Integrand size = 24, antiderivative size = 113 \[ \int e^{3 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2} \, dx=\frac {3 \sqrt {c-a^2 c x^2}}{a \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {x \sqrt {c-a^2 c x^2}}{2 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {4 \sqrt {c-a^2 c x^2} \log (1-a x)}{a^2 \sqrt {1-\frac {1}{a^2 x^2}} x} \]
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Time = 0.11 (sec) , antiderivative size = 113, 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} \[ \int e^{3 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2} \, dx=\frac {x \sqrt {c-a^2 c x^2}}{2 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {3 \sqrt {c-a^2 c x^2}}{a \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {4 \sqrt {c-a^2 c x^2} \log (1-a x)}{a^2 x \sqrt {1-\frac {1}{a^2 x^2}}} \]
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Rule 45
Rule 6327
Rule 6328
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {c-a^2 c x^2} \int e^{3 \coth ^{-1}(a x)} \sqrt {1-\frac {1}{a^2 x^2}} x \, dx}{\sqrt {1-\frac {1}{a^2 x^2}} x} \\ & = \frac {\sqrt {c-a^2 c x^2} \int \frac {(1+a x)^2}{-1+a x} \, dx}{a \sqrt {1-\frac {1}{a^2 x^2}} x} \\ & = \frac {\sqrt {c-a^2 c x^2} \int \left (3+a x+\frac {4}{-1+a x}\right ) \, dx}{a \sqrt {1-\frac {1}{a^2 x^2}} x} \\ & = \frac {3 \sqrt {c-a^2 c x^2}}{a \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {x \sqrt {c-a^2 c x^2}}{2 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {4 \sqrt {c-a^2 c x^2} \log (1-a x)}{a^2 \sqrt {1-\frac {1}{a^2 x^2}} x} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.52 \[ \int e^{3 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2} \, dx=\frac {\sqrt {c-a^2 c x^2} \left (\frac {3 x}{a}+\frac {x^2}{2}+\frac {4 \log (1-a x)}{a^2}\right )}{\sqrt {1-\frac {1}{a^2 x^2}} x} \]
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Time = 0.51 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.59
method | result | size |
default | \(\frac {\left (a^{2} x^{2}+6 a x +8 \ln \left (a x -1\right )\right ) \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (a x -1\right )}{2 a \left (a x +1\right )^{2} \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}\) | \(67\) |
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Time = 0.25 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.29 \[ \int e^{3 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2} \, dx=\frac {{\left (a^{2} x^{2} + 6 \, a x + 8 \, \log \left (a x - 1\right )\right )} \sqrt {-a^{2} c}}{2 \, a^{2}} \]
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\[ \int e^{3 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2} \, dx=\int \frac {\sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )}}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}\, dx \]
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\[ \int e^{3 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2} \, dx=\int { \frac {\sqrt {-a^{2} c x^{2} + c}}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} \,d x } \]
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\[ \int e^{3 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2} \, dx=\int { \frac {\sqrt {-a^{2} c x^{2} + c}}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int e^{3 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2} \, dx=\int \frac {\sqrt {c-a^2\,c\,x^2}}{{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}} \,d x \]
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