Integrand size = 22, antiderivative size = 38 \[ \int \frac {e^{\coth ^{-1}(a x)}}{\sqrt {c-a^2 c x^2}} \, dx=\frac {\sqrt {1-\frac {1}{a^2 x^2}} x \log (1-a x)}{\sqrt {c-a^2 c x^2}} \]
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Time = 0.13 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6327, 6328, 31} \[ \int \frac {e^{\coth ^{-1}(a x)}}{\sqrt {c-a^2 c x^2}} \, dx=\frac {x \sqrt {1-\frac {1}{a^2 x^2}} \log (1-a x)}{\sqrt {c-a^2 c x^2}} \]
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Rule 31
Rule 6327
Rule 6328
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {1-\frac {1}{a^2 x^2}} x\right ) \int \frac {e^{\coth ^{-1}(a x)}}{\sqrt {1-\frac {1}{a^2 x^2}} x} \, dx}{\sqrt {c-a^2 c x^2}} \\ & = \frac {\left (a \sqrt {1-\frac {1}{a^2 x^2}} x\right ) \int \frac {1}{-1+a x} \, dx}{\sqrt {c-a^2 c x^2}} \\ & = \frac {\sqrt {1-\frac {1}{a^2 x^2}} x \log (1-a x)}{\sqrt {c-a^2 c x^2}} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\coth ^{-1}(a x)}}{\sqrt {c-a^2 c x^2}} \, dx=\frac {\sqrt {1-\frac {1}{a^2 x^2}} x \log (1-a x)}{\sqrt {c-a^2 c x^2}} \]
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Time = 0.48 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.34
method | result | size |
default | \(-\frac {\ln \left (a x -1\right ) \sqrt {-c \left (a^{2} x^{2}-1\right )}}{c a \left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}\) | \(51\) |
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none
Time = 0.25 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.58 \[ \int \frac {e^{\coth ^{-1}(a x)}}{\sqrt {c-a^2 c x^2}} \, dx=-\frac {\sqrt {-a^{2} c} \log \left (a x - 1\right )}{a^{2} c} \]
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\[ \int \frac {e^{\coth ^{-1}(a x)}}{\sqrt {c-a^2 c x^2}} \, dx=\int \frac {1}{\sqrt {\frac {a x - 1}{a x + 1}} \sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]
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\[ \int \frac {e^{\coth ^{-1}(a x)}}{\sqrt {c-a^2 c x^2}} \, dx=\int { \frac {1}{\sqrt {-a^{2} c x^{2} + c} \sqrt {\frac {a x - 1}{a x + 1}}} \,d x } \]
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\[ \int \frac {e^{\coth ^{-1}(a x)}}{\sqrt {c-a^2 c x^2}} \, dx=\int { \frac {1}{\sqrt {-a^{2} c x^{2} + c} \sqrt {\frac {a x - 1}{a x + 1}}} \,d x } \]
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Timed out. \[ \int \frac {e^{\coth ^{-1}(a x)}}{\sqrt {c-a^2 c x^2}} \, dx=\int \frac {1}{\sqrt {c-a^2\,c\,x^2}\,\sqrt {\frac {a\,x-1}{a\,x+1}}} \,d x \]
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