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]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0200935, antiderivative size = 25, normalized size of antiderivative = 1., 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*}

Mathematica [A]  time = 0.0078259, size = 25, normalized size = 1. \[ \frac{\log \left (a x^n\right ) \sqrt{\log ^2\left (a x^n\right )}}{2 n} \]

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)

________________________________________________________________________________________

Maple [C]  time = 0.031, size = 21, normalized size = 0.8 \begin{align*}{\frac{{\it csgn} \left ( \ln \left ( a{x}^{n} \right ) \right ) \left ( \ln \left ( a{x}^{n} \right ) \right ) ^{2}}{2\,n}} \end{align*}

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

________________________________________________________________________________________

Maxima [A]  time = 1.04749, size = 27, normalized size = 1.08 \begin{align*} -\frac{1}{2} \, n \log \left (x\right )^{2} + \log \left (a\right ) \log \left (x\right ) + \log \left (x\right ) \log \left (x^{n}\right ) \end{align*}

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)

________________________________________________________________________________________

Fricas [A]  time = 2.09794, size = 43, normalized size = 1.72 \begin{align*} \frac{1}{2} \, n \log \left (x\right )^{2} + \log \left (a\right ) \log \left (x\right ) \end{align*}

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)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\log{\left (a x^{n} \right )}^{2}}}{x}\, dx \end{align*}

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)

________________________________________________________________________________________

Giac [A]  time = 1.35093, size = 36, normalized size = 1.44 \begin{align*} \frac{1}{2} \, n \log \left (x\right )^{2} \mathrm{sgn}\left (\log \left (a x^{n}\right )\right ) + \log \left (a\right ) \log \left (x\right ) \mathrm{sgn}\left (\log \left (a x^{n}\right )\right ) \end{align*}

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))