Integrand size = 27, antiderivative size = 186 \[ \int e^{3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^3 \, dx=\frac {4 \sqrt {c-\frac {c}{a^2 x^2}} x}{a^3 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {2 \sqrt {c-\frac {c}{a^2 x^2}} x^2}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {c-\frac {c}{a^2 x^2}} x^3}{a \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {c-\frac {c}{a^2 x^2}} x^4}{4 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {4 \sqrt {c-\frac {c}{a^2 x^2}} \log (1-a x)}{a^4 \sqrt {1-\frac {1}{a^2 x^2}}} \]
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Time = 0.21 (sec) , antiderivative size = 186, 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} \[ \int e^{3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^3 \, dx=\frac {2 x^2 \sqrt {c-\frac {c}{a^2 x^2}}}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {x^4 \sqrt {c-\frac {c}{a^2 x^2}}}{4 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {x^3 \sqrt {c-\frac {c}{a^2 x^2}}}{a \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {4 \sqrt {c-\frac {c}{a^2 x^2}} \log (1-a x)}{a^4 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {4 x \sqrt {c-\frac {c}{a^2 x^2}}}{a^3 \sqrt {1-\frac {1}{a^2 x^2}}} \]
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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 e^{3 \coth ^{-1}(a x)} \sqrt {1-\frac {1}{a^2 x^2}} x^3 \, dx}{\sqrt {1-\frac {1}{a^2 x^2}}} \\ & = \frac {\sqrt {c-\frac {c}{a^2 x^2}} \int \frac {x^2 (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^2}+\frac {4 x}{a}+3 x^2+a x^3+\frac {4}{a^2 (-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^3 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {2 \sqrt {c-\frac {c}{a^2 x^2}} x^2}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {c-\frac {c}{a^2 x^2}} x^3}{a \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {c-\frac {c}{a^2 x^2}} x^4}{4 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {4 \sqrt {c-\frac {c}{a^2 x^2}} \log (1-a x)}{a^4 \sqrt {1-\frac {1}{a^2 x^2}}} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.38 \[ \int e^{3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^3 \, dx=\frac {\sqrt {c-\frac {c}{a^2 x^2}} \left (\frac {4 x}{a^3}+\frac {2 x^2}{a^2}+\frac {x^3}{a}+\frac {x^4}{4}+\frac {4 \log (1-a x)}{a^4}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}} \]
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Time = 0.09 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.48
method | result | size |
default | \(\frac {\left (a^{4} x^{4}+4 a^{3} x^{3}+8 a^{2} x^{2}+16 a x +16 \ln \left (a x -1\right )\right ) x \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, \left (a x -1\right )}{4 a^{3} \left (a x +1\right )^{2} \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}\) | \(89\) |
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Time = 0.25 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.26 \[ \int e^{3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^3 \, dx=\frac {{\left (a^{4} x^{4} + 4 \, a^{3} x^{3} + 8 \, a^{2} x^{2} + 16 \, a x + 16 \, \log \left (a x - 1\right )\right )} \sqrt {a^{2} c}}{4 \, a^{5}} \]
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Timed out. \[ \int e^{3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^3 \, dx=\text {Timed out} \]
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\[ \int e^{3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^3 \, dx=\int { \frac {\sqrt {c - \frac {c}{a^{2} x^{2}}} x^{3}}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} \,d x } \]
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\[ \int e^{3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^3 \, dx=\int { \frac {\sqrt {c - \frac {c}{a^{2} x^{2}}} x^{3}}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int e^{3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^3 \, dx=\int \frac {x^3\,\sqrt {c-\frac {c}{a^2\,x^2}}}{{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}} \,d x \]
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