Integrand size = 8, antiderivative size = 42 \[ \int x^m \log ^2(x) \, dx=\frac {2 x^{1+m}}{(1+m)^3}-\frac {2 x^{1+m} \log (x)}{(1+m)^2}+\frac {x^{1+m} \log ^2(x)}{1+m} \]
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Time = 0.02 (sec) , antiderivative size = 42, 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} \[ \int x^m \log ^2(x) \, dx=\frac {2 x^{m+1}}{(m+1)^3}+\frac {x^{m+1} \log ^2(x)}{m+1}-\frac {2 x^{m+1} \log (x)}{(m+1)^2} \]
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Rule 2341
Rule 2342
Rubi steps \begin{align*} \text {integral}& = \frac {x^{1+m} \log ^2(x)}{1+m}-\frac {2 \int x^m \log (x) \, dx}{1+m} \\ & = \frac {2 x^{1+m}}{(1+m)^3}-\frac {2 x^{1+m} \log (x)}{(1+m)^2}+\frac {x^{1+m} \log ^2(x)}{1+m} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.71 \[ \int x^m \log ^2(x) \, dx=\frac {x^{1+m} \left (2-2 (1+m) \log (x)+(1+m)^2 \log ^2(x)\right )}{(1+m)^3} \]
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Time = 0.04 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.98
method | result | size |
risch | \(\frac {x \left (m^{2} \ln \left (x \right )^{2}+2 m \ln \left (x \right )^{2}-2 m \ln \left (x \right )+\ln \left (x \right )^{2}-2 \ln \left (x \right )+2\right ) x^{m}}{\left (1+m \right )^{3}}\) | \(41\) |
norman | \(\frac {x \ln \left (x \right )^{2} {\mathrm e}^{m \ln \left (x \right )}}{1+m}+\frac {2 x \,{\mathrm e}^{m \ln \left (x \right )}}{m^{3}+3 m^{2}+3 m +1}-\frac {2 x \ln \left (x \right ) {\mathrm e}^{m \ln \left (x \right )}}{m^{2}+2 m +1}\) | \(61\) |
parallelrisch | \(\frac {x \,x^{m} \ln \left (x \right )^{2} m^{2}+2 x \,x^{m} \ln \left (x \right )^{2} m +x^{m} \ln \left (x \right )^{2} x -2 x \,x^{m} \ln \left (x \right ) m -2 x^{m} \ln \left (x \right ) x +2 x \,x^{m}}{m^{3}+3 m^{2}+3 m +1}\) | \(73\) |
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Time = 0.25 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.07 \[ \int x^m \log ^2(x) \, dx=\frac {{\left ({\left (m^{2} + 2 \, m + 1\right )} x \log \left (x\right )^{2} - 2 \, {\left (m + 1\right )} x \log \left (x\right ) + 2 \, x\right )} x^{m}}{m^{3} + 3 \, m^{2} + 3 \, m + 1} \]
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Leaf count of result is larger than twice the leaf count of optimal. 155 vs. \(2 (39) = 78\).
Time = 0.36 (sec) , antiderivative size = 155, normalized size of antiderivative = 3.69 \[ \int x^m \log ^2(x) \, dx=\begin {cases} \frac {m^{2} x x^{m} \log {\left (x \right )}^{2}}{m^{3} + 3 m^{2} + 3 m + 1} + \frac {2 m x x^{m} \log {\left (x \right )}^{2}}{m^{3} + 3 m^{2} + 3 m + 1} - \frac {2 m x x^{m} \log {\left (x \right )}}{m^{3} + 3 m^{2} + 3 m + 1} + \frac {x x^{m} \log {\left (x \right )}^{2}}{m^{3} + 3 m^{2} + 3 m + 1} - \frac {2 x x^{m} \log {\left (x \right )}}{m^{3} + 3 m^{2} + 3 m + 1} + \frac {2 x x^{m}}{m^{3} + 3 m^{2} + 3 m + 1} & \text {for}\: m \neq -1 \\\frac {\log {\left (x \right )}^{3}}{3} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00 \[ \int x^m \log ^2(x) \, dx=\frac {x^{m + 1} \log \left (x\right )^{2}}{m + 1} - \frac {2 \, x^{m + 1} \log \left (x\right )}{{\left (m + 1\right )}^{2}} + \frac {2 \, x^{m + 1}}{{\left (m + 1\right )}^{3}} \]
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Time = 0.29 (sec) , antiderivative size = 84, normalized size of antiderivative = 2.00 \[ \int x^m \log ^2(x) \, dx=-\frac {2 \, m x x^{m} \log \left (x\right )}{{\left (m^{2} + 2 \, m + 1\right )} {\left (m + 1\right )}} + \frac {x^{m + 1} \log \left (x\right )^{2}}{m + 1} - \frac {2 \, x x^{m} \log \left (x\right )}{{\left (m^{2} + 2 \, m + 1\right )} {\left (m + 1\right )}} + \frac {2 \, x x^{m}}{{\left (m^{2} + 2 \, m + 1\right )} {\left (m + 1\right )}} \]
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Time = 0.40 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.02 \[ \int x^m \log ^2(x) \, dx=\left \{\begin {array}{cl} \frac {{\ln \left (x\right )}^3}{3} & \text {\ if\ \ }m=-1\\ \frac {x^{m+1}\,\left ({\ln \left (x\right )}^2\,{\left (m+1\right )}^2-2\,\ln \left (x\right )\,\left (m+1\right )+2\right )}{{\left (m+1\right )}^3} & \text {\ if\ \ }m\neq -1 \end {array}\right . \]
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