Integrand size = 22, antiderivative size = 127 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx=-\frac {16 e^{3 \coth ^{-1}(a x)}}{63 a c^4}-\frac {e^{3 \coth ^{-1}(a x)} (1-2 a x)}{9 a c^4 \left (1-a^2 x^2\right )^3}-\frac {10 e^{3 \coth ^{-1}(a x)} (3-4 a x)}{63 a c^4 \left (1-a^2 x^2\right )^2}+\frac {8 e^{3 \coth ^{-1}(a x)} (3-2 a x)}{21 a c^4 \left (1-a^2 x^2\right )} \]
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Time = 0.13 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {6320, 6318} \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx=-\frac {10 (3-4 a x) e^{3 \coth ^{-1}(a x)}}{63 a c^4 \left (1-a^2 x^2\right )^2}+\frac {8 (3-2 a x) e^{3 \coth ^{-1}(a x)}}{21 a c^4 \left (1-a^2 x^2\right )}-\frac {(1-2 a x) e^{3 \coth ^{-1}(a x)}}{9 a c^4 \left (1-a^2 x^2\right )^3}-\frac {16 e^{3 \coth ^{-1}(a x)}}{63 a c^4} \]
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Rule 6318
Rule 6320
Rubi steps \begin{align*} \text {integral}& = -\frac {e^{3 \coth ^{-1}(a x)} (1-2 a x)}{9 a c^4 \left (1-a^2 x^2\right )^3}+\frac {10 \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^3} \, dx}{9 c} \\ & = -\frac {e^{3 \coth ^{-1}(a x)} (1-2 a x)}{9 a c^4 \left (1-a^2 x^2\right )^3}-\frac {10 e^{3 \coth ^{-1}(a x)} (3-4 a x)}{63 a c^4 \left (1-a^2 x^2\right )^2}+\frac {40 \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx}{21 c^2} \\ & = -\frac {e^{3 \coth ^{-1}(a x)} (1-2 a x)}{9 a c^4 \left (1-a^2 x^2\right )^3}-\frac {10 e^{3 \coth ^{-1}(a x)} (3-4 a x)}{63 a c^4 \left (1-a^2 x^2\right )^2}+\frac {8 e^{3 \coth ^{-1}(a x)} (3-2 a x)}{21 a c^4 \left (1-a^2 x^2\right )}-\frac {16 \int \frac {e^{3 \coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx}{21 c^3} \\ & = -\frac {16 e^{3 \coth ^{-1}(a x)}}{63 a c^4}-\frac {e^{3 \coth ^{-1}(a x)} (1-2 a x)}{9 a c^4 \left (1-a^2 x^2\right )^3}-\frac {10 e^{3 \coth ^{-1}(a x)} (3-4 a x)}{63 a c^4 \left (1-a^2 x^2\right )^2}+\frac {8 e^{3 \coth ^{-1}(a x)} (3-2 a x)}{21 a c^4 \left (1-a^2 x^2\right )} \\ \end{align*}
Time = 0.49 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.65 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx=-\frac {\sqrt {1-\frac {1}{a^2 x^2}} x \left (19+6 a x-66 a^2 x^2+56 a^3 x^3+24 a^4 x^4-48 a^5 x^5+16 a^6 x^6\right )}{63 c^4 (-1+a x)^5 (1+a x)^2} \]
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Time = 0.49 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.64
method | result | size |
gosper | \(-\frac {16 a^{6} x^{6}-48 a^{5} x^{5}+24 a^{4} x^{4}+56 a^{3} x^{3}-66 a^{2} x^{2}+6 a x +19}{63 \left (a^{2} x^{2}-1\right )^{3} c^{4} \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} a}\) | \(81\) |
default | \(-\frac {16 a^{6} x^{6}-48 a^{5} x^{5}+24 a^{4} x^{4}+56 a^{3} x^{3}-66 a^{2} x^{2}+6 a x +19}{63 \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right )^{3} c^{4} \left (a x -1\right )^{3} a}\) | \(84\) |
trager | \(-\frac {\left (16 a^{6} x^{6}-48 a^{5} x^{5}+24 a^{4} x^{4}+56 a^{3} x^{3}-66 a^{2} x^{2}+6 a x +19\right ) \sqrt {-\frac {-a x +1}{a x +1}}}{63 a \,c^{4} \left (a x +1\right ) \left (a x -1\right )^{5}}\) | \(86\) |
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Time = 0.25 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.98 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx=-\frac {{\left (16 \, a^{6} x^{6} - 48 \, a^{5} x^{5} + 24 \, a^{4} x^{4} + 56 \, a^{3} x^{3} - 66 \, a^{2} x^{2} + 6 \, a x + 19\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{63 \, {\left (a^{7} c^{4} x^{6} - 4 \, a^{6} c^{4} x^{5} + 5 \, a^{5} c^{4} x^{4} - 5 \, a^{3} c^{4} x^{2} + 4 \, a^{2} c^{4} x - a c^{4}\right )}} \]
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Timed out. \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx=\text {Timed out} \]
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Time = 0.19 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.03 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx=\frac {1}{4032} \, a {\left (\frac {21 \, {\left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} - 18 \, \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{a^{2} c^{4}} + \frac {\frac {54 \, {\left (a x - 1\right )}}{a x + 1} - \frac {189 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac {420 \, {\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} - \frac {945 \, {\left (a x - 1\right )}^{4}}{{\left (a x + 1\right )}^{4}} - 7}{a^{2} c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {9}{2}}}\right )} \]
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\[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} - c\right )}^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} \,d x } \]
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Time = 3.98 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.60 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx=-\frac {16\,a^6\,x^6-48\,a^5\,x^5+24\,a^4\,x^4+56\,a^3\,x^3-66\,a^2\,x^2+6\,a\,x+19}{63\,a\,c^4\,{\left (a\,x+1\right )}^6\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{9/2}} \]
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