Integrand size = 22, antiderivative size = 152 \[ \int \frac {e^{\coth ^{-1}(a x)}}{\sqrt {c-\frac {c}{a x}}} \, dx=\frac {\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x}{\sqrt {c-\frac {c}{a x}}}+\frac {3 \sqrt {1-\frac {1}{a x}} \text {arctanh}\left (\sqrt {1+\frac {1}{a x}}\right )}{a \sqrt {c-\frac {c}{a x}}}-\frac {2 \sqrt {2} \sqrt {1-\frac {1}{a x}} \text {arctanh}\left (\frac {\sqrt {1+\frac {1}{a x}}}{\sqrt {2}}\right )}{a \sqrt {c-\frac {c}{a x}}} \]
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Time = 0.11 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {6317, 6314, 101, 162, 65, 214, 212} \[ \int \frac {e^{\coth ^{-1}(a x)}}{\sqrt {c-\frac {c}{a x}}} \, dx=\frac {3 \sqrt {1-\frac {1}{a x}} \text {arctanh}\left (\sqrt {\frac {1}{a x}+1}\right )}{a \sqrt {c-\frac {c}{a x}}}-\frac {2 \sqrt {2} \sqrt {1-\frac {1}{a x}} \text {arctanh}\left (\frac {\sqrt {\frac {1}{a x}+1}}{\sqrt {2}}\right )}{a \sqrt {c-\frac {c}{a x}}}+\frac {x \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}}{\sqrt {c-\frac {c}{a x}}} \]
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Rule 65
Rule 101
Rule 162
Rule 212
Rule 214
Rule 6314
Rule 6317
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1-\frac {1}{a x}} \int \frac {e^{\coth ^{-1}(a x)}}{\sqrt {1-\frac {1}{a x}}} \, dx}{\sqrt {c-\frac {c}{a x}}} \\ & = -\frac {\sqrt {1-\frac {1}{a x}} \text {Subst}\left (\int \frac {\sqrt {1+\frac {x}{a}}}{x^2 \left (1-\frac {x}{a}\right )} \, dx,x,\frac {1}{x}\right )}{\sqrt {c-\frac {c}{a x}}} \\ & = \frac {\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x}{\sqrt {c-\frac {c}{a x}}}-\frac {\sqrt {1-\frac {1}{a x}} \text {Subst}\left (\int \frac {\frac {3}{2 a}+\frac {x}{2 a^2}}{x \left (1-\frac {x}{a}\right ) \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{\sqrt {c-\frac {c}{a x}}} \\ & = \frac {\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x}{\sqrt {c-\frac {c}{a x}}}-\frac {\left (2 \sqrt {1-\frac {1}{a x}}\right ) \text {Subst}\left (\int \frac {1}{\left (1-\frac {x}{a}\right ) \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{a^2 \sqrt {c-\frac {c}{a x}}}-\frac {\left (3 \sqrt {1-\frac {1}{a x}}\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{2 a \sqrt {c-\frac {c}{a x}}} \\ & = \frac {\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x}{\sqrt {c-\frac {c}{a x}}}-\frac {\left (3 \sqrt {1-\frac {1}{a x}}\right ) \text {Subst}\left (\int \frac {1}{-a+a x^2} \, dx,x,\sqrt {1+\frac {1}{a x}}\right )}{\sqrt {c-\frac {c}{a x}}}-\frac {\left (4 \sqrt {1-\frac {1}{a x}}\right ) \text {Subst}\left (\int \frac {1}{2-x^2} \, dx,x,\sqrt {1+\frac {1}{a x}}\right )}{a \sqrt {c-\frac {c}{a x}}} \\ & = \frac {\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x}{\sqrt {c-\frac {c}{a x}}}+\frac {3 \sqrt {1-\frac {1}{a x}} \text {arctanh}\left (\sqrt {1+\frac {1}{a x}}\right )}{a \sqrt {c-\frac {c}{a x}}}-\frac {2 \sqrt {2} \sqrt {1-\frac {1}{a x}} \text {arctanh}\left (\frac {\sqrt {1+\frac {1}{a x}}}{\sqrt {2}}\right )}{a \sqrt {c-\frac {c}{a x}}} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.62 \[ \int \frac {e^{\coth ^{-1}(a x)}}{\sqrt {c-\frac {c}{a x}}} \, dx=\frac {\sqrt {1-\frac {1}{a x}} \left (\sqrt {1+\frac {1}{a x}} x+\frac {3 \text {arctanh}\left (\sqrt {1+\frac {1}{a x}}\right )}{a}-\frac {2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {1+\frac {1}{a x}}}{\sqrt {2}}\right )}{a}\right )}{\sqrt {c-\frac {c}{a x}}} \]
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Time = 0.26 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.99
method | result | size |
default | \(\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (2 \sqrt {\left (a x +1\right ) x}\, a^{\frac {3}{2}} \sqrt {\frac {1}{a}}-2 \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {\left (a x +1\right ) x}\, a +3 a x +1}{a x -1}\right ) \sqrt {a}+3 \ln \left (\frac {2 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right ) a \sqrt {\frac {1}{a}}\right )}{2 \sqrt {\frac {a x -1}{a x +1}}\, a^{\frac {3}{2}} c \sqrt {\left (a x +1\right ) x}\, \sqrt {\frac {1}{a}}}\) | \(151\) |
risch | \(\frac {a x -1}{a \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\frac {c \left (a x -1\right )}{a x}}}+\frac {\left (\frac {3 \ln \left (\frac {\frac {1}{2} a c +a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}+a c x}\right )}{2 a \sqrt {a^{2} c}}-\frac {\sqrt {2}\, \ln \left (\frac {4 c +3 \left (x -\frac {1}{a}\right ) a c +2 \sqrt {2}\, \sqrt {c}\, \sqrt {a^{2} c \left (x -\frac {1}{a}\right )^{2}+3 \left (x -\frac {1}{a}\right ) a c +2 c}}{x -\frac {1}{a}}\right )}{a^{2} \sqrt {c}}\right ) \sqrt {\left (a x +1\right ) a c x}\, \left (a x -1\right )}{\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, x}\) | \(223\) |
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Time = 0.33 (sec) , antiderivative size = 517, normalized size of antiderivative = 3.40 \[ \int \frac {e^{\coth ^{-1}(a x)}}{\sqrt {c-\frac {c}{a x}}} \, dx=\left [\frac {3 \, {\left (a x - 1\right )} \sqrt {c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x + 4 \, {\left (2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + a x\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) + 4 \, {\left (a^{2} x^{2} + a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}} + \frac {2 \, \sqrt {2} {\left (a c x - c\right )} \log \left (-\frac {17 \, a^{3} x^{3} - 3 \, a^{2} x^{2} - 13 \, a x - \frac {4 \, \sqrt {2} {\left (3 \, a^{3} x^{3} + 4 \, a^{2} x^{2} + a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{\sqrt {c}} - 1}{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right )}{\sqrt {c}}}{4 \, {\left (a^{2} c x - a c\right )}}, \frac {2 \, \sqrt {2} {\left (a c x - c\right )} \sqrt {-\frac {1}{c}} \arctan \left (\frac {2 \, \sqrt {2} {\left (a^{2} x^{2} + a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {-\frac {1}{c}} \sqrt {\frac {a c x - c}{a x}}}{3 \, a^{2} x^{2} - 2 \, a x - 1}\right ) - 3 \, {\left (a x - 1\right )} \sqrt {-c} \arctan \left (\frac {2 \, {\left (a^{2} x^{2} + a x\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) + 2 \, {\left (a^{2} x^{2} + a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, {\left (a^{2} c x - a c\right )}}\right ] \]
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\[ \int \frac {e^{\coth ^{-1}(a x)}}{\sqrt {c-\frac {c}{a x}}} \, dx=\int \frac {1}{\sqrt {\frac {a x - 1}{a x + 1}} \sqrt {- c \left (-1 + \frac {1}{a x}\right )}}\, dx \]
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\[ \int \frac {e^{\coth ^{-1}(a x)}}{\sqrt {c-\frac {c}{a x}}} \, dx=\int { \frac {1}{\sqrt {c - \frac {c}{a x}} \sqrt {\frac {a x - 1}{a x + 1}}} \,d x } \]
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\[ \int \frac {e^{\coth ^{-1}(a x)}}{\sqrt {c-\frac {c}{a x}}} \, dx=\int { \frac {1}{\sqrt {c - \frac {c}{a x}} \sqrt {\frac {a x - 1}{a x + 1}}} \,d x } \]
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Timed out. \[ \int \frac {e^{\coth ^{-1}(a x)}}{\sqrt {c-\frac {c}{a x}}} \, dx=\int \frac {1}{\sqrt {c-\frac {c}{a\,x}}\,\sqrt {\frac {a\,x-1}{a\,x+1}}} \,d x \]
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