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

Optimal. Leaf size=45 \[ a x+b x \log \left (c \log ^p\left (d x^n\right )\right )-b p x \left (d x^n\right )^{-1/n} \text{Ei}\left (\frac{\log \left (d x^n\right )}{n}\right ) \]

[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]

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Rubi [A]  time = 0.0300297, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2520, 2300, 2178} \[ a x+b x \log \left (c \log ^p\left (d x^n\right )\right )-b p x \left (d x^n\right )^{-1/n} \text{Ei}\left (\frac{\log \left (d x^n\right )}{n}\right ) \]

Antiderivative was successfully verified.

[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 2520

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]

Rule 2300

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 2178

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

Rubi steps

\begin{align*} \int \left (a+b \log \left (c \log ^p\left (d x^n\right )\right )\right ) \, dx &=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 ) \operatorname{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]  time = 0.0478073, size = 43, normalized size = 0.96 \[ x \left (a+b \log \left (c \log ^p\left (d x^n\right )\right )-b p \left (d x^n\right )^{-1/n} \text{Ei}\left (\frac{\log \left (d x^n\right )}{n}\right )\right ) \]

Antiderivative was successfully verified.

[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])

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Maple [F]  time = 0.066, size = 0, normalized size = 0. \begin{align*} \int a+b\ln \left ( c \left ( \ln \left ( d{x}^{n} \right ) \right ) ^{p} \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -{\left (n p \int \frac{1}{\log \left (d\right ) + \log \left (x^{n}\right )}\,{d x} - x \log \left (c\right ) - x \log \left ({\left (\log \left (d\right ) + \log \left (x^{n}\right )\right )}^{p}\right )\right )} b + a x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [A]  time = 1.97502, size = 144, normalized size = 3.2 \begin{align*} \frac{b d^{\left (\frac{1}{n}\right )} p x \log \left (n \log \left (x\right ) + \log \left (d\right )\right ) - b p \logintegral \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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

[Out]

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

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Giac [A]  time = 1.32369, size = 57, normalized size = 1.27 \begin{align*}{\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 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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