\(\int (a+b \log (c \log ^p(d x^n))) \, dx\) [50]

Optimal result

Integrand size = 15, antiderivative size = 45 \[ \int \left (a+b \log \left (c \log ^p\left (d x^n\right )\right )\right ) \, dx=a x-b p x \left (d x^n\right )^{-1/n} \operatorname {ExpIntegralEi}\left (\frac {\log \left (d x^n\right )}{n}\right )+b x \log \left (c \log ^p\left (d x^n\right )\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2600, 2337, 2209} \[ \int \left (a+b \log \left (c \log ^p\left (d x^n\right )\right )\right ) \, dx=a x+b x \log \left (c \log ^p\left (d x^n\right )\right )-b p x \left (d x^n\right )^{-1/n} \operatorname {ExpIntegralEi}\left (\frac {\log \left (d x^n\right )}{n}\right ) \]

[In]

Int[a + b*Log[c*Log[d*x^n]^p],x]

[Out]

a*x - (b*p*x*ExpIntegralEi[Log[d*x^n]/n])/(d*x^n)^n^(-1) + b*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}& = a x+b \int \log \left (c \log ^p\left (d x^n\right )\right ) \, dx \\ & = a x+b x \log \left (c \log ^p\left (d x^n\right )\right )-(b n p) \int \frac {1}{\log \left (d x^n\right )} \, dx \\ & = a x+b x \log \left (c \log ^p\left (d x^n\right )\right )-\left (b 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 ) \\ & = a x-b p x \left (d x^n\right )^{-1/n} \text {Ei}\left (\frac {\log \left (d x^n\right )}{n}\right )+b x \log \left (c \log ^p\left (d x^n\right )\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.96 \[ \int \left (a+b \log \left (c \log ^p\left (d x^n\right )\right )\right ) \, dx=x \left (a-b p \left (d x^n\right )^{-1/n} \operatorname {ExpIntegralEi}\left (\frac {\log \left (d x^n\right )}{n}\right )+b \log \left (c \log ^p\left (d x^n\right )\right )\right ) \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.18 \[ \int \left (a+b \log \left (c \log ^p\left (d x^n\right )\right )\right ) \, dx=\frac {b d^{\left (\frac {1}{n}\right )} p x \log \left (n \log \left (x\right ) + \log \left (d\right )\right ) - b p \operatorname {log\_integral}\left (d^{\left (\frac {1}{n}\right )} x\right ) + {\left (b x \log \left (c\right ) + a x\right )} d^{\left (\frac {1}{n}\right )}}{d^{\left (\frac {1}{n}\right )}} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

integrate(a+b*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))*b + a*x

Giac [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.93 \[ \int \left (a+b \log \left (c \log ^p\left (d x^n\right )\right )\right ) \, dx={\left (p x \log \left (n \log \left (x\right ) + \log \left (d\right )\right ) + x \log \left (c\right ) - \frac {p {\rm Ei}\left (\frac {\log \left (d\right )}{n} + \log \left (x\right )\right )}{d^{\left (\frac {1}{n}\right )}}\right )} b + a x \]

[In]

integrate(a+b*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))*b + a*x

Mupad [F(-1)]

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

[In]

int(a + b*log(c*log(d*x^n)^p),x)

[Out]

int(a + b*log(c*log(d*x^n)^p), x)