Integrand size = 24, antiderivative size = 171 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \, dx=\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{2 a c \sqrt {c-\frac {c}{a^2 x^2}} (1-a x)^2}+\frac {3 \sqrt {1-\frac {1}{a^2 x^2}}}{a c \sqrt {c-\frac {c}{a^2 x^2}} (1-a x)}+\frac {3 \sqrt {1-\frac {1}{a^2 x^2}} \log (1-a x)}{a c \sqrt {c-\frac {c}{a^2 x^2}}} \]
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Time = 0.11 (sec) , antiderivative size = 171, 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} \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \, dx=\frac {x \sqrt {1-\frac {1}{a^2 x^2}}}{c \sqrt {c-\frac {c}{a^2 x^2}}}+\frac {3 \sqrt {1-\frac {1}{a^2 x^2}}}{a c (1-a x) \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{2 a c (1-a x)^2 \sqrt {c-\frac {c}{a^2 x^2}}}+\frac {3 \sqrt {1-\frac {1}{a^2 x^2}} \log (1-a x)}{a c \sqrt {c-\frac {c}{a^2 x^2}}} \]
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Rule 45
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)}}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2}} \, dx}{c \sqrt {c-\frac {c}{a^2 x^2}}} \\ & = \frac {\left (a^3 \sqrt {1-\frac {1}{a^2 x^2}}\right ) \int \frac {x^3}{(-1+a x)^3} \, dx}{c \sqrt {c-\frac {c}{a^2 x^2}}} \\ & = \frac {\left (a^3 \sqrt {1-\frac {1}{a^2 x^2}}\right ) \int \left (\frac {1}{a^3}+\frac {1}{a^3 (-1+a x)^3}+\frac {3}{a^3 (-1+a x)^2}+\frac {3}{a^3 (-1+a x)}\right ) \, dx}{c \sqrt {c-\frac {c}{a^2 x^2}}} \\ & = \frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{2 a c \sqrt {c-\frac {c}{a^2 x^2}} (1-a x)^2}+\frac {3 \sqrt {1-\frac {1}{a^2 x^2}}}{a c \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 c \sqrt {c-\frac {c}{a^2 x^2}}} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.37 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \, dx=\frac {\left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (x+\frac {5-6 a x}{2 a (-1+a x)^2}+\frac {3 \log (1-a x)}{a}\right )}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \]
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Time = 0.05 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.60
method | result | size |
default | \(\frac {\left (a x -1\right ) \left (2 a^{3} x^{3}+6 a^{2} \ln \left (a x -1\right ) x^{2}-4 a^{2} x^{2}-12 a \ln \left (a x -1\right ) x -4 a x +6 \ln \left (a x -1\right )+5\right )}{2 \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} a^{4} x^{3} {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}\right )}^{\frac {3}{2}}}\) | \(102\) |
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Time = 0.24 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.47 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \, dx=\frac {{\left (2 \, a^{3} x^{3} - 4 \, a^{2} x^{2} - 4 \, a x + 6 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (a x - 1\right ) + 5\right )} \sqrt {a^{2} c}}{2 \, {\left (a^{4} c^{2} x^{2} - 2 \, a^{3} c^{2} x + a^{2} c^{2}\right )}} \]
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Timed out. \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \, dx=\int { \frac {1}{{\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {3}{2}} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \, dx=\int { \frac {1}{{\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {3}{2}} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \, dx=\int \frac {1}{{\left (c-\frac {c}{a^2\,x^2}\right )}^{3/2}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}} \,d x \]
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