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

Optimal. Leaf size=27 \[ \frac {\log \left (d x^n\right ) \log \left (c \log ^p\left (d x^n\right )\right )}{n}-p \log (x) \]

[Out]

-p*ln(x)+ln(d*x^n)*ln(c*ln(d*x^n)^p)/n

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Rubi [A]  time = 0.02, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2521} \[ \frac {\log \left (d x^n\right ) \log \left (c \log ^p\left (d x^n\right )\right )}{n}-p \log (x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]  time = 0.01, size = 34, normalized size = 1.26 \[ \frac {\log \left (d x^n\right ) \log \left (c \log ^p\left (d x^n\right )\right )}{n}-\frac {p \log \left (d x^n\right )}{n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.45, size = 37, normalized size = 1.37 \[ \frac {{\left (n p \log \relax (x) + p \log \relax (d)\right )} \log \left (n \log \relax (x) + \log \relax (d)\right ) - {\left (n p - n \log \relax (c)\right )} \log \relax (x)}{n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.19, size = 43, normalized size = 1.59 \[ \frac {{\left ({\left (n \log \relax (x) + \log \relax (d)\right )} \log \left (n \log \relax (x) + \log \relax (d)\right ) - n \log \relax (x) - \log \relax (d)\right )} p + {\left (n \log \relax (x) + \log \relax (d)\right )} \log \relax (c)}{n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.07, size = 35, normalized size = 1.30 \[ -\frac {p \ln \left (d \,x^{n}\right )}{n}+\frac {\ln \left (c \ln \left (d \,x^{n}\right )^{p}\right ) \ln \left (d \,x^{n}\right )}{n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(d*x^n)*ln(c*ln(d*x^n)^p)/n-1/n*p*ln(d*x^n)

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maxima [B]  time = 0.45, size = 55, normalized size = 2.04 \[ -p \log \relax (x) \log \left (\log \left (d x^{n}\right )\right ) + \log \left (c \log \left (d x^{n}\right )^{p}\right ) \log \relax (x) + \frac {{\left (\log \left (d x^{n}\right ) \log \left (\log \left (d x^{n}\right )\right ) - \log \left (d x^{n}\right )\right )} p}{n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.30, size = 27, normalized size = 1.00 \[ \frac {\ln \left (c\,{\ln \left (d\,x^n\right )}^p\right )\,\ln \left (d\,x^n\right )}{n}-p\,\ln \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(c*log(d*x^n)^p)/x,x)

[Out]

(log(c*log(d*x^n)^p)*log(d*x^n))/n - p*log(x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(log(c*log(d*x**n)**p)/x, x)

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