Integrand size = 25, antiderivative size = 37 \[ \int \frac {e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx=-\frac {2 a c^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{3 \left (c-\frac {c}{a x}\right )^{3/2}} \]
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Time = 0.11 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {6313, 663} \[ \int \frac {e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx=-\frac {2 a c^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{3 \left (c-\frac {c}{a x}\right )^{3/2}} \]
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Rule 663
Rule 6313
Rubi steps \begin{align*} \text {integral}& = -\left (c \text {Subst}\left (\int \frac {\sqrt {1-\frac {x^2}{a^2}}}{\sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )\right ) \\ & = -\frac {2 a c^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{3 \left (c-\frac {c}{a x}\right )^{3/2}} \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.22 \[ \int \frac {e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx=-\frac {2 a \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-\frac {c}{a x}} (1+a x)}{-3+3 a x} \]
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Time = 0.05 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.11
method | result | size |
gosper | \(-\frac {2 \left (a x +1\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}}{3 x \sqrt {\frac {a x -1}{a x +1}}}\) | \(41\) |
default | \(-\frac {2 \left (a x +1\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}}{3 x \sqrt {\frac {a x -1}{a x +1}}}\) | \(41\) |
risch | \(-\frac {2 \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \left (a^{2} x^{2}+2 a x +1\right )}{3 \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) x}\) | \(56\) |
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Time = 0.25 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.57 \[ \int \frac {e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx=-\frac {2 \, {\left (a^{2} x^{2} + 2 \, a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{3 \, {\left (a x^{2} - x\right )}} \]
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\[ \int \frac {e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx=\int \frac {\sqrt {- c \left (-1 + \frac {1}{a x}\right )}}{x^{2} \sqrt {\frac {a x - 1}{a x + 1}}}\, dx \]
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\[ \int \frac {e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx=\int { \frac {\sqrt {c - \frac {c}{a x}}}{x^{2} \sqrt {\frac {a x - 1}{a x + 1}}} \,d x } \]
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\[ \int \frac {e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx=\int { \frac {\sqrt {c - \frac {c}{a x}}}{x^{2} \sqrt {\frac {a x - 1}{a x + 1}}} \,d x } \]
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Time = 4.44 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.27 \[ \int \frac {e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx=-\frac {2\,\sqrt {c-\frac {c}{a\,x}}\,{\left (a\,x+1\right )}^2\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{3\,x\,\left (a\,x-1\right )} \]
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