Integrand size = 9, antiderivative size = 10 \[ \int \frac {1}{c^2+x^2} \, dx=\frac {\arctan \left (\frac {x}{c}\right )}{c} \]
[Out]
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} \[ \int \frac {1}{c^2+x^2} \, dx=\frac {\arctan \left (\frac {x}{c}\right )}{c} \]
[In]
[Out]
Rule 209
Rubi steps \begin{align*} \text {integral}& = \frac {\arctan \left (\frac {x}{c}\right )}{c} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {1}{c^2+x^2} \, dx=\frac {\arctan \left (\frac {x}{c}\right )}{c} \]
[In]
[Out]
Time = 0.11 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.10
method | result | size |
default | \(\frac {\arctan \left (\frac {x}{c}\right )}{c}\) | \(11\) |
risch | \(\frac {\arctan \left (\frac {x}{c}\right )}{c}\) | \(11\) |
parallelrisch | \(-\frac {i \ln \left (-i c +x \right )-i \ln \left (i c +x \right )}{2 c}\) | \(27\) |
[In]
[Out]
none
Time = 0.24 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {1}{c^2+x^2} \, dx=\frac {\arctan \left (\frac {x}{c}\right )}{c} \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 0.05 (sec) , antiderivative size = 20, normalized size of antiderivative = 2.00 \[ \int \frac {1}{c^2+x^2} \, dx=\frac {- \frac {i \log {\left (- i c + x \right )}}{2} + \frac {i \log {\left (i c + x \right )}}{2}}{c} \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {1}{c^2+x^2} \, dx=\frac {\arctan \left (\frac {x}{c}\right )}{c} \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {1}{c^2+x^2} \, dx=\frac {\arctan \left (\frac {x}{c}\right )}{c} \]
[In]
[Out]
Time = 0.04 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {1}{c^2+x^2} \, dx=\frac {\mathrm {atan}\left (\frac {x}{c}\right )}{c} \]
[In]
[Out]