Integrand size = 20, antiderivative size = 36 \[ \int \frac {a+b x}{\left (a^2-b^2 x^2\right )^2} \, dx=\frac {1}{2 a b (a-b x)}+\frac {\text {arctanh}\left (\frac {b x}{a}\right )}{2 a^2 b} \]
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Time = 0.02 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {641, 46, 214} \[ \int \frac {a+b x}{\left (a^2-b^2 x^2\right )^2} \, dx=\frac {\text {arctanh}\left (\frac {b x}{a}\right )}{2 a^2 b}+\frac {1}{2 a b (a-b x)} \]
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Rule 46
Rule 214
Rule 641
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{(a-b x)^2 (a+b x)} \, dx \\ & = \int \left (\frac {1}{2 a (a-b x)^2}+\frac {1}{2 a \left (a^2-b^2 x^2\right )}\right ) \, dx \\ & = \frac {1}{2 a b (a-b x)}+\frac {\int \frac {1}{a^2-b^2 x^2} \, dx}{2 a} \\ & = \frac {1}{2 a b (a-b x)}+\frac {\tanh ^{-1}\left (\frac {b x}{a}\right )}{2 a^2 b} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.39 \[ \int \frac {a+b x}{\left (a^2-b^2 x^2\right )^2} \, dx=\frac {2 a+(-a+b x) \log (a-b x)+(a-b x) \log (a+b x)}{4 a^2 b (a-b x)} \]
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Time = 2.19 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.31
method | result | size |
default | \(\frac {\ln \left (b x +a \right )}{4 a^{2} b}-\frac {\ln \left (-b x +a \right )}{4 a^{2} b}+\frac {1}{2 a b \left (-b x +a \right )}\) | \(47\) |
risch | \(\frac {\ln \left (b x +a \right )}{4 a^{2} b}-\frac {\ln \left (-b x +a \right )}{4 a^{2} b}+\frac {1}{2 a b \left (-b x +a \right )}\) | \(47\) |
norman | \(\frac {\frac {x}{2 a}+\frac {1}{2 b}}{-b^{2} x^{2}+a^{2}}-\frac {\ln \left (-b x +a \right )}{4 a^{2} b}+\frac {\ln \left (b x +a \right )}{4 a^{2} b}\) | \(58\) |
parallelrisch | \(-\frac {\ln \left (b x -a \right ) x b -b \ln \left (b x +a \right ) x -a \ln \left (b x -a \right )+a \ln \left (b x +a \right )+2 a}{4 a^{2} b \left (b x -a \right )}\) | \(62\) |
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Time = 0.32 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.50 \[ \int \frac {a+b x}{\left (a^2-b^2 x^2\right )^2} \, dx=\frac {{\left (b x - a\right )} \log \left (b x + a\right ) - {\left (b x - a\right )} \log \left (b x - a\right ) - 2 \, a}{4 \, {\left (a^{2} b^{2} x - a^{3} b\right )}} \]
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Time = 0.12 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.03 \[ \int \frac {a+b x}{\left (a^2-b^2 x^2\right )^2} \, dx=- \frac {1}{- 2 a^{2} b + 2 a b^{2} x} + \frac {- \frac {\log {\left (- \frac {a}{b} + x \right )}}{4} + \frac {\log {\left (\frac {a}{b} + x \right )}}{4}}{a^{2} b} \]
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Time = 0.20 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.33 \[ \int \frac {a+b x}{\left (a^2-b^2 x^2\right )^2} \, dx=-\frac {1}{2 \, {\left (a b^{2} x - a^{2} b\right )}} + \frac {\log \left (b x + a\right )}{4 \, a^{2} b} - \frac {\log \left (b x - a\right )}{4 \, a^{2} b} \]
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Time = 0.30 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.39 \[ \int \frac {a+b x}{\left (a^2-b^2 x^2\right )^2} \, dx=\frac {\log \left ({\left | b x + a \right |}\right )}{4 \, a^{2} b} - \frac {\log \left ({\left | b x - a \right |}\right )}{4 \, a^{2} b} - \frac {1}{2 \, {\left (b x - a\right )} a b} \]
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Time = 9.69 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.89 \[ \int \frac {a+b x}{\left (a^2-b^2 x^2\right )^2} \, dx=\frac {1}{2\,a\,b\,\left (a-b\,x\right )}+\frac {\mathrm {atanh}\left (\frac {b\,x}{a}\right )}{2\,a^2\,b} \]
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