Integrand size = 24, antiderivative size = 263 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{5/2}} \, dx=\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c^2 \sqrt {c-\frac {c}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{8 a c^2 \sqrt {c-\frac {c}{a^2 x^2}} (1-a x)}+\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{8 a c^2 \sqrt {c-\frac {c}{a^2 x^2}} (1+a x)^2}-\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{a c^2 \sqrt {c-\frac {c}{a^2 x^2}} (1+a x)}+\frac {7 \sqrt {1-\frac {1}{a^2 x^2}} \log (1-a x)}{16 a c^2 \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {23 \sqrt {1-\frac {1}{a^2 x^2}} \log (1+a x)}{16 a c^2 \sqrt {c-\frac {c}{a^2 x^2}}} \]
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Time = 0.13 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6332, 6328, 90} \[ \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{5/2}} \, dx=\frac {x \sqrt {1-\frac {1}{a^2 x^2}}}{c^2 \sqrt {c-\frac {c}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{8 a c^2 (1-a x) \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{a c^2 (a x+1) \sqrt {c-\frac {c}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{8 a c^2 (a x+1)^2 \sqrt {c-\frac {c}{a^2 x^2}}}+\frac {7 \sqrt {1-\frac {1}{a^2 x^2}} \log (1-a x)}{16 a c^2 \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {23 \sqrt {1-\frac {1}{a^2 x^2}} \log (a x+1)}{16 a c^2 \sqrt {c-\frac {c}{a^2 x^2}}} \]
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Rule 90
Rule 6328
Rule 6332
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1-\frac {1}{a^2 x^2}} \int \frac {e^{-\coth ^{-1}(a x)}}{\left (1-\frac {1}{a^2 x^2}\right )^{5/2}} \, dx}{c^2 \sqrt {c-\frac {c}{a^2 x^2}}} \\ & = \frac {\left (a^5 \sqrt {1-\frac {1}{a^2 x^2}}\right ) \int \frac {x^5}{(-1+a x)^2 (1+a x)^3} \, dx}{c^2 \sqrt {c-\frac {c}{a^2 x^2}}} \\ & = \frac {\left (a^5 \sqrt {1-\frac {1}{a^2 x^2}}\right ) \int \left (\frac {1}{a^5}+\frac {1}{8 a^5 (-1+a x)^2}+\frac {7}{16 a^5 (-1+a x)}-\frac {1}{4 a^5 (1+a x)^3}+\frac {1}{a^5 (1+a x)^2}-\frac {23}{16 a^5 (1+a x)}\right ) \, dx}{c^2 \sqrt {c-\frac {c}{a^2 x^2}}} \\ & = \frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c^2 \sqrt {c-\frac {c}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{8 a c^2 \sqrt {c-\frac {c}{a^2 x^2}} (1-a x)}+\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{8 a c^2 \sqrt {c-\frac {c}{a^2 x^2}} (1+a x)^2}-\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{a c^2 \sqrt {c-\frac {c}{a^2 x^2}} (1+a x)}+\frac {7 \sqrt {1-\frac {1}{a^2 x^2}} \log (1-a x)}{16 a c^2 \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {23 \sqrt {1-\frac {1}{a^2 x^2}} \log (1+a x)}{16 a c^2 \sqrt {c-\frac {c}{a^2 x^2}}} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.36 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{5/2}} \, dx=\frac {\left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (2 \left (8 x+\frac {1}{a (1+a x)^2}+\frac {1}{a-a^2 x}-\frac {8}{a+a^2 x}\right )+\frac {7 \log (1-a x)}{a}-\frac {23 \log (1+a x)}{a}\right )}{16 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}} \]
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Time = 0.06 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.67
method | result | size |
default | \(-\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) \left (a x -1\right ) \left (-16 a^{4} x^{4}+23 a^{3} \ln \left (a x +1\right ) x^{3}-7 a^{3} \ln \left (a x -1\right ) x^{3}-16 a^{3} x^{3}+23 a^{2} \ln \left (a x +1\right ) x^{2}-7 a^{2} \ln \left (a x -1\right ) x^{2}+34 a^{2} x^{2}-23 a \ln \left (a x +1\right ) x +7 a \ln \left (a x -1\right ) x +18 a x -23 \ln \left (a x +1\right )+7 \ln \left (a x -1\right )-12\right )}{16 a^{6} x^{5} {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}\right )}^{\frac {5}{2}}}\) | \(175\) |
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Time = 0.26 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.51 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{5/2}} \, dx=\frac {{\left (16 \, a^{4} x^{4} + 16 \, a^{3} x^{3} - 34 \, a^{2} x^{2} - 18 \, a x - 23 \, {\left (a^{3} x^{3} + a^{2} x^{2} - a x - 1\right )} \log \left (a x + 1\right ) + 7 \, {\left (a^{3} x^{3} + a^{2} x^{2} - a x - 1\right )} \log \left (a x - 1\right ) + 12\right )} \sqrt {a^{2} c}}{16 \, {\left (a^{5} c^{3} x^{3} + a^{4} c^{3} x^{2} - a^{3} c^{3} x - a^{2} c^{3}\right )}} \]
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Timed out. \[ \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{5/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{5/2}} \, dx=\int { \frac {\sqrt {\frac {a x - 1}{a x + 1}}}{{\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {5}{2}}} \,d x } \]
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Exception generated. \[ \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{5/2}} \, dx=\int \frac {\sqrt {\frac {a\,x-1}{a\,x+1}}}{{\left (c-\frac {c}{a^2\,x^2}\right )}^{5/2}} \,d x \]
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