Integrand size = 8, antiderivative size = 32 \[ \int x \log ^2(c x) \, dx=\frac {x^2}{4}-\frac {1}{2} x^2 \log (c x)+\frac {1}{2} x^2 \log ^2(c x) \]
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Time = 0.01 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2342, 2341} \[ \int x \log ^2(c x) \, dx=\frac {1}{2} x^2 \log ^2(c x)-\frac {1}{2} x^2 \log (c x)+\frac {x^2}{4} \]
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Rule 2341
Rule 2342
Rubi steps \begin{align*} \text {integral}& = \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*}
Time = 0.00 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00 \[ \int x \log ^2(c x) \, dx=\frac {x^2}{4}-\frac {1}{2} x^2 \log (c x)+\frac {1}{2} x^2 \log ^2(c x) \]
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Time = 0.02 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.84
method | result | size |
norman | \(\frac {x^{2}}{4}-\frac {x^{2} \ln \left (x c \right )}{2}+\frac {x^{2} \ln \left (x c \right )^{2}}{2}\) | \(27\) |
risch | \(\frac {x^{2}}{4}-\frac {x^{2} \ln \left (x c \right )}{2}+\frac {x^{2} \ln \left (x c \right )^{2}}{2}\) | \(27\) |
parallelrisch | \(\frac {x^{2}}{4}-\frac {x^{2} \ln \left (x c \right )}{2}+\frac {x^{2} \ln \left (x c \right )^{2}}{2}\) | \(27\) |
parts | \(\frac {x^{2} \ln \left (x c \right )^{2}}{2}-\frac {\frac {x^{2} c^{2} \ln \left (x c \right )}{2}-\frac {x^{2} c^{2}}{4}}{c^{2}}\) | \(39\) |
derivativedivides | \(\frac {\frac {x^{2} c^{2} \ln \left (x c \right )^{2}}{2}-\frac {x^{2} c^{2} \ln \left (x c \right )}{2}+\frac {x^{2} c^{2}}{4}}{c^{2}}\) | \(40\) |
default | \(\frac {\frac {x^{2} c^{2} \ln \left (x c \right )^{2}}{2}-\frac {x^{2} c^{2} \ln \left (x c \right )}{2}+\frac {x^{2} c^{2}}{4}}{c^{2}}\) | \(40\) |
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Time = 0.30 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.81 \[ \int x \log ^2(c x) \, dx=\frac {1}{2} \, x^{2} \log \left (c x\right )^{2} - \frac {1}{2} \, x^{2} \log \left (c x\right ) + \frac {1}{4} \, x^{2} \]
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Time = 0.04 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.81 \[ \int x \log ^2(c x) \, dx=\frac {x^{2} \log {\left (c x \right )}^{2}}{2} - \frac {x^{2} \log {\left (c x \right )}}{2} + \frac {x^{2}}{4} \]
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Time = 0.19 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.66 \[ \int x \log ^2(c x) \, dx=\frac {1}{4} \, {\left (2 \, \log \left (c x\right )^{2} - 2 \, \log \left (c x\right ) + 1\right )} x^{2} \]
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Time = 0.32 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.81 \[ \int x \log ^2(c x) \, dx=\frac {1}{2} \, x^{2} \log \left (c x\right )^{2} - \frac {1}{2} \, x^{2} \log \left (c x\right ) + \frac {1}{4} \, x^{2} \]
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Time = 0.29 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.66 \[ \int x \log ^2(c x) \, dx=\frac {x^2\,\left (2\,{\ln \left (c\,x\right )}^2-2\,\ln \left (c\,x\right )+1\right )}{4} \]
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