Integrand size = 24, antiderivative size = 187 \[ \int e^{-\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{7/2} \, dx=-\frac {8 (1-a x)^5 \left (c-a^2 c x^2\right )^{7/2}}{5 a^8 \left (1-\frac {1}{a^2 x^2}\right )^{7/2} x^7}+\frac {2 (1-a x)^6 \left (c-a^2 c x^2\right )^{7/2}}{a^8 \left (1-\frac {1}{a^2 x^2}\right )^{7/2} x^7}-\frac {6 (1-a x)^7 \left (c-a^2 c x^2\right )^{7/2}}{7 a^8 \left (1-\frac {1}{a^2 x^2}\right )^{7/2} x^7}+\frac {(1-a x)^8 \left (c-a^2 c x^2\right )^{7/2}}{8 a^8 \left (1-\frac {1}{a^2 x^2}\right )^{7/2} x^7} \]
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Time = 0.16 (sec) , antiderivative size = 187, 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} \[ \int e^{-\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{7/2} \, dx=\frac {(1-a x)^8 \left (c-a^2 c x^2\right )^{7/2}}{8 a^8 x^7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}-\frac {6 (1-a x)^7 \left (c-a^2 c x^2\right )^{7/2}}{7 a^8 x^7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}+\frac {2 (1-a x)^6 \left (c-a^2 c x^2\right )^{7/2}}{a^8 x^7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}-\frac {8 (1-a x)^5 \left (c-a^2 c x^2\right )^{7/2}}{5 a^8 x^7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}} \]
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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^{-\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)^4 (1+a x)^3 \, 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 (8 (-1+a x)^4+12 (-1+a x)^5+6 (-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 {8 (1-a x)^5 \left (c-a^2 c x^2\right )^{7/2}}{5 a^8 \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}}{a^8 \left (1-\frac {1}{a^2 x^2}\right )^{7/2} x^7}-\frac {6 (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 = 71, normalized size of antiderivative = 0.38 \[ \int e^{-\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{7/2} \, dx=-\frac {c^3 (-1+a x)^5 \sqrt {c-a^2 c x^2} \left (93+185 a x+135 a^2 x^2+35 a^3 x^3\right )}{280 a^2 \sqrt {1-\frac {1}{a^2 x^2}} x} \]
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Time = 0.51 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.52
method | result | size |
default | \(-\frac {\left (35 a^{7} x^{7}-40 a^{6} x^{6}-140 a^{5} x^{5}+168 a^{4} x^{4}+210 a^{3} x^{3}-280 a^{2} x^{2}-140 a x +280\right ) x \,c^{3} \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {\frac {a x -1}{a x +1}}}{280 \left (a x -1\right )}\) | \(97\) |
gosper | \(\frac {x \left (35 a^{7} x^{7}-40 a^{6} x^{6}-140 a^{5} x^{5}+168 a^{4} x^{4}+210 a^{3} x^{3}-280 a^{2} x^{2}-140 a x +280\right ) \left (-a^{2} c \,x^{2}+c \right )^{\frac {7}{2}} \sqrt {\frac {a x -1}{a x +1}}}{280 \left (a x -1\right )^{4} \left (a x +1\right )^{3}}\) | \(100\) |
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Time = 0.26 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.51 \[ \int e^{-\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{7/2} \, dx=-\frac {{\left (35 \, a^{7} c^{3} x^{8} - 40 \, a^{6} c^{3} x^{7} - 140 \, a^{5} c^{3} x^{6} + 168 \, a^{4} c^{3} x^{5} + 210 \, a^{3} c^{3} x^{4} - 280 \, a^{2} c^{3} x^{3} - 140 \, a c^{3} x^{2} + 280 \, c^{3} x\right )} \sqrt {-a^{2} c}}{280 \, a} \]
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Timed out. \[ \int e^{-\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{7/2} \, dx=\text {Timed out} \]
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\[ \int e^{-\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{7/2} \, dx=\int { {\left (-a^{2} c x^{2} + c\right )}^{\frac {7}{2}} \sqrt {\frac {a x - 1}{a x + 1}} \,d x } \]
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\[ \int e^{-\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{7/2} \, dx=\int { {\left (-a^{2} c x^{2} + c\right )}^{\frac {7}{2}} \sqrt {\frac {a x - 1}{a x + 1}} \,d x } \]
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Timed out. \[ \int e^{-\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{7/2} \, dx=\int {\left (c-a^2\,c\,x^2\right )}^{7/2}\,\sqrt {\frac {a\,x-1}{a\,x+1}} \,d x \]
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