Integrand size = 22, antiderivative size = 84 \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^3} \, dx=-\frac {1}{16 a c^3 (1-a x)}+\frac {1}{12 a c^3 (1+a x)^3}+\frac {1}{8 a c^3 (1+a x)^2}+\frac {3}{16 a c^3 (1+a x)}-\frac {\text {arctanh}(a x)}{4 a c^3} \]
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Time = 0.09 (sec) , antiderivative size = 84, 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} \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^3} \, dx=-\frac {\text {arctanh}(a x)}{4 a c^3}-\frac {1}{16 a c^3 (1-a x)}+\frac {3}{16 a c^3 (a x+1)}+\frac {1}{8 a c^3 (a x+1)^2}+\frac {1}{12 a c^3 (a x+1)^3} \]
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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 )^3} \, dx \\ & = -\frac {\int \frac {1}{(1-a x)^2 (1+a x)^4} \, dx}{c^3} \\ & = -\frac {\int \left (\frac {1}{16 (-1+a x)^2}+\frac {1}{4 (1+a x)^4}+\frac {1}{4 (1+a x)^3}+\frac {3}{16 (1+a x)^2}-\frac {1}{4 \left (-1+a^2 x^2\right )}\right ) \, dx}{c^3} \\ & = -\frac {1}{16 a c^3 (1-a x)}+\frac {1}{12 a c^3 (1+a x)^3}+\frac {1}{8 a c^3 (1+a x)^2}+\frac {3}{16 a c^3 (1+a x)}+\frac {\int \frac {1}{-1+a^2 x^2} \, dx}{4 c^3} \\ & = -\frac {1}{16 a c^3 (1-a x)}+\frac {1}{12 a c^3 (1+a x)^3}+\frac {1}{8 a c^3 (1+a x)^2}+\frac {3}{16 a c^3 (1+a x)}-\frac {\text {arctanh}(a x)}{4 a c^3} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.73 \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^3} \, dx=\frac {-4+a x+6 a^2 x^2+3 a^3 x^3-3 (-1+a x) (1+a x)^3 \text {arctanh}(a x)}{12 a (-1+a x) (c+a c x)^3} \]
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Time = 0.61 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.90
method | result | size |
default | \(\frac {\frac {1}{12 a \left (a x +1\right )^{3}}+\frac {1}{8 a \left (a x +1\right )^{2}}+\frac {3}{16 a \left (a x +1\right )}-\frac {\ln \left (a x +1\right )}{8 a}+\frac {1}{16 a \left (a x -1\right )}+\frac {\ln \left (a x -1\right )}{8 a}}{c^{3}}\) | \(76\) |
risch | \(\frac {\frac {a^{2} x^{3}}{4}+\frac {a \,x^{2}}{2}+\frac {x}{12}-\frac {1}{3 a}}{\left (a x +1\right )^{2} \left (a^{2} x^{2}-1\right ) c^{3}}+\frac {\ln \left (-a x +1\right )}{8 a \,c^{3}}-\frac {\ln \left (a x +1\right )}{8 a \,c^{3}}\) | \(76\) |
norman | \(\frac {-\frac {3 x}{4 c}+\frac {a \,x^{2}}{4 c}+\frac {11 a^{2} x^{3}}{12 c}-\frac {a^{3} x^{4}}{12 c}-\frac {a^{4} x^{5}}{3 c}}{\left (a x +1\right )^{3} c^{2} \left (a x -1\right )^{2}}+\frac {\ln \left (a x -1\right )}{8 a \,c^{3}}-\frac {\ln \left (a x +1\right )}{8 a \,c^{3}}\) | \(97\) |
parallelrisch | \(\frac {3 \ln \left (a x -1\right ) x^{4} a^{4}-3 \ln \left (a x +1\right ) x^{4} a^{4}-8 a^{4} x^{4}+6 a^{3} \ln \left (a x -1\right ) x^{3}-6 a^{3} \ln \left (a x +1\right ) x^{3}-10 a^{3} x^{3}+12 a^{2} x^{2}-6 a \ln \left (a x -1\right ) x +6 a \ln \left (a x +1\right ) x +18 a x -3 \ln \left (a x -1\right )+3 \ln \left (a x +1\right )}{24 c^{3} \left (a x +1\right )^{2} \left (a^{2} x^{2}-1\right ) a}\) | \(148\) |
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Time = 0.27 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.44 \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^3} \, dx=\frac {6 \, a^{3} x^{3} + 12 \, a^{2} x^{2} + 2 \, a x - 3 \, {\left (a^{4} x^{4} + 2 \, a^{3} x^{3} - 2 \, a x - 1\right )} \log \left (a x + 1\right ) + 3 \, {\left (a^{4} x^{4} + 2 \, a^{3} x^{3} - 2 \, a x - 1\right )} \log \left (a x - 1\right ) - 8}{24 \, {\left (a^{5} c^{3} x^{4} + 2 \, a^{4} c^{3} x^{3} - 2 \, a^{2} c^{3} x - a c^{3}\right )}} \]
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Time = 0.24 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.01 \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^3} \, dx=- \frac {- 3 a^{3} x^{3} - 6 a^{2} x^{2} - a x + 4}{12 a^{5} c^{3} x^{4} + 24 a^{4} c^{3} x^{3} - 24 a^{2} c^{3} x - 12 a c^{3}} - \frac {- \frac {\log {\left (x - \frac {1}{a} \right )}}{8} + \frac {\log {\left (x + \frac {1}{a} \right )}}{8}}{a c^{3}} \]
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Time = 0.21 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.08 \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^3} \, dx=\frac {3 \, a^{3} x^{3} + 6 \, a^{2} x^{2} + a x - 4}{12 \, {\left (a^{5} c^{3} x^{4} + 2 \, a^{4} c^{3} x^{3} - 2 \, a^{2} c^{3} x - a c^{3}\right )}} - \frac {\log \left (a x + 1\right )}{8 \, a c^{3}} + \frac {\log \left (a x - 1\right )}{8 \, a c^{3}} \]
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Time = 0.27 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.88 \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^3} \, dx=-\frac {\log \left ({\left | a x + 1 \right |}\right )}{8 \, a c^{3}} + \frac {\log \left ({\left | a x - 1 \right |}\right )}{8 \, a c^{3}} + \frac {3 \, a^{3} x^{3} + 6 \, a^{2} x^{2} + a x - 4}{12 \, {\left (a x + 1\right )}^{3} {\left (a x - 1\right )} a c^{3}} \]
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Time = 4.00 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.87 \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^3} \, dx=-\frac {\frac {x}{12}+\frac {a\,x^2}{2}-\frac {1}{3\,a}+\frac {a^2\,x^3}{4}}{-a^4\,c^3\,x^4-2\,a^3\,c^3\,x^3+2\,a\,c^3\,x+c^3}-\frac {\mathrm {atanh}\left (a\,x\right )}{4\,a\,c^3} \]
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