Integrand size = 18, antiderivative size = 17 \[ \int \frac {2+2 x}{(-1+x)^3 \left (1+x^2\right )} \, dx=-\frac {1}{(1-x)^2}+\frac {1}{-1+x}+\arctan (x) \]
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Time = 0.01 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {815, 209} \[ \int \frac {2+2 x}{(-1+x)^3 \left (1+x^2\right )} \, dx=\arctan (x)+\frac {1}{x-1}-\frac {1}{(1-x)^2} \]
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Rule 209
Rule 815
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {2}{(-1+x)^3}-\frac {1}{(-1+x)^2}+\frac {1}{1+x^2}\right ) \, dx \\ & = -\frac {1}{(1-x)^2}+\frac {1}{-1+x}+\int \frac {1}{1+x^2} \, dx \\ & = -\frac {1}{(1-x)^2}+\frac {1}{-1+x}+\tan ^{-1}(x) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \frac {2+2 x}{(-1+x)^3 \left (1+x^2\right )} \, dx=\frac {-2+x+(-1+x)^2 \arctan (x)}{(-1+x)^2} \]
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Time = 0.86 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76
method | result | size |
risch | \(\frac {x -2}{\left (x -1\right )^{2}}+\arctan \left (x \right )\) | \(13\) |
default | \(\arctan \left (x \right )-\frac {1}{\left (x -1\right )^{2}}+\frac {1}{x -1}\) | \(16\) |
parallelrisch | \(-\frac {i \ln \left (x -i\right ) x^{2}-i \ln \left (x +i\right ) x^{2}-2 i \ln \left (x -i\right ) x +2 i \ln \left (x +i\right ) x +3+i \ln \left (x -i\right )-i \ln \left (x +i\right )-x^{2}}{2 \left (x -1\right )^{2}}\) | \(71\) |
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Time = 0.25 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.47 \[ \int \frac {2+2 x}{(-1+x)^3 \left (1+x^2\right )} \, dx=\frac {{\left (x^{2} - 2 \, x + 1\right )} \arctan \left (x\right ) + x - 2}{x^{2} - 2 \, x + 1} \]
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Time = 0.06 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82 \[ \int \frac {2+2 x}{(-1+x)^3 \left (1+x^2\right )} \, dx=\frac {x - 2}{x^{2} - 2 x + 1} + \operatorname {atan}{\left (x \right )} \]
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Time = 0.28 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \frac {2+2 x}{(-1+x)^3 \left (1+x^2\right )} \, dx=\frac {x - 2}{x^{2} - 2 \, x + 1} + \arctan \left (x\right ) \]
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Time = 0.28 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.71 \[ \int \frac {2+2 x}{(-1+x)^3 \left (1+x^2\right )} \, dx=\frac {x - 2}{{\left (x - 1\right )}^{2}} + \arctan \left (x\right ) \]
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Time = 0.02 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \frac {2+2 x}{(-1+x)^3 \left (1+x^2\right )} \, dx=\mathrm {atan}\left (x\right )+\frac {x-2}{x^2-2\,x+1} \]
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