Optimal. Leaf size=77 \[ \frac {x^{1-n} (d x)^{n-1} \text {li}\left (c x^n\right )}{2 c n}-\frac {(d x)^n}{2 d n \log ^2\left (c x^n\right )}-\frac {(d x)^n}{2 d n \log \left (c x^n\right )} \]
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Rubi [A] time = 0.08, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2306, 2308, 2307, 2298} \[ \frac {x^{1-n} (d x)^{n-1} \text {li}\left (c x^n\right )}{2 c n}-\frac {(d x)^n}{2 d n \log ^2\left (c x^n\right )}-\frac {(d x)^n}{2 d n \log \left (c x^n\right )} \]
Antiderivative was successfully verified.
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Rule 2298
Rule 2306
Rule 2307
Rule 2308
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*}
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Mathematica [A] time = 0.02, 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.
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fricas [A] time = 0.44, size = 84, normalized size = 1.09 \[ -\frac {{\left (n \log \relax (x) + \log \relax (c) + 1\right )} d^{n - 1} x^{n} - \frac {{\left (n^{2} \log \relax (x)^{2} + 2 \, n \log \relax (c) \log \relax (x) + \log \relax (c)^{2}\right )} d^{n - 1} {\rm Ei}\left (n \log \relax (x) + \log \relax (c)\right )}{c}}{2 \, {\left (n^{3} \log \relax (x)^{2} + 2 \, n^{2} \log \relax (c) \log \relax (x) + n \log \relax (c)^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (d x\right )^{n - 1}}{\log \left (c x^{n}\right )^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.75, size = 0, normalized size = 0.00 \[ \int \frac {\left (d x \right )^{n -1}}{\ln \left (c \,x^{n}\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ d^{n} \int \frac {x^{n}}{2 \, {\left (d x \log \relax (c) + d x \log \left (x^{n}\right )\right )}}\,{d x} - \frac {d^{n} x^{n} \log \left (x^{n}\right ) + {\left (d^{n} \log \relax (c) + d^{n}\right )} x^{n}}{2 \, {\left (d n \log \relax (c)^{2} + 2 \, d n \log \relax (c) \log \left (x^{n}\right ) + d n \log \left (x^{n}\right )^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (d\,x\right )}^{n-1}}{{\ln \left (c\,x^n\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (d x\right )^{n - 1}}{\log {\left (c x^{n} \right )}^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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