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

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 32, normalized size of antiderivative = 1.00, 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 = {2601} \begin {gather*} \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) \end {gather*}

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 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*} \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*}

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Mathematica [A]
time = 0.01, size = 40, normalized size = 1.25 \begin {gather*} 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} \end {gather*}

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 = 0.12, size = 43, normalized size = 1.34

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.28, size = 64, normalized size = 2.00 \begin {gather*} 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 {gather*}

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 = 0.36, size = 45, normalized size = 1.41 \begin {gather*} \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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a + b \log {\left (c \log {\left (d x^{n} \right )}^{p} \right )}}{x}\, dx \end {gather*}

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 = 3.40, size = 54, normalized size = 1.69 \begin {gather*} \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 {gather*}

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

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Mupad [B]
time = 0.32, size = 32, normalized size = 1.00 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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