Integrand size = 22, antiderivative size = 183 \[ \int e^{\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{7/2} \, dx=-\frac {8 (1+a x)^5 \left (c-a^2 c x^2\right )^{7/2}}{5 a^8 \left (1-\frac {1}{a^2 x^2}\right )^{7/2} x^7}+\frac {2 (1+a x)^6 \left (c-a^2 c x^2\right )^{7/2}}{a^8 \left (1-\frac {1}{a^2 x^2}\right )^{7/2} x^7}-\frac {6 (1+a x)^7 \left (c-a^2 c x^2\right )^{7/2}}{7 a^8 \left (1-\frac {1}{a^2 x^2}\right )^{7/2} x^7}+\frac {(1+a x)^8 \left (c-a^2 c x^2\right )^{7/2}}{8 a^8 \left (1-\frac {1}{a^2 x^2}\right )^{7/2} x^7} \]
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Time = 0.15 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6327, 6328, 45} \[ \int e^{\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{7/2} \, dx=\frac {(a x+1)^8 \left (c-a^2 c x^2\right )^{7/2}}{8 a^8 x^7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}-\frac {6 (a x+1)^7 \left (c-a^2 c x^2\right )^{7/2}}{7 a^8 x^7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}+\frac {2 (a x+1)^6 \left (c-a^2 c x^2\right )^{7/2}}{a^8 x^7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}-\frac {8 (a x+1)^5 \left (c-a^2 c x^2\right )^{7/2}}{5 a^8 x^7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}} \]
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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)^3 (1+a x)^4 \, 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 = 0.07 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.39 \[ \int e^{\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{7/2} \, dx=-\frac {c^3 (1+a x)^5 \sqrt {c-a^2 c x^2} \left (-93+185 a x-135 a^2 x^2+35 a^3 x^3\right )}{280 a^2 \sqrt {1-\frac {1}{a^2 x^2}} x} \]
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Time = 0.50 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.53
method | result | size |
default | \(-\frac {\left (35 a^{7} x^{7}+40 a^{6} x^{6}-140 a^{5} x^{5}-168 a^{4} x^{4}+210 a^{3} x^{3}+280 a^{2} x^{2}-140 a x -280\right ) x \,c^{3} \sqrt {-c \left (a^{2} x^{2}-1\right )}}{280 \left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}\) | \(97\) |
gosper | \(\frac {x \left (35 a^{7} x^{7}+40 a^{6} x^{6}-140 a^{5} x^{5}-168 a^{4} x^{4}+210 a^{3} x^{3}+280 a^{2} x^{2}-140 a x -280\right ) \left (-a^{2} c \,x^{2}+c \right )^{\frac {7}{2}}}{280 \left (a x -1\right )^{3} \left (a x +1\right )^{4} \sqrt {\frac {a x -1}{a x +1}}}\) | \(100\) |
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Time = 0.24 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.52 \[ \int e^{\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{7/2} \, dx=-\frac {{\left (35 \, a^{7} c^{3} x^{8} + 40 \, a^{6} c^{3} x^{7} - 140 \, a^{5} c^{3} x^{6} - 168 \, a^{4} c^{3} x^{5} + 210 \, a^{3} c^{3} x^{4} + 280 \, a^{2} c^{3} x^{3} - 140 \, a c^{3} x^{2} - 280 \, c^{3} x\right )} \sqrt {-a^{2} c}}{280 \, a} \]
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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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\[ \int e^{\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{7/2} \, dx=\int { \frac {{\left (-a^{2} c x^{2} + c\right )}^{\frac {7}{2}}}{\sqrt {\frac {a x - 1}{a x + 1}}} \,d x } \]
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\[ \int e^{\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{7/2} \, dx=\int { \frac {{\left (-a^{2} c x^{2} + c\right )}^{\frac {7}{2}}}{\sqrt {\frac {a x - 1}{a x + 1}}} \,d x } \]
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Timed out. \[ \int e^{\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{7/2} \, dx=\int \frac {{\left (c-a^2\,c\,x^2\right )}^{7/2}}{\sqrt {\frac {a\,x-1}{a\,x+1}}} \,d x \]
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