Integrand size = 20, antiderivative size = 108 \[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right ) \, dx=\frac {2 c \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}{a}+c \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}} x+\frac {c \csc ^{-1}(a x)}{a}-\frac {c \text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{a} \]
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Time = 0.07 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {6329, 99, 159, 21, 132, 41, 222, 94, 214} \[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right ) \, dx=-\frac {c \text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )}{a}+c x \sqrt {\frac {1}{a x}+1} \left (1-\frac {1}{a x}\right )^{3/2}+\frac {2 c \sqrt {\frac {1}{a x}+1} \sqrt {1-\frac {1}{a x}}}{a}+\frac {c \csc ^{-1}(a x)}{a} \]
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Rule 21
Rule 41
Rule 94
Rule 99
Rule 132
Rule 159
Rule 214
Rule 222
Rule 6329
Rubi steps \begin{align*} \text {integral}& = -\left (c \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{3/2} \sqrt {1+\frac {x}{a}}}{x^2} \, dx,x,\frac {1}{x}\right )\right ) \\ & = c \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}} x-c \text {Subst}\left (\int \frac {\left (-\frac {1}{a}-\frac {2 x}{a^2}\right ) \sqrt {1-\frac {x}{a}}}{x \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {2 c \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}{a}+c \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}} x-(a c) \text {Subst}\left (\int \frac {-\frac {1}{a^2}-\frac {x}{a^3}}{x \sqrt {1-\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {2 c \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}{a}+c \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}} x+\frac {c \text {Subst}\left (\int \frac {\sqrt {1+\frac {x}{a}}}{x \sqrt {1-\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{a} \\ & = \frac {2 c \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}{a}+c \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}} x+\frac {c \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{a^2}+\frac {c \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{a} \\ & = \frac {2 c \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}{a}+c \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}} x+\frac {c \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{a^2}-\frac {c \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} \\ & = \frac {2 c \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}{a}+c \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}} x+\frac {c \csc ^{-1}(a x)}{a}-\frac {c \text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{a} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.51 \[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right ) \, dx=\frac {c \left (\sqrt {1-\frac {1}{a^2 x^2}} (1+a x)+\arcsin \left (\frac {1}{a x}\right )-\log \left (\left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )\right )}{a} \]
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Time = 0.08 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.18
method | result | size |
risch | \(\frac {\left (a x +1\right ) c \sqrt {\frac {a x -1}{a x +1}}}{x \,a^{2}}+\frac {\left (-\frac {a \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{\sqrt {a^{2}}}+\sqrt {\left (a x -1\right ) \left (a x +1\right )}+\arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )\right ) c \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{a \left (a x -1\right )}\) | \(127\) |
default | \(-\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) c \left (-\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{2} x^{2}+\left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}-\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a x -a x \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )+\ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{2} x \right )}{\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{2} x \sqrt {a^{2}}}\) | \(166\) |
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Time = 0.26 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.99 \[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right ) \, dx=-\frac {2 \, a c x \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) + a c x \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - a c x \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) - {\left (a^{2} c x^{2} + 2 \, a c x + c\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} x} \]
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\[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right ) \, dx=\frac {c \left (\int a^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}\, dx + \int \left (- \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{x^{2}}\right )\, dx\right )}{a^{2}} \]
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Time = 0.30 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.08 \[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right ) \, dx=-a {\left (\frac {4 \, c \sqrt {\frac {a x - 1}{a x + 1}}}{\frac {{\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} - a^{2}} + \frac {2 \, c \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right )}{a^{2}} + \frac {c \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac {c \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}}\right )} \]
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Time = 0.28 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.12 \[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right ) \, dx=-\frac {2 \, c \arctan \left (-x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1}\right ) \mathrm {sgn}\left (a x + 1\right )}{a} + \frac {c \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right ) \mathrm {sgn}\left (a x + 1\right )}{{\left | a \right |}} + \frac {\sqrt {a^{2} x^{2} - 1} c \mathrm {sgn}\left (a x + 1\right )}{a} + \frac {2 \, c \mathrm {sgn}\left (a x + 1\right )}{{\left ({\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{2} + 1\right )} {\left | a \right |}} \]
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Time = 3.87 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.78 \[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right ) \, dx=\frac {4\,c\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{a-\frac {a\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}}-\frac {2\,c\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a}-\frac {2\,c\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a} \]
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