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

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Rubi [A]  time = 0.01, antiderivative size = 15, normalized size of antiderivative = 1.00, 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 30

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

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]

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*}

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Mathematica [A]  time = 0.00, size = 15, normalized size = 1.00 \[ \frac {2 \sqrt {\log \left (a x^n\right )}}{n} \]

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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fricas [A]  time = 0.41, size = 14, normalized size = 0.93 \[ \frac {2 \, \sqrt {n \log \relax (x) + \log \relax (a)}}{n} \]

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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giac [A]  time = 0.21, size = 14, normalized size = 0.93 \[ \frac {2 \, \sqrt {n \log \relax (x) + \log \relax (a)}}{n} \]

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

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maple [A]  time = 0.03, size = 14, normalized size = 0.93 \[ \frac {2 \sqrt {\ln \left (a \,x^{n}\right )}}{n} \]

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 = 0.55, size = 13, normalized size = 0.87 \[ \frac {2 \, \sqrt {\log \left (a x^{n}\right )}}{n} \]

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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mupad [B]  time = 3.58, size = 13, normalized size = 0.87 \[ \frac {2\,\sqrt {\ln \left (a\,x^n\right )}}{n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 2.08, size = 24, normalized size = 1.60 \[ \begin {cases} \frac {2 \sqrt {n \log {\relax (x )} + \log {\relax (a )}}}{n} & \text {for}\: n \neq 0 \\\frac {\log {\relax (x )}}{\sqrt {\log {\relax (a )}}} & \text {otherwise} \end {cases} \]

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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