Optimal. Leaf size=32 \[ \frac{1}{2} x^2 \log ^2(c x)-\frac{1}{2} x^2 \log (c x)+\frac{x^2}{4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0108939, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2305, 2304} \[ \frac{1}{2} x^2 \log ^2(c x)-\frac{1}{2} x^2 \log (c x)+\frac{x^2}{4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2305
Rule 2304
Rubi steps
\begin{align*} \int x \log ^2(c x) \, dx &=\frac{1}{2} x^2 \log ^2(c x)-\int x \log (c x) \, dx\\ &=\frac{x^2}{4}-\frac{1}{2} x^2 \log (c x)+\frac{1}{2} x^2 \log ^2(c x)\\ \end{align*}
Mathematica [A] time = 0.0009494, size = 32, normalized size = 1. \[ \frac{1}{2} x^2 \log ^2(c x)-\frac{1}{2} x^2 \log (c x)+\frac{x^2}{4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.036, size = 27, normalized size = 0.8 \begin{align*}{\frac{{x}^{2}}{4}}-{\frac{{x}^{2}\ln \left ( cx \right ) }{2}}+{\frac{{x}^{2} \left ( \ln \left ( cx \right ) \right ) ^{2}}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.977959, size = 28, normalized size = 0.88 \begin{align*} \frac{1}{4} \,{\left (2 \, \log \left (c x\right )^{2} - 2 \, \log \left (c x\right ) + 1\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 0.744731, size = 66, normalized size = 2.06 \begin{align*} \frac{1}{2} \, x^{2} \log \left (c x\right )^{2} - \frac{1}{2} \, x^{2} \log \left (c x\right ) + \frac{1}{4} \, x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.112876, size = 26, normalized size = 0.81 \begin{align*} \frac{x^{2} \log{\left (c x \right )}^{2}}{2} - \frac{x^{2} \log{\left (c x \right )}}{2} + \frac{x^{2}}{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.10867, size = 35, normalized size = 1.09 \begin{align*} \frac{1}{2} \, x^{2} \log \left (c x\right )^{2} - \frac{1}{2} \, x^{2} \log \left (c x\right ) + \frac{1}{4} \, x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]