3.132 \(\int \frac{1}{x \sqrt{\log (a x^n)}} \, dx\)

Optimal. Leaf size=15 \[ \frac{2 \sqrt{\log \left (a x^n\right )}}{n} \]

[Out]

(2*Sqrt[Log[a*x^n]])/n

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Rubi [A]  time = 0.013232, antiderivative size = 15, 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 = {2302, 30} \[ \frac{2 \sqrt{\log \left (a x^n\right )}}{n} \]

Antiderivative was successfully verified.

[In]

Int[1/(x*Sqrt[Log[a*x^n]]),x]

[Out]

(2*Sqrt[Log[a*x^n]])/n

Rule 2302

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.0013985, size = 15, normalized size = 1. \[ \frac{2 \sqrt{\log \left (a x^n\right )}}{n} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*Sqrt[Log[a*x^n]]),x]

[Out]

(2*Sqrt[Log[a*x^n]])/n

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Maple [A]  time = 0.039, size = 14, normalized size = 0.9 \begin{align*} 2\,{\frac{\sqrt{\ln \left ( a{x}^{n} \right ) }}{n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/ln(a*x^n)^(1/2),x)

[Out]

2*ln(a*x^n)^(1/2)/n

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Maxima [A]  time = 1.1009, size = 18, normalized size = 1.2 \begin{align*} \frac{2 \, \sqrt{\log \left (a x^{n}\right )}}{n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/log(a*x^n)^(1/2),x, algorithm="maxima")

[Out]

2*sqrt(log(a*x^n))/n

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Fricas [A]  time = 0.943296, size = 39, normalized size = 2.6 \begin{align*} \frac{2 \, \sqrt{n \log \left (x\right ) + \log \left (a\right )}}{n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/log(a*x^n)^(1/2),x, algorithm="fricas")

[Out]

2*sqrt(n*log(x) + log(a))/n

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Sympy [A]  time = 3.26988, size = 24, normalized size = 1.6 \begin{align*} \begin{cases} \frac{2 \sqrt{n \log{\left (x \right )} + \log{\left (a \right )}}}{n} & \text{for}\: n \neq 0 \\\frac{\log{\left (x \right )}}{\sqrt{\log{\left (a \right )}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/ln(a*x**n)**(1/2),x)

[Out]

Piecewise((2*sqrt(n*log(x) + log(a))/n, Ne(n, 0)), (log(x)/sqrt(log(a)), True))

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Giac [A]  time = 1.28396, size = 19, normalized size = 1.27 \begin{align*} \frac{2 \, \sqrt{n \log \left (x\right ) + \log \left (a\right )}}{n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/log(a*x^n)^(1/2),x, algorithm="giac")

[Out]

2*sqrt(n*log(x) + log(a))/n