\(\int \frac {a+b \log (c \log ^p(d x^n))}{x} \, dx\) [51]

Optimal result

Integrand size = 19, antiderivative size = 32 \[ \int \frac {a+b \log \left (c \log ^p\left (d x^n\right )\right )}{x} \, dx=-b p \log (x)+\frac {\log \left (d x^n\right ) \left (a+b \log \left (c \log ^p\left (d x^n\right )\right )\right )}{n} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2601} \[ \int \frac {a+b \log \left (c \log ^p\left (d x^n\right )\right )}{x} \, dx=\frac {\log \left (d x^n\right ) \left (a+b \log \left (c \log ^p\left (d x^n\right )\right )\right )}{n}-b p \log (x) \]

[In]

Int[(a + b*Log[c*Log[d*x^n]^p])/x,x]

[Out]

-(b*p*Log[x]) + (Log[d*x^n]*(a + b*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}& = -b p \log (x)+\frac {\log \left (d x^n\right ) \left (a+b \log \left (c \log ^p\left (d x^n\right )\right )\right )}{n} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.25 \[ \int \frac {a+b \log \left (c \log ^p\left (d x^n\right )\right )}{x} \, dx=a \log (x)-\frac {b p \log \left (d x^n\right )}{n}+\frac {b \log \left (d x^n\right ) \log \left (c \log ^p\left (d x^n\right )\right )}{n} \]

[In]

Integrate[(a + b*Log[c*Log[d*x^n]^p])/x,x]

[Out]

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

Maple [A] (verified)

Time = 1.66 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.22

method	result	size
parts	\(\ln \left (x \right ) a +\frac {b \left (\ln \left (c \ln \left (d \,x^{n}\right )^{p}\right ) \ln \left (d \,x^{n}\right )-p \ln \left (d \,x^{n}\right )\right )}{n}\)	\(39\)
derivativedivides	\(\frac {\ln \left (d \,x^{n}\right ) a +\ln \left (d \,x^{n}\right ) \ln \left (c \ln \left (d \,x^{n}\right )^{p}\right ) b -b p \ln \left (d \,x^{n}\right )}{n}\)	\(43\)
default	\(\frac {\ln \left (d \,x^{n}\right ) a +\ln \left (d \,x^{n}\right ) \ln \left (c \ln \left (d \,x^{n}\right )^{p}\right ) b -b p \ln \left (d \,x^{n}\right )}{n}\)	\(43\)
parallelrisch	\(\frac {\ln \left (d \,x^{n}\right ) a +\ln \left (d \,x^{n}\right ) \ln \left (c \ln \left (d \,x^{n}\right )^{p}\right ) b -b p \ln \left (d \,x^{n}\right )}{n}\)	\(43\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.41 \[ \int \frac {a+b \log \left (c \log ^p\left (d x^n\right )\right )}{x} \, dx=\frac {{\left (b n p \log \left (x\right ) + b p \log \left (d\right )\right )} \log \left (n \log \left (x\right ) + \log \left (d\right )\right ) - {\left (b n p - b n \log \left (c\right ) - a n\right )} \log \left (x\right )}{n} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

Integral((a + b*log(c*log(d*x**n)**p))/x, x)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.00 \[ \int \frac {a+b \log \left (c \log ^p\left (d x^n\right )\right )}{x} \, dx=b \log \left (c \log \left (d x^{n}\right )^{p}\right ) \log \left (x\right ) - {\left (p \log \left (x\right ) \log \left (\log \left (d x^{n}\right )\right ) - \frac {{\left (\log \left (d x^{n}\right ) \log \left (\log \left (d x^{n}\right )\right ) - \log \left (d x^{n}\right )\right )} p}{n}\right )} b + a \log \left (x\right ) \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.69 \[ \int \frac {a+b \log \left (c \log ^p\left (d x^n\right )\right )}{x} \, dx=\frac {{\left ({\left (n \log \left (x\right ) + \log \left (d\right )\right )} \log \left (n \log \left (x\right ) + \log \left (d\right )\right ) - n \log \left (x\right ) - \log \left (d\right )\right )} b p + {\left (n \log \left (x\right ) + \log \left (d\right )\right )} b \log \left (c\right ) + {\left (n \log \left (x\right ) + \log \left (d\right )\right )} a}{n} \]

[In]

integrate((a+b*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))*b*p + (n*log(x) + log(d))*b*log(c) + (n*log(
x) + log(d))*a)/n

Mupad [B] (verification not implemented)

Time = 1.69 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00 \[ \int \frac {a+b \log \left (c \log ^p\left (d x^n\right )\right )}{x} \, dx=\ln \left (x\right )\,\left (a-b\,p\right )+\frac {b\,\ln \left (c\,{\ln \left (d\,x^n\right )}^p\right )\,\ln \left (d\,x^n\right )}{n} \]

[In]

int((a + b*log(c*log(d*x^n)^p))/x,x)

[Out]

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