Integrand size = 22, antiderivative size = 105 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{c-\frac {c}{a^2 x^2}} \, dx=\frac {2 \sqrt {1-\frac {1}{a x}}}{a c \sqrt {1+\frac {1}{a x}}}+\frac {\sqrt {1-\frac {1}{a x}} x}{c \sqrt {1+\frac {1}{a x}}}-\frac {\text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{a c} \]
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Time = 0.06 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {6329, 105, 21, 96, 94, 214} \[ \int \frac {e^{-\coth ^{-1}(a x)}}{c-\frac {c}{a^2 x^2}} \, dx=-\frac {\text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )}{a c}+\frac {x \sqrt {1-\frac {1}{a x}}}{c \sqrt {\frac {1}{a x}+1}}+\frac {2 \sqrt {1-\frac {1}{a x}}}{a c \sqrt {\frac {1}{a x}+1}} \]
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Rule 21
Rule 94
Rule 96
Rule 105
Rule 214
Rule 6329
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {1}{x^2 \sqrt {1-\frac {x}{a}} \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{c} \\ & = \frac {\sqrt {1-\frac {1}{a x}} x}{c \sqrt {1+\frac {1}{a x}}}+\frac {\text {Subst}\left (\int \frac {\frac {1}{a}-\frac {x}{a^2}}{x \sqrt {1-\frac {x}{a}} \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{c} \\ & = \frac {\sqrt {1-\frac {1}{a x}} x}{c \sqrt {1+\frac {1}{a x}}}+\frac {\text {Subst}\left (\int \frac {\sqrt {1-\frac {x}{a}}}{x \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{a c} \\ & = \frac {2 \sqrt {1-\frac {1}{a x}}}{a c \sqrt {1+\frac {1}{a x}}}+\frac {\sqrt {1-\frac {1}{a x}} x}{c \sqrt {1+\frac {1}{a x}}}+\frac {\text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{a c} \\ & = \frac {2 \sqrt {1-\frac {1}{a x}}}{a c \sqrt {1+\frac {1}{a x}}}+\frac {\sqrt {1-\frac {1}{a x}} x}{c \sqrt {1+\frac {1}{a x}}}-\frac {\text {Subst}\left (\int \frac {1}{\frac {1}{a}-\frac {x^2}{a}} \, dx,x,\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{a^2 c} \\ & = \frac {2 \sqrt {1-\frac {1}{a x}}}{a c \sqrt {1+\frac {1}{a x}}}+\frac {\sqrt {1-\frac {1}{a x}} x}{c \sqrt {1+\frac {1}{a x}}}-\frac {\text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{a c} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.54 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{c-\frac {c}{a^2 x^2}} \, dx=\frac {\frac {\sqrt {1-\frac {1}{a^2 x^2}} x (2+a x)}{1+a x}-\frac {\log \left (\left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )}{a}}{c} \]
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Time = 0.20 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.31
method | result | size |
risch | \(\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}{a c}+\frac {\left (-\frac {\ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{a^{2} \sqrt {a^{2}}}+\frac {\sqrt {a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a \left (x +\frac {1}{a}\right )}}{a^{4} \left (x +\frac {1}{a}\right )}\right ) a^{2} \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{c \left (a x -1\right )}\) | \(138\) |
default | \(-\frac {\left (-3 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{2} x^{2}+2 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}+\left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}-6 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a x +4 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{2} x -3 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}+2 a \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right )\right ) \sqrt {\frac {a x -1}{a x +1}}}{2 a \sqrt {a^{2}}\, \left (a x +1\right ) c \sqrt {\left (a x -1\right ) \left (a x +1\right )}}\) | \(250\) |
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Time = 0.25 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.64 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{c-\frac {c}{a^2 x^2}} \, dx=\frac {{\left (a x + 2\right )} \sqrt {\frac {a x - 1}{a x + 1}} - \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) + \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a c} \]
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\[ \int \frac {e^{-\coth ^{-1}(a x)}}{c-\frac {c}{a^2 x^2}} \, dx=\frac {a^{2} \int \frac {x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a^{2} x^{2} - 1}\, dx}{c} \]
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Time = 0.20 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.15 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{c-\frac {c}{a^2 x^2}} \, dx=-a {\left (\frac {2 \, \sqrt {\frac {a x - 1}{a x + 1}}}{\frac {{\left (a x - 1\right )} a^{2} c}{a x + 1} - a^{2} c} + \frac {\log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c} - \frac {\log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2} c} - \frac {\sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c}\right )} \]
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\[ \int \frac {e^{-\coth ^{-1}(a x)}}{c-\frac {c}{a^2 x^2}} \, dx=\int { \frac {\sqrt {\frac {a x - 1}{a x + 1}}}{c - \frac {c}{a^{2} x^{2}}} \,d x } \]
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Time = 0.06 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.82 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{c-\frac {c}{a^2 x^2}} \, dx=\frac {2\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{a\,c-\frac {a\,c\,\left (a\,x-1\right )}{a\,x+1}}+\frac {\sqrt {\frac {a\,x-1}{a\,x+1}}}{a\,c}-\frac {2\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a\,c} \]
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