Integrand size = 20, antiderivative size = 29 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{\sqrt {c-a c x}} \, dx=\frac {2 e^{-\coth ^{-1}(a x)} (1+a x)}{a \sqrt {c-a c x}} \]
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Time = 0.03 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {6309} \[ \int \frac {e^{-\coth ^{-1}(a x)}}{\sqrt {c-a c x}} \, dx=\frac {2 (a x+1) e^{-\coth ^{-1}(a x)}}{a \sqrt {c-a c x}} \]
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Rule 6309
Rubi steps \begin{align*} \text {integral}& = \frac {2 e^{-\coth ^{-1}(a x)} (1+a x)}{a \sqrt {c-a c x}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.97 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{\sqrt {c-a c x}} \, dx=\frac {2 \sqrt {1-\frac {1}{a^2 x^2}} x}{\sqrt {c-a c x}} \]
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Time = 0.41 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.21
method | result | size |
gosper | \(\frac {2 \left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}{a \sqrt {-a c x +c}}\) | \(35\) |
risch | \(\frac {2 \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}{\sqrt {-c \left (a x -1\right )}\, a}\) | \(36\) |
default | \(-\frac {2 \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) \sqrt {-c \left (a x -1\right )}}{\left (a x -1\right ) c a}\) | \(46\) |
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none
Time = 0.25 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.52 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{\sqrt {c-a c x}} \, dx=-\frac {2 \, \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c x - a c} \]
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\[ \int \frac {e^{-\coth ^{-1}(a x)}}{\sqrt {c-a c x}} \, dx=\int \frac {\sqrt {\frac {a x - 1}{a x + 1}}}{\sqrt {- c \left (a x - 1\right )}}\, dx \]
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none
Time = 0.21 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{\sqrt {c-a c x}} \, dx=-\frac {2 \, {\left (a \sqrt {-c} x + \sqrt {-c}\right )}}{\sqrt {a x + 1} a c} \]
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Exception generated. \[ \int \frac {e^{-\coth ^{-1}(a x)}}{\sqrt {c-a c x}} \, dx=\text {Exception raised: TypeError} \]
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Time = 3.99 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.17 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{\sqrt {c-a c x}} \, dx=\frac {\left (2\,x+\frac {2}{a}\right )\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{\sqrt {c-a\,c\,x}} \]
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