Integrand size = 60, antiderivative size = 23 \[ \int \frac {9+6 x (i \pi +\log (25))+\left (-1+x^2\right ) (i \pi +\log (25))^2}{36+24 x (i \pi +\log (25))+4 x^2 (i \pi +\log (25))^2} \, dx=\frac {1}{4} \left (-3+x+\frac {1}{x+\frac {3}{i \pi +\log (25)}}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.52, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {27, 6, 12, 1864} \[ \int \frac {9+6 x (i \pi +\log (25))+\left (-1+x^2\right ) (i \pi +\log (25))^2}{36+24 x (i \pi +\log (25))+4 x^2 (i \pi +\log (25))^2} \, dx=\frac {x}{4}-\frac {\pi -i \log (25)}{4 (3 i-x (\pi -i \log (25)))} \]
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Rule 6
Rule 12
Rule 27
Rule 1864
Rubi steps \begin{align*} \text {integral}& = \int -\frac {9+6 x (i \pi +\log (25))+\left (-1+x^2\right ) (i \pi +\log (25))^2}{4 (-3 i+\pi x-i x \log (25))^2} \, dx \\ & = \int \frac {-9-6 x (i \pi +\log (25))-\left (-1+x^2\right ) (i \pi +\log (25))^2}{4 (-3 i+x (\pi -i \log (25)))^2} \, dx \\ & = \frac {1}{4} \int \frac {-9-6 x (i \pi +\log (25))-\left (-1+x^2\right ) (i \pi +\log (25))^2}{(-3 i+x (\pi -i \log (25)))^2} \, dx \\ & = \frac {1}{4} \int \left (1-\frac {(\pi -i \log (25))^2}{(3 i-x (\pi -i \log (25)))^2}\right ) \, dx \\ & = \frac {x}{4}-\frac {\pi -i \log (25)}{4 (3 i-x (\pi -i \log (25)))} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.48 \[ \int \frac {9+6 x (i \pi +\log (25))+\left (-1+x^2\right ) (i \pi +\log (25))^2}{36+24 x (i \pi +\log (25))+4 x^2 (i \pi +\log (25))^2} \, dx=\frac {1}{4} \left (x-\frac {-\pi +i \log (25)}{-3 i+\pi x-i x \log (25)}\right ) \]
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Time = 1.16 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.78
method | result | size |
risch | \(\frac {x}{4}+\frac {\pi }{-8 i x \ln \left (5\right )+4 \pi x -12 i}-\frac {i \ln \left (5\right )}{2 \left (-2 i x \ln \left (5\right )+\pi x -3 i\right )}\) | \(41\) |
default | \(\frac {x}{4}-\frac {-\pi ^{2}+4 i \pi \ln \left (5\right )+4 \ln \left (5\right )^{2}}{4 \left (\pi -2 i \ln \left (5\right )\right ) \left (-2 i x \ln \left (5\right )+\pi x -3 i\right )}\) | \(48\) |
parallelrisch | \(-\frac {-3 \pi \,x^{2}+i x \,\pi ^{2}-4 i x \ln \left (5\right )^{2}+6 i \ln \left (5\right ) x^{2}+4 \pi \ln \left (5\right ) x +9 i x}{12 \left (-2 i x \ln \left (5\right )+\pi x -3 i\right )}\) | \(57\) |
norman | \(\frac {\left (\frac {\pi ^{2}}{4}+\ln \left (5\right )^{2}\right ) x^{3}+\left (\frac {9}{4}+\frac {\pi ^{2}}{4}-\ln \left (5\right )^{2}-i \pi \ln \left (5\right )\right ) x +\left (-\frac {i \pi ^{3}}{12}-\frac {i \pi \ln \left (5\right )^{2}}{3}-\frac {\pi ^{2} \ln \left (5\right )}{6}-\frac {2 \ln \left (5\right )^{3}}{3}+3 \ln \left (5\right )\right ) x^{2}}{\pi ^{2} x^{2}+4 x^{2} \ln \left (5\right )^{2}+12 x \ln \left (5\right )+9}\) | \(99\) |
gosper | \(-\frac {\left (-2 i x \ln \left (5\right )+\pi x -3 i\right ) x \left (4 \pi \ln \left (5\right )+i \pi ^{2}-4 i \ln \left (5\right )^{2}+6 i x \ln \left (5\right )-3 \pi x +9 i\right ) \left (-4 i \pi \ln \left (5\right ) x^{2}+\pi ^{2} x^{2}-4 x^{2} \ln \left (5\right )^{2}+4 i \pi \ln \left (5\right )-6 i \pi x -\pi ^{2}+4 \ln \left (5\right )^{2}-12 x \ln \left (5\right )-9\right )}{12 \left (-2 i x \ln \left (5\right )+\pi x +2 i \ln \left (5\right )-\pi -3 i\right ) \left (-2 i x \ln \left (5\right )+\pi x -2 i \ln \left (5\right )+\pi -3 i\right ) \left (-4 i \pi \ln \left (5\right ) x^{2}+\pi ^{2} x^{2}-4 x^{2} \ln \left (5\right )^{2}-6 i \pi x -12 x \ln \left (5\right )-9\right )}\) | \(182\) |
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Time = 0.25 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.57 \[ \int \frac {9+6 x (i \pi +\log (25))+\left (-1+x^2\right ) (i \pi +\log (25))^2}{36+24 x (i \pi +\log (25))+4 x^2 (i \pi +\log (25))^2} \, dx=\frac {i \, \pi + i \, \pi x^{2} + 2 \, {\left (x^{2} + 1\right )} \log \left (5\right ) + 3 \, x}{4 i \, \pi x + 8 \, x \log \left (5\right ) + 12} \]
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Time = 0.27 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04 \[ \int \frac {9+6 x (i \pi +\log (25))+\left (-1+x^2\right ) (i \pi +\log (25))^2}{36+24 x (i \pi +\log (25))+4 x^2 (i \pi +\log (25))^2} \, dx=\frac {x}{4} + \frac {2 \log {\left (5 \right )} + i \pi }{x \left (8 \log {\left (5 \right )} + 4 i \pi \right ) + 12} \]
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Time = 0.19 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.22 \[ \int \frac {9+6 x (i \pi +\log (25))+\left (-1+x^2\right ) (i \pi +\log (25))^2}{36+24 x (i \pi +\log (25))+4 x^2 (i \pi +\log (25))^2} \, dx=\frac {1}{4} \, x + \frac {i \, \pi + 2 \, \log \left (5\right )}{4 \, {\left ({\left (i \, \pi + 2 \, \log \left (5\right )\right )} x + 3\right )}} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (23) = 46\).
Time = 0.26 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.70 \[ \int \frac {9+6 x (i \pi +\log (25))+\left (-1+x^2\right ) (i \pi +\log (25))^2}{36+24 x (i \pi +\log (25))+4 x^2 (i \pi +\log (25))^2} \, dx=\frac {\pi ^{2} x - 4 i \, \pi x \log \left (5\right ) - 4 \, x \log \left (5\right )^{2}}{4 \, {\left (\pi ^{2} - 4 i \, \pi \log \left (5\right ) - 4 \, \log \left (5\right )^{2}\right )}} + \frac {-i \, \pi - 2 \, \log \left (5\right )}{4 \, {\left (-i \, \pi x - 2 \, x \log \left (5\right ) - 3\right )}} \]
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Time = 0.69 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.87 \[ \int \frac {9+6 x (i \pi +\log (25))+\left (-1+x^2\right ) (i \pi +\log (25))^2}{36+24 x (i \pi +\log (25))+4 x^2 (i \pi +\log (25))^2} \, dx=\frac {-\Pi +x\,3{}\mathrm {i}+\ln \left (5\right )\,2{}\mathrm {i}-\Pi \,x^2+x^2\,\ln \left (5\right )\,2{}\mathrm {i}}{4\,\left (-\Pi \,x+x\,\ln \left (5\right )\,2{}\mathrm {i}+3{}\mathrm {i}\right )} \]
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