Integrand size = 25, antiderivative size = 46 \[ \int \frac {e^{n \coth ^{-1}(a x)} x}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\frac {e^{n \coth ^{-1}(a x)} (1-a n x)}{a^2 c \left (1-n^2\right ) \sqrt {c-a^2 c x^2}} \]
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Time = 0.06 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {6321} \[ \int \frac {e^{n \coth ^{-1}(a x)} x}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\frac {(1-a n x) e^{n \coth ^{-1}(a x)}}{a^2 c \left (1-n^2\right ) \sqrt {c-a^2 c x^2}} \]
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Rule 6321
Rubi steps \begin{align*} \text {integral}& = \frac {e^{n \coth ^{-1}(a x)} (1-a n x)}{a^2 c \left (1-n^2\right ) \sqrt {c-a^2 c x^2}} \\ \end{align*}
Time = 0.37 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.93 \[ \int \frac {e^{n \coth ^{-1}(a x)} x}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\frac {e^{n \coth ^{-1}(a x)} (-1+a n x)}{a^2 c \left (-1+n^2\right ) \sqrt {c-a^2 c x^2}} \]
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Time = 0.54 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.07
method | result | size |
gosper | \(-\frac {\left (a x -1\right ) \left (a x +1\right ) \left (a n x -1\right ) {\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )}}{a^{2} \left (n^{2}-1\right ) \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}\) | \(49\) |
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Time = 0.25 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.78 \[ \int \frac {e^{n \coth ^{-1}(a x)} x}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\frac {\sqrt {-a^{2} c x^{2} + c} {\left (a n x - 1\right )} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{a^{2} c^{2} n^{2} - a^{2} c^{2} - {\left (a^{4} c^{2} n^{2} - a^{4} c^{2}\right )} x^{2}} \]
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\[ \int \frac {e^{n \coth ^{-1}(a x)} x}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\int \frac {x e^{n \operatorname {acoth}{\left (a x \right )}}}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {e^{n \coth ^{-1}(a x)} x}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\int { \frac {x \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {e^{n \coth ^{-1}(a x)} x}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\int { \frac {x \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \]
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Time = 4.13 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.76 \[ \int \frac {e^{n \coth ^{-1}(a x)} x}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=-\frac {\left (\frac {1}{a^2\,c\,\left (n^2-1\right )}-\frac {n\,x}{a\,c\,\left (n^2-1\right )}\right )\,{\left (\frac {a\,x+1}{a\,x}\right )}^{n/2}}{\sqrt {c-a^2\,c\,x^2}\,{\left (\frac {a\,x-1}{a\,x}\right )}^{n/2}} \]
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