Integrand size = 27, antiderivative size = 227 \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x^5} \, dx=-\frac {\sqrt {c-a^2 c x^2}}{4 a \sqrt {1-\frac {1}{a^2 x^2}} x^5}+\frac {\sqrt {c-a^2 c x^2}}{\sqrt {1-\frac {1}{a^2 x^2}} x^4}-\frac {2 a \sqrt {c-a^2 c x^2}}{\sqrt {1-\frac {1}{a^2 x^2}} x^3}+\frac {4 a^2 \sqrt {c-a^2 c x^2}}{\sqrt {1-\frac {1}{a^2 x^2}} x^2}+\frac {4 a^3 \sqrt {c-a^2 c x^2} \log (x)}{\sqrt {1-\frac {1}{a^2 x^2}} x}-\frac {4 a^3 \sqrt {c-a^2 c x^2} \log (1+a x)}{\sqrt {1-\frac {1}{a^2 x^2}} x} \]
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Time = 0.18 (sec) , antiderivative size = 227, 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 = {6327, 6328, 90} \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x^5} \, dx=\frac {4 a^2 \sqrt {c-a^2 c x^2}}{x^2 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {\sqrt {c-a^2 c x^2}}{4 a x^5 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {c-a^2 c x^2}}{x^4 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {2 a \sqrt {c-a^2 c x^2}}{x^3 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {4 a^3 \log (x) \sqrt {c-a^2 c x^2}}{x \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {4 a^3 \sqrt {c-a^2 c x^2} \log (a x+1)}{x \sqrt {1-\frac {1}{a^2 x^2}}} \]
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Rule 90
Rule 6327
Rule 6328
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {c-a^2 c x^2} \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {1-\frac {1}{a^2 x^2}}}{x^4} \, dx}{\sqrt {1-\frac {1}{a^2 x^2}} x} \\ & = \frac {\sqrt {c-a^2 c x^2} \int \frac {(-1+a x)^2}{x^5 (1+a x)} \, dx}{a \sqrt {1-\frac {1}{a^2 x^2}} x} \\ & = \frac {\sqrt {c-a^2 c x^2} \int \left (\frac {1}{x^5}-\frac {3 a}{x^4}+\frac {4 a^2}{x^3}-\frac {4 a^3}{x^2}+\frac {4 a^4}{x}-\frac {4 a^5}{1+a x}\right ) \, dx}{a \sqrt {1-\frac {1}{a^2 x^2}} x} \\ & = -\frac {\sqrt {c-a^2 c x^2}}{4 a \sqrt {1-\frac {1}{a^2 x^2}} x^5}+\frac {\sqrt {c-a^2 c x^2}}{\sqrt {1-\frac {1}{a^2 x^2}} x^4}-\frac {2 a \sqrt {c-a^2 c x^2}}{\sqrt {1-\frac {1}{a^2 x^2}} x^3}+\frac {4 a^2 \sqrt {c-a^2 c x^2}}{\sqrt {1-\frac {1}{a^2 x^2}} x^2}+\frac {4 a^3 \sqrt {c-a^2 c x^2} \log (x)}{\sqrt {1-\frac {1}{a^2 x^2}} x}-\frac {4 a^3 \sqrt {c-a^2 c x^2} \log (1+a x)}{\sqrt {1-\frac {1}{a^2 x^2}} x} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.35 \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x^5} \, dx=\frac {\sqrt {c-a^2 c x^2} \left (-\frac {1}{4 a x^4}+\frac {1}{x^3}-\frac {2 a}{x^2}+\frac {4 a^2}{x}+4 a^3 \log (x)-4 a^3 \log (1+a x)\right )}{\sqrt {1-\frac {1}{a^2 x^2}} x} \]
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Time = 0.54 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.41
method | result | size |
default | \(-\frac {\left (16 \ln \left (a x +1\right ) x^{4} a^{4}-16 \ln \left (x \right ) x^{4} a^{4}-16 a^{3} x^{3}+8 a^{2} x^{2}-4 a x +1\right ) \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (a x +1\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{4 x^{4} \left (a x -1\right )^{2}}\) | \(93\) |
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Time = 0.25 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.46 \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x^5} \, dx=\frac {16 \, a^{5} \sqrt {-c} x^{4} \log \left (\frac {2 \, a^{3} c x^{2} + 2 \, a^{2} c x + \sqrt {-a^{2} c} {\left (2 \, a x + 1\right )} \sqrt {-c} + a c}{a x^{2} + x}\right ) + {\left (16 \, a^{3} x^{3} - 8 \, a^{2} x^{2} + 4 \, a x - 1\right )} \sqrt {-a^{2} c}}{4 \, a x^{4}} \]
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Timed out. \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x^5} \, dx=\text {Timed out} \]
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\[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x^5} \, dx=\int { \frac {\sqrt {-a^{2} c x^{2} + c} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}{x^{5}} \,d x } \]
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\[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x^5} \, dx=\int { \frac {\sqrt {-a^{2} c x^{2} + c} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}{x^{5}} \,d x } \]
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Timed out. \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x^5} \, dx=\int \frac {\sqrt {c-a^2\,c\,x^2}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{x^5} \,d x \]
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