Integrand size = 69, antiderivative size = 36 \[ \int \frac {30 x-6 e^4 x+6 x^2+9 x^4+\left (-25+5 e^4-30 x^3\right ) \log (4)+25 x^2 \log ^2(4)}{9 x^4-30 x^3 \log (4)+25 x^2 \log ^2(4)} \, dx=\frac {-2 x+x^2+\frac {5-e^4+2 x}{2 x+5 (-x+\log (4))}}{x} \]
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Time = 0.07 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.39, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.058, Rules used = {6, 1608, 27, 1634} \[ \int \frac {30 x-6 e^4 x+6 x^2+9 x^4+\left (-25+5 e^4-30 x^3\right ) \log (4)+25 x^2 \log ^2(4)}{9 x^4-30 x^3 \log (4)+25 x^2 \log ^2(4)} \, dx=x+\frac {3 e^4-5 (3+\log (16))}{5 \log (4) (3 x-5 \log (4))}+\frac {5-e^4}{5 x \log (4)} \]
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Rule 6
Rule 27
Rule 1608
Rule 1634
Rubi steps \begin{align*} \text {integral}& = \int \frac {\left (30-6 e^4\right ) x+6 x^2+9 x^4+\left (-25+5 e^4-30 x^3\right ) \log (4)+25 x^2 \log ^2(4)}{9 x^4-30 x^3 \log (4)+25 x^2 \log ^2(4)} \, dx \\ & = \int \frac {\left (30-6 e^4\right ) x+9 x^4+\left (-25+5 e^4-30 x^3\right ) \log (4)+x^2 \left (6+25 \log ^2(4)\right )}{9 x^4-30 x^3 \log (4)+25 x^2 \log ^2(4)} \, dx \\ & = \int \frac {\left (30-6 e^4\right ) x+9 x^4+\left (-25+5 e^4-30 x^3\right ) \log (4)+x^2 \left (6+25 \log ^2(4)\right )}{x^2 \left (9 x^2-30 x \log (4)+25 \log ^2(4)\right )} \, dx \\ & = \int \frac {\left (30-6 e^4\right ) x+9 x^4+\left (-25+5 e^4-30 x^3\right ) \log (4)+x^2 \left (6+25 \log ^2(4)\right )}{x^2 (3 x-5 \log (4))^2} \, dx \\ & = \int \left (1+\frac {-5+e^4}{5 x^2 \log (4)}-\frac {3 \left (-15+3 e^4-5 \log (16)\right )}{5 (3 x-5 \log (4))^2 \log (4)}\right ) \, dx \\ & = x+\frac {5-e^4}{5 x \log (4)}+\frac {3 e^4-5 (3+\log (16))}{5 (3 x-5 \log (4)) \log (4)} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.94 \[ \int \frac {30 x-6 e^4 x+6 x^2+9 x^4+\left (-25+5 e^4-30 x^3\right ) \log (4)+25 x^2 \log ^2(4)}{9 x^4-30 x^3 \log (4)+25 x^2 \log ^2(4)} \, dx=\frac {-5+e^4-2 x+3 x^3-5 x^2 \log (4)}{x (3 x-5 \log (4))} \]
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Time = 0.09 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.75
method | result | size |
risch | \(x +\frac {2 x -{\mathrm e}^{4}+5}{x \left (10 \ln \left (2\right )-3 x \right )}\) | \(27\) |
norman | \(\frac {-3 x^{3}+5+\left (\frac {100 \ln \left (2\right )^{2}}{3}+2\right ) x -{\mathrm e}^{4}}{x \left (10 \ln \left (2\right )-3 x \right )}\) | \(36\) |
gosper | \(-\frac {-100 x \ln \left (2\right )^{2}+9 x^{3}+3 \,{\mathrm e}^{4}-6 x -15}{3 x \left (10 \ln \left (2\right )-3 x \right )}\) | \(37\) |
parallelrisch | \(-\frac {-100 x \ln \left (2\right )^{2}+9 x^{3}+3 \,{\mathrm e}^{4}-6 x -15}{3 x \left (10 \ln \left (2\right )-3 x \right )}\) | \(37\) |
default | \(x -\frac {{\mathrm e}^{4}-5}{10 \ln \left (2\right ) x}-\frac {-9 \,{\mathrm e}^{4}+60 \ln \left (2\right )+45}{30 \ln \left (2\right ) \left (-10 \ln \left (2\right )+3 x \right )}\) | \(42\) |
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Time = 0.25 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.92 \[ \int \frac {30 x-6 e^4 x+6 x^2+9 x^4+\left (-25+5 e^4-30 x^3\right ) \log (4)+25 x^2 \log ^2(4)}{9 x^4-30 x^3 \log (4)+25 x^2 \log ^2(4)} \, dx=\frac {3 \, x^{3} - 10 \, x^{2} \log \left (2\right ) - 2 \, x + e^{4} - 5}{3 \, x^{2} - 10 \, x \log \left (2\right )} \]
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Time = 0.47 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.56 \[ \int \frac {30 x-6 e^4 x+6 x^2+9 x^4+\left (-25+5 e^4-30 x^3\right ) \log (4)+25 x^2 \log ^2(4)}{9 x^4-30 x^3 \log (4)+25 x^2 \log ^2(4)} \, dx=x + \frac {- 2 x - 5 + e^{4}}{3 x^{2} - 10 x \log {\left (2 \right )}} \]
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Time = 0.19 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.72 \[ \int \frac {30 x-6 e^4 x+6 x^2+9 x^4+\left (-25+5 e^4-30 x^3\right ) \log (4)+25 x^2 \log ^2(4)}{9 x^4-30 x^3 \log (4)+25 x^2 \log ^2(4)} \, dx=x - \frac {2 \, x - e^{4} + 5}{3 \, x^{2} - 10 \, x \log \left (2\right )} \]
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Time = 0.27 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.72 \[ \int \frac {30 x-6 e^4 x+6 x^2+9 x^4+\left (-25+5 e^4-30 x^3\right ) \log (4)+25 x^2 \log ^2(4)}{9 x^4-30 x^3 \log (4)+25 x^2 \log ^2(4)} \, dx=x - \frac {2 \, x - e^{4} + 5}{3 \, x^{2} - 10 \, x \log \left (2\right )} \]
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Time = 0.17 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.72 \[ \int \frac {30 x-6 e^4 x+6 x^2+9 x^4+\left (-25+5 e^4-30 x^3\right ) \log (4)+25 x^2 \log ^2(4)}{9 x^4-30 x^3 \log (4)+25 x^2 \log ^2(4)} \, dx=x-\frac {2\,x-{\mathrm {e}}^4+5}{x\,\left (3\,x-10\,\ln \left (2\right )\right )} \]
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