Integrand size = 22, antiderivative size = 87 \[ \int \frac {e^{4 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^3} \, dx=\frac {1}{8 a c^3 (1-a x)^4}+\frac {1}{12 a c^3 (1-a x)^3}+\frac {1}{16 a c^3 (1-a x)^2}+\frac {1}{16 a c^3 (1-a x)}+\frac {\text {arctanh}(a x)}{16 a c^3} \]
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Time = 0.13 (sec) , antiderivative size = 87, 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} \[ \int \frac {e^{4 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^3} \, dx=\frac {\text {arctanh}(a x)}{16 a c^3}+\frac {1}{16 a c^3 (1-a x)}+\frac {1}{16 a c^3 (1-a x)^2}+\frac {1}{12 a c^3 (1-a x)^3}+\frac {1}{8 a c^3 (1-a x)^4} \]
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Rule 46
Rule 213
Rule 6275
Rule 6302
Rubi steps \begin{align*} \text {integral}& = \int \frac {e^{4 \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^3} \, dx \\ & = \frac {\int \frac {1}{(1-a x)^5 (1+a x)} \, dx}{c^3} \\ & = \frac {\int \left (-\frac {1}{2 (-1+a x)^5}+\frac {1}{4 (-1+a x)^4}-\frac {1}{8 (-1+a x)^3}+\frac {1}{16 (-1+a x)^2}-\frac {1}{16 \left (-1+a^2 x^2\right )}\right ) \, dx}{c^3} \\ & = \frac {1}{8 a c^3 (1-a x)^4}+\frac {1}{12 a c^3 (1-a x)^3}+\frac {1}{16 a c^3 (1-a x)^2}+\frac {1}{16 a c^3 (1-a x)}-\frac {\int \frac {1}{-1+a^2 x^2} \, dx}{16 c^3} \\ & = \frac {1}{8 a c^3 (1-a x)^4}+\frac {1}{12 a c^3 (1-a x)^3}+\frac {1}{16 a c^3 (1-a x)^2}+\frac {1}{16 a c^3 (1-a x)}+\frac {\text {arctanh}(a x)}{16 a c^3} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.60 \[ \int \frac {e^{4 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^3} \, dx=\frac {16-19 a x+12 a^2 x^2-3 a^3 x^3+3 (-1+a x)^4 \text {arctanh}(a x)}{48 a c^3 (-1+a x)^4} \]
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Time = 0.62 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.75
method | result | size |
risch | \(\frac {-\frac {a^{2} x^{3}}{16}+\frac {a \,x^{2}}{4}-\frac {19 x}{48}+\frac {1}{3 a}}{c^{3} \left (a x -1\right )^{4}}+\frac {\ln \left (-a x -1\right )}{32 a \,c^{3}}-\frac {\ln \left (a x -1\right )}{32 a \,c^{3}}\) | \(65\) |
default | \(\frac {\frac {\ln \left (a x +1\right )}{32 a}+\frac {1}{8 a \left (a x -1\right )^{4}}-\frac {1}{12 a \left (a x -1\right )^{3}}+\frac {1}{16 \left (a x -1\right )^{2} a}-\frac {1}{16 a \left (a x -1\right )}-\frac {\ln \left (a x -1\right )}{32 a}}{c^{3}}\) | \(76\) |
norman | \(\frac {\frac {15 x}{16 c}+\frac {a \,x^{2}}{8 c}-\frac {31 a^{2} x^{3}}{24 c}+\frac {11 a^{3} x^{4}}{24 c}+\frac {29 a^{4} x^{5}}{48 c}-\frac {a^{5} x^{6}}{3 c}}{\left (a x +1\right )^{2} \left (a x -1\right )^{4} c^{2}}-\frac {\ln \left (a x -1\right )}{32 a \,c^{3}}+\frac {\ln \left (a x +1\right )}{32 a \,c^{3}}\) | \(108\) |
parallelrisch | \(\frac {-3 \ln \left (a x -1\right ) x^{4} a^{4}+3 \ln \left (a x +1\right ) x^{4} a^{4}-32 a^{4} x^{4}+12 a^{3} \ln \left (a x -1\right ) x^{3}-12 a^{3} \ln \left (a x +1\right ) x^{3}+122 a^{3} x^{3}-18 a^{2} \ln \left (a x -1\right ) x^{2}+18 a^{2} \ln \left (a x +1\right ) x^{2}-168 a^{2} x^{2}+12 a \ln \left (a x -1\right ) x -12 a \ln \left (a x +1\right ) x +90 a x -3 \ln \left (a x -1\right )+3 \ln \left (a x +1\right )}{96 \left (a x -1\right )^{4} c^{3} a}\) | \(165\) |
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Leaf count of result is larger than twice the leaf count of optimal. 147 vs. \(2 (73) = 146\).
Time = 0.25 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.69 \[ \int \frac {e^{4 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^3} \, dx=-\frac {6 \, a^{3} x^{3} - 24 \, a^{2} x^{2} + 38 \, a x - 3 \, {\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right )} \log \left (a x + 1\right ) + 3 \, {\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right )} \log \left (a x - 1\right ) - 32}{96 \, {\left (a^{5} c^{3} x^{4} - 4 \, a^{4} c^{3} x^{3} + 6 \, a^{3} c^{3} x^{2} - 4 \, a^{2} c^{3} x + a c^{3}\right )}} \]
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Time = 0.25 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.14 \[ \int \frac {e^{4 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^3} \, dx=- \frac {3 a^{3} x^{3} - 12 a^{2} x^{2} + 19 a x - 16}{48 a^{5} c^{3} x^{4} - 192 a^{4} c^{3} x^{3} + 288 a^{3} c^{3} x^{2} - 192 a^{2} c^{3} x + 48 a c^{3}} - \frac {\frac {\log {\left (x - \frac {1}{a} \right )}}{32} - \frac {\log {\left (x + \frac {1}{a} \right )}}{32}}{a c^{3}} \]
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Time = 0.19 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.17 \[ \int \frac {e^{4 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^3} \, dx=-\frac {3 \, a^{3} x^{3} - 12 \, a^{2} x^{2} + 19 \, a x - 16}{48 \, {\left (a^{5} c^{3} x^{4} - 4 \, a^{4} c^{3} x^{3} + 6 \, a^{3} c^{3} x^{2} - 4 \, a^{2} c^{3} x + a c^{3}\right )}} + \frac {\log \left (a x + 1\right )}{32 \, a c^{3}} - \frac {\log \left (a x - 1\right )}{32 \, a c^{3}} \]
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Time = 0.28 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.05 \[ \int \frac {e^{4 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^3} \, dx=\frac {\log \left ({\left | -\frac {2}{a x - 1} - 1 \right |}\right )}{32 \, a c^{3}} - \frac {\frac {3 \, a^{3} c^{9}}{a x - 1} - \frac {3 \, a^{3} c^{9}}{{\left (a x - 1\right )}^{2}} + \frac {4 \, a^{3} c^{9}}{{\left (a x - 1\right )}^{3}} - \frac {6 \, a^{3} c^{9}}{{\left (a x - 1\right )}^{4}}}{48 \, a^{4} c^{12}} \]
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Time = 0.10 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.95 \[ \int \frac {e^{4 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^3} \, dx=\frac {\mathrm {atanh}\left (a\,x\right )}{16\,a\,c^3}-\frac {\frac {19\,x}{48}-\frac {a\,x^2}{4}-\frac {1}{3\,a}+\frac {a^2\,x^3}{16}}{a^4\,c^3\,x^4-4\,a^3\,c^3\,x^3+6\,a^2\,c^3\,x^2-4\,a\,c^3\,x+c^3} \]
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