Integrand size = 9, antiderivative size = 10 \[ \int \frac {1}{a^2+x^2} \, dx=\frac {\arctan \left (\frac {x}{a}\right )}{a} \]
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Time = 0.00 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {209} \[ \int \frac {1}{a^2+x^2} \, dx=\frac {\arctan \left (\frac {x}{a}\right )}{a} \]
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Rule 209
Rubi steps \begin{align*} \text {integral}& = \frac {\tan ^{-1}\left (\frac {x}{a}\right )}{a} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {1}{a^2+x^2} \, dx=\frac {\arctan \left (\frac {x}{a}\right )}{a} \]
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Time = 0.84 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.10
method | result | size |
default | \(\frac {\arctan \left (\frac {x}{a}\right )}{a}\) | \(11\) |
risch | \(\frac {\arctan \left (\frac {x}{a}\right )}{a}\) | \(11\) |
parallelrisch | \(-\frac {i \ln \left (-i a +x \right )-i \ln \left (i a +x \right )}{2 a}\) | \(27\) |
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none
Time = 0.24 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {1}{a^2+x^2} \, dx=\frac {\arctan \left (\frac {x}{a}\right )}{a} \]
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Result contains complex when optimal does not.
Time = 0.05 (sec) , antiderivative size = 20, normalized size of antiderivative = 2.00 \[ \int \frac {1}{a^2+x^2} \, dx=\frac {- \frac {i \log {\left (- i a + x \right )}}{2} + \frac {i \log {\left (i a + x \right )}}{2}}{a} \]
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none
Time = 0.28 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {1}{a^2+x^2} \, dx=\frac {\arctan \left (\frac {x}{a}\right )}{a} \]
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none
Time = 0.29 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {1}{a^2+x^2} \, dx=\frac {\arctan \left (\frac {x}{a}\right )}{a} \]
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Time = 0.02 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {1}{a^2+x^2} \, dx=\frac {\mathrm {atan}\left (\frac {x}{a}\right )}{a} \]
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