Integrand size = 58, antiderivative size = 18 \[ \int \frac {4 e^4 x+24 e^2 x \log (5)+32 x \log ^2(5)+\left (6 e^4 x+16 e^2 x \log (5)\right ) \log (2 x)+2 e^4 x \log ^2(2 x)}{e^4} \, dx=\left (x+\frac {4 x \log (5)}{e^2}+x \log (2 x)\right )^2 \]
[Out]
Leaf count is larger than twice the leaf count of optimal. \(80\) vs. \(2(18)=36\).
Time = 0.04 (sec) , antiderivative size = 80, normalized size of antiderivative = 4.44, number of steps used = 9, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {6, 12, 2341, 2342} \[ \int \frac {4 e^4 x+24 e^2 x \log (5)+32 x \log ^2(5)+\left (6 e^4 x+16 e^2 x \log (5)\right ) \log (2 x)+2 e^4 x \log ^2(2 x)}{e^4} \, dx=\frac {x^2}{2}+x^2 \log ^2(2 x)+x^2 \left (3+\frac {8 \log (5)}{e^2}\right ) \log (2 x)-x^2 \log (2 x)+\frac {2 x^2 \left (e^2+\log (25)\right ) \left (e^2+\log (625)\right )}{e^4}-\frac {1}{2} x^2 \left (3+\frac {8 \log (5)}{e^2}\right ) \]
[In]
[Out]
Rule 6
Rule 12
Rule 2341
Rule 2342
Rubi steps \begin{align*} \text {integral}& = \int \frac {32 x \log ^2(5)+x \left (4 e^4+24 e^2 \log (5)\right )+\left (6 e^4 x+16 e^2 x \log (5)\right ) \log (2 x)+2 e^4 x \log ^2(2 x)}{e^4} \, dx \\ & = \int \frac {x \left (4 e^4+24 e^2 \log (5)+32 \log ^2(5)\right )+\left (6 e^4 x+16 e^2 x \log (5)\right ) \log (2 x)+2 e^4 x \log ^2(2 x)}{e^4} \, dx \\ & = \frac {\int \left (x \left (4 e^4+24 e^2 \log (5)+32 \log ^2(5)\right )+\left (6 e^4 x+16 e^2 x \log (5)\right ) \log (2 x)+2 e^4 x \log ^2(2 x)\right ) \, dx}{e^4} \\ & = \frac {2 x^2 \left (e^2+\log (25)\right ) \left (e^2+\log (625)\right )}{e^4}+2 \int x \log ^2(2 x) \, dx+\frac {\int \left (6 e^4 x+16 e^2 x \log (5)\right ) \log (2 x) \, dx}{e^4} \\ & = \frac {2 x^2 \left (e^2+\log (25)\right ) \left (e^2+\log (625)\right )}{e^4}+x^2 \log ^2(2 x)-2 \int x \log (2 x) \, dx+\frac {\int x \left (6 e^4+16 e^2 \log (5)\right ) \log (2 x) \, dx}{e^4} \\ & = \frac {x^2}{2}+\frac {2 x^2 \left (e^2+\log (25)\right ) \left (e^2+\log (625)\right )}{e^4}-x^2 \log (2 x)+x^2 \log ^2(2 x)+\left (2 \left (3+\frac {8 \log (5)}{e^2}\right )\right ) \int x \log (2 x) \, dx \\ & = \frac {x^2}{2}-\frac {1}{2} x^2 \left (3+\frac {8 \log (5)}{e^2}\right )+\frac {2 x^2 \left (e^2+\log (25)\right ) \left (e^2+\log (625)\right )}{e^4}-x^2 \log (2 x)+x^2 \left (3+\frac {8 \log (5)}{e^2}\right ) \log (2 x)+x^2 \log ^2(2 x) \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(76\) vs. \(2(18)=36\).
Time = 0.05 (sec) , antiderivative size = 76, normalized size of antiderivative = 4.22 \[ \int \frac {4 e^4 x+24 e^2 x \log (5)+32 x \log ^2(5)+\left (6 e^4 x+16 e^2 x \log (5)\right ) \log (2 x)+2 e^4 x \log ^2(2 x)}{e^4} \, dx=\frac {2 \left (\frac {e^4 x^2}{2}+4 e^2 x^2 \log (5)+8 x^2 \log ^2(5)+e^4 x^2 \log (2 x)+4 e^2 x^2 \log (5) \log (2 x)+\frac {1}{2} e^4 x^2 \log ^2(2 x)\right )}{e^4} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(56\) vs. \(2(19)=38\).
Time = 0.22 (sec) , antiderivative size = 57, normalized size of antiderivative = 3.17
method | result | size |
risch | \(16 \ln \left (5\right )^{2} {\mathrm e}^{-4} x^{2}+8 \,{\mathrm e}^{-2} x^{2} \ln \left (2 x \right ) \ln \left (5\right )+8 x^{2} \ln \left (5\right ) {\mathrm e}^{-2}+2 x^{2} \ln \left (2 x \right )+x^{2} \ln \left (2 x \right )^{2}+x^{2}\) | \(57\) |
norman | \(\left (x^{2} {\mathrm e}^{2} \ln \left (2 x \right )^{2}+\left (8 \ln \left (5\right )+2 \,{\mathrm e}^{2}\right ) x^{2} \ln \left (2 x \right )+\left (16 \ln \left (5\right )^{2}+8 \,{\mathrm e}^{2} \ln \left (5\right )+{\mathrm e}^{4}\right ) {\mathrm e}^{-2} x^{2}\right ) {\mathrm e}^{-2}\) | \(61\) |
parallelrisch | \({\mathrm e}^{-4} \left (x^{2} {\mathrm e}^{4} \ln \left (2 x \right )^{2}+8 \ln \left (5\right ) {\mathrm e}^{2} x^{2} \ln \left (2 x \right )+2 \,{\mathrm e}^{4} x^{2} \ln \left (2 x \right )+16 x^{2} \ln \left (5\right )^{2}+8 \,{\mathrm e}^{2} \ln \left (5\right ) x^{2}+x^{2} {\mathrm e}^{4}\right )\) | \(73\) |
parts | \(x^{2} \ln \left (2 x \right )^{2}-x^{2} \ln \left (2 x \right )+\frac {5 x^{2}}{2}+16 \ln \left (5\right )^{2} {\mathrm e}^{-4} x^{2}+12 x^{2} \ln \left (5\right ) {\mathrm e}^{-2}+2 \,{\mathrm e}^{-2} \left (8 \ln \left (5\right )+3 \,{\mathrm e}^{2}\right ) \left (\frac {x^{2} \ln \left (2 x \right )}{2}-\frac {x^{2}}{4}\right )\) | \(80\) |
derivativedivides | \(\frac {{\mathrm e}^{-4} \left (8 \,{\mathrm e}^{2} \ln \left (5\right ) \left (2 x^{2} \ln \left (2 x \right )-x^{2}\right )+3 \,{\mathrm e}^{4} \left (2 x^{2} \ln \left (2 x \right )-x^{2}\right )+{\mathrm e}^{4} \left (2 x^{2} \ln \left (2 x \right )^{2}-2 x^{2} \ln \left (2 x \right )+x^{2}\right )+4 x^{2} {\mathrm e}^{4}+32 x^{2} \ln \left (5\right )^{2}+24 \,{\mathrm e}^{2} \ln \left (5\right ) x^{2}\right )}{2}\) | \(106\) |
default | \({\mathrm e}^{-4} \left (4 \,{\mathrm e}^{2} \ln \left (5\right ) \left (2 x^{2} \ln \left (2 x \right )-x^{2}\right )+\frac {3 \,{\mathrm e}^{4} \left (2 x^{2} \ln \left (2 x \right )-x^{2}\right )}{2}+2 x^{2} {\mathrm e}^{4}+16 x^{2} \ln \left (5\right )^{2}+12 \,{\mathrm e}^{2} \ln \left (5\right ) x^{2}+2 \,{\mathrm e}^{4} \left (\frac {x^{2} \ln \left (2 x \right )^{2}}{2}-\frac {x^{2} \ln \left (2 x \right )}{2}+\frac {x^{2}}{4}\right )\right )\) | \(108\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (17) = 34\).
Time = 0.23 (sec) , antiderivative size = 62, normalized size of antiderivative = 3.44 \[ \int \frac {4 e^4 x+24 e^2 x \log (5)+32 x \log ^2(5)+\left (6 e^4 x+16 e^2 x \log (5)\right ) \log (2 x)+2 e^4 x \log ^2(2 x)}{e^4} \, dx={\left (x^{2} e^{4} \log \left (2 \, x\right )^{2} + 8 \, x^{2} e^{2} \log \left (5\right ) + 16 \, x^{2} \log \left (5\right )^{2} + x^{2} e^{4} + 2 \, {\left (4 \, x^{2} e^{2} \log \left (5\right ) + x^{2} e^{4}\right )} \log \left (2 \, x\right )\right )} e^{\left (-4\right )} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (19) = 38\).
Time = 0.12 (sec) , antiderivative size = 60, normalized size of antiderivative = 3.33 \[ \int \frac {4 e^4 x+24 e^2 x \log (5)+32 x \log ^2(5)+\left (6 e^4 x+16 e^2 x \log (5)\right ) \log (2 x)+2 e^4 x \log ^2(2 x)}{e^4} \, dx=x^{2} \log {\left (2 x \right )}^{2} + \frac {x^{2} \cdot \left (16 \log {\left (5 \right )}^{2} + e^{4} + 8 e^{2} \log {\left (5 \right )}\right )}{e^{4}} + \frac {\left (8 x^{2} \log {\left (5 \right )} + 2 x^{2} e^{2}\right ) \log {\left (2 x \right )}}{e^{2}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 91 vs. \(2 (17) = 34\).
Time = 0.20 (sec) , antiderivative size = 91, normalized size of antiderivative = 5.06 \[ \int \frac {4 e^4 x+24 e^2 x \log (5)+32 x \log ^2(5)+\left (6 e^4 x+16 e^2 x \log (5)\right ) \log (2 x)+2 e^4 x \log ^2(2 x)}{e^4} \, dx=\frac {1}{2} \, {\left ({\left (2 \, \log \left (2 \, x\right )^{2} - 2 \, \log \left (2 \, x\right ) + 1\right )} x^{2} e^{4} + 24 \, x^{2} e^{2} \log \left (5\right ) + 32 \, x^{2} \log \left (5\right )^{2} - {\left (8 \, e^{2} \log \left (5\right ) + 3 \, e^{4}\right )} x^{2} + 4 \, x^{2} e^{4} + 2 \, {\left (8 \, x^{2} e^{2} \log \left (5\right ) + 3 \, x^{2} e^{4}\right )} \log \left (2 \, x\right )\right )} e^{\left (-4\right )} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (17) = 34\).
Time = 0.30 (sec) , antiderivative size = 80, normalized size of antiderivative = 4.44 \[ \int \frac {4 e^4 x+24 e^2 x \log (5)+32 x \log ^2(5)+\left (6 e^4 x+16 e^2 x \log (5)\right ) \log (2 x)+2 e^4 x \log ^2(2 x)}{e^4} \, dx=\frac {1}{2} \, {\left (16 \, x^{2} e^{2} \log \left (5\right ) \log \left (2 \, x\right ) + 16 \, x^{2} e^{2} \log \left (5\right ) + 32 \, x^{2} \log \left (5\right )^{2} + 6 \, x^{2} e^{4} \log \left (2 \, x\right ) + x^{2} e^{4} + {\left (2 \, x^{2} \log \left (2 \, x\right )^{2} - 2 \, x^{2} \log \left (2 \, x\right ) + x^{2}\right )} e^{4}\right )} e^{\left (-4\right )} \]
[In]
[Out]
Time = 12.11 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {4 e^4 x+24 e^2 x \log (5)+32 x \log ^2(5)+\left (6 e^4 x+16 e^2 x \log (5)\right ) \log (2 x)+2 e^4 x \log ^2(2 x)}{e^4} \, dx=x^2\,{\mathrm {e}}^{-4}\,{\left ({\mathrm {e}}^2+\ln \left (625\right )+\ln \left (2\,x\right )\,{\mathrm {e}}^2\right )}^2 \]
[In]
[Out]