\(\int \log (c \log ^p(d x^n)) \, dx\) [57]

Optimal result

Integrand size = 11, antiderivative size = 40 \[ \int \log \left (c \log ^p\left (d x^n\right )\right ) \, dx=-p x \left (d x^n\right )^{-1/n} \operatorname {ExpIntegralEi}\left (\frac {\log \left (d x^n\right )}{n}\right )+x \log \left (c \log ^p\left (d x^n\right )\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {2600, 2337, 2209} \[ \int \log \left (c \log ^p\left (d x^n\right )\right ) \, dx=x \log \left (c \log ^p\left (d x^n\right )\right )-p x \left (d x^n\right )^{-1/n} \operatorname {ExpIntegralEi}\left (\frac {\log \left (d x^n\right )}{n}\right ) \]

[In]

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

[Out]

-((p*x*ExpIntegralEi[Log[d*x^n]/n])/(d*x^n)^n^(-1)) + x*Log[c*Log[d*x^n]^p]

Rule 2209

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - c*(f/d)))/d)*ExpInteg
ralEi[f*g*(c + d*x)*(Log[F]/d)], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]

Rule 2337

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Dist[x/(n*(c*x^n)^(1/n)), Subst[Int[E^(x/n)*(a +
b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rule 2600

Int[Log[Log[(d_.)*(x_)^(n_.)]^(p_.)*(c_.)], x_Symbol] :> Simp[x*Log[c*Log[d*x^n]^p], x] - Dist[n*p, Int[1/Log[
d*x^n], x], x] /; FreeQ[{c, d, n, p}, x]

Rubi steps \begin{align*} \text {integral}& = x \log \left (c \log ^p\left (d x^n\right )\right )-(n p) \int \frac {1}{\log \left (d x^n\right )} \, dx \\ & = x \log \left (c \log ^p\left (d x^n\right )\right )-\left (p x \left (d x^n\right )^{-1/n}\right ) \text {Subst}\left (\int \frac {e^{\frac {x}{n}}}{x} \, dx,x,\log \left (d x^n\right )\right ) \\ & = -p x \left (d x^n\right )^{-1/n} \text {Ei}\left (\frac {\log \left (d x^n\right )}{n}\right )+x \log \left (c \log ^p\left (d x^n\right )\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.98 \[ \int \log \left (c \log ^p\left (d x^n\right )\right ) \, dx=x \left (-p \left (d x^n\right )^{-1/n} \operatorname {ExpIntegralEi}\left (\frac {\log \left (d x^n\right )}{n}\right )+\log \left (c \log ^p\left (d x^n\right )\right )\right ) \]

[In]

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

[Out]

x*(-((p*ExpIntegralEi[Log[d*x^n]/n])/(d*x^n)^n^(-1)) + Log[c*Log[d*x^n]^p])

Maple [F]

\[\int \ln \left (c \ln \left (d \,x^{n}\right )^{p}\right )d x\]

[In]

int(ln(c*ln(d*x^n)^p),x)

[Out]

int(ln(c*ln(d*x^n)^p),x)

Fricas [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.12 \[ \int \log \left (c \log ^p\left (d x^n\right )\right ) \, dx=\frac {d^{\left (\frac {1}{n}\right )} p x \log \left (n \log \left (x\right ) + \log \left (d\right )\right ) + d^{\left (\frac {1}{n}\right )} x \log \left (c\right ) - p \operatorname {log\_integral}\left (d^{\left (\frac {1}{n}\right )} x\right )}{d^{\left (\frac {1}{n}\right )}} \]

[In]

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

[Out]

(d^(1/n)*p*x*log(n*log(x) + log(d)) + d^(1/n)*x*log(c) - p*log_integral(d^(1/n)*x))/d^(1/n)

Sympy [F]

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

[In]

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

[Out]

Integral(log(c*log(d*x**n)**p), x)

Maxima [F]

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

[In]

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

[Out]

-n*p*integrate(1/(log(d) + log(x^n)), x) + x*log(c) + x*log((log(d) + log(x^n))^p)

Giac [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.90 \[ \int \log \left (c \log ^p\left (d x^n\right )\right ) \, dx=p x \log \left (n \log \left (x\right ) + \log \left (d\right )\right ) + x \log \left (c\right ) - \frac {p {\rm Ei}\left (\frac {\log \left (d\right )}{n} + \log \left (x\right )\right )}{d^{\left (\frac {1}{n}\right )}} \]

[In]

integrate(log(c*log(d*x^n)^p),x, algorithm="giac")

[Out]

p*x*log(n*log(x) + log(d)) + x*log(c) - p*Ei(log(d)/n + log(x))/d^(1/n)

Mupad [F(-1)]

Timed out. \[ \int \log \left (c \log ^p\left (d x^n\right )\right ) \, dx=\int \ln \left (c\,{\ln \left (d\,x^n\right )}^p\right ) \,d x \]

[In]

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

[Out]

int(log(c*log(d*x^n)^p), x)