\(\int \frac {\log (c \log ^p(d x))}{x} \, dx\) [56]

Optimal result

Integrand size = 13, antiderivative size = 20 \[ \int \frac {\log \left (c \log ^p(d x)\right )}{x} \, dx=-p \log (x)+\log (d x) \log \left (c \log ^p(d x)\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2601} \[ \int \frac {\log \left (c \log ^p(d x)\right )}{x} \, dx=\log (d x) \log \left (c \log ^p(d x)\right )-p \log (x) \]

[In]

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

[Out]

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

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)+\log (d x) \log \left (c \log ^p(d x)\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {\log \left (c \log ^p(d x)\right )}{x} \, dx=-p \log (d x)+\log (d x) \log \left (c \log ^p(d x)\right ) \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.55 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.15

method	result	size
derivativedivides	\(\ln \left (d x \right ) \ln \left (c \ln \left (d x \right )^{p}\right )-\ln \left (d x \right ) p\)	\(23\)
default	\(\ln \left (d x \right ) \ln \left (c \ln \left (d x \right )^{p}\right )-\ln \left (d x \right ) p\)	\(23\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.20 \[ \int \frac {\log \left (c \log ^p(d x)\right )}{x} \, dx=p \log \left (d x\right ) \log \left (\log \left (d x\right )\right ) - {\left (p - \log \left (c\right )\right )} \log \left (d x\right ) \]

[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)

Sympy [F]

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

[In]

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

[Out]

Integral(log(c*log(d*x)**p)/x, x)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {\log \left (c \log ^p(d x)\right )}{x} \, dx=-p \log \left (d x\right ) + \log \left (d x\right ) \log \left (c \log \left (d x\right )^{p}\right ) \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.60 \[ \int \frac {\log \left (c \log ^p(d x)\right )}{x} \, dx={\left ({\left (\log \left (d\right ) + \log \left (x\right )\right )} \log \left (\log \left (d\right ) + \log \left (x\right )\right ) - \log \left (d\right ) - \log \left (x\right )\right )} p + {\left (\log \left (d\right ) + \log \left (x\right )\right )} \log \left (c\right ) \]

[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)

Mupad [B] (verification not implemented)

Time = 1.49 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\log \left (c \log ^p(d x)\right )}{x} \, dx=\ln \left (c\,{\ln \left (d\,x\right )}^p\right )\,\ln \left (d\,x\right )-p\,\ln \left (x\right ) \]

[In]

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

[Out]

log(c*log(d*x)^p)*log(d*x) - p*log(x)