Integrand size = 19, antiderivative size = 41 \[ \int \frac {1}{(a+b x)^2 (a c-b c x)} \, dx=-\frac {1}{2 a b c (a+b x)}+\frac {\text {arctanh}\left (\frac {b x}{a}\right )}{2 a^2 b c} \]
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Time = 0.02 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {46, 214} \[ \int \frac {1}{(a+b x)^2 (a c-b c x)} \, dx=\frac {\text {arctanh}\left (\frac {b x}{a}\right )}{2 a^2 b c}-\frac {1}{2 a b c (a+b x)} \]
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Rule 46
Rule 214
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{2 a c (a+b x)^2}+\frac {1}{2 a c \left (a^2-b^2 x^2\right )}\right ) \, dx \\ & = -\frac {1}{2 a b c (a+b x)}+\frac {\int \frac {1}{a^2-b^2 x^2} \, dx}{2 a c} \\ & = -\frac {1}{2 a b c (a+b x)}+\frac {\tanh ^{-1}\left (\frac {b x}{a}\right )}{2 a^2 b c} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.22 \[ \int \frac {1}{(a+b x)^2 (a c-b c x)} \, dx=\frac {-2 a-(a+b x) \log (a-b x)+(a+b x) \log (a+b x)}{4 a^2 b c (a+b x)} \]
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Time = 0.19 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.22
method | result | size |
default | \(\frac {\frac {\ln \left (b x +a \right )}{4 a^{2} b}-\frac {1}{2 b a \left (b x +a \right )}-\frac {\ln \left (-b x +a \right )}{4 a^{2} b}}{c}\) | \(50\) |
norman | \(-\frac {1}{2 a b c \left (b x +a \right )}-\frac {\ln \left (-b x +a \right )}{4 a^{2} b c}+\frac {\ln \left (b x +a \right )}{4 a^{2} b c}\) | \(55\) |
risch | \(-\frac {1}{2 a b c \left (b x +a \right )}-\frac {\ln \left (-b x +a \right )}{4 a^{2} b c}+\frac {\ln \left (b x +a \right )}{4 a^{2} b c}\) | \(55\) |
parallelrisch | \(\frac {-\ln \left (b x -a \right ) x b +b \ln \left (b x +a \right ) x -a \ln \left (b x -a \right )+a \ln \left (b x +a \right )-2 a}{4 a^{2} b c \left (b x +a \right )}\) | \(63\) |
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Time = 0.23 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.24 \[ \int \frac {1}{(a+b x)^2 (a c-b c x)} \, dx=\frac {{\left (b x + a\right )} \log \left (b x + a\right ) - {\left (b x + a\right )} \log \left (b x - a\right ) - 2 \, a}{4 \, {\left (a^{2} b^{2} c x + a^{3} b c\right )}} \]
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Time = 0.29 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.07 \[ \int \frac {1}{(a+b x)^2 (a c-b c x)} \, dx=- \frac {1}{2 a^{2} b c + 2 a b^{2} c x} - \frac {\frac {\log {\left (- \frac {a}{b} + x \right )}}{4} - \frac {\log {\left (\frac {a}{b} + x \right )}}{4}}{a^{2} b c} \]
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Time = 0.20 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.34 \[ \int \frac {1}{(a+b x)^2 (a c-b c x)} \, dx=-\frac {1}{2 \, {\left (a b^{2} c x + a^{2} b c\right )}} + \frac {\log \left (b x + a\right )}{4 \, a^{2} b c} - \frac {\log \left (b x - a\right )}{4 \, a^{2} b c} \]
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Time = 0.30 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.07 \[ \int \frac {1}{(a+b x)^2 (a c-b c x)} \, dx=-\frac {\log \left ({\left | -\frac {2 \, a}{b x + a} + 1 \right |}\right )}{4 \, a^{2} b c} - \frac {1}{2 \, {\left (b x + a\right )} a b c} \]
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Time = 0.21 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.90 \[ \int \frac {1}{(a+b x)^2 (a c-b c x)} \, dx=\frac {\mathrm {atanh}\left (\frac {b\,x}{a}\right )}{2\,a^2\,b\,c}-\frac {1}{2\,a\,b\,\left (a\,c+b\,c\,x\right )} \]
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