Integrand size = 24, antiderivative size = 77 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\sqrt {c-a^2 c x^2}} \, dx=\frac {2 \sqrt {1-\frac {1}{a^2 x^2}} x}{(1+a x) \sqrt {c-a^2 c x^2}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} x \log (1+a x)}{\sqrt {c-a^2 c x^2}} \]
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Time = 0.12 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6327, 6328, 45} \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\sqrt {c-a^2 c x^2}} \, dx=\frac {2 x \sqrt {1-\frac {1}{a^2 x^2}}}{(a x+1) \sqrt {c-a^2 c x^2}}+\frac {x \sqrt {1-\frac {1}{a^2 x^2}} \log (a x+1)}{\sqrt {c-a^2 c x^2}} \]
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Rule 45
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^{-3 \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+a x}{(1+a x)^2} \, dx}{\sqrt {c-a^2 c x^2}} \\ & = \frac {\left (a \sqrt {1-\frac {1}{a^2 x^2}} x\right ) \int \left (-\frac {2}{(1+a x)^2}+\frac {1}{1+a x}\right ) \, dx}{\sqrt {c-a^2 c x^2}} \\ & = \frac {2 \sqrt {1-\frac {1}{a^2 x^2}} x}{(1+a x) \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 = 0.05 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.61 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\sqrt {c-a^2 c x^2}} \, dx=\frac {\sqrt {1-\frac {1}{a^2 x^2}} x \left (\frac {2}{1+a x}+\log (1+a x)\right )}{\sqrt {c-a^2 c x^2}} \]
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Time = 0.53 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.81
method | result | size |
default | \(-\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (a \ln \left (a x +1\right ) x +\ln \left (a x +1\right )+2\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{a c \left (a x -1\right )^{2}}\) | \(62\) |
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none
Time = 0.24 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.49 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\sqrt {c-a^2 c x^2}} \, dx=-\frac {\sqrt {-a^{2} c} {\left ({\left (a x + 1\right )} \log \left (a x + 1\right ) + 2\right )}}{a^{3} c x + a^{2} c} \]
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\[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\sqrt {c-a^2 c x^2}} \, dx=\int \frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}{\sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]
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\[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\sqrt {c-a^2 c x^2}} \, dx=\int { \frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}{\sqrt {-a^{2} c x^{2} + c}} \,d x } \]
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\[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\sqrt {c-a^2 c x^2}} \, dx=\int { \frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}{\sqrt {-a^{2} c x^{2} + c}} \,d x } \]
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Timed out. \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\sqrt {c-a^2 c x^2}} \, dx=\int \frac {{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{\sqrt {c-a^2\,c\,x^2}} \,d x \]
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