Integrand size = 24, antiderivative size = 95 \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{5/2} \, dx=-\frac {2 (1-a x)^5 \left (c-a^2 c x^2\right )^{5/2}}{5 a^6 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5}+\frac {(1-a x)^6 \left (c-a^2 c x^2\right )^{5/2}}{6 a^6 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5} \]
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Time = 0.13 (sec) , antiderivative size = 95, 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} \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{5/2} \, dx=\frac {(1-a x)^6 \left (c-a^2 c x^2\right )^{5/2}}{6 a^6 x^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {2 (1-a x)^5 \left (c-a^2 c x^2\right )^{5/2}}{5 a^6 x^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}} \]
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Rule 45
Rule 6327
Rule 6328
Rubi steps \begin{align*} \text {integral}& = \frac {\left (c-a^2 c x^2\right )^{5/2} \int e^{-3 \coth ^{-1}(a x)} \left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5 \, dx}{\left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5} \\ & = \frac {\left (c-a^2 c x^2\right )^{5/2} \int (-1+a x)^4 (1+a x) \, dx}{a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5} \\ & = \frac {\left (c-a^2 c x^2\right )^{5/2} \int \left (2 (-1+a x)^4+(-1+a x)^5\right ) \, dx}{a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5} \\ & = -\frac {2 (1-a x)^5 \left (c-a^2 c x^2\right )^{5/2}}{5 a^6 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5}+\frac {(1-a x)^6 \left (c-a^2 c x^2\right )^{5/2}}{6 a^6 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.58 \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{5/2} \, dx=\frac {c^2 (-1+a x)^5 (7+5 a x) \sqrt {c-a^2 c x^2}}{30 a^2 \sqrt {1-\frac {1}{a^2 x^2}} x} \]
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Time = 0.52 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.88
method | result | size |
gosper | \(\frac {x \left (5 a^{5} x^{5}-18 a^{4} x^{4}+15 a^{3} x^{3}+20 a^{2} x^{2}-45 a x +30\right ) \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}} \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{30 \left (a x -1\right )^{4} \left (a x +1\right )}\) | \(84\) |
default | \(\frac {\left (5 a^{5} x^{5}-18 a^{4} x^{4}+15 a^{3} x^{3}+20 a^{2} x^{2}-45 a x +30\right ) x \,c^{2} \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (a x +1\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{30 \left (a x -1\right )^{2}}\) | \(86\) |
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Time = 0.24 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.77 \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{5/2} \, dx=\frac {{\left (5 \, a^{5} c^{2} x^{6} - 18 \, a^{4} c^{2} x^{5} + 15 \, a^{3} c^{2} x^{4} + 20 \, a^{2} c^{2} x^{3} - 45 \, a c^{2} x^{2} + 30 \, c^{2} x\right )} \sqrt {-a^{2} c}}{30 \, a} \]
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Timed out. \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{5/2} \, dx=\text {Timed out} \]
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Time = 0.21 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.47 \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{5/2} \, dx=\frac {{\left (5 \, a^{7} \sqrt {-c} c^{2} x^{7} - 13 \, a^{6} \sqrt {-c} c^{2} x^{6} - 3 \, a^{5} \sqrt {-c} c^{2} x^{5} + 35 \, a^{4} \sqrt {-c} c^{2} x^{4} - 25 \, a^{3} \sqrt {-c} c^{2} x^{3} - 15 \, a^{2} \sqrt {-c} c^{2} x^{2} - 30 \, \sqrt {-c} c^{2}\right )} {\left (a x - 1\right )}^{2}}{30 \, {\left (a^{3} x^{2} - 2 \, a^{2} x + a\right )} {\left (a x + 1\right )}} \]
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\[ \int e^{-3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{5/2} \, dx=\int { {\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} \,d x } \]
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Timed out. \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{5/2} \, dx=\int {\left (c-a^2\,c\,x^2\right )}^{5/2}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2} \,d x \]
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