Integrand size = 27, antiderivative size = 151 \[ \int e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^2 \, dx=\frac {4 \sqrt {c-\frac {c}{a^2 x^2}} x}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {3 \sqrt {c-\frac {c}{a^2 x^2}} x^2}{2 a \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {c-\frac {c}{a^2 x^2}} x^3}{3 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {4 \sqrt {c-\frac {c}{a^2 x^2}} \log (1+a x)}{a^3 \sqrt {1-\frac {1}{a^2 x^2}}} \]
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Time = 0.21 (sec) , antiderivative size = 151, 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, 78} \[ \int e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^2 \, dx=-\frac {3 x^2 \sqrt {c-\frac {c}{a^2 x^2}}}{2 a \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {4 x \sqrt {c-\frac {c}{a^2 x^2}}}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {x^3 \sqrt {c-\frac {c}{a^2 x^2}}}{3 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {4 \sqrt {c-\frac {c}{a^2 x^2}} \log (a x+1)}{a^3 \sqrt {1-\frac {1}{a^2 x^2}}} \]
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Rule 78
Rule 6328
Rule 6332
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {c-\frac {c}{a^2 x^2}} \int e^{-3 \coth ^{-1}(a x)} \sqrt {1-\frac {1}{a^2 x^2}} x^2 \, dx}{\sqrt {1-\frac {1}{a^2 x^2}}} \\ & = \frac {\sqrt {c-\frac {c}{a^2 x^2}} \int \frac {x (-1+a x)^2}{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 {4}{a}-3 x+a x^2-\frac {4}{a (1+a x)}\right ) \, dx}{a \sqrt {1-\frac {1}{a^2 x^2}}} \\ & = \frac {4 \sqrt {c-\frac {c}{a^2 x^2}} x}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {3 \sqrt {c-\frac {c}{a^2 x^2}} x^2}{2 a \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {c-\frac {c}{a^2 x^2}} x^3}{3 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {4 \sqrt {c-\frac {c}{a^2 x^2}} \log (1+a x)}{a^3 \sqrt {1-\frac {1}{a^2 x^2}}} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.41 \[ \int e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^2 \, dx=\frac {\sqrt {c-\frac {c}{a^2 x^2}} \left (a x \left (24-9 a x+2 a^2 x^2\right )-24 \log (1+a x)\right )}{6 a^3 \sqrt {1-\frac {1}{a^2 x^2}}} \]
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Time = 0.05 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.54
method | result | size |
default | \(-\frac {\left (-2 a^{3} x^{3}+9 a^{2} x^{2}-24 a x +24 \ln \left (a x +1\right )\right ) x \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, \left (a x +1\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{6 a^{2} \left (a x -1\right )^{2}}\) | \(82\) |
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Time = 0.24 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.27 \[ \int e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^2 \, dx=\frac {{\left (2 \, a^{3} x^{3} - 9 \, a^{2} x^{2} + 24 \, a x - 24 \, \log \left (a x + 1\right )\right )} \sqrt {a^{2} c}}{6 \, a^{4}} \]
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Timed out. \[ \int e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^2 \, dx=\text {Timed out} \]
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\[ \int e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^2 \, dx=\int { \sqrt {c - \frac {c}{a^{2} x^{2}}} x^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} \,d x } \]
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\[ \int e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^2 \, dx=\int { \sqrt {c - \frac {c}{a^{2} x^{2}}} x^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} \,d x } \]
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Timed out. \[ \int e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^2 \, dx=\int x^2\,\sqrt {c-\frac {c}{a^2\,x^2}}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2} \,d x \]
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