3.151 \(\int \frac {\sqrt {\log ^2(a x^n)}}{x} \, dx\)

Optimal. Leaf size=25 \[ \frac {\log \left (a x^n\right ) \sqrt {\log ^2\left (a x^n\right )}}{2 n} \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {15, 30} \[ \frac {\log \left (a x^n\right ) \sqrt {\log ^2\left (a x^n\right )}}{2 n} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[Log[a*x^n]^2]/x,x]

[Out]

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

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[(a^IntPart[m]*(a*x^n)^FracPart[m])/x^(n*FracPart[m]), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

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

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Mathematica [A]  time = 0.01, size = 25, normalized size = 1.00 \[ \frac {\log \left (a x^n\right ) \sqrt {\log ^2\left (a x^n\right )}}{2 n} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[Log[a*x^n]^2]/x,x]

[Out]

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

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fricas [A]  time = 1.55, size = 13, normalized size = 0.52 \[ \frac {1}{2} \, n \log \relax (x)^{2} + \log \relax (a) \log \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*n*log(x)^2 + log(a)*log(x)

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giac [A]  time = 0.20, size = 27, normalized size = 1.08 \[ \frac {1}{2} \, n \log \relax (x)^{2} \mathrm {sgn}\left (\log \left (a x^{n}\right )\right ) + \log \relax (a) \log \relax (x) \mathrm {sgn}\left (\log \left (a x^{n}\right )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.35, size = 21, normalized size = 0.84 \[ \frac {\mathrm {csgn}\left (\ln \left (a \,x^{n}\right )\right ) \ln \left (a \,x^{n}\right )^{2}}{2 n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.63, size = 20, normalized size = 0.80 \[ -\frac {1}{2} \, n \log \relax (x)^{2} + \log \relax (a) \log \relax (x) + \log \relax (x) \log \left (x^{n}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.40, size = 21, normalized size = 0.84 \[ \frac {\ln \left (a\,x^n\right )\,\sqrt {{\ln \left (a\,x^n\right )}^2}}{2\,n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\log {\left (a x^{n} \right )}^{2}}}{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(log(a*x**n)**2)/x, x)

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