Integrand size = 27, antiderivative size = 42 \[ \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx=4 a \sqrt {c-\frac {c}{a x}}-\frac {2 a \left (c-\frac {c}{a x}\right )^{3/2}}{3 c} \]
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Time = 0.24 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6302, 6268, 25, 528, 455, 45} \[ \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx=4 a \sqrt {c-\frac {c}{a x}}-\frac {2 a \left (c-\frac {c}{a x}\right )^{3/2}}{3 c} \]
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Rule 25
Rule 45
Rule 455
Rule 528
Rule 6268
Rule 6302
Rubi steps \begin{align*} \text {integral}& = -\int \frac {e^{2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx \\ & = -\int \frac {\sqrt {c-\frac {c}{a x}} (1+a x)}{x^2 (1-a x)} \, dx \\ & = \frac {c \int \frac {1+a x}{\sqrt {c-\frac {c}{a x}} x^3} \, dx}{a} \\ & = \frac {c \int \frac {a+\frac {1}{x}}{\sqrt {c-\frac {c}{a x}} x^2} \, dx}{a} \\ & = -\frac {c \text {Subst}\left (\int \frac {a+x}{\sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )}{a} \\ & = -\frac {c \text {Subst}\left (\int \left (\frac {2 a}{\sqrt {c-\frac {c x}{a}}}-\frac {a \sqrt {c-\frac {c x}{a}}}{c}\right ) \, dx,x,\frac {1}{x}\right )}{a} \\ & = 4 a \sqrt {c-\frac {c}{a x}}-\frac {2 a \left (c-\frac {c}{a x}\right )^{3/2}}{3 c} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.67 \[ \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx=\frac {2 \sqrt {c-\frac {c}{a x}} (1+5 a x)}{3 x} \]
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Time = 0.48 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.64
method | result | size |
gosper | \(\frac {2 \left (5 a x +1\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}}{3 x}\) | \(27\) |
trager | \(\frac {2 \left (5 a x +1\right ) \sqrt {-\frac {-a c x +c}{a x}}}{3 x}\) | \(29\) |
risch | \(\frac {2 \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \left (5 a^{2} x^{2}-4 a x -1\right )}{3 \left (a x -1\right ) x}\) | \(42\) |
default | \(-\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, \left (-6 \sqrt {a \,x^{2}-x}\, a^{\frac {5}{2}} x^{3}-6 a^{\frac {5}{2}} \sqrt {\left (a x -1\right ) x}\, x^{3}+12 a^{\frac {3}{2}} \left (a \,x^{2}-x \right )^{\frac {3}{2}} x +3 \ln \left (\frac {2 \sqrt {a \,x^{2}-x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) a^{2} x^{3}-3 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) a^{2} x^{3}+2 \left (a \,x^{2}-x \right )^{\frac {3}{2}} \sqrt {a}\right )}{3 x^{2} \sqrt {\left (a x -1\right ) x}\, \sqrt {a}}\) | \(173\) |
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Time = 0.26 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.67 \[ \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx=\frac {2 \, {\left (5 \, a x + 1\right )} \sqrt {\frac {a c x - c}{a x}}}{3 \, x} \]
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\[ \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx=\int \frac {\sqrt {- c \left (-1 + \frac {1}{a x}\right )} \left (a x + 1\right )}{x^{2} \left (a x - 1\right )}\, dx \]
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\[ \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx=\int { \frac {{\left (a x + 1\right )} \sqrt {c - \frac {c}{a x}}}{{\left (a x - 1\right )} x^{2}} \,d x } \]
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Exception generated. \[ \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx=\text {Exception raised: TypeError} \]
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Time = 4.04 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.57 \[ \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx=\frac {2\,\sqrt {c-\frac {c}{a\,x}}\,\left (5\,a\,x+1\right )}{3\,x} \]
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