Integrand size = 20, antiderivative size = 12 \[ \int \frac {a+b x}{a^2-b^2 x^2} \, dx=-\frac {\log (a-b x)}{b} \]
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Time = 0.00 (sec) , antiderivative size = 12, 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 = {641, 31} \[ \int \frac {a+b x}{a^2-b^2 x^2} \, dx=-\frac {\log (a-b x)}{b} \]
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Rule 31
Rule 641
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{a-b x} \, dx \\ & = -\frac {\log (a-b x)}{b} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \frac {a+b x}{a^2-b^2 x^2} \, dx=-\frac {\log (a-b x)}{b} \]
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Time = 2.63 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.08
method | result | size |
default | \(-\frac {\ln \left (-b x +a \right )}{b}\) | \(13\) |
norman | \(-\frac {\ln \left (-b x +a \right )}{b}\) | \(13\) |
risch | \(-\frac {\ln \left (-b x +a \right )}{b}\) | \(13\) |
parallelrisch | \(-\frac {\ln \left (b x -a \right )}{b}\) | \(14\) |
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none
Time = 0.27 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.08 \[ \int \frac {a+b x}{a^2-b^2 x^2} \, dx=-\frac {\log \left (b x - a\right )}{b} \]
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Time = 0.02 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.67 \[ \int \frac {a+b x}{a^2-b^2 x^2} \, dx=- \frac {\log {\left (- a + b x \right )}}{b} \]
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none
Time = 0.19 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.08 \[ \int \frac {a+b x}{a^2-b^2 x^2} \, dx=-\frac {\log \left (b x - a\right )}{b} \]
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none
Time = 0.29 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {a+b x}{a^2-b^2 x^2} \, dx=-\frac {\log \left ({\left | b x - a \right |}\right )}{b} \]
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Time = 0.03 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.08 \[ \int \frac {a+b x}{a^2-b^2 x^2} \, dx=-\frac {\ln \left (b\,x-a\right )}{b} \]
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