Integrand size = 24, antiderivative size = 95 \[ \int e^{-\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{3/2} \, dx=-\frac {2 (1-a x)^3 \left (c-a^2 c x^2\right )^{3/2}}{3 a^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3}+\frac {(1-a x)^4 \left (c-a^2 c x^2\right )^{3/2}}{4 a^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3} \]
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Time = 0.13 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6327, 6328, 45} \[ \int e^{-\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{3/2} \, dx=\frac {(1-a x)^4 \left (c-a^2 c x^2\right )^{3/2}}{4 a^4 x^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {2 (1-a x)^3 \left (c-a^2 c x^2\right )^{3/2}}{3 a^4 x^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}} \]
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Rule 45
Rule 6327
Rule 6328
Rubi steps \begin{align*} \text {integral}& = \frac {\left (c-a^2 c x^2\right )^{3/2} \int e^{-\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)^2 (1+a x) \, dx}{a^3 \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 \left (2 (-1+a x)^2+(-1+a x)^3\right ) \, dx}{a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3} \\ & = -\frac {2 (1-a x)^3 \left (c-a^2 c x^2\right )^{3/2}}{3 a^4 \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 = 53, normalized size of antiderivative = 0.56 \[ \int e^{-\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{3/2} \, dx=-\frac {c (-1+a x)^3 (5+3 a x) \sqrt {c-a^2 c x^2}}{12 a^2 \sqrt {1-\frac {1}{a^2 x^2}} x} \]
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Time = 0.51 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.66
method | result | size |
default | \(-\frac {\left (3 a^{3} x^{3}-4 a^{2} x^{2}-6 a x +12\right ) x c \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {\frac {a x -1}{a x +1}}}{12 \left (a x -1\right )}\) | \(63\) |
gosper | \(\frac {x \left (3 a^{3} x^{3}-4 a^{2} x^{2}-6 a x +12\right ) \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}} \sqrt {\frac {a x -1}{a x +1}}}{12 \left (a x +1\right ) \left (a x -1\right )^{2}}\) | \(68\) |
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Time = 0.25 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.45 \[ \int e^{-\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{3/2} \, dx=-\frac {{\left (3 \, a^{3} c x^{4} - 4 \, a^{2} c x^{3} - 6 \, a c x^{2} + 12 \, c x\right )} \sqrt {-a^{2} c}}{12 \, a} \]
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Timed out. \[ \int e^{-\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{3/2} \, dx=\text {Timed out} \]
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\[ \int e^{-\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{3/2} \, dx=\int { {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \sqrt {\frac {a x - 1}{a x + 1}} \,d x } \]
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\[ \int e^{-\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{3/2} \, dx=\int { {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \sqrt {\frac {a x - 1}{a x + 1}} \,d x } \]
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Timed out. \[ \int e^{-\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{3/2} \, dx=\int {\left (c-a^2\,c\,x^2\right )}^{3/2}\,\sqrt {\frac {a\,x-1}{a\,x+1}} \,d x \]
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