Integrand size = 22, antiderivative size = 49 \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx=\frac {1}{4 a c^2 (1+a x)^2}+\frac {1}{4 a c^2 (1+a x)}-\frac {\text {arctanh}(a x)}{4 a c^2} \]
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Time = 0.07 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {6302, 6275, 46, 213} \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx=-\frac {\text {arctanh}(a x)}{4 a c^2}+\frac {1}{4 a c^2 (a x+1)}+\frac {1}{4 a c^2 (a x+1)^2} \]
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Rule 46
Rule 213
Rule 6275
Rule 6302
Rubi steps \begin{align*} \text {integral}& = -\int \frac {e^{-2 \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx \\ & = -\frac {\int \frac {1}{(1-a x) (1+a x)^3} \, dx}{c^2} \\ & = -\frac {\int \left (\frac {1}{2 (1+a x)^3}+\frac {1}{4 (1+a x)^2}-\frac {1}{4 \left (-1+a^2 x^2\right )}\right ) \, dx}{c^2} \\ & = \frac {1}{4 a c^2 (1+a x)^2}+\frac {1}{4 a c^2 (1+a x)}+\frac {\int \frac {1}{-1+a^2 x^2} \, dx}{4 c^2} \\ & = \frac {1}{4 a c^2 (1+a x)^2}+\frac {1}{4 a c^2 (1+a x)}-\frac {\text {arctanh}(a x)}{4 a c^2} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.67 \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx=\frac {2+a x-(1+a x)^2 \text {arctanh}(a x)}{4 a (c+a c x)^2} \]
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Time = 0.59 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.04
method | result | size |
risch | \(\frac {\frac {x}{4}+\frac {1}{2 a}}{\left (a x +1\right )^{2} c^{2}}+\frac {\ln \left (-a x +1\right )}{8 a \,c^{2}}-\frac {\ln \left (a x +1\right )}{8 a \,c^{2}}\) | \(51\) |
default | \(\frac {\frac {1}{4 a \left (a x +1\right )^{2}}+\frac {1}{4 a \left (a x +1\right )}-\frac {\ln \left (a x +1\right )}{8 a}+\frac {\ln \left (a x -1\right )}{8 a}}{c^{2}}\) | \(52\) |
norman | \(\frac {-\frac {3}{4 a c}+\frac {a \,x^{2}}{2 c}+\frac {a^{2} x^{3}}{4 c}}{c \left (a x +1\right )^{2} \left (a x -1\right )}+\frac {\ln \left (a x -1\right )}{8 a \,c^{2}}-\frac {\ln \left (a x +1\right )}{8 a \,c^{2}}\) | \(77\) |
parallelrisch | \(\frac {a^{2} \ln \left (a x -1\right ) x^{2}-a^{2} \ln \left (a x +1\right ) x^{2}-4 a^{2} x^{2}+2 a \ln \left (a x -1\right ) x -2 a \ln \left (a x +1\right ) x -6 a x +\ln \left (a x -1\right )-\ln \left (a x +1\right )}{8 c^{2} \left (a x +1\right )^{2} a}\) | \(90\) |
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Time = 0.24 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.55 \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx=\frac {2 \, a x - {\left (a^{2} x^{2} + 2 \, a x + 1\right )} \log \left (a x + 1\right ) + {\left (a^{2} x^{2} + 2 \, a x + 1\right )} \log \left (a x - 1\right ) + 4}{8 \, {\left (a^{3} c^{2} x^{2} + 2 \, a^{2} c^{2} x + a c^{2}\right )}} \]
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Time = 0.16 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.10 \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx=\frac {a x + 2}{4 a^{3} c^{2} x^{2} + 8 a^{2} c^{2} x + 4 a c^{2}} + \frac {\frac {\log {\left (x - \frac {1}{a} \right )}}{8} - \frac {\log {\left (x + \frac {1}{a} \right )}}{8}}{a c^{2}} \]
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Time = 0.20 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.29 \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx=\frac {a x + 2}{4 \, {\left (a^{3} c^{2} x^{2} + 2 \, a^{2} c^{2} x + a c^{2}\right )}} - \frac {\log \left (a x + 1\right )}{8 \, a c^{2}} + \frac {\log \left (a x - 1\right )}{8 \, a c^{2}} \]
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Time = 0.28 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.04 \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx=-\frac {\log \left ({\left | a x + 1 \right |}\right )}{8 \, a c^{2}} + \frac {\log \left ({\left | a x - 1 \right |}\right )}{8 \, a c^{2}} + \frac {a x + 2}{4 \, {\left (a x + 1\right )}^{2} a c^{2}} \]
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Time = 0.07 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.94 \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx=\frac {\frac {x}{4}+\frac {1}{2\,a}}{a^2\,c^2\,x^2+2\,a\,c^2\,x+c^2}-\frac {\mathrm {atanh}\left (a\,x\right )}{4\,a\,c^2} \]
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