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

Optimal. Leaf size=27 \[ \frac{x^{1-n} (d x)^{n-1} \text{li}\left (c x^n\right )}{c n} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0373724, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {2308, 2307, 2298} \[ \frac{x^{1-n} (d x)^{n-1} \text{li}\left (c x^n\right )}{c n} \]

Antiderivative was successfully verified.

[In]

Int[(d*x)^(-1 + n)/Log[c*x^n],x]

[Out]

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

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

Mathematica [A]  time = 0.0079955, size = 27, normalized size = 1. \[ \frac{x^{1-n} (d x)^{n-1} \text{li}\left (c x^n\right )}{c n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 0.254, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( dx \right ) ^{-1+n}}{\ln \left ( c{x}^{n} \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.01431, size = 53, normalized size = 1.96 \begin{align*} \frac{d^{n - 1}{\rm Ei}\left (n \log \left (x\right ) + \log \left (c\right )\right )}{c n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d^(n - 1)*Ei(n*log(x) + log(c))/(c*n)

________________________________________________________________________________________

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 )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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 )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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