Integrand size = 24, antiderivative size = 140 \[ \int e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} \, dx=\frac {9 \sqrt {c-\frac {c}{a x}}}{a \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {\sqrt {c-\frac {c}{a x}} x}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}-\frac {7 \sqrt {c-\frac {c}{a x}} \text {arctanh}\left (\sqrt {1+\frac {1}{a x}}\right )}{a \sqrt {1-\frac {1}{a x}}} \]
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Time = 0.08 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6317, 6314, 91, 79, 65, 214} \[ \int e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} \, dx=-\frac {7 \text {arctanh}\left (\sqrt {\frac {1}{a x}+1}\right ) \sqrt {c-\frac {c}{a x}}}{a \sqrt {1-\frac {1}{a x}}}+\frac {x \sqrt {c-\frac {c}{a x}}}{\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}}+\frac {9 \sqrt {c-\frac {c}{a x}}}{a \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}} \]
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Rule 65
Rule 79
Rule 91
Rule 214
Rule 6314
Rule 6317
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {c-\frac {c}{a x}} \int e^{-3 \coth ^{-1}(a x)} \sqrt {1-\frac {1}{a x}} \, dx}{\sqrt {1-\frac {1}{a x}}} \\ & = -\frac {\sqrt {c-\frac {c}{a x}} \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^2}{x^2 \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a x}}} \\ & = \frac {\sqrt {c-\frac {c}{a x}} x}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}-\frac {\sqrt {c-\frac {c}{a x}} \text {Subst}\left (\int \frac {-\frac {7}{2 a}+\frac {x}{a^2}}{x \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a x}}} \\ & = \frac {9 \sqrt {c-\frac {c}{a x}}}{a \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {\sqrt {c-\frac {c}{a x}} x}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {\left (7 \sqrt {c-\frac {c}{a x}}\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{2 a \sqrt {1-\frac {1}{a x}}} \\ & = \frac {9 \sqrt {c-\frac {c}{a x}}}{a \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {\sqrt {c-\frac {c}{a x}} x}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {\left (7 \sqrt {c-\frac {c}{a x}}\right ) \text {Subst}\left (\int \frac {1}{-a+a x^2} \, dx,x,\sqrt {1+\frac {1}{a x}}\right )}{\sqrt {1-\frac {1}{a x}}} \\ & = \frac {9 \sqrt {c-\frac {c}{a x}}}{a \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {\sqrt {c-\frac {c}{a x}} x}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}-\frac {7 \sqrt {c-\frac {c}{a x}} \text {arctanh}\left (\sqrt {1+\frac {1}{a x}}\right )}{a \sqrt {1-\frac {1}{a x}}} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.48 \[ \int e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} \, dx=\frac {\sqrt {c-\frac {c}{a x}} \left (9+a x-7 \sqrt {1+\frac {1}{a x}} \text {arctanh}\left (\sqrt {1+\frac {1}{a x}}\right )\right )}{a \sqrt {1-\frac {1}{a^2 x^2}}} \]
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Time = 0.12 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.04
method | result | size |
default | \(\frac {\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (2 a^{\frac {3}{2}} x \sqrt {\left (a x +1\right ) x}-7 \ln \left (\frac {2 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right ) a x +18 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}-7 \ln \left (\frac {2 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right )\right )}{2 \left (a x -1\right )^{2} \sqrt {a}\, \sqrt {\left (a x +1\right ) x}}\) | \(146\) |
risch | \(\frac {\left (a x +1\right ) x \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\frac {c \left (a x -1\right )}{a x}}}{a x -1}+\frac {\left (-\frac {7 \ln \left (\frac {\frac {1}{2} a c +a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}+a c x}\right )}{2 \sqrt {a^{2} c}}+\frac {8 \sqrt {a^{2} c \left (x +\frac {1}{a}\right )^{2}-\left (x +\frac {1}{a}\right ) a c}}{a^{2} c \left (x +\frac {1}{a}\right )}\right ) \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\left (a x +1\right ) a c x}}{a x -1}\) | \(180\) |
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Time = 0.28 (sec) , antiderivative size = 299, normalized size of antiderivative = 2.14 \[ \int e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} \, dx=\left [\frac {7 \, {\left (a x - 1\right )} \sqrt {c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x - 4 \, {\left (2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + a x\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) + 4 \, {\left (a^{2} x^{2} + 9 \, a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{4 \, {\left (a^{2} x - a\right )}}, \frac {7 \, {\left (a x - 1\right )} \sqrt {-c} \arctan \left (\frac {2 \, {\left (a^{2} x^{2} + a x\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) + 2 \, {\left (a^{2} x^{2} + 9 \, a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, {\left (a^{2} x - a\right )}}\right ] \]
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Timed out. \[ \int e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} \, dx=\text {Timed out} \]
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\[ \int e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} \, dx=\int { \sqrt {c - \frac {c}{a x}} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} \,d x } \]
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Exception generated. \[ \int e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} \, dx=\int \sqrt {c-\frac {c}{a\,x}}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2} \,d x \]
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