Integrand size = 20, antiderivative size = 21 \[ \int e^{-2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right ) \, dx=-\frac {c}{a^2 x}+c x-\frac {2 c \log (x)}{a} \]
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Time = 0.07 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6302, 6292, 6285, 45} \[ \int e^{-2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right ) \, dx=-\frac {c}{a^2 x}-\frac {2 c \log (x)}{a}+c x \]
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Rule 45
Rule 6285
Rule 6292
Rule 6302
Rubi steps \begin{align*} \text {integral}& = -\int e^{-2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right ) \, dx \\ & = \frac {c \int \frac {e^{-2 \text {arctanh}(a x)} \left (1-a^2 x^2\right )}{x^2} \, dx}{a^2} \\ & = \frac {c \int \frac {(1-a x)^2}{x^2} \, dx}{a^2} \\ & = \frac {c \int \left (a^2+\frac {1}{x^2}-\frac {2 a}{x}\right ) \, dx}{a^2} \\ & = -\frac {c}{a^2 x}+c x-\frac {2 c \log (x)}{a} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int e^{-2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right ) \, dx=-\frac {c}{a^2 x}+c x-\frac {2 c \log (x)}{a} \]
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Time = 0.62 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.05
method | result | size |
default | \(\frac {c \left (a^{2} x -2 a \ln \left (x \right )-\frac {1}{x}\right )}{a^{2}}\) | \(22\) |
risch | \(-\frac {c}{a^{2} x}+c x -\frac {2 c \ln \left (x \right )}{a}\) | \(22\) |
parallelrisch | \(-\frac {-a^{2} c \,x^{2}+2 c \ln \left (x \right ) a x +c}{a^{2} x}\) | \(27\) |
norman | \(\frac {a c \,x^{2}-\frac {c}{a}}{a x}-\frac {2 c \ln \left (x \right )}{a}\) | \(30\) |
meijerg | \(\frac {c \left (a x -\ln \left (a x +1\right )\right )}{a}-\frac {c \ln \left (a x +1\right )}{a}-\frac {c \left (-\ln \left (a x +1\right )+\ln \left (x \right )+\ln \left (a \right )\right )}{a}+\frac {c \left (\ln \left (a x +1\right )-\ln \left (x \right )-\ln \left (a \right )-\frac {1}{a x}\right )}{a}\) | \(78\) |
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Time = 0.23 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.24 \[ \int e^{-2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right ) \, dx=\frac {a^{2} c x^{2} - 2 \, a c x \log \left (x\right ) - c}{a^{2} x} \]
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Time = 0.06 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95 \[ \int e^{-2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right ) \, dx=\frac {a^{2} c x - 2 a c \log {\left (x \right )} - \frac {c}{x}}{a^{2}} \]
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Time = 0.20 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int e^{-2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right ) \, dx=c x - \frac {2 \, c \log \left (x\right )}{a} - \frac {c}{a^{2} x} \]
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Time = 0.27 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.05 \[ \int e^{-2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right ) \, dx=c x - \frac {2 \, c \log \left ({\left | x \right |}\right )}{a} - \frac {c}{a^{2} x} \]
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Time = 0.06 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.19 \[ \int e^{-2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right ) \, dx=-\frac {c\,\left (2\,a\,x\,\ln \left (x\right )-a^2\,x^2+1\right )}{a^2\,x} \]
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