Optimal. Leaf size=49 \[ \frac {x^{1-n} (d x)^{n-1} \text {li}\left (c x^n\right )}{c n}-\frac {(d x)^n}{d n \log \left (c x^n\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2306, 2308, 2307, 2298} \[ \frac {x^{1-n} (d x)^{n-1} \text {li}\left (c x^n\right )}{c n}-\frac {(d x)^n}{d n \log \left (c x^n\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2298
Rule 2306
Rule 2307
Rule 2308
Rubi steps
\begin {align*} \int \frac {(d x)^{-1+n}}{\log ^2\left (c x^n\right )} \, dx &=-\frac {(d x)^n}{d n \log \left (c x^n\right )}+\int \frac {(d x)^{-1+n}}{\log \left (c x^n\right )} \, dx\\ &=-\frac {(d x)^n}{d n \log \left (c x^n\right )}+\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}{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 )}{n}\\ &=-\frac {(d x)^n}{d n \log \left (c x^n\right )}+\frac {x^{1-n} (d x)^{-1+n} \text {li}\left (c x^n\right )}{c n}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 49, normalized size = 1.00 \[ \frac {x^{1-n} (d x)^{n-1} \text {li}\left (c x^n\right )}{c n}-\frac {x (d x)^{n-1}}{n \log \left (c x^n\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.44, size = 50, normalized size = 1.02 \[ -\frac {d^{n - 1} x^{n} - \frac {{\left (n \log \relax (x) + \log \relax (c)\right )} d^{n - 1} {\rm Ei}\left (n \log \relax (x) + \log \relax (c)\right )}{c}}{n^{2} \log \relax (x) + n \log \relax (c)} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (d x\right )^{n - 1}}{\log \left (c x^{n}\right )^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 1.84, size = 0, normalized size = 0.00 \[ \int \frac {\left (d x \right )^{n -1}}{\ln \left (c \,x^{n}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ d^{n} \int \frac {x^{n}}{d x \log \relax (c) + d x \log \left (x^{n}\right )}\,{d x} - \frac {d^{n} x^{n}}{d n \log \relax (c) + d n \log \left (x^{n}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (d\,x\right )}^{n-1}}{{\ln \left (c\,x^n\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (d x\right )^{n - 1}}{\log {\left (c x^{n} \right )}^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________