\(\int \frac {(d x)^{-1+n}}{\log ^3(c x^n)} \, dx\) [161]

Optimal result

Integrand size = 16, antiderivative size = 77 \[ \int \frac {(d x)^{-1+n}}{\log ^3\left (c x^n\right )} \, dx=-\frac {(d x)^n}{2 d n \log ^2\left (c x^n\right )}-\frac {(d x)^n}{2 d n \log \left (c x^n\right )}+\frac {x^{1-n} (d x)^{-1+n} \operatorname {LogIntegral}\left (c x^n\right )}{2 c n} \]

[Out]

1/2*x^(1-n)*(d*x)^(-1+n)*Li(c*x^n)/c/n-1/2*(d*x)^n/d/n/ln(c*x^n)^2-1/2*(d*x)^n/d/n/ln(c*x^n)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2343, 2345, 2344, 2335} \[ \int \frac {(d x)^{-1+n}}{\log ^3\left (c x^n\right )} \, dx=\frac {x^{1-n} (d x)^{n-1} \operatorname {LogIntegral}\left (c x^n\right )}{2 c n}-\frac {(d x)^n}{2 d n \log ^2\left (c x^n\right )}-\frac {(d x)^n}{2 d n \log \left (c x^n\right )} \]

[In]

Int[(d*x)^(-1 + n)/Log[c*x^n]^3,x]

[Out]

-1/2*(d*x)^n/(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 2335

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

Rule 2343

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 2344

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 2345

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]

Rubi steps \begin{align*} \text {integral}& = -\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 ) \text {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] (verified)

Time = 0.04 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.79 \[ \int \frac {(d x)^{-1+n}}{\log ^3\left (c x^n\right )} \, dx=\frac {x^{-n} (d x)^n \left (-c x^n \left (1+\log \left (c x^n\right )\right )+\log ^2\left (c x^n\right ) \operatorname {LogIntegral}\left (c x^n\right )\right )}{2 c d n \log ^2\left (c x^n\right )} \]

[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)

Maple [F]

\[\int \frac {\left (d x \right )^{n -1}}{\ln \left (c \,x^{n}\right )^{3}}d x\]

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.09 \[ \int \frac {(d x)^{-1+n}}{\log ^3\left (c x^n\right )} \, dx=-\frac {{\left (n \log \left (x\right ) + \log \left (c\right ) + 1\right )} d^{n - 1} x^{n} - \frac {{\left (n^{2} \log \left (x\right )^{2} + 2 \, n \log \left (c\right ) \log \left (x\right ) + \log \left (c\right )^{2}\right )} d^{n - 1} {\rm Ei}\left (n \log \left (x\right ) + \log \left (c\right )\right )}{c}}{2 \, {\left (n^{3} \log \left (x\right )^{2} + 2 \, n^{2} \log \left (c\right ) \log \left (x\right ) + n \log \left (c\right )^{2}\right )}} \]

[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)

Sympy [F]

\[ \int \frac {(d x)^{-1+n}}{\log ^3\left (c x^n\right )} \, dx=\int \frac {\left (d x\right )^{n - 1}}{\log {\left (c x^{n} \right )}^{3}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {(d x)^{-1+n}}{\log ^3\left (c x^n\right )} \, dx=\int { \frac {\left (d x\right )^{n - 1}}{\log \left (c x^{n}\right )^{3}} \,d x } \]

[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)

Giac [F]

\[ \int \frac {(d x)^{-1+n}}{\log ^3\left (c x^n\right )} \, dx=\int { \frac {\left (d x\right )^{n - 1}}{\log \left (c x^{n}\right )^{3}} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {(d x)^{-1+n}}{\log ^3\left (c x^n\right )} \, dx=\int \frac {{\left (d\,x\right )}^{n-1}}{{\ln \left (c\,x^n\right )}^3} \,d x \]

[In]

int((d*x)^(n - 1)/log(c*x^n)^3,x)

[Out]

int((d*x)^(n - 1)/log(c*x^n)^3, x)