Integrand size = 24, antiderivative size = 47 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=-\frac {a^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3}{2 (1-a x)^2 \left (c-a^2 c x^2\right )^{3/2}} \]
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Time = 0.13 (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} \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=-\frac {a^2 x^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{2 (1-a x)^2 \left (c-a^2 c x^2\right )^{3/2}} \]
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Rule 32
Rule 6327
Rule 6328
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3\right ) \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3} \, dx}{\left (c-a^2 c x^2\right )^{3/2}} \\ & = \frac {\left (a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3\right ) \int \frac {1}{(-1+a x)^3} \, dx}{\left (c-a^2 c x^2\right )^{3/2}} \\ & = -\frac {a^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3}{2 (1-a x)^2 \left (c-a^2 c x^2\right )^{3/2}} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.09 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=-\frac {\sqrt {1-\frac {1}{a^2 x^2}} x \sqrt {c-a^2 c x^2}}{2 c^2 (-1+a x)^3 (1+a x)} \]
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Time = 0.52 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.83
method | result | size |
gosper | \(-\frac {a x -1}{2 a \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}\) | \(39\) |
default | \(-\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}}{2 \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) \left (a^{2} x^{2}-1\right ) c^{2} a}\) | \(56\) |
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Time = 0.24 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.83 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=-\frac {\sqrt {-a^{2} c}}{2 \, {\left (a^{4} c^{2} x^{2} - 2 \, a^{3} c^{2} x + a^{2} c^{2}\right )}} \]
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Timed out. \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} \,d x } \]
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Time = 4.19 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.91 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\frac {\left (\frac {1}{2\,a^3\,c}+\frac {x}{2\,a^2\,c}\right )\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{\frac {\sqrt {c-a^2\,c\,x^2}}{a^2}+x^2\,\sqrt {c-a^2\,c\,x^2}-\frac {2\,x\,\sqrt {c-a^2\,c\,x^2}}{a}} \]
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