Integrand size = 27, antiderivative size = 125 \[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx=\frac {2 a c^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{3 \left (c-\frac {c}{a x}\right )^{3/2}}+\frac {4 a c \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}-4 \sqrt {2} a \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {2} \sqrt {c-\frac {c}{a x}}}\right ) \]
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Time = 0.17 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {6313, 679, 675, 214} \[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx=-4 \sqrt {2} a \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {2} \sqrt {c-\frac {c}{a x}}}\right )+\frac {2 a c^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{3 \left (c-\frac {c}{a x}\right )^{3/2}}+\frac {4 a c \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}} \]
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Rule 214
Rule 675
Rule 679
Rule 6313
Rubi steps \begin{align*} \text {integral}& = -\left (c^3 \text {Subst}\left (\int \frac {\left (1-\frac {x^2}{a^2}\right )^{3/2}}{\left (c-\frac {c x}{a}\right )^{5/2}} \, 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}}-\left (2 c^2\right ) \text {Subst}\left (\int \frac {\sqrt {1-\frac {x^2}{a^2}}}{\left (c-\frac {c x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\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}}+\frac {4 a c \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}-(4 c) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {c x}{a}} \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\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}}+\frac {4 a c \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}+\frac {\left (8 c^2\right ) \text {Subst}\left (\int \frac {1}{-\frac {2 c}{a^2}+\frac {c^2 x^2}{a^2}} \, dx,x,\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right )}{a} \\ & = \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}}+\frac {4 a c \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}-4 \sqrt {2} a \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {2} \sqrt {c-\frac {c}{a x}}}\right ) \\ \end{align*}
Time = 0.38 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.24 \[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx=\frac {2 a \left (\sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-\frac {c}{a x}} (1+7 a x)+3 \sqrt {2} \sqrt {c} (-1+a x) \log \left ((-1+a x)^2\right )-3 \sqrt {2} \sqrt {c} (-1+a x) \log \left (2 \sqrt {2} a^2 \sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-\frac {c}{a x}} x^2+c \left (-1-2 a x+3 a^2 x^2\right )\right )\right )}{-3+3 a x} \]
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Time = 0.25 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.12
method | result | size |
default | \(\frac {2 \left (a x -1\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \left (-3 a \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {\left (a x +1\right ) x}\, a +3 a x +1}{a x -1}\right ) x^{2}+7 x \sqrt {\left (a x +1\right ) x}\, a \sqrt {\frac {1}{a}}+\sqrt {\left (a x +1\right ) x}\, \sqrt {\frac {1}{a}}\right )}{3 \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) x \sqrt {\left (a x +1\right ) x}\, \sqrt {\frac {1}{a}}}\) | \(140\) |
risch | \(\frac {2 \left (7 a^{2} x^{2}+8 a x +1\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}}{3 x \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}-\frac {2 a \sqrt {2}\, \ln \left (\frac {4 c +3 \left (x -\frac {1}{a}\right ) a c +2 \sqrt {2}\, \sqrt {c}\, \sqrt {a^{2} c \left (x -\frac {1}{a}\right )^{2}+3 \left (x -\frac {1}{a}\right ) a c +2 c}}{x -\frac {1}{a}}\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\left (a x +1\right ) a c x}}{\sqrt {c}\, \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}\) | \(180\) |
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Time = 0.30 (sec) , antiderivative size = 353, normalized size of antiderivative = 2.82 \[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx=\left [\frac {3 \, \sqrt {2} {\left (a^{2} x^{2} - a x\right )} \sqrt {c} \log \left (-\frac {17 \, a^{3} c x^{3} - 3 \, a^{2} c x^{2} - 13 \, a c x - 4 \, \sqrt {2} {\left (3 \, a^{3} x^{3} + 4 \, a^{2} x^{2} + a x\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}} - c}{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right ) + 2 \, {\left (7 \, a^{2} x^{2} + 8 \, a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{3 \, {\left (a x^{2} - x\right )}}, \frac {2 \, {\left (3 \, \sqrt {2} {\left (a^{2} x^{2} - a x\right )} \sqrt {-c} \arctan \left (\frac {2 \, \sqrt {2} {\left (a^{2} x^{2} + a x\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{3 \, a^{2} c x^{2} - 2 \, a c x - c}\right ) + {\left (7 \, a^{2} x^{2} + 8 \, a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}\right )}}{3 \, {\left (a x^{2} - x\right )}}\right ] \]
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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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\[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx=\int { \frac {\sqrt {c - \frac {c}{a x}}}{x^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} \,d x } \]
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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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Timed out. \[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx=\int \frac {\sqrt {c-\frac {c}{a\,x}}}{x^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}} \,d x \]
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