Optimal. Leaf size=41 \[ \frac {9 \text {Ei}(3 \log (c x))}{2 c^3}-\frac {x^3}{2 \log ^2(c x)}-\frac {3 x^3}{2 \log (c x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {2306, 2309, 2178} \[ \frac {9 \text {Ei}(3 \log (c x))}{2 c^3}-\frac {x^3}{2 \log ^2(c x)}-\frac {3 x^3}{2 \log (c x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2178
Rule 2306
Rule 2309
Rubi steps
\begin {align*} \int \frac {x^2}{\log ^3(c x)} \, dx &=-\frac {x^3}{2 \log ^2(c x)}+\frac {3}{2} \int \frac {x^2}{\log ^2(c x)} \, dx\\ &=-\frac {x^3}{2 \log ^2(c x)}-\frac {3 x^3}{2 \log (c x)}+\frac {9}{2} \int \frac {x^2}{\log (c x)} \, dx\\ &=-\frac {x^3}{2 \log ^2(c x)}-\frac {3 x^3}{2 \log (c x)}+\frac {9 \operatorname {Subst}\left (\int \frac {e^{3 x}}{x} \, dx,x,\log (c x)\right )}{2 c^3}\\ &=\frac {9 \text {Ei}(3 \log (c x))}{2 c^3}-\frac {x^3}{2 \log ^2(c x)}-\frac {3 x^3}{2 \log (c x)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 41, normalized size = 1.00 \[ \frac {9 \text {Ei}(3 \log (c x))}{2 c^3}-\frac {x^3}{2 \log ^2(c x)}-\frac {3 x^3}{2 \log (c x)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.60, size = 47, normalized size = 1.15 \[ -\frac {3 \, c^{3} x^{3} \log \left (c x\right ) + c^{3} x^{3} - 9 \, \log \left (c x\right )^{2} \operatorname {log\_integral}\left (c^{3} x^{3}\right )}{2 \, c^{3} \log \left (c x\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.22, size = 35, normalized size = 0.85 \[ -\frac {3 \, x^{3}}{2 \, \log \left (c x\right )} - \frac {x^{3}}{2 \, \log \left (c x\right )^{2}} + \frac {9 \, {\rm Ei}\left (3 \, \log \left (c x\right )\right )}{2 \, c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.03, size = 37, normalized size = 0.90 \[ -\frac {3 x^{3}}{2 \ln \left (c x \right )}-\frac {x^{3}}{2 \ln \left (c x \right )^{2}}-\frac {9 \Ei \left (1, -3 \ln \left (c x \right )\right )}{2 c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.87, size = 13, normalized size = 0.32 \[ -\frac {9 \, \Gamma \left (-2, -3 \, \log \left (c x\right )\right )}{c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x^2}{{\ln \left (c\,x\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {- 3 x^{3} \log {\left (c x \right )} - x^{3}}{2 \log {\left (c x \right )}^{2}} + \frac {9 \int \frac {x^{2}}{\log {\left (c x \right )}}\, dx}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________