Integrand size = 24, antiderivative size = 142 \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{7/2} \, dx=\frac {2 (1-a x)^6 \left (c-a^2 c x^2\right )^{7/2}}{3 a^8 \left (1-\frac {1}{a^2 x^2}\right )^{7/2} x^7}-\frac {4 (1-a x)^7 \left (c-a^2 c x^2\right )^{7/2}}{7 a^8 \left (1-\frac {1}{a^2 x^2}\right )^{7/2} x^7}+\frac {(1-a x)^8 \left (c-a^2 c x^2\right )^{7/2}}{8 a^8 \left (1-\frac {1}{a^2 x^2}\right )^{7/2} x^7} \]
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Time = 0.14 (sec) , antiderivative size = 142, 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} \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{7/2} \, dx=\frac {(1-a x)^8 \left (c-a^2 c x^2\right )^{7/2}}{8 a^8 x^7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}-\frac {4 (1-a x)^7 \left (c-a^2 c x^2\right )^{7/2}}{7 a^8 x^7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}+\frac {2 (1-a x)^6 \left (c-a^2 c x^2\right )^{7/2}}{3 a^8 x^7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}} \]
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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 )^{7/2} \int e^{-3 \coth ^{-1}(a x)} \left (1-\frac {1}{a^2 x^2}\right )^{7/2} x^7 \, dx}{\left (1-\frac {1}{a^2 x^2}\right )^{7/2} x^7} \\ & = \frac {\left (c-a^2 c x^2\right )^{7/2} \int (-1+a x)^5 (1+a x)^2 \, dx}{a^7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2} x^7} \\ & = \frac {\left (c-a^2 c x^2\right )^{7/2} \int \left (4 (-1+a x)^5+4 (-1+a x)^6+(-1+a x)^7\right ) \, dx}{a^7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2} x^7} \\ & = \frac {2 (1-a x)^6 \left (c-a^2 c x^2\right )^{7/2}}{3 a^8 \left (1-\frac {1}{a^2 x^2}\right )^{7/2} x^7}-\frac {4 (1-a x)^7 \left (c-a^2 c x^2\right )^{7/2}}{7 a^8 \left (1-\frac {1}{a^2 x^2}\right )^{7/2} x^7}+\frac {(1-a x)^8 \left (c-a^2 c x^2\right )^{7/2}}{8 a^8 \left (1-\frac {1}{a^2 x^2}\right )^{7/2} x^7} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.44 \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{7/2} \, dx=-\frac {c^3 (-1+a x)^6 \left (37+54 a x+21 a^2 x^2\right ) \sqrt {c-a^2 c x^2}}{168 a^2 \sqrt {1-\frac {1}{a^2 x^2}} x} \]
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Time = 0.52 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.70
method | result | size |
gosper | \(\frac {x \left (21 a^{7} x^{7}-72 a^{6} x^{6}+28 a^{5} x^{5}+168 a^{4} x^{4}-210 a^{3} x^{3}-56 a^{2} x^{2}+252 a x -168\right ) \left (-a^{2} c \,x^{2}+c \right )^{\frac {7}{2}} \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{168 \left (a x +1\right )^{2} \left (a x -1\right )^{5}}\) | \(100\) |
default | \(-\frac {\left (21 a^{7} x^{7}-72 a^{6} x^{6}+28 a^{5} x^{5}+168 a^{4} x^{4}-210 a^{3} x^{3}-56 a^{2} x^{2}+252 a x -168\right ) x \,c^{3} \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (a x +1\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{168 \left (a x -1\right )^{2}}\) | \(102\) |
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Time = 0.24 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.67 \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{7/2} \, dx=-\frac {{\left (21 \, a^{7} c^{3} x^{8} - 72 \, a^{6} c^{3} x^{7} + 28 \, a^{5} c^{3} x^{6} + 168 \, a^{4} c^{3} x^{5} - 210 \, a^{3} c^{3} x^{4} - 56 \, a^{2} c^{3} x^{3} + 252 \, a c^{3} x^{2} - 168 \, c^{3} x\right )} \sqrt {-a^{2} c}}{168 \, a} \]
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Timed out. \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{7/2} \, dx=\text {Timed out} \]
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Time = 0.24 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.21 \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{7/2} \, dx=-\frac {{\left (21 \, a^{9} \sqrt {-c} c^{3} x^{9} - 51 \, a^{8} \sqrt {-c} c^{3} x^{8} - 44 \, a^{7} \sqrt {-c} c^{3} x^{7} + 196 \, a^{6} \sqrt {-c} c^{3} x^{6} - 42 \, a^{5} \sqrt {-c} c^{3} x^{5} - 266 \, a^{4} \sqrt {-c} c^{3} x^{4} + 196 \, a^{3} \sqrt {-c} c^{3} x^{3} + 84 \, a^{2} \sqrt {-c} c^{3} x^{2} + 168 \, \sqrt {-c} c^{3}\right )} {\left (a x - 1\right )}^{2}}{168 \, {\left (a^{3} x^{2} - 2 \, a^{2} x + a\right )} {\left (a x + 1\right )}} \]
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\[ \int e^{-3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{7/2} \, dx=\int { {\left (-a^{2} c x^{2} + c\right )}^{\frac {7}{2}} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} \,d x } \]
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Timed out. \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{7/2} \, dx=\int {\left (c-a^2\,c\,x^2\right )}^{7/2}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2} \,d x \]
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