Integrand size = 27, antiderivative size = 109 \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx=\frac {64 a \sqrt {c-\frac {c}{a x}}}{3 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {16 a \left (c-\frac {c}{a x}\right )^{3/2}}{3 c \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {2 a \left (c-\frac {c}{a x}\right )^{5/2}}{3 c^2 \sqrt {1-\frac {1}{a^2 x^2}}} \]
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Time = 0.15 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6313, 671, 663} \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx=-\frac {2 a \left (c-\frac {c}{a x}\right )^{5/2}}{3 c^2 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {16 a \left (c-\frac {c}{a x}\right )^{3/2}}{3 c \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {64 a \sqrt {c-\frac {c}{a x}}}{3 \sqrt {1-\frac {1}{a^2 x^2}}} \]
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Rule 663
Rule 671
Rule 6313
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {\left (c-\frac {c x}{a}\right )^{7/2}}{\left (1-\frac {x^2}{a^2}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{c^3} \\ & = -\frac {2 a \left (c-\frac {c}{a x}\right )^{5/2}}{3 c^2 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {8 \text {Subst}\left (\int \frac {\left (c-\frac {c x}{a}\right )^{5/2}}{\left (1-\frac {x^2}{a^2}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{3 c^2} \\ & = -\frac {16 a \left (c-\frac {c}{a x}\right )^{3/2}}{3 c \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {2 a \left (c-\frac {c}{a x}\right )^{5/2}}{3 c^2 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {32 \text {Subst}\left (\int \frac {\left (c-\frac {c x}{a}\right )^{3/2}}{\left (1-\frac {x^2}{a^2}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{3 c} \\ & = \frac {64 a \sqrt {c-\frac {c}{a x}}}{3 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {16 a \left (c-\frac {c}{a x}\right )^{3/2}}{3 c \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {2 a \left (c-\frac {c}{a x}\right )^{5/2}}{3 c^2 \sqrt {1-\frac {1}{a^2 x^2}}} \\ \end{align*}
Time = 0.22 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.53 \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx=\frac {2 a \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-\frac {c}{a x}} \left (-1+10 a x+23 a^2 x^2\right )}{-3+3 a^2 x^2} \]
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Time = 0.15 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.57
method | result | size |
gosper | \(\frac {2 \left (a x +1\right ) \left (23 a^{2} x^{2}+10 a x -1\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{3 x \left (a x -1\right )^{2}}\) | \(62\) |
default | \(\frac {2 \left (a x +1\right ) \left (23 a^{2} x^{2}+10 a x -1\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{3 x \left (a x -1\right )^{2}}\) | \(62\) |
risch | \(\frac {2 \left (11 a^{2} x^{2}+10 a x -1\right ) \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\frac {c \left (a x -1\right )}{a x}}}{3 x \left (a x -1\right )}+\frac {8 a^{2} x \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\frac {c \left (a x -1\right )}{a x}}}{a x -1}\) | \(101\) |
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Time = 0.26 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.54 \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx=\frac {2 \, {\left (23 \, a^{2} x^{2} + 10 \, a x - 1\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{3 \, {\left (a x^{2} - x\right )}} \]
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Timed out. \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx=\text {Timed out} \]
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\[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx=\int { \frac {\sqrt {c - \frac {c}{a x}} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}{x^{2}} \,d x } \]
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Exception generated. \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx=\text {Exception raised: TypeError} \]
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Time = 4.19 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.50 \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx=\frac {2\,\sqrt {c-\frac {c}{a\,x}}\,\sqrt {\frac {a\,x-1}{a\,x+1}}\,\left (23\,a^2\,x^2+10\,a\,x-1\right )}{3\,x\,\left (a\,x-1\right )} \]
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