Optimal. Leaf size=45 \[ \frac{1}{4} x^4 \log ^3(c x)-\frac{3}{16} x^4 \log ^2(c x)+\frac{3}{32} x^4 \log (c x)-\frac{3 x^4}{128} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03594, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2305, 2304} \[ \frac{1}{4} x^4 \log ^3(c x)-\frac{3}{16} x^4 \log ^2(c x)+\frac{3}{32} x^4 \log (c x)-\frac{3 x^4}{128} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2305
Rule 2304
Rubi steps
\begin{align*} \int x^3 \log ^3(c x) \, dx &=\frac{1}{4} x^4 \log ^3(c x)-\frac{3}{4} \int x^3 \log ^2(c x) \, dx\\ &=-\frac{3}{16} x^4 \log ^2(c x)+\frac{1}{4} x^4 \log ^3(c x)+\frac{3}{8} \int x^3 \log (c x) \, dx\\ &=-\frac{3 x^4}{128}+\frac{3}{32} x^4 \log (c x)-\frac{3}{16} x^4 \log ^2(c x)+\frac{1}{4} x^4 \log ^3(c x)\\ \end{align*}
Mathematica [A] time = 0.0014539, size = 45, normalized size = 1. \[ \frac{1}{4} x^4 \log ^3(c x)-\frac{3}{16} x^4 \log ^2(c x)+\frac{3}{32} x^4 \log (c x)-\frac{3 x^4}{128} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.036, size = 38, normalized size = 0.8 \begin{align*} -{\frac{3\,{x}^{4}}{128}}+{\frac{3\,{x}^{4}\ln \left ( cx \right ) }{32}}-{\frac{3\,{x}^{4} \left ( \ln \left ( cx \right ) \right ) ^{2}}{16}}+{\frac{{x}^{4} \left ( \ln \left ( cx \right ) \right ) ^{3}}{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.99127, size = 39, normalized size = 0.87 \begin{align*} \frac{1}{128} \,{\left (32 \, \log \left (c x\right )^{3} - 24 \, \log \left (c x\right )^{2} + 12 \, \log \left (c x\right ) - 3\right )} x^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 0.766697, size = 100, normalized size = 2.22 \begin{align*} \frac{1}{4} \, x^{4} \log \left (c x\right )^{3} - \frac{3}{16} \, x^{4} \log \left (c x\right )^{2} + \frac{3}{32} \, x^{4} \log \left (c x\right ) - \frac{3}{128} \, x^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.123327, size = 42, normalized size = 0.93 \begin{align*} \frac{x^{4} \log{\left (c x \right )}^{3}}{4} - \frac{3 x^{4} \log{\left (c x \right )}^{2}}{16} + \frac{3 x^{4} \log{\left (c x \right )}}{32} - \frac{3 x^{4}}{128} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.10661, size = 50, normalized size = 1.11 \begin{align*} \frac{1}{4} \, x^{4} \log \left (c x\right )^{3} - \frac{3}{16} \, x^{4} \log \left (c x\right )^{2} + \frac{3}{32} \, x^{4} \log \left (c x\right ) - \frac{3}{128} \, x^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]