Integrand size = 20, antiderivative size = 25 \[ \int \frac {a c+b c x^2}{\left (a+b x^2\right )^2} \, dx=\frac {c \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b}} \]
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Time = 0.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {21, 211} \[ \int \frac {a c+b c x^2}{\left (a+b x^2\right )^2} \, dx=\frac {c \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b}} \]
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Rule 21
Rule 211
Rubi steps \begin{align*} \text {integral}& = c \int \frac {1}{a+b x^2} \, dx \\ & = \frac {c \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b}} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {a c+b c x^2}{\left (a+b x^2\right )^2} \, dx=\frac {c \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b}} \]
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Time = 2.53 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.68
method | result | size |
default | \(\frac {c \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}}\) | \(17\) |
risch | \(-\frac {c \ln \left (b x +\sqrt {-a b}\right )}{2 \sqrt {-a b}}+\frac {c \ln \left (-b x +\sqrt {-a b}\right )}{2 \sqrt {-a b}}\) | \(43\) |
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Time = 0.28 (sec) , antiderivative size = 69, normalized size of antiderivative = 2.76 \[ \int \frac {a c+b c x^2}{\left (a+b x^2\right )^2} \, dx=\left [-\frac {\sqrt {-a b} c \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right )}{2 \, a b}, \frac {\sqrt {a b} c \arctan \left (\frac {\sqrt {a b} x}{a}\right )}{a b}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (24) = 48\).
Time = 0.06 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.16 \[ \int \frac {a c+b c x^2}{\left (a+b x^2\right )^2} \, dx=c \left (- \frac {\sqrt {- \frac {1}{a b}} \log {\left (- a \sqrt {- \frac {1}{a b}} + x \right )}}{2} + \frac {\sqrt {- \frac {1}{a b}} \log {\left (a \sqrt {- \frac {1}{a b}} + x \right )}}{2}\right ) \]
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Time = 0.27 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.64 \[ \int \frac {a c+b c x^2}{\left (a+b x^2\right )^2} \, dx=\frac {c \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}} \]
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Time = 0.27 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.64 \[ \int \frac {a c+b c x^2}{\left (a+b x^2\right )^2} \, dx=\frac {c \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}} \]
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Time = 5.15 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.68 \[ \int \frac {a c+b c x^2}{\left (a+b x^2\right )^2} \, dx=\frac {c\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{\sqrt {a}\,\sqrt {b}} \]
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