Integrand size = 23, antiderivative size = 124 \[ \int e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x \, dx=\frac {c \sqrt {1-\frac {1}{a^2 x^2}} x}{4 a \sqrt {c-\frac {c}{a x}}}+\frac {c \sqrt {1-\frac {1}{a^2 x^2}} x^2}{2 \sqrt {c-\frac {c}{a x}}}-\frac {\sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right )}{4 a^2} \]
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Time = 0.18 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {6313, 877, 887, 889, 214} \[ \int e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x \, dx=-\frac {\sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right )}{4 a^2}+\frac {c x^2 \sqrt {1-\frac {1}{a^2 x^2}}}{2 \sqrt {c-\frac {c}{a x}}}+\frac {c x \sqrt {1-\frac {1}{a^2 x^2}}}{4 a \sqrt {c-\frac {c}{a x}}} \]
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Rule 214
Rule 877
Rule 887
Rule 889
Rule 6313
Rubi steps \begin{align*} \text {integral}& = -\left (c \text {Subst}\left (\int \frac {\sqrt {1-\frac {x^2}{a^2}}}{x^3 \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )\right ) \\ & = \frac {c \sqrt {1-\frac {1}{a^2 x^2}} x^2}{2 \sqrt {c-\frac {c}{a x}}}-\frac {\text {Subst}\left (\int \frac {\sqrt {c-\frac {c x}{a}}}{x^2 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{4 a} \\ & = \frac {c \sqrt {1-\frac {1}{a^2 x^2}} x}{4 a \sqrt {c-\frac {c}{a x}}}+\frac {c \sqrt {1-\frac {1}{a^2 x^2}} x^2}{2 \sqrt {c-\frac {c}{a x}}}+\frac {\text {Subst}\left (\int \frac {\sqrt {c-\frac {c x}{a}}}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{8 a^2} \\ & = \frac {c \sqrt {1-\frac {1}{a^2 x^2}} x}{4 a \sqrt {c-\frac {c}{a x}}}+\frac {c \sqrt {1-\frac {1}{a^2 x^2}} x^2}{2 \sqrt {c-\frac {c}{a x}}}+\frac {c^2 \text {Subst}\left (\int \frac {1}{-\frac {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 )}{4 a^4} \\ & = \frac {c \sqrt {1-\frac {1}{a^2 x^2}} x}{4 a \sqrt {c-\frac {c}{a x}}}+\frac {c \sqrt {1-\frac {1}{a^2 x^2}} x^2}{2 \sqrt {c-\frac {c}{a x}}}-\frac {\sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right )}{4 a^2} \\ \end{align*}
Time = 1.43 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.19 \[ \int e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x \, dx=\frac {2 a^2 \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-\frac {c}{a x}} x^2 (1+2 a x)+\sqrt {c} (-1+a x) \log (1-a x)+\sqrt {c} (1-a x) \log \left (2 a^2 \sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-\frac {c}{a x}} x^2+c \left (-1-a x+2 a^2 x^2\right )\right )}{8 a^2 (-1+a x)} \]
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Time = 0.07 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.82
method | result | size |
default | \(-\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (-4 a^{\frac {3}{2}} x \sqrt {\left (a x +1\right ) x}-2 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}+\ln \left (\frac {2 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right )\right )}{8 \sqrt {\frac {a x -1}{a x +1}}\, a^{\frac {3}{2}} \sqrt {\left (a x +1\right ) x}}\) | \(102\) |
risch | \(\frac {\left (2 a x +1\right ) x \sqrt {\frac {c \left (a x -1\right )}{a x}}}{4 a \sqrt {\frac {a x -1}{a x +1}}}-\frac {\ln \left (\frac {\frac {1}{2} a c +a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}+a c x}\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\left (a x +1\right ) a c x}}{8 a \sqrt {a^{2} c}\, \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}\) | \(140\) |
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Time = 0.29 (sec) , antiderivative size = 317, normalized size of antiderivative = 2.56 \[ \int e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x \, dx=\left [\frac {{\left (a x - 1\right )} \sqrt {c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x - 4 \, {\left (2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + a x\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) + 4 \, {\left (2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{16 \, {\left (a^{3} x - a^{2}\right )}}, \frac {{\left (a x - 1\right )} \sqrt {-c} \arctan \left (\frac {2 \, {\left (a^{2} x^{2} + a x\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) + 2 \, {\left (2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{8 \, {\left (a^{3} x - a^{2}\right )}}\right ] \]
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\[ \int e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x \, dx=\int \frac {x \sqrt {- c \left (-1 + \frac {1}{a x}\right )}}{\sqrt {\frac {a x - 1}{a x + 1}}}\, dx \]
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\[ \int e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x \, dx=\int { \frac {\sqrt {c - \frac {c}{a x}} x}{\sqrt {\frac {a x - 1}{a x + 1}}} \,d x } \]
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\[ \int e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x \, dx=\int { \frac {\sqrt {c - \frac {c}{a x}} x}{\sqrt {\frac {a x - 1}{a x + 1}}} \,d x } \]
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Timed out. \[ \int e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x \, dx=\int \frac {x\,\sqrt {c-\frac {c}{a\,x}}}{\sqrt {\frac {a\,x-1}{a\,x+1}}} \,d x \]
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