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

Optimal. Leaf size=32 \[ \frac{\log \left (d x^n\right ) \left (a+b \log \left (c \log ^p\left (d x^n\right )\right )\right )}{n}-b p \log (x) \]

[Out]

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

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Rubi [A]  time = 0.0171733, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {2521} \[ \frac{\log \left (d x^n\right ) \left (a+b \log \left (c \log ^p\left (d x^n\right )\right )\right )}{n}-b p \log (x) \]

Antiderivative was successfully verified.

[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 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{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}\\ \end{align*}

Mathematica [A]  time = 0.0113264, size = 40, normalized size = 1.25 \[ a \log (x)+\frac{b \log \left (d x^n\right ) \log \left (c \log ^p\left (d x^n\right )\right )}{n}-\frac{b p \log \left (d x^n\right )}{n} \]

Antiderivative was successfully verified.

[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

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Maple [A]  time = 0.01, size = 48, normalized size = 1.5 \begin{align*}{\frac{\ln \left ( d{x}^{n} \right ) \ln \left ( c \left ( \ln \left ( d{x}^{n} \right ) \right ) ^{p} \right ) b}{n}}-{\frac{\ln \left ( d{x}^{n} \right ) bp}{n}}+{\frac{\ln \left ( d{x}^{n} \right ) a}{n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.02756, size = 86, normalized size = 2.69 \begin{align*} 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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]  time = 1.8738, size = 124, normalized size = 3.88 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a + b \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((a+b*ln(c*ln(d*x**n)**p))/x,x)

[Out]

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

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Giac [A]  time = 1.30007, size = 73, normalized size = 2.28 \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 )} 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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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