Integrand size = 24, antiderivative size = 148 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{3/2} \, dx=-\frac {c \sqrt {c-\frac {c}{a^2 x^2}}}{2 a^3 \sqrt {1-\frac {1}{a^2 x^2}} x^2}-\frac {3 c \sqrt {c-\frac {c}{a^2 x^2}}}{a^2 \sqrt {1-\frac {1}{a^2 x^2}} x}+\frac {c \sqrt {c-\frac {c}{a^2 x^2}} x}{\sqrt {1-\frac {1}{a^2 x^2}}}+\frac {3 c \sqrt {c-\frac {c}{a^2 x^2}} \log (x)}{a \sqrt {1-\frac {1}{a^2 x^2}}} \]
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Time = 0.10 (sec) , antiderivative size = 148, 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, 45} \[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{3/2} \, dx=\frac {c x \sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {1-\frac {1}{a^2 x^2}}}-\frac {3 c \sqrt {c-\frac {c}{a^2 x^2}}}{a^2 x \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {3 c \log (x) \sqrt {c-\frac {c}{a^2 x^2}}}{a \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {c \sqrt {c-\frac {c}{a^2 x^2}}}{2 a^3 x^2 \sqrt {1-\frac {1}{a^2 x^2}}} \]
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Rule 45
Rule 6328
Rule 6332
Rubi steps \begin{align*} \text {integral}& = \frac {\left (c \sqrt {c-\frac {c}{a^2 x^2}}\right ) \int e^{3 \coth ^{-1}(a x)} \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \, dx}{\sqrt {1-\frac {1}{a^2 x^2}}} \\ & = \frac {\left (c \sqrt {c-\frac {c}{a^2 x^2}}\right ) \int \frac {(1+a x)^3}{x^3} \, dx}{a^3 \sqrt {1-\frac {1}{a^2 x^2}}} \\ & = \frac {\left (c \sqrt {c-\frac {c}{a^2 x^2}}\right ) \int \left (a^3+\frac {1}{x^3}+\frac {3 a}{x^2}+\frac {3 a^2}{x}\right ) \, dx}{a^3 \sqrt {1-\frac {1}{a^2 x^2}}} \\ & = -\frac {c \sqrt {c-\frac {c}{a^2 x^2}}}{2 a^3 \sqrt {1-\frac {1}{a^2 x^2}} x^2}-\frac {3 c \sqrt {c-\frac {c}{a^2 x^2}}}{a^2 \sqrt {1-\frac {1}{a^2 x^2}} x}+\frac {c \sqrt {c-\frac {c}{a^2 x^2}} x}{\sqrt {1-\frac {1}{a^2 x^2}}}+\frac {3 c \sqrt {c-\frac {c}{a^2 x^2}} \log (x)}{a \sqrt {1-\frac {1}{a^2 x^2}}} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.39 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{3/2} \, dx=\frac {\left (c-\frac {c}{a^2 x^2}\right )^{3/2} \left (-\frac {1}{2 a^3 x^2}-\frac {3}{a^2 x}+x+\frac {3 \log (x)}{a}\right )}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2}} \]
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Time = 0.05 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.47
method | result | size |
default | \(\frac {\left (2 a^{3} x^{3}+6 a^{2} \ln \left (x \right ) x^{2}-6 a x -1\right ) {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}\right )}^{\frac {3}{2}} x}{2 \left (a x +1\right )^{3} \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}\) | \(69\) |
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Time = 0.25 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.30 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{3/2} \, dx=\frac {{\left (2 \, a^{3} c x^{3} + 6 \, a^{2} c x^{2} \log \left (x\right ) - 6 \, a c x - c\right )} \sqrt {a^{2} c}}{2 \, a^{4} x^{2}} \]
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Timed out. \[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{3/2} \, dx=\text {Timed out} \]
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\[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{3/2} \, dx=\int { \frac {{\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {3}{2}}}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} \,d x } \]
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\[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{3/2} \, dx=\int { \frac {{\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {3}{2}}}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{3/2} \, dx=\int \frac {{\left (c-\frac {c}{a^2\,x^2}\right )}^{3/2}}{{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}} \,d x \]
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