3.56 \(\int \frac{\log (c \log ^p(d x))}{x} \, dx\)

Optimal. Leaf size=20 \[ \log (d x) \log \left (c \log ^p(d x)\right )-p \log (x) \]

[Out]

-(p*Log[x]) + Log[d*x]*Log[c*Log[d*x]^p]

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Rubi [A]  time = 0.0198224, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {2521} \[ \log (d x) \log \left (c \log ^p(d x)\right )-p \log (x) \]

Antiderivative was successfully verified.

[In]

Int[Log[c*Log[d*x]^p]/x,x]

[Out]

-(p*Log[x]) + Log[d*x]*Log[c*Log[d*x]^p]

Rule 2521

Int[((a_.) + Log[Log[(d_.)*(x_)^(n_.)]^(p_.)*(c_.)]*(b_.))/(x_), x_Symbol] :> Simp[(Log[d*x^n]*(a + b*Log[c*Lo
g[d*x^n]^p]))/n, x] - Simp[b*p*Log[x], x] /; FreeQ[{a, b, c, d, n, p}, x]

Rubi steps

\begin{align*} \int \frac{\log \left (c \log ^p(d x)\right )}{x} \, dx &=-p \log (x)+\log (d x) \log \left (c \log ^p(d x)\right )\\ \end{align*}

Mathematica [A]  time = 0.0065151, size = 22, normalized size = 1.1 \[ \log (d x) \log \left (c \log ^p(d x)\right )-p \log (d x) \]

Antiderivative was successfully verified.

[In]

Integrate[Log[c*Log[d*x]^p]/x,x]

[Out]

-(p*Log[d*x]) + Log[d*x]*Log[c*Log[d*x]^p]

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Maple [A]  time = 0.006, size = 23, normalized size = 1.2 \begin{align*} \ln \left ( dx \right ) \ln \left ( c \left ( \ln \left ( dx \right ) \right ) ^{p} \right ) -p\ln \left ( dx \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(c*ln(d*x)^p)/x,x)

[Out]

ln(d*x)*ln(c*ln(d*x)^p)-p*ln(d*x)

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Maxima [A]  time = 0.993782, size = 30, normalized size = 1.5 \begin{align*} -p \log \left (d x\right ) + \log \left (d x\right ) \log \left (c \log \left (d x\right )^{p}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*log(d*x)^p)/x,x, algorithm="maxima")

[Out]

-p*log(d*x) + log(d*x)*log(c*log(d*x)^p)

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Fricas [A]  time = 1.63894, size = 68, normalized size = 3.4 \begin{align*} p \log \left (d x\right ) \log \left (\log \left (d x\right )\right ) -{\left (p - \log \left (c\right )\right )} \log \left (d x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*log(d*x)^p)/x,x, algorithm="fricas")

[Out]

p*log(d*x)*log(log(d*x)) - (p - log(c))*log(d*x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(c*ln(d*x)**p)/x,x)

[Out]

Integral(log(c*log(d*x)**p)/x, x)

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Giac [A]  time = 1.30266, size = 43, normalized size = 2.15 \begin{align*}{\left ({\left (\log \left (d\right ) + \log \left (x\right )\right )} \log \left (\log \left (d\right ) + \log \left (x\right )\right ) - \log \left (d\right ) - \log \left (x\right )\right )} p +{\left (\log \left (d\right ) + \log \left (x\right )\right )} \log \left (c\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*log(d*x)^p)/x,x, algorithm="giac")

[Out]

((log(d) + log(x))*log(log(d) + log(x)) - log(d) - log(x))*p + (log(d) + log(x))*log(c)