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

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Rubi [A]  time = 0.0211121, antiderivative size = 27, normalized size of antiderivative = 1., 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*}

Mathematica [A]  time = 0.0101253, 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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Maple [A]  time = 0.006, size = 35, normalized size = 1.3 \begin{align*}{\frac{\ln \left ( c \left ( \ln \left ( d{x}^{n} \right ) \right ) ^{p} \right ) \ln \left ( d{x}^{n} \right ) }{n}}-{\frac{p\ln \left ( d{x}^{n} \right ) }{n}} \end{align*}

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 = 1.01719, size = 74, normalized size = 2.74 \begin{align*} -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} \end{align*}

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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Fricas [A]  time = 1.52428, size = 105, normalized size = 3.89 \begin{align*} \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} \end{align*}

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

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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Giac [A]  time = 1.36748, size = 58, normalized size = 2.15 \begin{align*} \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} \end{align*}

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