3.291 \(\int \frac{\log (x)}{x+4 x \log ^2(x)} \, dx\)

Optimal. Leaf size=13 \[ \frac{1}{8} \log \left (4 \log ^2(x)+1\right ) \]

[Out]

Log[1 + 4*Log[x]^2]/8

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Rubi [A]  time = 0.0243466, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {203, 260} \[ \frac{1}{8} \log \left (4 \log ^2(x)+1\right ) \]

Antiderivative was successfully verified.

[In]

Int[Log[x]/(x + 4*x*Log[x]^2),x]

[Out]

Log[1 + 4*Log[x]^2]/8

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 260

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

Rubi steps

\begin{align*} \int \frac{\log (x)}{x+4 x \log ^2(x)} \, dx &=\operatorname{Subst}\left (\int \frac{x}{1+4 x^2} \, dx,x,\log (x)\right )\\ &=\frac{1}{8} \log \left (1+4 \log ^2(x)\right )\\ \end{align*}

Mathematica [A]  time = 0.0082652, size = 13, normalized size = 1. \[ \frac{1}{8} \log \left (4 \log ^2(x)+1\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Log[1 + 4*Log[x]^2]/8

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Maple [A]  time = 0.003, size = 12, normalized size = 0.9 \begin{align*}{\frac{\ln \left ( 1+4\, \left ( \ln \left ( x \right ) \right ) ^{2} \right ) }{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(x)/(x+4*x*ln(x)^2),x)

[Out]

1/8*ln(1+4*ln(x)^2)

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Maxima [A]  time = 1.19732, size = 12, normalized size = 0.92 \begin{align*} \frac{1}{8} \, \log \left (\log \left (x\right )^{2} + \frac{1}{4}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*log(log(x)^2 + 1/4)

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Fricas [A]  time = 2.12501, size = 34, normalized size = 2.62 \begin{align*} \frac{1}{8} \, \log \left (4 \, \log \left (x\right )^{2} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*log(4*log(x)^2 + 1)

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Sympy [A]  time = 0.115593, size = 10, normalized size = 0.77 \begin{align*} \frac{\log{\left (\log{\left (x \right )}^{2} + \frac{1}{4} \right )}}{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(log(x)**2 + 1/4)/8

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Giac [A]  time = 1.33486, size = 15, normalized size = 1.15 \begin{align*} \frac{1}{8} \, \log \left (4 \, \log \left (x\right )^{2} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*log(4*log(x)^2 + 1)