Integrand size = 24, antiderivative size = 47 \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{3/2} \, dx=\frac {(1-a x)^4 \left (c-a^2 c x^2\right )^{3/2}}{4 a^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3} \]
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Time = 0.12 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6327, 6328, 32} \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{3/2} \, dx=\frac {(1-a x)^4 \left (c-a^2 c x^2\right )^{3/2}}{4 a^4 x^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}} \]
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Rule 32
Rule 6327
Rule 6328
Rubi steps \begin{align*} \text {integral}& = \frac {\left (c-a^2 c x^2\right )^{3/2} \int e^{-3 \coth ^{-1}(a x)} \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3 \, dx}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3} \\ & = \frac {\left (c-a^2 c x^2\right )^{3/2} \int (-1+a x)^3 \, dx}{a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3} \\ & = \frac {(1-a x)^4 \left (c-a^2 c x^2\right )^{3/2}}{4 a^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.23 \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{3/2} \, dx=-\frac {c \sqrt {c-a^2 c x^2} \left (-4+6 a x-4 a^2 x^2+a^3 x^3\right )}{4 a \sqrt {1-\frac {1}{a^2 x^2}}} \]
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Time = 0.52 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.02
method | result | size |
default | \(-\frac {\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) \left (a x -1\right )^{2} \sqrt {-c \left (a^{2} x^{2}-1\right )}\, c}{4 a}\) | \(48\) |
gosper | \(\frac {x \left (a^{3} x^{3}-4 a^{2} x^{2}+6 a x -4\right ) \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}} \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{4 \left (a x -1\right )^{3}}\) | \(60\) |
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none
Time = 0.24 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.89 \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{3/2} \, dx=-\frac {{\left (a^{3} c x^{4} - 4 \, a^{2} c x^{3} + 6 \, a c x^{2} - 4 \, c x\right )} \sqrt {-a^{2} c}}{4 \, a} \]
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Timed out. \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{3/2} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 97 vs. \(2 (40) = 80\).
Time = 0.21 (sec) , antiderivative size = 97, normalized size of antiderivative = 2.06 \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{3/2} \, dx=-\frac {{\left (a^{5} \sqrt {-c} c x^{5} - 3 \, a^{4} \sqrt {-c} c x^{4} + 2 \, a^{3} \sqrt {-c} c x^{3} + 2 \, a^{2} \sqrt {-c} c x^{2} + 4 \, \sqrt {-c} c\right )} {\left (a x - 1\right )}^{2}}{4 \, {\left (a^{3} x^{2} - 2 \, a^{2} x + a\right )} {\left (a x + 1\right )}} \]
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\[ \int e^{-3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{3/2} \, dx=\int { {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} \,d x } \]
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Timed out. \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{3/2} \, dx=\int {\left (c-a^2\,c\,x^2\right )}^{3/2}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2} \,d x \]
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