]

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 33, normalized size of antiderivative = 1.00, 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 = {2498, 327, 209} \begin {gather*} 2 \sqrt {\frac {2}{3}} \text {ArcTan}\left (\sqrt {\frac {3}{2}} x\right )+x \log \left (3 x^2+2\right )-2 x \end {gather*}

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 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 327

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 2498

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]

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*}

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Mathematica [A]
time = 0.01, size = 33, normalized size = 1.00 \begin {gather*} -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 {gather*}

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]

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Maple [A]
time = 0.02, size = 27, normalized size = 0.82

method	result	size
default	\(-2 x +x \ln \left (3 x^{2}+2\right )+\frac {2 \arctan \left (\frac {x \sqrt {6}}{2}\right ) \sqrt {6}}{3}\)	\(27\)
risch	\(-2 x +x \ln \left (3 x^{2}+2\right )+\frac {2 \arctan \left (\frac {x \sqrt {6}}{2}\right ) \sqrt {6}}{3}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(3*x^2+2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 2.17, size = 26, normalized size = 0.79 \begin {gather*} 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 {gather*}

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

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Fricas [A]
time = 0.66, size = 32, normalized size = 0.97 \begin {gather*} \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 {gather*}

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

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Sympy [A]
time = 0.05, size = 31, normalized size = 0.94 \begin {gather*} 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 {gather*}

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

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Giac [A]
time = 0.44, size = 26, normalized size = 0.79 \begin {gather*} 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 {gather*}

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

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Mupad [B]
time = 0.06, size = 26, normalized size = 0.79 \begin {gather*} \frac {2\,\sqrt {6}\,\mathrm {atan}\left (\frac {\sqrt {6}\,x}{2}\right )}{3}-2\,x+x\,\ln \left (3\,x^2+2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(3*x^2 + 2),x)

[Out]

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

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