3.161 \(\int \frac{(d x)^{-1+n}}{\log ^3(c x^n)} \, dx\)

Optimal. Leaf size=77 \[ \frac{x^{1-n} (d x)^{n-1} \text{li}\left (c x^n\right )}{2 c n}-\frac{(d x)^n}{2 d n \log ^2\left (c x^n\right )}-\frac{(d x)^n}{2 d n \log \left (c x^n\right )} \]

[Out]

-(d*x)^n/(2*d*n*Log[c*x^n]^2) - (d*x)^n/(2*d*n*Log[c*x^n]) + (x^(1 - n)*(d*x)^(-1 + n)*LogIntegral[c*x^n])/(2*
c*n)

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Rubi [A]  time = 0.075966, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2306, 2308, 2307, 2298} \[ \frac{x^{1-n} (d x)^{n-1} \text{li}\left (c x^n\right )}{2 c n}-\frac{(d x)^n}{2 d n \log ^2\left (c x^n\right )}-\frac{(d x)^n}{2 d n \log \left (c x^n\right )} \]

Antiderivative was successfully verified.

[In]

Int[(d*x)^(-1 + n)/Log[c*x^n]^3,x]

[Out]

-(d*x)^n/(2*d*n*Log[c*x^n]^2) - (d*x)^n/(2*d*n*Log[c*x^n]) + (x^(1 - n)*(d*x)^(-1 + n)*LogIntegral[c*x^n])/(2*
c*n)

Rule 2306

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

Rule 2308

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

Rule 2307

Int[(x_)^(m_.)/Log[(c_.)*(x_)^(n_)], x_Symbol] :> Dist[1/n, Subst[Int[1/Log[c*x], x], x, x^n], x] /; FreeQ[{c,
m, n}, x] && EqQ[m, n - 1]

Rule 2298

Int[Log[(c_.)*(x_)]^(-1), x_Symbol] :> Simp[LogIntegral[c*x]/c, x] /; FreeQ[c, x]

Rubi steps

\begin{align*} \int \frac{(d x)^{-1+n}}{\log ^3\left (c x^n\right )} \, dx &=-\frac{(d x)^n}{2 d n \log ^2\left (c x^n\right )}+\frac{1}{2} \int \frac{(d x)^{-1+n}}{\log ^2\left (c x^n\right )} \, dx\\ &=-\frac{(d x)^n}{2 d n \log ^2\left (c x^n\right )}-\frac{(d x)^n}{2 d n \log \left (c x^n\right )}+\frac{1}{2} \int \frac{(d x)^{-1+n}}{\log \left (c x^n\right )} \, dx\\ &=-\frac{(d x)^n}{2 d n \log ^2\left (c x^n\right )}-\frac{(d x)^n}{2 d n \log \left (c x^n\right )}+\frac{1}{2} \left (x^{1-n} (d x)^{-1+n}\right ) \int \frac{x^{-1+n}}{\log \left (c x^n\right )} \, dx\\ &=-\frac{(d x)^n}{2 d n \log ^2\left (c x^n\right )}-\frac{(d x)^n}{2 d n \log \left (c x^n\right )}+\frac{\left (x^{1-n} (d x)^{-1+n}\right ) \operatorname{Subst}\left (\int \frac{1}{\log (c x)} \, dx,x,x^n\right )}{2 n}\\ &=-\frac{(d x)^n}{2 d n \log ^2\left (c x^n\right )}-\frac{(d x)^n}{2 d n \log \left (c x^n\right )}+\frac{x^{1-n} (d x)^{-1+n} \text{li}\left (c x^n\right )}{2 c n}\\ \end{align*}

Mathematica [A]  time = 0.0238329, size = 61, normalized size = 0.79 \[ \frac{x^{-n} (d x)^n \left (\text{li}\left (c x^n\right ) \log ^2\left (c x^n\right )-c x^n \left (\log \left (c x^n\right )+1\right )\right )}{2 c d n \log ^2\left (c x^n\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[(d*x)^(-1 + n)/Log[c*x^n]^3,x]

[Out]

((d*x)^n*(-(c*x^n*(1 + Log[c*x^n])) + Log[c*x^n]^2*LogIntegral[c*x^n]))/(2*c*d*n*x^n*Log[c*x^n]^2)

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Maple [F]  time = 1.195, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( dx \right ) ^{-1+n}}{ \left ( \ln \left ( c{x}^{n} \right ) \right ) ^{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^(-1+n)/ln(c*x^n)^3,x)

[Out]

int((d*x)^(-1+n)/ln(c*x^n)^3,x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(-1+n)/log(c*x^n)^3,x, algorithm="maxima")

[Out]

d^n*integrate(1/2*x^n/(d*x*log(c) + d*x*log(x^n)), x) - 1/2*(d^n*x^n*log(x^n) + (d^n*log(c) + d^n)*x^n)/(d*n*l
og(c)^2 + 2*d*n*log(c)*log(x^n) + d*n*log(x^n)^2)

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Fricas [A]  time = 1.01033, size = 240, normalized size = 3.12 \begin{align*} -\frac{{\left (n \log \left (x\right ) + \log \left (c\right ) + 1\right )} d^{n - 1} x^{n} - \frac{{\left (n^{2} \log \left (x\right )^{2} + 2 \, n \log \left (c\right ) \log \left (x\right ) + \log \left (c\right )^{2}\right )} d^{n - 1}{\rm Ei}\left (n \log \left (x\right ) + \log \left (c\right )\right )}{c}}{2 \,{\left (n^{3} \log \left (x\right )^{2} + 2 \, n^{2} \log \left (c\right ) \log \left (x\right ) + n \log \left (c\right )^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(-1+n)/log(c*x^n)^3,x, algorithm="fricas")

[Out]

-1/2*((n*log(x) + log(c) + 1)*d^(n - 1)*x^n - (n^2*log(x)^2 + 2*n*log(c)*log(x) + log(c)^2)*d^(n - 1)*Ei(n*log
(x) + log(c))/c)/(n^3*log(x)^2 + 2*n^2*log(c)*log(x) + n*log(c)^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d x\right )^{n - 1}}{\log{\left (c x^{n} \right )}^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)**(-1+n)/ln(c*x**n)**3,x)

[Out]

Integral((d*x)**(n - 1)/log(c*x**n)**3, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d x\right )^{n - 1}}{\log \left (c x^{n}\right )^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(-1+n)/log(c*x^n)^3,x, algorithm="giac")

[Out]

integrate((d*x)^(n - 1)/log(c*x^n)^3, x)