Integrand size = 14, antiderivative size = 14 \[ \int \frac {1}{a^2-b^2 x^2} \, dx=\frac {\text {arctanh}\left (\frac {b x}{a}\right )}{a b} \]
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Time = 0.01 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {214} \[ \int \frac {1}{a^2-b^2 x^2} \, dx=\frac {\text {arctanh}\left (\frac {b x}{a}\right )}{a b} \]
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Rule 214
Rubi steps \begin{align*} \text {integral}& = \frac {\text {arctanh}\left (\frac {b x}{a}\right )}{a b} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {1}{a^2-b^2 x^2} \, dx=\frac {\text {arctanh}\left (\frac {b x}{a}\right )}{a b} \]
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Time = 0.19 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.86
method | result | size |
parallelrisch | \(-\frac {\ln \left (b x -a \right )-\ln \left (b x +a \right )}{2 a b}\) | \(26\) |
default | \(-\frac {\ln \left (-b x +a \right )}{2 b a}+\frac {\ln \left (b x +a \right )}{2 a b}\) | \(31\) |
norman | \(-\frac {\ln \left (-b x +a \right )}{2 b a}+\frac {\ln \left (b x +a \right )}{2 a b}\) | \(31\) |
risch | \(-\frac {\ln \left (-b x +a \right )}{2 b a}+\frac {\ln \left (b x +a \right )}{2 a b}\) | \(31\) |
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none
Time = 0.25 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.79 \[ \int \frac {1}{a^2-b^2 x^2} \, dx=\frac {\log \left (b x + a\right ) - \log \left (b x - a\right )}{2 \, a b} \]
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Leaf count of result is larger than twice the leaf count of optimal. 20 vs. \(2 (8) = 16\).
Time = 0.06 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.43 \[ \int \frac {1}{a^2-b^2 x^2} \, dx=- \frac {\frac {\log {\left (- \frac {a}{b} + x \right )}}{2} - \frac {\log {\left (\frac {a}{b} + x \right )}}{2}}{a b} \]
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Leaf count of result is larger than twice the leaf count of optimal. 31 vs. \(2 (14) = 28\).
Time = 0.20 (sec) , antiderivative size = 31, normalized size of antiderivative = 2.21 \[ \int \frac {1}{a^2-b^2 x^2} \, dx=\frac {\log \left (b x + a\right )}{2 \, a b} - \frac {\log \left (b x - a\right )}{2 \, a b} \]
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Leaf count of result is larger than twice the leaf count of optimal. 33 vs. \(2 (14) = 28\).
Time = 0.28 (sec) , antiderivative size = 33, normalized size of antiderivative = 2.36 \[ \int \frac {1}{a^2-b^2 x^2} \, dx=\frac {\log \left ({\left | b x + a \right |}\right )}{2 \, a b} - \frac {\log \left ({\left | b x - a \right |}\right )}{2 \, a b} \]
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Time = 0.06 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {1}{a^2-b^2 x^2} \, dx=\frac {\mathrm {atanh}\left (\frac {b\,x}{a}\right )}{a\,b} \]
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