Integrand size = 11, antiderivative size = 21 \[ \int \frac {1}{c+(a+b x)^2} \, dx=\frac {\arctan \left (\frac {a+b x}{\sqrt {c}}\right )}{b \sqrt {c}} \]
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Time = 0.01 (sec) , antiderivative size = 21, 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} \[ \int \frac {1}{c+(a+b x)^2} \, dx=\frac {\arctan \left (\frac {a+b x}{\sqrt {c}}\right )}{b \sqrt {c}} \]
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Rule 209
Rule 253
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{c+x^2} \, dx,x,a+b x\right )}{b} \\ & = \frac {\tan ^{-1}\left (\frac {a+b x}{\sqrt {c}}\right )}{b \sqrt {c}} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {1}{c+(a+b x)^2} \, dx=\frac {\arctan \left (\frac {a+b x}{\sqrt {c}}\right )}{b \sqrt {c}} \]
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Time = 0.83 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.33
method | result | size |
default | \(\frac {\arctan \left (\frac {2 b^{2} x +2 a b}{2 \sqrt {c}\, b}\right )}{\sqrt {c}\, b}\) | \(28\) |
risch | \(-\frac {\ln \left (b x +\sqrt {-c}+a \right )}{2 \sqrt {-c}\, b}+\frac {\ln \left (-b x +\sqrt {-c}-a \right )}{2 \sqrt {-c}\, b}\) | \(47\) |
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none
Time = 0.27 (sec) , antiderivative size = 83, normalized size of antiderivative = 3.95 \[ \int \frac {1}{c+(a+b x)^2} \, dx=\left [-\frac {\sqrt {-c} \log \left (\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - 2 \, {\left (b x + a\right )} \sqrt {-c} - c}{b^{2} x^{2} + 2 \, a b x + a^{2} + c}\right )}{2 \, b c}, \frac {\arctan \left (\frac {b x + a}{\sqrt {c}}\right )}{b \sqrt {c}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (17) = 34\).
Time = 0.09 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.57 \[ \int \frac {1}{c+(a+b x)^2} \, dx=\frac {- \frac {\sqrt {- \frac {1}{c}} \log {\left (x + \frac {a - c \sqrt {- \frac {1}{c}}}{b} \right )}}{2} + \frac {\sqrt {- \frac {1}{c}} \log {\left (x + \frac {a + c \sqrt {- \frac {1}{c}}}{b} \right )}}{2}}{b} \]
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Time = 0.28 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.14 \[ \int \frac {1}{c+(a+b x)^2} \, dx=\frac {\arctan \left (\frac {b^{2} x + a b}{b \sqrt {c}}\right )}{b \sqrt {c}} \]
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Time = 0.29 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int \frac {1}{c+(a+b x)^2} \, dx=\frac {\arctan \left (\frac {b x + a}{\sqrt {c}}\right )}{b \sqrt {c}} \]
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Time = 0.03 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int \frac {1}{c+(a+b x)^2} \, dx=\frac {\mathrm {atan}\left (\frac {a+b\,x}{\sqrt {c}}\right )}{b\,\sqrt {c}} \]
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