Integrand size = 27, antiderivative size = 264 \[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^5} \, dx=\frac {\sqrt {c-\frac {c}{a^2 x^2}}}{5 a \sqrt {1-\frac {1}{a^2 x^2}} x^5}+\frac {3 \sqrt {c-\frac {c}{a^2 x^2}}}{4 \sqrt {1-\frac {1}{a^2 x^2}} x^4}+\frac {4 a \sqrt {c-\frac {c}{a^2 x^2}}}{3 \sqrt {1-\frac {1}{a^2 x^2}} x^3}+\frac {2 a^2 \sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {1-\frac {1}{a^2 x^2}} x^2}+\frac {4 a^3 \sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {1-\frac {1}{a^2 x^2}} x}-\frac {4 a^4 \sqrt {c-\frac {c}{a^2 x^2}} \log (x)}{\sqrt {1-\frac {1}{a^2 x^2}}}+\frac {4 a^4 \sqrt {c-\frac {c}{a^2 x^2}} \log (1-a x)}{\sqrt {1-\frac {1}{a^2 x^2}}} \]
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Time = 0.21 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6332, 6328, 90} \[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^5} \, dx=\frac {2 a^2 \sqrt {c-\frac {c}{a^2 x^2}}}{x^2 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {c-\frac {c}{a^2 x^2}}}{5 a x^5 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {3 \sqrt {c-\frac {c}{a^2 x^2}}}{4 x^4 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {4 a \sqrt {c-\frac {c}{a^2 x^2}}}{3 x^3 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {4 a^4 \log (x) \sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {1-\frac {1}{a^2 x^2}}}+\frac {4 a^4 \sqrt {c-\frac {c}{a^2 x^2}} \log (1-a x)}{\sqrt {1-\frac {1}{a^2 x^2}}}+\frac {4 a^3 \sqrt {c-\frac {c}{a^2 x^2}}}{x \sqrt {1-\frac {1}{a^2 x^2}}} \]
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Rule 90
Rule 6328
Rule 6332
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {c-\frac {c}{a^2 x^2}} \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {1-\frac {1}{a^2 x^2}}}{x^5} \, dx}{\sqrt {1-\frac {1}{a^2 x^2}}} \\ & = \frac {\sqrt {c-\frac {c}{a^2 x^2}} \int \frac {(1+a x)^2}{x^6 (-1+a x)} \, dx}{a \sqrt {1-\frac {1}{a^2 x^2}}} \\ & = \frac {\sqrt {c-\frac {c}{a^2 x^2}} \int \left (-\frac {1}{x^6}-\frac {3 a}{x^5}-\frac {4 a^2}{x^4}-\frac {4 a^3}{x^3}-\frac {4 a^4}{x^2}-\frac {4 a^5}{x}+\frac {4 a^6}{-1+a x}\right ) \, dx}{a \sqrt {1-\frac {1}{a^2 x^2}}} \\ & = \frac {\sqrt {c-\frac {c}{a^2 x^2}}}{5 a \sqrt {1-\frac {1}{a^2 x^2}} x^5}+\frac {3 \sqrt {c-\frac {c}{a^2 x^2}}}{4 \sqrt {1-\frac {1}{a^2 x^2}} x^4}+\frac {4 a \sqrt {c-\frac {c}{a^2 x^2}}}{3 \sqrt {1-\frac {1}{a^2 x^2}} x^3}+\frac {2 a^2 \sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {1-\frac {1}{a^2 x^2}} x^2}+\frac {4 a^3 \sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {1-\frac {1}{a^2 x^2}} x}-\frac {4 a^4 \sqrt {c-\frac {c}{a^2 x^2}} \log (x)}{\sqrt {1-\frac {1}{a^2 x^2}}}+\frac {4 a^4 \sqrt {c-\frac {c}{a^2 x^2}} \log (1-a x)}{\sqrt {1-\frac {1}{a^2 x^2}}} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.34 \[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^5} \, dx=\frac {\sqrt {c-\frac {c}{a^2 x^2}} \left (\frac {1}{5 a x^5}+\frac {3}{4 x^4}+\frac {4 a}{3 x^3}+\frac {2 a^2}{x^2}+\frac {4 a^3}{x}-4 a^4 \log (x)+4 a^4 \log (1-a x)\right )}{\sqrt {1-\frac {1}{a^2 x^2}}} \]
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Time = 0.06 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.40
method | result | size |
default | \(-\frac {\left (240 a^{5} \ln \left (x \right ) x^{5}-240 \ln \left (a x -1\right ) x^{5} a^{5}-240 a^{4} x^{4}-120 a^{3} x^{3}-80 a^{2} x^{2}-45 a x -12\right ) \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, \left (a x -1\right )}{60 \left (a x +1\right )^{2} x^{4} \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}\) | \(106\) |
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Time = 0.26 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.41 \[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^5} \, dx=\frac {240 \, a^{6} \sqrt {c} x^{5} \log \left (\frac {2 \, a^{3} c x^{2} - 2 \, a^{2} c x - \sqrt {a^{2} c} {\left (2 \, a x - 1\right )} \sqrt {c} + a c}{a x^{2} - x}\right ) + {\left (240 \, a^{4} x^{4} + 120 \, a^{3} x^{3} + 80 \, a^{2} x^{2} + 45 \, a x + 12\right )} \sqrt {a^{2} c}}{60 \, a^{2} x^{5}} \]
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Timed out. \[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^5} \, dx=\text {Timed out} \]
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\[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^5} \, dx=\int { \frac {\sqrt {c - \frac {c}{a^{2} x^{2}}}}{x^{5} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} \,d x } \]
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Exception generated. \[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^5} \, dx=\text {Exception raised: RuntimeError} \]
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Timed out. \[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^5} \, dx=\int \frac {\sqrt {c-\frac {c}{a^2\,x^2}}}{x^5\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}} \,d x \]
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