Integrand size = 136, antiderivative size = 36 \[ \int \frac {450-141855 x+93650 x^2-15476 x^3+\left (-11250 x+7470 x^2-1240 x^3\right ) \log \left (\frac {e^{2 x}}{x}\right )+\left (-225 x+150 x^2-25 x^3\right ) \log ^2\left (\frac {e^{2 x}}{x}\right )}{562500 x-372000 x^2+61504 x^3+\left (45000 x-29880 x^2+4960 x^3\right ) \log \left (\frac {e^{2 x}}{x}\right )+\left (900 x-600 x^2+100 x^3\right ) \log ^2\left (\frac {e^{2 x}}{x}\right )} \, dx=\frac {1}{4} \left (-x+\frac {2}{25+\frac {x}{5 (3-x)}+\log \left (\frac {e^{2 x}}{x}\right )}\right ) \]
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\[ \int \frac {450-141855 x+93650 x^2-15476 x^3+\left (-11250 x+7470 x^2-1240 x^3\right ) \log \left (\frac {e^{2 x}}{x}\right )+\left (-225 x+150 x^2-25 x^3\right ) \log ^2\left (\frac {e^{2 x}}{x}\right )}{562500 x-372000 x^2+61504 x^3+\left (45000 x-29880 x^2+4960 x^3\right ) \log \left (\frac {e^{2 x}}{x}\right )+\left (900 x-600 x^2+100 x^3\right ) \log ^2\left (\frac {e^{2 x}}{x}\right )} \, dx=\int \frac {450-141855 x+93650 x^2-15476 x^3+\left (-11250 x+7470 x^2-1240 x^3\right ) \log \left (\frac {e^{2 x}}{x}\right )+\left (-225 x+150 x^2-25 x^3\right ) \log ^2\left (\frac {e^{2 x}}{x}\right )}{562500 x-372000 x^2+61504 x^3+\left (45000 x-29880 x^2+4960 x^3\right ) \log \left (\frac {e^{2 x}}{x}\right )+\left (900 x-600 x^2+100 x^3\right ) \log ^2\left (\frac {e^{2 x}}{x}\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {450-141855 x+93650 x^2-15476 x^3-10 x \left (1125-747 x+124 x^2\right ) \log \left (\frac {e^{2 x}}{x}\right )-25 (-3+x)^2 x \log ^2\left (\frac {e^{2 x}}{x}\right )}{4 x \left (375-124 x-5 (-3+x) \log \left (\frac {e^{2 x}}{x}\right )\right )^2} \, dx \\ & = \frac {1}{4} \int \frac {450-141855 x+93650 x^2-15476 x^3-10 x \left (1125-747 x+124 x^2\right ) \log \left (\frac {e^{2 x}}{x}\right )-25 (-3+x)^2 x \log ^2\left (\frac {e^{2 x}}{x}\right )}{x \left (375-124 x-5 (-3+x) \log \left (\frac {e^{2 x}}{x}\right )\right )^2} \, dx \\ & = \frac {1}{4} \int \left (-1-\frac {10 \left (-45+123 x-65 x^2+10 x^3\right )}{x \left (-375+124 x-15 \log \left (\frac {e^{2 x}}{x}\right )+5 x \log \left (\frac {e^{2 x}}{x}\right )\right )^2}\right ) \, dx \\ & = -\frac {x}{4}-\frac {5}{2} \int \frac {-45+123 x-65 x^2+10 x^3}{x \left (-375+124 x-15 \log \left (\frac {e^{2 x}}{x}\right )+5 x \log \left (\frac {e^{2 x}}{x}\right )\right )^2} \, dx \\ & = -\frac {x}{4}-\frac {5}{2} \int \frac {-45+123 x-65 x^2+10 x^3}{x \left (375-124 x-5 (-3+x) \log \left (\frac {e^{2 x}}{x}\right )\right )^2} \, dx \\ & = -\frac {x}{4}-\frac {5}{2} \int \left (\frac {123}{\left (-375+124 x-15 \log \left (\frac {e^{2 x}}{x}\right )+5 x \log \left (\frac {e^{2 x}}{x}\right )\right )^2}-\frac {45}{x \left (-375+124 x-15 \log \left (\frac {e^{2 x}}{x}\right )+5 x \log \left (\frac {e^{2 x}}{x}\right )\right )^2}-\frac {65 x}{\left (-375+124 x-15 \log \left (\frac {e^{2 x}}{x}\right )+5 x \log \left (\frac {e^{2 x}}{x}\right )\right )^2}+\frac {10 x^2}{\left (-375+124 x-15 \log \left (\frac {e^{2 x}}{x}\right )+5 x \log \left (\frac {e^{2 x}}{x}\right )\right )^2}\right ) \, dx \\ & = -\frac {x}{4}-25 \int \frac {x^2}{\left (-375+124 x-15 \log \left (\frac {e^{2 x}}{x}\right )+5 x \log \left (\frac {e^{2 x}}{x}\right )\right )^2} \, dx+\frac {225}{2} \int \frac {1}{x \left (-375+124 x-15 \log \left (\frac {e^{2 x}}{x}\right )+5 x \log \left (\frac {e^{2 x}}{x}\right )\right )^2} \, dx+\frac {325}{2} \int \frac {x}{\left (-375+124 x-15 \log \left (\frac {e^{2 x}}{x}\right )+5 x \log \left (\frac {e^{2 x}}{x}\right )\right )^2} \, dx-\frac {615}{2} \int \frac {1}{\left (-375+124 x-15 \log \left (\frac {e^{2 x}}{x}\right )+5 x \log \left (\frac {e^{2 x}}{x}\right )\right )^2} \, dx \\ & = -\frac {x}{4}-25 \int \frac {x^2}{\left (375-124 x-5 (-3+x) \log \left (\frac {e^{2 x}}{x}\right )\right )^2} \, dx+\frac {225}{2} \int \frac {1}{x \left (375-124 x-5 (-3+x) \log \left (\frac {e^{2 x}}{x}\right )\right )^2} \, dx+\frac {325}{2} \int \frac {x}{\left (375-124 x-5 (-3+x) \log \left (\frac {e^{2 x}}{x}\right )\right )^2} \, dx-\frac {615}{2} \int \frac {1}{\left (375-124 x-5 (-3+x) \log \left (\frac {e^{2 x}}{x}\right )\right )^2} \, dx \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.44 \[ \int \frac {450-141855 x+93650 x^2-15476 x^3+\left (-11250 x+7470 x^2-1240 x^3\right ) \log \left (\frac {e^{2 x}}{x}\right )+\left (-225 x+150 x^2-25 x^3\right ) \log ^2\left (\frac {e^{2 x}}{x}\right )}{562500 x-372000 x^2+61504 x^3+\left (45000 x-29880 x^2+4960 x^3\right ) \log \left (\frac {e^{2 x}}{x}\right )+\left (900 x-600 x^2+100 x^3\right ) \log ^2\left (\frac {e^{2 x}}{x}\right )} \, dx=-\frac {30-385 x+124 x^2+5 (-3+x) x \log \left (\frac {e^{2 x}}{x}\right )}{4 \left (-375+124 x+5 (-3+x) \log \left (\frac {e^{2 x}}{x}\right )\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(67\) vs. \(2(28)=56\).
Time = 0.74 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.89
method | result | size |
parallelrisch | \(\frac {5475-25 \ln \left (\frac {{\mathrm e}^{2 x}}{x}\right ) x^{2}-620 x^{2}+65 x +225 \ln \left (\frac {{\mathrm e}^{2 x}}{x}\right )}{100 \ln \left (\frac {{\mathrm e}^{2 x}}{x}\right ) x -300 \ln \left (\frac {{\mathrm e}^{2 x}}{x}\right )+2480 x -7500}\) | \(68\) |
default | \(-\frac {13 x +\left (-124-5 \ln \left (\frac {{\mathrm e}^{2 x}}{x}\right )-5 \ln \left (x \right )+10 x \right ) x^{2}-10 x^{3}+5 x^{2} \ln \left (x \right )+1095+45 \ln \left (\frac {{\mathrm e}^{2 x}}{x}\right )}{4 \left (5 x \ln \left (x \right )-10 x^{2}-5 x \left (\ln \left (\frac {{\mathrm e}^{2 x}}{x}\right )-2 \ln \left ({\mathrm e}^{x}\right )+\ln \left (x \right )\right )-10 x \left (\ln \left ({\mathrm e}^{x}\right )-x \right )-124 x +15 \ln \left (\frac {{\mathrm e}^{2 x}}{x}\right )+375\right )}\) | \(113\) |
risch | \(-\frac {x}{4}+\frac {5 i \left (-3+x \right )}{5 \pi x \operatorname {csgn}\left (i {\mathrm e}^{2 x}\right )^{3}-10 \pi x \operatorname {csgn}\left (i {\mathrm e}^{2 x}\right )^{2} \operatorname {csgn}\left (i {\mathrm e}^{x}\right )+5 \pi x \,\operatorname {csgn}\left (i {\mathrm e}^{2 x}\right ) \operatorname {csgn}\left (i {\mathrm e}^{x}\right )^{2}-5 \pi x \,\operatorname {csgn}\left (i {\mathrm e}^{2 x}\right ) \operatorname {csgn}\left (\frac {i {\mathrm e}^{2 x}}{x}\right )^{2}+5 \pi x \,\operatorname {csgn}\left (i {\mathrm e}^{2 x}\right ) \operatorname {csgn}\left (\frac {i {\mathrm e}^{2 x}}{x}\right ) \operatorname {csgn}\left (\frac {i}{x}\right )+5 \pi x \operatorname {csgn}\left (\frac {i {\mathrm e}^{2 x}}{x}\right )^{3}-5 \pi x \operatorname {csgn}\left (\frac {i {\mathrm e}^{2 x}}{x}\right )^{2} \operatorname {csgn}\left (\frac {i}{x}\right )-15 \pi \operatorname {csgn}\left (i {\mathrm e}^{2 x}\right )^{3}+30 \pi \operatorname {csgn}\left (i {\mathrm e}^{2 x}\right )^{2} \operatorname {csgn}\left (i {\mathrm e}^{x}\right )-15 \pi \,\operatorname {csgn}\left (i {\mathrm e}^{2 x}\right ) \operatorname {csgn}\left (i {\mathrm e}^{x}\right )^{2}+15 \pi \,\operatorname {csgn}\left (i {\mathrm e}^{2 x}\right ) \operatorname {csgn}\left (\frac {i {\mathrm e}^{2 x}}{x}\right )^{2}-15 \pi \,\operatorname {csgn}\left (i {\mathrm e}^{2 x}\right ) \operatorname {csgn}\left (\frac {i {\mathrm e}^{2 x}}{x}\right ) \operatorname {csgn}\left (\frac {i}{x}\right )-15 \pi \operatorname {csgn}\left (\frac {i {\mathrm e}^{2 x}}{x}\right )^{3}+15 \pi \operatorname {csgn}\left (\frac {i {\mathrm e}^{2 x}}{x}\right )^{2} \operatorname {csgn}\left (\frac {i}{x}\right )+20 i x \ln \left ({\mathrm e}^{x}\right )-60 i \ln \left ({\mathrm e}^{x}\right )+30 i \ln \left (x \right )+248 i x -10 i x \ln \left (x \right )-750 i}\) | \(337\) |
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Time = 0.25 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.42 \[ \int \frac {450-141855 x+93650 x^2-15476 x^3+\left (-11250 x+7470 x^2-1240 x^3\right ) \log \left (\frac {e^{2 x}}{x}\right )+\left (-225 x+150 x^2-25 x^3\right ) \log ^2\left (\frac {e^{2 x}}{x}\right )}{562500 x-372000 x^2+61504 x^3+\left (45000 x-29880 x^2+4960 x^3\right ) \log \left (\frac {e^{2 x}}{x}\right )+\left (900 x-600 x^2+100 x^3\right ) \log ^2\left (\frac {e^{2 x}}{x}\right )} \, dx=-\frac {124 \, x^{2} + 5 \, {\left (x^{2} - 3 \, x\right )} \log \left (\frac {e^{\left (2 \, x\right )}}{x}\right ) - 385 \, x + 30}{4 \, {\left (5 \, {\left (x - 3\right )} \log \left (\frac {e^{\left (2 \, x\right )}}{x}\right ) + 124 \, x - 375\right )}} \]
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Time = 0.14 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.72 \[ \int \frac {450-141855 x+93650 x^2-15476 x^3+\left (-11250 x+7470 x^2-1240 x^3\right ) \log \left (\frac {e^{2 x}}{x}\right )+\left (-225 x+150 x^2-25 x^3\right ) \log ^2\left (\frac {e^{2 x}}{x}\right )}{562500 x-372000 x^2+61504 x^3+\left (45000 x-29880 x^2+4960 x^3\right ) \log \left (\frac {e^{2 x}}{x}\right )+\left (900 x-600 x^2+100 x^3\right ) \log ^2\left (\frac {e^{2 x}}{x}\right )} \, dx=- \frac {x}{4} + \frac {5 x - 15}{248 x + \left (10 x - 30\right ) \log {\left (\frac {e^{2 x}}{x} \right )} - 750} \]
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Time = 0.24 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.31 \[ \int \frac {450-141855 x+93650 x^2-15476 x^3+\left (-11250 x+7470 x^2-1240 x^3\right ) \log \left (\frac {e^{2 x}}{x}\right )+\left (-225 x+150 x^2-25 x^3\right ) \log ^2\left (\frac {e^{2 x}}{x}\right )}{562500 x-372000 x^2+61504 x^3+\left (45000 x-29880 x^2+4960 x^3\right ) \log \left (\frac {e^{2 x}}{x}\right )+\left (900 x-600 x^2+100 x^3\right ) \log ^2\left (\frac {e^{2 x}}{x}\right )} \, dx=-\frac {10 \, x^{3} + 94 \, x^{2} - 5 \, {\left (x^{2} - 3 \, x\right )} \log \left (x\right ) - 385 \, x + 30}{4 \, {\left (10 \, x^{2} - 5 \, {\left (x - 3\right )} \log \left (x\right ) + 94 \, x - 375\right )}} \]
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Time = 0.29 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.83 \[ \int \frac {450-141855 x+93650 x^2-15476 x^3+\left (-11250 x+7470 x^2-1240 x^3\right ) \log \left (\frac {e^{2 x}}{x}\right )+\left (-225 x+150 x^2-25 x^3\right ) \log ^2\left (\frac {e^{2 x}}{x}\right )}{562500 x-372000 x^2+61504 x^3+\left (45000 x-29880 x^2+4960 x^3\right ) \log \left (\frac {e^{2 x}}{x}\right )+\left (900 x-600 x^2+100 x^3\right ) \log ^2\left (\frac {e^{2 x}}{x}\right )} \, dx=-\frac {1}{4} \, x + \frac {5 \, {\left (x - 3\right )}}{2 \, {\left (10 \, x^{2} - 5 \, x \log \left (x\right ) + 94 \, x + 15 \, \log \left (x\right ) - 375\right )}} \]
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Timed out. \[ \int \frac {450-141855 x+93650 x^2-15476 x^3+\left (-11250 x+7470 x^2-1240 x^3\right ) \log \left (\frac {e^{2 x}}{x}\right )+\left (-225 x+150 x^2-25 x^3\right ) \log ^2\left (\frac {e^{2 x}}{x}\right )}{562500 x-372000 x^2+61504 x^3+\left (45000 x-29880 x^2+4960 x^3\right ) \log \left (\frac {e^{2 x}}{x}\right )+\left (900 x-600 x^2+100 x^3\right ) \log ^2\left (\frac {e^{2 x}}{x}\right )} \, dx=\int -\frac {141855\,x+{\ln \left (\frac {{\mathrm {e}}^{2\,x}}{x}\right )}^2\,\left (25\,x^3-150\,x^2+225\,x\right )-93650\,x^2+15476\,x^3+\ln \left (\frac {{\mathrm {e}}^{2\,x}}{x}\right )\,\left (1240\,x^3-7470\,x^2+11250\,x\right )-450}{562500\,x+{\ln \left (\frac {{\mathrm {e}}^{2\,x}}{x}\right )}^2\,\left (100\,x^3-600\,x^2+900\,x\right )-372000\,x^2+61504\,x^3+\ln \left (\frac {{\mathrm {e}}^{2\,x}}{x}\right )\,\left (4960\,x^3-29880\,x^2+45000\,x\right )} \,d x \]
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