\(\int \frac {\log (c \log ^p(d x^n))}{x} \, dx\) [58]

Optimal result

Integrand size = 15, antiderivative size = 27 \[ \int \frac {\log \left (c \log ^p\left (d x^n\right )\right )}{x} \, dx=-p \log (x)+\frac {\log \left (d x^n\right ) \log \left (c \log ^p\left (d x^n\right )\right )}{n} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2601} \[ \int \frac {\log \left (c \log ^p\left (d x^n\right )\right )}{x} \, dx=\frac {\log \left (d x^n\right ) \log \left (c \log ^p\left (d x^n\right )\right )}{n}-p \log (x) \]

[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 2601

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.26 \[ \int \frac {\log \left (c \log ^p\left (d x^n\right )\right )}{x} \, dx=-\frac {p \log \left (d x^n\right )}{n}+\frac {\log \left (d x^n\right ) \log \left (c \log ^p\left (d x^n\right )\right )}{n} \]

[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

Maple [A] (verified)

Time = 1.29 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.22

method	result	size
derivativedivides	\(\frac {\ln \left (c \ln \left (d \,x^{n}\right )^{p}\right ) \ln \left (d \,x^{n}\right )-p \ln \left (d \,x^{n}\right )}{n}\)	\(33\)
default	\(\frac {\ln \left (c \ln \left (d \,x^{n}\right )^{p}\right ) \ln \left (d \,x^{n}\right )-p \ln \left (d \,x^{n}\right )}{n}\)	\(33\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.37 \[ \int \frac {\log \left (c \log ^p\left (d x^n\right )\right )}{x} \, dx=\frac {{\left (n p \log \left (x\right ) + p \log \left (d\right )\right )} \log \left (n \log \left (x\right ) + \log \left (d\right )\right ) - {\left (n p - n \log \left (c\right )\right )} \log \left (x\right )}{n} \]

[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

Sympy [F]

\[ \int \frac {\log \left (c \log ^p\left (d x^n\right )\right )}{x} \, dx=\int \frac {\log {\left (c \log {\left (d x^{n} \right )}^{p} \right )}}{x}\, dx \]

[In]

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

[Out]

Integral(log(c*log(d*x**n)**p)/x, x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 55 vs. \(2 (27) = 54\).

Time = 0.19 (sec) , antiderivative size = 55, normalized size of antiderivative = 2.04 \[ \int \frac {\log \left (c \log ^p\left (d x^n\right )\right )}{x} \, dx=-p \log \left (x\right ) \log \left (\log \left (d x^{n}\right )\right ) + \log \left (c \log \left (d x^{n}\right )^{p}\right ) \log \left (x\right ) + \frac {{\left (\log \left (d x^{n}\right ) \log \left (\log \left (d x^{n}\right )\right ) - \log \left (d x^{n}\right )\right )} p}{n} \]

[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

Giac [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.59 \[ \int \frac {\log \left (c \log ^p\left (d x^n\right )\right )}{x} \, dx=\frac {{\left ({\left (n \log \left (x\right ) + \log \left (d\right )\right )} \log \left (n \log \left (x\right ) + \log \left (d\right )\right ) - n \log \left (x\right ) - \log \left (d\right )\right )} p + {\left (n \log \left (x\right ) + \log \left (d\right )\right )} \log \left (c\right )}{n} \]

[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

Mupad [B] (verification not implemented)

Time = 1.47 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {\log \left (c \log ^p\left (d x^n\right )\right )}{x} \, dx=\frac {\ln \left (c\,{\ln \left (d\,x^n\right )}^p\right )\,\ln \left (d\,x^n\right )}{n}-p\,\ln \left (x\right ) \]

[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)