Integrand size = 27, antiderivative size = 69 \[ \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^3} \, dx=4 a^2 \sqrt {c-\frac {c}{a x}}-\frac {2 a^2 \left (c-\frac {c}{a x}\right )^{3/2}}{c}+\frac {2 a^2 \left (c-\frac {c}{a x}\right )^{5/2}}{5 c^2} \]
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Time = 0.25 (sec) , antiderivative size = 69, 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, 457, 78} \[ \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^3} \, dx=\frac {2 a^2 \left (c-\frac {c}{a x}\right )^{5/2}}{5 c^2}-\frac {2 a^2 \left (c-\frac {c}{a x}\right )^{3/2}}{c}+4 a^2 \sqrt {c-\frac {c}{a x}} \]
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Rule 25
Rule 78
Rule 457
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^3} \, dx \\ & = -\int \frac {\sqrt {c-\frac {c}{a x}} (1+a x)}{x^3 (1-a x)} \, dx \\ & = \frac {c \int \frac {1+a x}{\sqrt {c-\frac {c}{a x}} x^4} \, dx}{a} \\ & = \frac {c \int \frac {a+\frac {1}{x}}{\sqrt {c-\frac {c}{a x}} x^3} \, dx}{a} \\ & = -\frac {c \text {Subst}\left (\int \frac {x (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^2}{\sqrt {c-\frac {c x}{a}}}-\frac {3 a^2 \sqrt {c-\frac {c x}{a}}}{c}+\frac {a^2 \left (c-\frac {c x}{a}\right )^{3/2}}{c^2}\right ) \, dx,x,\frac {1}{x}\right )}{a} \\ & = 4 a^2 \sqrt {c-\frac {c}{a x}}-\frac {2 a^2 \left (c-\frac {c}{a x}\right )^{3/2}}{c}+\frac {2 a^2 \left (c-\frac {c}{a x}\right )^{5/2}}{5 c^2} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.52 \[ \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^3} \, dx=\frac {2 \sqrt {c-\frac {c}{a x}} \left (1+3 a x+6 a^2 x^2\right )}{5 x^2} \]
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Time = 0.48 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.51
method | result | size |
gosper | \(\frac {2 \left (6 a^{2} x^{2}+3 a x +1\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}}{5 x^{2}}\) | \(35\) |
trager | \(\frac {2 \left (6 a^{2} x^{2}+3 a x +1\right ) \sqrt {-\frac {-a c x +c}{a x}}}{5 x^{2}}\) | \(37\) |
risch | \(\frac {2 \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \left (6 a^{3} x^{3}-3 a^{2} x^{2}-2 a x -1\right )}{5 \left (a x -1\right ) x^{2}}\) | \(50\) |
default | \(-\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, \left (-10 \sqrt {a \,x^{2}-x}\, a^{\frac {7}{2}} x^{4}-10 a^{\frac {7}{2}} \sqrt {\left (a x -1\right ) x}\, x^{4}+20 a^{\frac {5}{2}} \left (a \,x^{2}-x \right )^{\frac {3}{2}} x^{2}+5 \ln \left (\frac {2 \sqrt {a \,x^{2}-x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) a^{3} x^{4}-5 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) a^{3} x^{4}+8 a^{\frac {3}{2}} \left (a \,x^{2}-x \right )^{\frac {3}{2}} x +2 \left (a \,x^{2}-x \right )^{\frac {3}{2}} \sqrt {a}\right )}{5 x^{3} \sqrt {\left (a x -1\right ) x}\, \sqrt {a}}\) | \(192\) |
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Time = 0.25 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.52 \[ \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^3} \, dx=\frac {2 \, {\left (6 \, a^{2} x^{2} + 3 \, a x + 1\right )} \sqrt {\frac {a c x - c}{a x}}}{5 \, x^{2}} \]
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\[ \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^3} \, dx=\int \frac {\sqrt {- c \left (-1 + \frac {1}{a x}\right )} \left (a x + 1\right )}{x^{3} \left (a x - 1\right )}\, dx \]
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\[ \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^3} \, dx=\int { \frac {{\left (a x + 1\right )} \sqrt {c - \frac {c}{a x}}}{{\left (a x - 1\right )} x^{3}} \,d x } \]
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Exception generated. \[ \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^3} \, dx=\text {Exception raised: TypeError} \]
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Time = 3.90 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.46 \[ \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^3} \, dx=\frac {2\,\sqrt {c-\frac {c}{a\,x}}\,\left (6\,a^2\,x^2+3\,a\,x+1\right )}{5\,x^2} \]
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