Optimal. Leaf size=15 \[ -\frac {2}{n \sqrt {\log \left (a x^n\right )}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2302, 30} \[ -\frac {2}{n \sqrt {\log \left (a x^n\right )}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 30
Rule 2302
Rubi steps
\begin {align*} \int \frac {1}{x \log ^{\frac {3}{2}}\left (a x^n\right )} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{x^{3/2}} \, dx,x,\log \left (a x^n\right )\right )}{n}\\ &=-\frac {2}{n \sqrt {\log \left (a x^n\right )}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.00, size = 15, normalized size = 1.00 \[ -\frac {2}{n \sqrt {\log \left (a x^n\right )}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.46, size = 24, normalized size = 1.60 \[ -\frac {2 \, \sqrt {n \log \relax (x) + \log \relax (a)}}{n^{2} \log \relax (x) + n \log \relax (a)} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.27, size = 14, normalized size = 0.93 \[ -\frac {2}{\sqrt {n \log \relax (x) + \log \relax (a)} n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.03, size = 14, normalized size = 0.93 \[ -\frac {2}{n \sqrt {\ln \left (a \,x^{n}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.70, size = 13, normalized size = 0.87 \[ -\frac {2}{n \sqrt {\log \left (a x^{n}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 3.45, size = 13, normalized size = 0.87 \[ -\frac {2}{n\,\sqrt {\ln \left (a\,x^n\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 93.98, size = 41, normalized size = 2.73 \[ \begin {cases} \tilde {\infty } \log {\relax (x )} & \text {for}\: a = 1 \wedge n = 0 \\\frac {\log {\relax (x )}}{\log {\relax (a )}^{\frac {3}{2}}} & \text {for}\: n = 0 \\\tilde {\infty } \log {\relax (x )} & \text {for}\: a = e^{- n \log {\relax (x )}} \\- \frac {2}{n \sqrt {n \log {\relax (x )} + \log {\relax (a )}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________