Integrand size = 24, antiderivative size = 115 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\sqrt {c-\frac {c}{a^2 x^2}}} \, dx=\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{\sqrt {c-\frac {c}{a^2 x^2}}}+\frac {2 \sqrt {1-\frac {1}{a^2 x^2}}}{a \sqrt {c-\frac {c}{a^2 x^2}} (1-a x)}+\frac {3 \sqrt {1-\frac {1}{a^2 x^2}} \log (1-a x)}{a \sqrt {c-\frac {c}{a^2 x^2}}} \]
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Time = 0.09 (sec) , antiderivative size = 115, 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 = {6332, 6328, 78} \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\sqrt {c-\frac {c}{a^2 x^2}}} \, dx=\frac {x \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a^2 x^2}}}+\frac {2 \sqrt {1-\frac {1}{a^2 x^2}}}{a (1-a x) \sqrt {c-\frac {c}{a^2 x^2}}}+\frac {3 \sqrt {1-\frac {1}{a^2 x^2}} \log (1-a x)}{a \sqrt {c-\frac {c}{a^2 x^2}}} \]
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Rule 78
Rule 6328
Rule 6332
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1-\frac {1}{a^2 x^2}} \int \frac {e^{3 \coth ^{-1}(a x)}}{\sqrt {1-\frac {1}{a^2 x^2}}} \, dx}{\sqrt {c-\frac {c}{a^2 x^2}}} \\ & = \frac {\left (a \sqrt {1-\frac {1}{a^2 x^2}}\right ) \int \frac {x (1+a x)}{(-1+a x)^2} \, dx}{\sqrt {c-\frac {c}{a^2 x^2}}} \\ & = \frac {\left (a \sqrt {1-\frac {1}{a^2 x^2}}\right ) \int \left (\frac {1}{a}+\frac {2}{a (-1+a x)^2}+\frac {3}{a (-1+a x)}\right ) \, dx}{\sqrt {c-\frac {c}{a^2 x^2}}} \\ & = \frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{\sqrt {c-\frac {c}{a^2 x^2}}}+\frac {2 \sqrt {1-\frac {1}{a^2 x^2}}}{a \sqrt {c-\frac {c}{a^2 x^2}} (1-a x)}+\frac {3 \sqrt {1-\frac {1}{a^2 x^2}} \log (1-a x)}{a \sqrt {c-\frac {c}{a^2 x^2}}} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.50 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\sqrt {c-\frac {c}{a^2 x^2}}} \, dx=\frac {\sqrt {1-\frac {1}{a^2 x^2}} \left (x+\frac {2}{a (1-a x)}+\frac {3 \log (1-a x)}{a}\right )}{\sqrt {c-\frac {c}{a^2 x^2}}} \]
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Time = 0.05 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.74
method | result | size |
default | \(\frac {\left (a x -1\right ) \left (a^{2} x^{2}+3 a \ln \left (a x -1\right ) x -a x -3 \ln \left (a x -1\right )-2\right )}{\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, x \,a^{2}}\) | \(85\) |
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Time = 0.24 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.43 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\sqrt {c-\frac {c}{a^2 x^2}}} \, dx=\frac {{\left (a^{2} x^{2} - a x + 3 \, {\left (a x - 1\right )} \log \left (a x - 1\right ) - 2\right )} \sqrt {a^{2} c}}{a^{3} c x - a^{2} c} \]
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Timed out. \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\sqrt {c-\frac {c}{a^2 x^2}}} \, dx=\text {Timed out} \]
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\[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\sqrt {c-\frac {c}{a^2 x^2}}} \, dx=\int { \frac {1}{\sqrt {c - \frac {c}{a^{2} x^{2}}} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\sqrt {c-\frac {c}{a^2 x^2}}} \, dx=\int { \frac {1}{\sqrt {c - \frac {c}{a^{2} x^{2}}} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\sqrt {c-\frac {c}{a^2 x^2}}} \, dx=\int \frac {1}{\sqrt {c-\frac {c}{a^2\,x^2}}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}} \,d x \]
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