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 = 0.11 (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} \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx=\frac {(2 a x+1) e^{-3 \coth ^{-1}(a x)}}{9 a c^4 \left (1-a^2 x^2\right )^3}-\frac {8 (2 a x+3) e^{-3 \coth ^{-1}(a x)}}{21 a c^4 \left (1-a^2 x^2\right )}+\frac {10 (4 a x+3) e^{-3 \coth ^{-1}(a x)}}{63 a c^4 \left (1-a^2 x^2\right )^2}+\frac {16 e^{-3 \coth ^{-1}(a x)}}{63 a c^4} \]
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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 = 0.51 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.65 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx=\frac {\sqrt {1-\frac {1}{a^2 x^2}} x \left (19-6 a x-66 a^2 x^2-56 a^3 x^3+24 a^4 x^4+48 a^5 x^5+16 a^6 x^6\right )}{63 c^4 (-1+a x)^2 (1+a x)^5} \]
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Time = 0.52 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.64
method | result | size |
gosper | \(\frac {\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (16 a^{6} x^{6}+48 a^{5} x^{5}+24 a^{4} x^{4}-56 a^{3} x^{3}-66 a^{2} x^{2}-6 a x +19\right )}{63 \left (a^{2} x^{2}-1\right )^{3} c^{4} a}\) | \(81\) |
default | \(\frac {\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (16 a^{6} x^{6}+48 a^{5} x^{5}+24 a^{4} x^{4}-56 a^{3} x^{3}-66 a^{2} x^{2}-6 a x +19\right )}{63 \left (a x -1\right )^{3} c^{4} \left (a x +1\right )^{3} a}\) | \(84\) |
trager | \(\frac {\left (16 a^{6} x^{6}+48 a^{5} x^{5}+24 a^{4} x^{4}-56 a^{3} x^{3}-66 a^{2} x^{2}-6 a x +19\right ) \sqrt {-\frac {-a x +1}{a x +1}}}{63 a \,c^{4} \left (a x -1\right )^{2} \left (a x +1\right )^{4}}\) | \(86\) |
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Time = 0.25 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.06 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx=\frac {{\left (16 \, a^{6} x^{6} + 48 \, a^{5} x^{5} + 24 \, a^{4} x^{4} - 56 \, a^{3} x^{3} - 66 \, a^{2} x^{2} - 6 \, a x + 19\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{63 \, {\left (a^{7} c^{4} x^{6} + 2 \, a^{6} c^{4} x^{5} - a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} + 2 \, a^{2} c^{4} x + a c^{4}\right )}} \]
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\[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx=\frac {\int \left (- \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a^{9} x^{9} + a^{8} x^{8} - 4 a^{7} x^{7} - 4 a^{6} x^{6} + 6 a^{5} x^{5} + 6 a^{4} x^{4} - 4 a^{3} x^{3} - 4 a^{2} x^{2} + a x + 1}\right )\, dx + \int \frac {a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a^{9} x^{9} + a^{8} x^{8} - 4 a^{7} x^{7} - 4 a^{6} x^{6} + 6 a^{5} x^{5} + 6 a^{4} x^{4} - 4 a^{3} x^{3} - 4 a^{2} x^{2} + a x + 1}\, dx}{c^{4}} \]
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Time = 0.21 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.07 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx=\frac {1}{4032} \, a {\left (\frac {7 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {9}{2}} - 54 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} + 189 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} - 420 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} + 945 \, \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c^{4}} + \frac {21 \, {\left (\frac {18 \, {\left (a x - 1\right )}}{a x + 1} - 1\right )}}{a^{2} c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}\right )} \]
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\[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx=\int { \frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}{{\left (a^{2} c x^{2} - c\right )}^{4}} \,d x } \]
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Time = 0.04 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.22 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx=\frac {15\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{64\,a\,c^4}-\frac {5\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{48\,a\,c^4}+\frac {3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{64\,a\,c^4}-\frac {3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}}{224\,a\,c^4}+\frac {{\left (\frac {a\,x-1}{a\,x+1}\right )}^{9/2}}{576\,a\,c^4}+\frac {\frac {6\,\left (a\,x-1\right )}{a\,x+1}-\frac {1}{3}}{64\,a\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}} \]
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