3.52 \(\int \frac{a+b \log (c \log ^p(d x^n))}{x^2} \, dx\)

Optimal. Leaf size=48 \[ \frac{b p \left (d x^n\right )^{\frac{1}{n}} \text{Ei}\left (-\frac{\log \left (d x^n\right )}{n}\right )}{x}-\frac{a+b \log \left (c \log ^p\left (d x^n\right )\right )}{x} \]

[Out]

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

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Rubi [A]  time = 0.046958, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {2522, 2310, 2178} \[ \frac{b p \left (d x^n\right )^{\frac{1}{n}} \text{Ei}\left (-\frac{\log \left (d x^n\right )}{n}\right )}{x}-\frac{a+b \log \left (c \log ^p\left (d x^n\right )\right )}{x} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*Log[d*x^n]^p])/x^2,x]

[Out]

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

Rule 2522

Int[((a_.) + Log[Log[(d_.)*(x_)^(n_.)]^(p_.)*(c_.)]*(b_.))*((e_.)*(x_))^(m_.), x_Symbol] :> Simp[((e*x)^(m + 1
)*(a + b*Log[c*Log[d*x^n]^p]))/(e*(m + 1)), x] - Dist[(b*n*p)/(m + 1), Int[(e*x)^m/Log[d*x^n], x], x] /; FreeQ
[{a, b, c, d, e, m, n, p}, x] && NeQ[m, -1]

Rule 2310

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_)*((d_.)*(x_))^(m_.), x_Symbol] :> Dist[(d*x)^(m + 1)/(d*n*(c*x^n
)^((m + 1)/n)), Subst[Int[E^(((m + 1)*x)/n)*(a + b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, d, m, 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 \frac{a+b \log \left (c \log ^p\left (d x^n\right )\right )}{x^2} \, dx &=-\frac{a+b \log \left (c \log ^p\left (d x^n\right )\right )}{x}+(b n p) \int \frac{1}{x^2 \log \left (d x^n\right )} \, dx\\ &=-\frac{a+b \log \left (c \log ^p\left (d x^n\right )\right )}{x}+\frac{\left (b p \left (d x^n\right )^{\frac{1}{n}}\right ) \operatorname{Subst}\left (\int \frac{e^{-\frac{x}{n}}}{x} \, dx,x,\log \left (d x^n\right )\right )}{x}\\ &=\frac{b p \left (d x^n\right )^{\frac{1}{n}} \text{Ei}\left (-\frac{\log \left (d x^n\right )}{n}\right )}{x}-\frac{a+b \log \left (c \log ^p\left (d x^n\right )\right )}{x}\\ \end{align*}

Mathematica [A]  time = 0.0405497, size = 45, normalized size = 0.94 \[ -\frac{a+b \log \left (c \log ^p\left (d x^n\right )\right )-b p \left (d x^n\right )^{\frac{1}{n}} \text{Ei}\left (-\frac{\log \left (d x^n\right )}{n}\right )}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Log[c*Log[d*x^n]^p])/x^2,x]

[Out]

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

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Maple [F]  time = 0.066, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b\ln \left ( c \left ( \ln \left ( d{x}^{n} \right ) \right ) ^{p} \right ) }{{x}^{2}}}\, 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^2,x)

[Out]

int((a+b*ln(c*ln(d*x^n)^p))/x^2,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\left (n p \int \frac{1}{x^{2} \log \left (d\right ) + x^{2} \log \left (x^{n}\right )}\,{d x} - \frac{\log \left (c\right ) + \log \left ({\left (\log \left (d\right ) + \log \left (x^{n}\right )\right )}^{p}\right )}{x}\right )} b - \frac{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^2,x, algorithm="maxima")

[Out]

(n*p*integrate(1/(x^2*log(d) + x^2*log(x^n)), x) - (log(c) + log((log(d) + log(x^n))^p))/x)*b - a/x

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Fricas [A]  time = 1.89519, size = 123, normalized size = 2.56 \begin{align*} \frac{b d^{\left (\frac{1}{n}\right )} p x \logintegral \left (\frac{1}{d^{\left (\frac{1}{n}\right )} x}\right ) - b p \log \left (n \log \left (x\right ) + \log \left (d\right )\right ) - b \log \left (c\right ) - 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^2,x, algorithm="fricas")

[Out]

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

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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^{2}}\, 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**2,x)

[Out]

Integral((a + b*log(c*log(d*x**n)**p))/x**2, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \log \left (c \log \left (d x^{n}\right )^{p}\right ) + a}{x^{2}}\,{d 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^2,x, algorithm="giac")

[Out]

integrate((b*log(c*log(d*x^n)^p) + a)/x^2, x)