Integrand size = 13, antiderivative size = 31 \[ \int \frac {1}{a+b (c+d x)^2} \, dx=\frac {\arctan \left (\frac {\sqrt {b} (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} d} \]
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Time = 0.01 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {253, 211} \[ \int \frac {1}{a+b (c+d x)^2} \, dx=\frac {\arctan \left (\frac {\sqrt {b} (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} d} \]
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Rule 211
Rule 253
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,c+d x\right )}{d} \\ & = \frac {\tan ^{-1}\left (\frac {\sqrt {b} (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} d} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int \frac {1}{a+b (c+d x)^2} \, dx=\frac {\arctan \left (\frac {\sqrt {b} (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} d} \]
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Time = 1.11 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.10
method | result | size |
default | \(\frac {\arctan \left (\frac {2 b \,d^{2} x +2 b c d}{2 d \sqrt {a b}}\right )}{d \sqrt {a b}}\) | \(34\) |
risch | \(-\frac {\ln \left (b d x +b c +\sqrt {-a b}\right )}{2 \sqrt {-a b}\, d}+\frac {\ln \left (-b d x -b c +\sqrt {-a b}\right )}{2 \sqrt {-a b}\, d}\) | \(56\) |
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Time = 0.27 (sec) , antiderivative size = 109, normalized size of antiderivative = 3.52 \[ \int \frac {1}{a+b (c+d x)^2} \, dx=\left [-\frac {\sqrt {-a b} \log \left (\frac {b d^{2} x^{2} + 2 \, b c d x + b c^{2} - 2 \, \sqrt {-a b} {\left (d x + c\right )} - a}{b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a}\right )}{2 \, a b d}, \frac {\sqrt {a b} \arctan \left (\frac {\sqrt {a b} {\left (d x + c\right )}}{a}\right )}{a b d}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (27) = 54\).
Time = 0.09 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.97 \[ \int \frac {1}{a+b (c+d x)^2} \, dx=\frac {- \frac {\sqrt {- \frac {1}{a b}} \log {\left (x + \frac {- a \sqrt {- \frac {1}{a b}} + c}{d} \right )}}{2} + \frac {\sqrt {- \frac {1}{a b}} \log {\left (x + \frac {a \sqrt {- \frac {1}{a b}} + c}{d} \right )}}{2}}{d} \]
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Time = 0.28 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.97 \[ \int \frac {1}{a+b (c+d x)^2} \, dx=\frac {\arctan \left (\frac {b d^{2} x + b c d}{\sqrt {a b} d}\right )}{\sqrt {a b} d} \]
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Time = 0.30 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.77 \[ \int \frac {1}{a+b (c+d x)^2} \, dx=\frac {\arctan \left (\frac {b d x + b c}{\sqrt {a b}}\right )}{\sqrt {a b} d} \]
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Time = 0.04 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.87 \[ \int \frac {1}{a+b (c+d x)^2} \, dx=\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,c+\sqrt {b}\,d\,x}{\sqrt {a}}\right )}{\sqrt {a}\,\sqrt {b}\,d} \]
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