Integrand size = 11, antiderivative size = 10 \[ \int \frac {1}{1+(c+d x)^2} \, dx=\frac {\arctan (c+d x)}{d} \]
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Time = 0.00 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {253, 209} \[ \int \frac {1}{1+(c+d x)^2} \, dx=\frac {\arctan (c+d x)}{d} \]
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Rule 209
Rule 253
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,c+d x\right )}{d} \\ & = \frac {\tan ^{-1}(c+d x)}{d} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {1}{1+(c+d x)^2} \, dx=\frac {\arctan (c+d x)}{d} \]
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Time = 0.98 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.10
method | result | size |
default | \(\frac {\arctan \left (d x +c \right )}{d}\) | \(11\) |
risch | \(\frac {\arctan \left (d x +c \right )}{d}\) | \(11\) |
parallelrisch | \(-\frac {i \ln \left (d x +c -i\right )-i \ln \left (d x +c +i\right )}{2 d}\) | \(29\) |
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none
Time = 0.25 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {1}{1+(c+d x)^2} \, dx=\frac {\arctan \left (d x + c\right )}{d} \]
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Result contains complex when optimal does not.
Time = 0.08 (sec) , antiderivative size = 24, normalized size of antiderivative = 2.40 \[ \int \frac {1}{1+(c+d x)^2} \, dx=\frac {- \frac {i \log {\left (x + \frac {c - i}{d} \right )}}{2} + \frac {i \log {\left (x + \frac {c + i}{d} \right )}}{2}}{d} \]
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none
Time = 0.27 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.80 \[ \int \frac {1}{1+(c+d x)^2} \, dx=\frac {\arctan \left (\frac {d^{2} x + c d}{d}\right )}{d} \]
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none
Time = 0.29 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {1}{1+(c+d x)^2} \, dx=\frac {\arctan \left (d x + c\right )}{d} \]
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Time = 0.03 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {1}{1+(c+d x)^2} \, dx=\frac {\mathrm {atan}\left (c+d\,x\right )}{d} \]
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